Macroscopic quantum tunnelling. effects in Josephson junctions. Suzanne Gildert

Transcription

1

2 Macroscopic quantum tunnelling effects in Josephson junctions by Suzanne Gildert A thesis submitted to the FACULTY OF SCIENCE, UNIVERSITY OF BIRMINGHAM for the degree of DOCTOR OF PHILOSOPHY School of Physics and Astronomy The University Of Birmingham January 2008

3 Abstract Macroscopic quantum tunnelling and thermal activation in Josephson junctions are investigated by measurements of fluctuations in the critical current of single junctions. The data are presented as current switching histograms and escape rate characteristics for a range of temperatures and magnetic fields. A well shielded dilution refrigerator is used to vary the sample temperature between 30mK 1K. Custom software to analyse the results is developed and documented. The system is tested with several Niobium tunnel junctions in order to optimise the performance, and studies into the behaviour of high temperature superconducting YBCO grain boundary junctions are performed. The results are compared to the theoretical predictions from thermal activation and quantum tunnelling theory, and results of similar experimental efforts in the field are also reviewed. It is found that the Low T c junctions fit the theory for thermal activation when other sources of noise are accounted for, and also that excess noise is a problem in the YBCO samples. This has implications for the future of High T c qubit technology, the feasibility of which is also assessed. An investigation into phase slips in narrow tracks of YBCO is also undertaken.

4 Acknowledgements Having been a part of the Condensed Matter group for nearly six years, I have come to know a lot of people, past and present! I would like to thank everyone I have met during the past six years for their support during that time, especially those people who have patiently listened to my somewhat over-enthusiastic reviews of quantum computing, popular science and physics and electronics in general. I would like to specifically thank Edd, for being there through everything, and my family for supporting me in my pursuit of an academic path from a very young age. I would also like to thank my supervisors Mark Colclough and Chris Muirhead, and many other important mentors: Ted Forgan, Ed Tarte, Colin Gough and Joe Vinen. A special mention goes to Alastair Rae, for first sparking my interest in the subject many years ago, through his book Quantum Physics: Illusion or Reality?. I should also mention my office colleagues (and friends) Charlotte, Silvia, Rich, Jo, Jonny, Dom, Christian, Xuefeng, and all the people who have humoured me by tolerating my many post-7pm physics related bar conversations. Thanks also go to Steve Brown, Tim Barraclough, Mugdha Joshi, Daniel Ucko, Jane Ireland, Jon Fenton and Emma Jones, for welcoming me into the group, and to Pavel Mikheenko (for providing constant enthusiasm and chocolate), Gary Walsh, Michael Parkes, and all the guys from the workshop. I would also like to thank the Simply Coffee company for providing the best takeaway caffeinated beverages in the known universe and fuelling my working day!

5 Commonly used abbreviations T c - Critical temperature of a superconductor I c - Critical current of a superconductor J c - Critical current density of a superconductor HTS/HTC - High Temperature Superconductor CMR - Colossal MagnetoResistance MQT - Macroscopic Quantum Tunnelling MQC - Macroscopic Quantum Coherence RSJ - Resistively Shunted Junction RCSJ - Resistively and Capacitively Shunted Junction RF - Radio Frequency µ-wave - Microwave CPR - Current Phase Relation PSC - Phase Slip Centre IJJ - Intrinsic Josephson Junction SQUID - Superconducting QUantum Interference Device QUBIT - QUantum BIT RSFQ - Rapid Single Flux Quantum FIB - Focussed Ion Beam (microscope) SEM - Scanning Electron Microscope GBJ - Grain Boundary Junction YBCO - Y BaCu 3 O 7, a High Temperature Superconductor BSCCO - Bi 2 Sr 2 CaCu 2 O 2, a High Temperature Superconductor

6 Contents 1 Introduction Introducing the project Motivation Basic Superconductivity Quantum effects in bulk superconductors Josephson physics The Josephson junction The RCSJ model Energy of the Josephson Junction Damping in the Josephson Junction Junctions in an applied magnetic field The Current-Phase Relation Macroscopic Quantum Tunnelling The qubit The cuprate superconductors Crystal properties Symmetry of the order parameter Tunnelling in HTS Intrinsic Josephson Junctions Dimensional Superconducting tracks Grain Boundary Junctions MQT in High Temperature Superconductors The HTS Flux Qubit Principles of the experiment The MQT experimental technique Data analysis techniques Previous experiments ii

7 4.4 High T c MQT measurements Apparatus design Sample fabrication Low temperature techniques Cryostat wiring The measurement system Computer Interfacing System testing Results Measurement procedures Preliminary studies Low T c Junctions High T c junctions Difficulties with the measurements Additional work Fabrication of Single Crystal BSCCO whiskers The Focussed Ion Beam (FIB) Microscope Contacts to BSCCO whiskers Manipulation of BSCCO using the FIB Phase slips in Narrow tracks Conclusions and future prospects Implications of this work Future proposals A wider perspective Conclusion

8 List of Figures 1.1 Energy distribution amongst quasiparticle excitations, where f(e) denotes the Fermi function at a finite temperature Semiconductor model of an NS tunnelling system Schematic representation of Andreev reflection Effect of barrier transparency on conductance measurements. a.) Z 0; A low barrier transparency b.) Z ; A high barrier transparency Andreev bound states in superconducting systems The appearance of a ZBCP upon cooling in the tunnelling characteristic of an YBCO/Au junction. From [1] Schematic representation of a Josephson Junction A schematic of the underdamped Josephson IV characteristic Circuit diagram illustrating the RCSJ model The washboard potential for a Josephson Junction The phase diffusion branch of a Josephson junction IV curve Schematic of a Fiske resonance in a Josephson junction Left: Schematic of a SQUID Right: The magnetic diffraction pattern for an ideal SQUID, analogous to two-slit interference. For a real SQUID I MAX usually remains greater than zero for all Φ/Φ Resulting modification of the CPR in a weak link as the barrier transparency is adjusted. Γ refers to the coupling strength between the two superconducting regions, with a low Γ value corresponding to strong coupling. From [3] Illustration of the various regions under consideration in the MQT regime a.) Stochastic switching of a Josephson junction b.) Probability of a switching event as a function of applied current Diagram illustrating how the width of the switching histogram changes with Temperature iv

9 2.12 Effect of dissipation on the escape rate from a cubic potential. The value of α is proportional to the frequency-independent dissipation in the system. From Grabert et al. [4] Schematic of the energy wells in a flux qubit Schematic of the Bloch sphere representation of a qubit SEM image of a 3 junction flux qubit. From Chiorescu et al. [5] Rabi oscillations of a phase qubit detected with a DC SQUID readout technique, from Bertet et al. [6]. The two traces show two different measurements methods: Light grey measured using a DC current pulse, dark grey measured using a resonant activation technique Perovskite crystal structure of YBCO a.) Schematic polar plots showing the properties of electrons in a d-wave superconductor a.) The magnitude and phase of the wavefunction, Ψ; b.) The magnitude of the energy gap, Measurement of the intrinsic phase shift in a d-wave SQUID a.) Experimental setup b.) Magnetic diffraction patterns. From Wollman et al. [7] AFM image of a grain boundary junction. From Mannhart et al. [8] Order parameter in a 0-45 bicrystal GBJ. From Lindstrom et al. [74] Lorentzian modulation of the noise spectrum due to dominant trapping sites. From [9] Temperature dependence of a GBJ critical current (triangles), and the Josephson current harmonic components I 1 (squares) and I 2 (circles). The inset shows the theoretically expected dependence. From Il ichev et al. [10] Schematic of the angle-resolved tunnelling measurement performed by Smilde et al. [11] From [12] Block diagram of the measurement system The voltage ramp applied to the junction and the resulting switching signal Martinis et al. results demonstrating the thermal-quantum crossover (1987) Results of MQT in 1 µm Nb junctions, from Voss et al., [13] Results of MQT measurements on an RF SQUID, from Corato et al., [14]

10 4.6 Diagram illustrating the effect of microwave radiation on tunnelling from the washboard potential. E 1 denotes the ground state, E 2 the excited state Results from Inomata et al. (2003) demonstrating the first successful Macroscopic Quantum Tunnelling experiments on a HTS sample (BSCCO) Jin et al. [15] show that the tunnelling rate in a stack of N Josephson Junctions is enhanced by a factor of N Results from Bauch et al. (2005) illustrating features in the temperature dependence of the activation processes in a YBCO sample [16] Lithography steps for the preparation of a YBCO Grain Boundary Junction sample. The grey line indicates an invisible GB, the red line an exposed GB The chip carrier, installed into the dilution refrigerator. Good thermal contact is ensured by a metallic contact between the sample substrate and the Cu backing plate Schematic of the fridge thermometry The fridge insert with wiring (below 4.2K stage) Wiring schematic of the fridge insert Custom designed MCX connector assemblies, installed on each of the 4 DC signal lines The Cu sample box Microwave response of -20dB (blue line), -10dB (yellow line) and -6dB (red line) attenuators compared with a microwave short (top line) Microwave response of low pass filter (blue line) and coil filter (pink line) compared with a microwave short (top line) Microwave response of a section of co-axial cable (blue line) compared with a microwave short (top line) Microwave response of the current leads (blue and pink lines) from the top of the cryostat to the sample box including all inline filtering, compared with a microwave short (top line) Microwave response of the voltage leads (blue and pink lines) from the top of the cryostat to the sample box including all inline filtering, compared with a microwave short (top line) The current circuit for the attenuator network Block diagram of the measurement system

11 5.15 Final circuit diagram for the low noise preamplifier. The initial 6231 amplifiers were later replaced with the superior 8620 package Photograph of the low noise preamplifier illustrating the daughterboard concept Current noise at the junction from the attenuator network calculated using SPICE simulation software (total current noise is shown as the pink curve) The proposed circuit for future MQT measurements Testing the auxiliary circuit. Top trace shows comparator 1 output; Bottom trace shows comparator 2 output Block diagram of the MQT Control Centre software The User interface for the MQT Control Centre software Simulated current switching histograms in Niobium junction, I c = 10.0µA Simulated drift on mean in Niobium junction, I c = 10.0µA Fitting histograms using the MQT control centre software Example escape rates from data values and theoretical calculation Testing the histogram modelling using theoretically generated data Testing the escape rate model using theoretically generated data Testing the escape temperature model using theoretically generated data Example of a graphical histogram output from the SR The effects of changing the trigger level on the width of the histogram Circuit diagram showing the dummy attenuator network for noise testing The drift on the histogram mean and sigma at the start of the data acquisition The drift on the histogram mean and sigma several hours into the data acquisition Demonstration that sigma is accurate even for low numbers of samples. The x-axis shows the natural logarithm of the number of samples Circuit diagram of the Josephson junction simulator IV characteristics of the artificial Josephson Junction at different values of I c Current switching histograms of the artificial Josephson Junction Monitoring switching events using an oscilloscope, sample IPHT, I c = 9.63µA. Horizontal scale = 3.7µA/div, Vertical scale = 1mV/div.. 131

12 6.2 An example of a series of switching histograms taken at different temperatures Example of an escape temperature plot taken from the grain boundary sample P395, illustrating the fit to the theoretical thermal and quantum regimes Example of a sigma plot IV characteristic of NBARR junction array, 4.2K IPHT junction IV characteristic measured at 4.2K in a continuous flow cryostat. The critical current is suppressed due to poor filtering of the system Escape rate, sample IPHT, I c0 = 9.63uA Escape temperatures, sample IPHT, I c0 = 9.63uA Histograms, sample IPHT, I c0 = 9.63uA IV characteristic, sample PTB, 4.2K Current switching histograms, sample PTB, I c = 43.9µA Escape rates, sample PTB I c = 43.9µA Sigma plot, sample PTB, I c = 43.9µA Escape temperatures, sample PTB, I c = 43.9µA Mean drift, sample PTB, I c = 43.9µA Fiske steps observes on the IV curve, sample CAM03, Junction Multivalued I c observed in CAM03 sample. The white vertical lines have been added as a guide to the eye, indicating the positions of I c Escape rates, CAM03 sample, duty cycle 1.3% Escape rates, CAM03 sample, duty cycle 35.8% Escape rates, CAM03 sample, duty cycle 2.6% Current switching histogram, CAM03 sample, duty cycle 1.3%. The two anomalous points arise due to the autoranging of the SR620 and should be disregarded Sigma plot, sample CAM03, I c = 67.7µA. A clean crossover can clearly be seen at around 300mK, which is of the correct order of magnitude for the estimated I c0 and C values Escape temperature, sample CAM03, I c = 67.7µA. The red line shows the theoretically expected thermal escape, the purple line shows the expected escape due to quantum tunneling Mean Drift, sample CAM03, I c = 67.7µA Fluctuations of I c as a function of time, sample CAM03, 1K, demonstrating telegraph noise on the critical current. There are at least three distinct stable values, with fluctuations between two of the levels appearing to dominate

13 6.26 Frequency analysis of fluctuations with 1/f fit, sample CAM03, 1K Fitted histograms, CAM03 sample, I c0 = 58.8µA showing temperature effects Current switching histograms, CAM03 sample, I c0 = 24.0µA Escape rates, CAM03 sample, I c = 24.0µA Escape temperatures, CAM03 sample, I c0 = 24.0µA Mean drift with fit, CAM03 sample, I c0 = 24.0µA IV characteristic, sample P395 JJ1, 10K Escape temperature, sample P395 JJ1, I c0 = 25.4µA Current switching histograms, sample P395 JJ1, I c0 = 25.4µA. It should be noted that the histograms look significantly more Gaussian than theoretically expected Mean drift, sample P395 JJ1, I c0 = 25.4µA Escape rates, Sample P395, JJ1, I c0 = 35.4µA Escape temperature, Sample P395, JJ1, I c0 = 35.4µA Mean drift, Sample P395, JJ1, I c0 = 35.4µA Evidence for phase diffusion behaviour in sigma plot, Sample P395, JJ1, I c0 = 35.4µA Regimes of escape from the potential well. From Kivioja et al. [17] Fitted histograms, sample P395, JJ2, I c0 = 4.26µA Fitted escape temperature, sample P395, JJ2, I c0 = 4.26µA Time Domain noise, sample P395, JJ2 50mK Frequency Domain noise, sample P395, JJ2 50mK. The purple line illustrates the (1/f 2 ) fit Magnetic characteristic, sample P395 JJ2, 50mK. The right hand side of the trace is potentially increasing toward the first maximum of the Fraunhofer pattern Schematic of thermal resistances at low temperature stage Self heating effects of the junction in the voltage state, sample IPHT Self heating effects of the junction in the voltage state, sample CAM Fitted histograms with variable temperature parameter, sample CAM Fitted Temperature value vs. measured value, sample CAM Schematic of the FIB system operation SEM image of a BSCCO device illustrating Platinum deposition (horizontal strips) and precision FIB milling (dark trenches) Resistance vs. Temperature curve from the 4-terminal measurement of a non-superconducting BSCCO whisker

14 7.4 Schematic of the isolation of a stack of Josephson junctions in a layered superconductor using an undercutting technique TRIM simulation of 30keV Ga Ions incident on a BSCCO sample Undercut in a BSCCO whisker demonstrating 3D patterning with the FIB A narrow track of YBCO produced by FIB milling Evidence for phase slips in YBCO narrow tracks Track 4, 500nm YBCO, resistance below T c. The solid line is a fit to the LAMH theory Underetching in Nb thin film microbridges Results of in-situ monitoring of resistance during the Ion beam milling process YB784b Resistance Temperature characteristic detail. The track has been subject to chemical etching. The inset shows the full characteristic YB724b track Resistance Temperature characteristic detail. Track has been subject to chemical etching and ion beam milling. The inset shows the full characteristic YB721b Resistance Temperature characteristic detail. The track has been subject to chemical etching. The inset shows the full characteristic Assembly of the custom filter circuits Assembly of the custom filter modules

15 Chapter 1 Introduction This chapter provides an overview of the project and gives an introduction to the layout of the report. It explains the motivation behind the studies presented here, and introduces some of the key principles of superconductivity and the field of device physics in general, in order to set the scene for the following chapters. 1

16 CHAPTER 1. INTRODUCTION 1.1 Introducing the project The project involves the design and building of a system capable of characterising various superconducting devices. The system is used to measure the properties of low temperature (conventional) superconducting Josephson Junctions. These experiments will firstly ensure that the results agree with those found in the literature, by comparison with results taken on well understood junction systems. The report goes on to investigate the properties of novel junction systems, including various high temperature superconducting configurations. This report will concentrate on the design of the system, and on the analysis and interpretation of experimental data gathered from several different samples at temperatures below 1K. It will also give an overview of similar experiments and how these have contributed to an overall progress of the field. The structure of the report is as follows: The first three chapters will introduce the concepts of superconductivity and the areas of Josephson Physics which are of key importance to the project. The principles of the experiment are discussed in the fourth chapter, which explains the justification for the the experimental procedures used, and serves as a literature review of other work undertaken in this field. The fifth chapter will document the apparatus design, building and data acquisition. The results chapter will analyse the data gathered on several different Josephson Junction samples, and includes a discussion of problems encountered during the measurements and analysis. In addition to summarising the achievements, the conclusions chapter will emphasise the perspective of the work presented here by discussing the results in the light of several future experiments. The remainder of this chapter will explain the motivation and aims of the project, and also introduce some of the basic concepts of superconductivity, as a groundwork for the detailed physics described in the later sections. The chapter is presented with a bias toward the physics of superconducting devices and their applications. 2

17 CHAPTER 1. INTRODUCTION 1.2 Motivation There have been many reports recently of interesting quantum effects in superconducting devices, and this field has in the past proven to yield interesting applications. It is important to continue experimentally investigating some of the more exotic effects in superconductors, as many theories of superconductivity are as of yet unverified by experiment. Some of the effects seen in superconducting devices also provide a building block for device applications, as will be discussed in this report. The devices group at the University of Birmingham has enjoyed a fruitful yield from this area in the past. In 1987 the quantisation of flux was demonstrated for the first time in a High T c superconductor [18]. The group continued to investigate effects in these interesting systems, with work on Spin Injection effects and the interaction of superconducting materials with Colossal Magnetoresistive (CMR) materials [19]. Research into the properties and applications of the Josephson Junction has been intensely pursued over the past 50 years, and continues to provide a fascinating way of gaining an insight into the basic mechanisms of superconductivity and quantum mechanics. The quantum properties of the system are easily accessible using modern measurement techniques, which allows experimental physicists to explore (and resolve some of the mystery surrounding) the area of High Temperature Superconductivity, in which there are numerous and sometimes contradictory theories of the underlying mechanisms. The Josephson junction also has several practical applications. Much work has focussed on the use of the junction as a replacement for transistor switches in silicon technology. This field has been progressing steadily (see for example [20]). The superconducting system is unique in its ability to demonstrate quantum effects on a micron scale, as opposed to single atom or single photon systems. By starting with superconducting devices of micron dimensions, reducing the size of the device then 3

18 CHAPTER 1. INTRODUCTION has the added benefit of spatially confining the wavefunction, hence increasing the spacing between the energy levels, and allowing well defined quantum states. With conventional silicon technology, scaling down the devices to a size where quantum effects begin to dominate proves disadvantageous. This is due to several factors: The tunneling of electrons creating a leakage current, the electromigration of device material over time causing both shorts and open circuits in device regions, and extensive adverse heating effects. The work described in this report provide strong evidence that the superconducting devices under investigation may eventually prove useful for quantum information applications, in particular as potential qubit candidates. These experiments are the first steps towards a system which will enable the Birmingham group to characterise such devices. It is hoped that the new experiments proposed and undertaken here will help maintain the group s position amongst the leaders in the field of high temperature superconductivity and quantum devices. 1.3 Basic Superconductivity It is important for the reader to have a background in the main concepts in the area of superconductivity, and so will follow a short discussion of some of the key aspects of the field relevant to the work presented in this report. For a more thorough treatment of fundamental properties of superconductors, there are many popular texts available, for example M. Tinkham Introduction to Superconductivity [21], or J. R. Waldram Superconductivity of metals and cuprates [22], which provide a broad overview of the subject area. Electromagnetic properties of superconductors The most easily observable feature of a superconductor is the appearance of perfect conductivity below a critical temperature, T c. This property allows the transport of 4

19 CHAPTER 1. INTRODUCTION electrical current without dissipation or heating, which immediately yields obvious commercial applications. The resistance of a superconducting sample as a function of temperature falls rapidly to zero at a particular temperature denoted the critical temperature, T c. The transition is sharp, which gave the first suggestion of a cooperative behaviour of the electrons; a phase transition. The zero resistance state has been confirmed to a high accuracy by monitoring the decay of persistent currents in a superconducting sample. Perhaps more interesting from a scientific point of view however are the magnetic properties of such a system. A striking feature of the zero resistance state below T c is the appearance of perfect diamagnetic behaviour. Unlike a perfect conductor, for example a pure metal at T = 0K, a superconductor will actively expel magnetic field from its bulk. This effect was first observed by Meissner and Ochsenfeld in 1933, and the expulsion of field in this way is termed the Meissner effect. The magnetic field is screened from the interior of the superconductor by currents flowing along the surface. In this way, the magnetic field is only allowed to enter the superconductor up to a characteristic length, known as the London penetration depth, λ L. Under the application of a strong magnetic field, the superconducting state will no longer be able to support the high field density at the surface of the superconductor, and will disappear altogether. The field at which this occurs is denoted the critical field, H c. The London equations In 1935, London and London proposed a model for the electrodynamics of the superconducting state, by treating the electrons as accelerating under the influence of an electric field. The two main findings can be summarised in the London equations, (1.1) and (1.2) E = t (ΛJ s) (1.1) 5

20 CHAPTER 1. INTRODUCTION H = Λ( J s ) (1.2) where Λ = m/n s e 2, with m denoting the mass of the electron, and n s the number density of superconducting electrons. The first of these equations predicts the perfect conductivity, and the second predicts the existence of the Meissner state. The London penetration depth, λ L, is also defined by the model to be equal to Λ/µ 0. It is also useful at this stage to define another parameter, the coherence length, ξ, which is analogous to the distance over which interactions between superconducting electrons occur. The critical current of a bulk superconductor An external magnetic field is not necessarily required to destroy superconductivity. The supercurrent itself produces a self-field which, when high enough, causes the same effect. This was first established by Silsbee in Therefore it is also possible to define a maximum value of the current density, known as the critical current density J c, in the superconductor. Given specific sample dimensions, one can calculate the critical current of the superconducting sample, I c. Type I and Type II superconductors A type I superconductor is dominated by the Meissner state, as mentioned previously. In this state, no magnetic field is allowed to penetrate the bulk of the sample, by the screening currents which are set up at the surface of the superconductor. However, in some cases it is energetically favourable for magnetic field to enter the superconductor. This occurs when the magnetic field density at the surface exceeds the critical field, whereby the energy required to allow in a small amount of field and screen it inside the superconductor becomes equal to the energy required to maintain screening currents against a high field density at the surface. 6

21 CHAPTER 1. INTRODUCTION The relation between the coherence length and the penetration depth gives a measure of how favourable it is for flux to enter the superconductor, whilst remaining superconducting (known as the mixed state). The ratio is commonly known as the Ginzburg-Landau parameter, κ = λ/ξ. The conditions under which a superconductor will entertain the mixed state as the field is increased can be neatly summarised by their value of κ. The superconductors can therefore be classified into two main types, I and II. For Type I superconductors κ < 1/ 2 and for type II κ > 1/ 2. Flux lines In a type II superconductor, the flux is allowed to enter in such a way as to maximise the normal-superconducting boundary area, as if κ < 1/ 2, it is energetically favourable to create such boundaries. The flux passes into the superconductor in the form of a thin line, known as a flux line or vortex. Each line contains the minimum non-zero amount of flux which can be bounded inside the bulk of a superconductor by circulating currents. This quantity is known as the flux quantum, with a value of Φ 0 = Wb. In an ideal superconducting sample, flux lines are free to move through the bulk. They form a regular array, of triangular (lowest energy configuration) or square symmetry due to their mutual repulsion. Upon the application of a current through a superconducting wire, flux lines will pass from one side of the superconductor to the other, an effect known as flux flow. The resulting motion of the flux lines causes dissipation of energy, and as such there is a resistance associated with the flow. In the presence of sample defects, for example impurities, physical holes, crystal dislocations or grain boundaries, the flux lines may become trapped, or pinned, and flux flow will be impeded to some extent. Flux lines will be pinned at these locations because the energy required to turn the core of the flux line normal is lower. Superconductors in which the flux can be easily pinned offer better prospects for practical applications, as the zero resistance state can be exploited under higher 7

22 CHAPTER 1. INTRODUCTION magnetic fields and currents. 1.4 Quantum effects in bulk superconductors Microscopic theories The main underlying physics of the superconducting system is the tendency of the electrons within the material to form a condensate with a lower energy than their normal counterparts. In order for the properties of the superconducting state to be derived, a microscopic theory of the effect was needed. In 1957, such a theory was proposed by Bardeen, Cooper and Schrieffer [23]. Their main model was based around the electrons forming pairs, which subsequently became known as Cooper pairs. Electrons pair from diametrically opposite sides of k-space, meaning that their momenta are equal and opposite. The net momentum of the pair is therefore zero, corresponding to the lowest energy state. The coherence length, ξ, can be thought of as the size of a Cooper pair. In BCS theory, the pairing arises due to a lattice-mediated interaction in which a phonon interacts with the condensate to form an attractive potential between electrons. Several important properties of the superconducting state were able to be both explained and predicted by the BCS theory, including the appearance of an energy gap,, between the superconducting electron condensate and the normal electrons, the temperature dependence of this gap, and the critical temperature T c of a superconducting material. The theory also correctly predicts the dependence of the critical temperature on the mass of the lattice ion, T c 1/ M, (known as the isotope effect), which strengthens the picture of the lattice playing an important part in the superconducting state. The theory, however, does not correctly explain the properties of some of the more exotic varieties of superconductors which were subsequently discovered, such as the High Temperature Superconductors (HTS). These materials will be discussed in Section 3. 8

23 CHAPTER 1. INTRODUCTION By applying a fundamental quantum mechanical treatment to the collective behaviour of the electrons, several important properties can be derived. In this section a few selected topics will be introduced which will help the reader become familiar with the ideas that motivate the discussion on Josephson Physics in the next section. Flux quantisation Consider a bulk superconductor, in which the wavefunction of the Cooper Pair condensate can be defined as: Ψ(r) = Ψ(r) e iφ(r) (1.3) where φ(r) is the macroscopic phase of the condensate. In a bulk, simply connected sample, there will be no currents flowing in the interior of the sample. Upon taking a line integral around a path one finds that the superconducting phase must change by zero, or by a factor of 2πn, where n is an integer, for Ψ(r) to remain single-valued. This is still valid if the superconductor is made into a ring, as long as the path is taken inside the bulk, such that there are no currents flowing. Thus if a field is applied to the superconductor before cooling below T c, the amount of field that becomes trapped in the hole must be such as to maintain this condition. Thus the flux inside the hole is quantised in units of Φ 0. This phenomenon will be important for the discussion of SQUIDs and qubits later in the report. Quasiparticle excitations As mentioned previously, one of the main features of the superconducting state is the appearance of an energy gap. Above the energy gap, excitations of the superconducting state can exist. These excitations are known as quasiparticles, as they are not real physical particles, rather they can share energy from both the positive and negative regions of k-space. At a temperature above 0K, the energy of the 9

24 CHAPTER 1. INTRODUCTION Figure 1.1: Energy distribution amongst quasiparticle excitations, where f(e) denotes the Fermi function at a finite temperature. quasiparticle can have contributions from both sides of k-space, due to the smearing of the Fermi surface. Thus the excitations are known as electron-like and hole-like quasiparticles correspondingly. This is illustrated in Figure 1.1. If a normal metal is connected to a superconductor, tunnelling from the normal side into the quasiparticle states of the superconductor is possible, and the semiconductor model is used to describe this process. Figure 1.2 shows schematically the way in which the energy bands of the two materials behave under an applied bias voltage (at T = 0K). There can be no tunnelling until the gap voltage is reached, beyond which point, quasiparticles can tunnel across the interface. All quasiparticles tunnelling across the barrier are injected at the energy gap edge, where the density of states is extremely high, and therefore the conductance, or di/dv characteristic of the system is high once this point is exceeded. This is illustrated in Figure 1.4a. The conductance is proportional to the transmission co-efficients of the quasiparticles across the barrier. 10

25 CHAPTER 1. INTRODUCTION Figure 1.2: Semiconductor model of an NS tunnelling system Figure 1.3: Schematic representation of Andreev reflection 11

26 CHAPTER 1. INTRODUCTION Figure 1.4: Effect of barrier transparency on conductance measurements. a.) Z 0; A low barrier transparency b.) Z ; A high barrier transparency Andreev effects There is another mechanism by which quasiparticles can be transported across the barrier. In the case where a superconductor is connected to a normal metal and the barrier is transparent, an electron can pass across the interface as a Cooper pair plus a reflected hole travelling with the opposite momentum, thus conserving both charge, q, and momentum, k. This process is known as Andreev reflection, and the concept is illustrated in Figure 1.3. In the case of Andreev reflection, quasiparticles can be transported across the barrier even if ev <. A conductance due to this mechanism of charge transfer is therefore present below ev =. As the barrier transparency, Z, the Andreev reflection becomes dominant, and the conductance in this region is enhanced, as shown in Figure 1.4b. Resonances of the quasiparticle energy spectrum can occur when the energy (due to the application of a voltage) is equal to that bound energy states formed by treating an electron in the barrier as a particle in a box. These resonances are known as Andreev bound states, and are illustrated in Figure 1.5. They are important because they provide a mechanism for quasiparticles to exist in the barrier even when no bias is applied to the device. The result of bound states on the energy spectrum is to cause a Zero Bias Conductance Peak (ZBCP), which manifests in the measurement of di/dv. Conversely, the appearance of such a peak in an experimental measurement 12

27 CHAPTER 1. INTRODUCTION Figure 1.5: Andreev bound states in superconducting systems Figure 1.6: The appearance of a ZBCP upon cooling in the tunnelling characteristic of an YBCO/Au junction. From [1]. 13

28 CHAPTER 1. INTRODUCTION is a good indicator of the barrier transparency. An example of an experimentally observed ZBCP from an YBCO-Au interface is shown in Figure 1.6. In this example the gap is not sharply defined; this is a feature of the non-ideal properties of the High T c superconductor (YBCO), which will be discussed further in Section

29 Chapter 2 Josephson physics Josephson junctions are useful both from the viewpoint of fundamental physics research and device applications. This chapter will document a range of properties of such junctions and explore theoretically the dynamics of the devices which will be measured later on in the report. 15

30 CHAPTER 2. JOSEPHSON PHYSICS 2.1 The Josephson junction The Josephson Junction system consists of two regions of superconducting material separated by a region with reduced superconducting properties. The effect was first predicted by Josephson in 1962, [24], to occur between two superconductors separated by a thin insulating layer. The useful properties of the system result from the interaction between the wavefunctions of the Cooper Pairs on either side of the tunnel barrier. The phases on either side of the barrier can be driven out of equilibrium due to the weak connection. The theory was refined by subsequent authors, see for example [25]. Figure 2.1: Schematic representation of a Josephson Junction By recalling equation (1.3), we can treat each region of superconductor as in possession of a macroscopic quantum wavefunction, and several properties of the junction can be derived. The Josephson Junction is one of very few physical systems where a quantum mechanical variable can be investigated by macroscopic techniques, due to the existence of a single wavefunction extending throughout the entire bulk superconducting sample. In addition to revealing some interesting fundamental physics, the system has a number of applications, such as its use as a voltage standard and as a sensitive magnetic sensor. For an interesting overview of the applications of Josephson devices, see for example [26]. 16

31 CHAPTER 2. JOSEPHSON PHYSICS The Josephson relations Josephson predicted that the current flowing through such a junction would be proportional to the sine of the difference in phase between the two wavefunctions Ψ 1 (r) = Ψ 1 (r) e iφ1(r) and Ψ 2 (r) = Ψ 2 (r) e iφ2(r) of the superconducting areas. By adding a coupling term to the Schrodinger equation, as performed in [27], one can derive the following relationship: I = I 0 sin(φ) where φ = φ 2 φ 1 (2.1) This is generally known as the first Josephson relation, and for an ideal junction describes the Current-Phase Relation (CPR). As will be shown later, deviations from this relation can result in interesting physical effects, some of which are potentially useful for device applications. The application of a voltage across the junction results in a constantly evolving phase difference between the two superconducting electrodes. This is known as the second Josephson relation: dφ dt = 2eV h (2.2) Thus the rate of evolution of the phase defines a unique voltage to frequency relation. This yields a constant of 484M Hz/µV and indeed large series arrays of junctions under the application of microwave frequencies can be used to provide accurate voltage standards. It should be noted that the phase difference across the junction is not a physical (gauge invariant) property of the system, as it cannot be directly measured in the same way as the current, or voltage. The phase difference can be converted into a gauge invariant version by the transformation: 17

32 CHAPTER 2. JOSEPHSON PHYSICS γ = φ 2π Φ 0 A ds (2.3) where A is the vector potential. Further use of the phase term in this report is to be taken as representing the gauge-invariant phase. This is an important concept when considering the magnetic properties of the junction. The Josephson effect was subsequently generalised to systems involving any type of weak link. Josephson s original junction model is now known as the tunnel junction. Some examples of other weak link systems include a point contact, where the superconductivity is reduced by the dimensions of the point becoming comparable to the coherence length in the material, a normal metal junction (SNS), where the insulating layer is replaced by a normal metal, other proximity effect devices (SS S) consisting of dissimilar superconductors, and Grain Boundary Junctions, (GBJ), where a change in orientation of the crystal lattice causes a weak link. Many combinations of the above effects can be achieved, giving a wide variety of junctions. The different families of junction are classified according to their structure or method of fabrication. For example, SIS, denoting two superconducting electrodes separated by an insulating barrier, or SNS, relating to a normal metal barrier. Each junction has unique properties. Many variants on the standard junction are seen in the literature, for example junctions involving semiconducting or even magnetic materials as the barrier. The Josephson critical current A feature of a Josephson junction is the appearance of a critical current, I c, above which the system will no longer remain in the superconducting, zero voltage state. At T = 0K the critical current is denoted I c0. The critical current is temperature dependent, and follows the Ambegaokar-Baratoff relation [25] for BCS superconductors. 18

33 CHAPTER 2. JOSEPHSON PHYSICS I c R N = π ( ) 2e tanh 2kT (2.4) The I c R N product is an invariant quantity, which does not depend on the dimensions of the junction, and as such is commonly used as an indicator of the ideality of the junction. At T=0 the above result reduces to I c R N = π (0)/2e. The gap dependence (T ) can be estimated by BCS theory near to T c and therefore the I c dependence can be calculated in this regime. To determine the dependence over the full range T = 0 to T = T c requires a numerical computation of (T ). The Ambegaokar-Baratoff relation applies to ideal tunnel junctions, however for other types of junction the dependence may differ. Kulik and Omel yanchuk derive expressions for constriction junctions (for a review see [28]), taking into account the electron mean free path, l. The results show different temperature dependences for junctions in the clean and dirty limits. For other types of junction such as those discussed in Section 3.6, I c (T ) may also differ from the ideal case, which should be kept in mind when fitting to the data using I c as a fixed parameter. 2.2 The RCSJ model In order to describe the experimentally accessible current-voltage (IV) characteristic of a Junction, it is useful to introduce a physical model of the junction to which a standard circuit analysis technique can be applied. The ideal Josephson Junction can be modelled as a switch, with a voltage of zero below I c0, and a range of finite voltages above I c0, which follow the Ohmic response of a resistance, denoted R N. A schematic of such a characteristic is shown in Figure 2.2. The characteristic voltage, V c of the junction, is defined as the voltage generated at I c due to the normal resistance R N. 19

34 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.2: A schematic of the underdamped Josephson IV characteristic V c = I c R N (2.5) In our ideal junction V c corresponds to the gap voltage of the superconducting material, V c = V G. In a real junction, Ohm s law will still apply above the gap voltage, V G, however V c can fall below V G, i.e. the junction may switch to a voltage value lower than the V G. See Figure 2.2 for a schematic of a near-ideal Josephson junction IV characteristic. Real tunnel junctions are fabricated from a layer of insulator or normal metal between two superconducting plates. The electrical response of the junction will therefore be subject to an intrinsic capacitance term, in parallel with the ideal junction. In addition, the junction normal resistance, R N must be considered as a further parallel term, present at all values of current. A circuit diagram illustrating this is shown in Figure 2.3. The junction can now be modelled by including these extra current terms: I J = I s + C dv dt + V R (2.6) Where I J is the total current, and I s is the supercurrent flowing through an ideal 20

35 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.3: Circuit diagram illustrating the RCSJ model junction. Substituting in the first and second Josephson relations yields I J = I 0 sin(φ) + hc 2e φ + h 2eR φ (2.7) The system is now described as the resistively and capacitively shunted junction (RCSJ), and corresponds to a damped oscillator system. addressed in the literature by Stewart [2] and McCumber [29]. This system was first It is interesting to note that the inclusion of these terms vastly complicates the dynamics of the junction, and many different regimes of operation become available. One main feature to note is that the resistive state approaches the Ohmic behaviour of the ideal junction as the current is further increased past I c0. A few terms will now be introduced in order to further investigate the dynamics of the RCSJ model. 2.3 Energy of the Josephson Junction Although the Josephson Junction has no resistance in the superconducting state, and therefore no energy dissipation, the coupling energy between the two junction regions can be modified, and thus the junction can be thought of as storing energy. By considering the work done during an arbitrary change of phase across the junction, the energy-phase relation can be derived, which will prove to be a very important concept for later discussions. 21

36 CHAPTER 2. JOSEPHSON PHYSICS The work done to change the phase across the junction from φ 1 to φ 2 is given by W s = φ2 φ 1 I s V dt (2.8) Substituting the first and second Josephson equations (2.1), (2.2) yields W s = hi c 2e φ2 φ 1 sinφ dφ (2.9) The work done can also be considered as a change in potential energy of the Junction, U s (φ 2 ) U s (φ 1 ). This provides us with a way of expressing the potential energy of the junction system, U s = E c (1 cosφ) const where E C = hi c 2e (2.10) The washboard potential By considering the Gibbs free energy of the system, we can also model the junction as a current is applied. The general form of the Gibbs free energy is G = E F a, where the energy of a system is altered by the application of a force, F, to the system. In our case, F is equivalent to the bias current, I. To define the position variable a, it is useful to note that F ȧ can be described by the power, IV into the junction. Our variable, a, can thus be described as a = 1 F ȧ = V dt = hφ 2e + const (2.11) The Gibbs free energy of the system can thus be expressed as a combination of the energy stored in the junction and force term introduced by the bias current, U(φ) = U S (φ) I hφ 2e 22 + const (2.12)

37 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.4: The washboard potential for a Josephson Junction Using our derived expression 2.10 for the energy stored, the total potential energy of the junction is now U(φ) = E C (1 cos φ xφ) + const where x = I I c0 (2.13) Upon the application of a bias current through the Junction, the potential described in (2.10) is therefore tilted. Due to its shape, the resulting U φ is often referred to as the washboard potential. A schematic diagram of the washboard potential for 3 different values of I/I c0 is shown in Figure 2.4. The phase across the Josephson Junction can be localised in one of the wells, and is commonly conceptualised as a fictitious particle in such a potential. This corresponds to an analogous situation where a physical mass is located in a gravitational potential, which is similar in shape to U φ. In a similar fashion, in the static (superconducting) case, the phase of the wavefunction is localised in one of the wells. As the current is increased, the tilt of the potential becomes greater and the height of the potential barrier between adjacent wells is reduced. The relationship between 23

38 CHAPTER 2. JOSEPHSON PHYSICS barrier height and applied current is given by U = I c0φ 0 2π [(1 x2 ) 1/2 xcos 1 x] (2.14) Upon tilting the potential, until the gradient of U φ becomes positive for all values of φ, the mass (phase) will no longer be constrained to the local minimum, and will be free to move down the resulting slope. At this point the phase begins to increase with respect to time. This corresponds to the junction entering the voltage state, and the second Josephson relation applies (2.2). As the current is decreased and the minima reappear, the particle will slow down until it becomes retrapped in a potential well. The retrapping current, I r, is dependent upon the damping of the system. The addition of capacitance into the system is equivalent to the mass of the particle in this gravitational analogy, and the resistance to the drag force, or damping, as the particle moves down the potential whilst in the voltage state. 2.4 Damping in the Josephson Junction It is now important to introduce two relevant frequency scales within our argument. The first is the characteristic frequency of the junction, ω c, which is determined by the characteristic voltage of the junction. The characteristic frequency can be visualised as corresponding to the minimum response time of the junction to a change in current, i.e. how quickly the onset, or rise time, of the voltage state occurs. ω c = 2e h V c (2.15) The second is the plasma frequency, ω p of the junction, which can be thought of as the frequency at which the superconducting phase oscillates within a single potential well in the washboard potential. For an untilted potential, the plasma frequency is 24

39 CHAPTER 2. JOSEPHSON PHYSICS ω p = 1 LC = 2eIc hc (2.16) Where L is the sum of the geometric and kinetic inductances of the Josephson junction. Upon the application of a steady bias current I, the washboard potential is modified, and therefore the oscillation frequency is also dependent upon the applied current. From this consideration we obtain a generalised relation for ω p, ω 0 = ω p (1 x 2 ) 1/4 where x = I I c0 (2.17) The quality factor of the junction is given by Q = ω p RC, and will be of use in some subsequent derivations. The damping in the junction can be expressed as the ratio of the characteristic frequency to the plasma frequency. This is commonly referred to as the McCumber parameter, β c β c = ( ωc ω p ) 2 (2.18) When the damping is low, β c >> 1, the junctions exhibit hysteretic effects. To visualise this effect, again it is useful to think in terms of the mechanical analogue to the washboard potential. The hysteresis occurs because once the junction has entered the voltage state, the mass acquires a finite momentum as it moves down the potential, and the washboard must therefore be tilted back well beyond the point of inflection in order to retrap the state. In the case where β c < 1, the junction is overdamped. The case of overdamping will not be considered in detail in this report, as the models under consideration work best when the damping in the junctions is in the high-β limit. For a detailed treatment of overdamped junctions see Ambegaokar [30]. The RCSJ junction model has had great success in explaining the shape of the IV curves of numerous different types of Josephson junctions. However, there are many 25

40 CHAPTER 2. JOSEPHSON PHYSICS subtleties involved with the properties of real junctions. Some of these properties will be discussed in Section 3.6. They are particularly relevant to the case of the High T c superconductors, whose properties deviate quite substantially from the ideal models described here. The retrapping current In a hysteretic Josephson Junction, the current at which the system returns to the zero-voltage state, I r0, differs from the critical current, I c0, as previously illustrated in Figure 2.2. The retrapping current occurs at I r0 = 4I c0 πq (2.19) This expression is derived from the consideration of a mass rolling down the washboard potential; by considering the potential energy versus kinetic energy, one can derive an expression for a point along the return current curve, where the particle will not be able to reach the next maximum. The McCumber parameter can then be expressed in terms of this retrapping current. β c = β can be redefined from the previous expressions to yield 4Ic0 πi r0 (2.20) β c = 2πI cr 2 N C Φ 0 (2.21) where C is the the total capacitance, including any stray capacitance in parallel with the junction, for example any coupling through the substrate. The capacitance of the junction can therefore be estimated from the other physical parameters. 26

41 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.5: The phase diffusion branch of a Josephson junction IV curve Phase Diffusion and retrapping In cases where the junction is damped at high frequencies, β c 1, the system may enter an additional regime known as the phase diffusion branch. In this case, the onset of the voltage state is suppressed. Instead of the system being free to move down the wasboard potential, it loses energy due to the viscous drag force and can be retrapped in a subsequent minimum. At this point the combination of thermal activation and retrapping results in the particle undergoing a random walk behaviour, and can be viewed as diffusing down the potential. This manifests as an additional resistive slope observed on the critical current, as shown in Figure 2.5. Once the potential becomes steep, the junction enters the fully running voltage state. The effect is most noticeable in overdamped Josephson Junctions, however it can also manifest in junctions with hysteresis, if the damping is still present at high frequencies. A related damping effect is that of thermal retrapping. When the temperature is high enough, there becomes more of an opportunity for the particle to move backwards, up the washboard potential, and thus the rate of escape from the potential may become less if k B T becomes high compared to the barrier height in both directions. 27

42 CHAPTER 2. JOSEPHSON PHYSICS 2.5 Junctions in an applied magnetic field Upon the application of a magnetic field in the plane of a Josephson junction, it is possible to suppress the critical current, I c. Specifically, the value of the junction critical current follows a Fraunhofer type behaviour as a function of field. For the purpose of this report, it is useful to note that the suppression of I c is a reliable way of changing a fundamental property of the Junction, as will be explained in Section 4.1. A magnetic field will penetrate into the junction over a distance of λ J, [ ] 1/2 cφ 0 λ J = (2.22) 8π 2 J c (2λ L + d) which is greater than the London penetration depth, λ L, for a bulk superconductor. If the junction width W is smaller than 2λ J, it can be considered to be in the small junction limit. The physics of larger Josephson Junctions (sometimes denoted long Josephson junctions) is different, as circulating currents can be set up along the length of the junction, and the phase difference can vary along the length of the junction, as well as across it. All the devices considered in this report are in the small junction limit. Fiske resonances In the voltage state, a Josephson Junction contains moving flux lines. Upon the application of a magnetic field, certain modes of moving flux can be enhanced. These cause the appearance of steps in the IV characteristic, known as Fiske resonances or Fiske steps [31]. The presence of such steps is a useful tool in determining some fundamental properties of the junction. From the voltage at which the resonances occur V f and the width of the junction W, the speed of light in the junction c J can be calculated. 28

43 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.6: Schematic of a Fiske resonance in a Josephson junction This is equivalent to the inverse square root of the LC product. c J = 2V f1w Φ 0 = 1 L C (2.23) L = ( ) µ0 λ L coth h (2.24) h 2λ L The inductance of the junction per unit length, L, can then be estimated by treating the junction as a thin film resonator, where h is the film thickness, and λ L is the London penetration depth (see Section 1.3). From equations (2.23) and (2.24), a value of the capacitance of the junction per unit length can be estimated. This can help to confirm the approximate value of capacitance calculated from hysteresis measurements. The SQUID A superconducting weak link in a loop provides an interesting system, which yields many useful applications. As already discussed in Section 1.4, in a loop configuration the total flux within the loop is quantised. However, with a weak link in the system, it is possible to maintain a phase difference and therefore a voltage across the Josephson barriers, and as such the critical current of such a system is sensitive 29

44 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.7: Left: Schematic of a SQUID Right: The magnetic diffraction pattern for an ideal SQUID, analogous to two-slit interference. For a real SQUID I MAX usually remains greater than zero for all Φ/Φ 0 to the total flux, Φ, in the loop. Such a system is known as a Superconducting QUantum Interference Device (SQUID). In the case of the DC SQUID there are 2 junctions in the loop, and this system behaves as a two-slit interferometer. In the same way as the single junction exhibited a single slit magnetic diffraction pattern, the standard Fraunhofer diffraction pattern for a 2-slit system is observed in the critical current of the SQUID as the field is varied, as shown in Figure The Current-Phase Relation Josephson s first equation (2.1) introduces the idea of the Current Phase Relation (CPR) of an ideal Josephson Junction, which follows a sinusoidal relationship. However, in a real junction, there are deviations from this ideal, which can be expressed by a Fourier series of sinusoids: I(φ) = I c1 sin(φ) + I c2 sin2φ +... (2.25) Consider two superconducting regions (islands) placed in close proximity. In the case of the ideal junction, the link is weak enough that the superconductivity can fall to zero between the two regions, and the phase can then easily adjust. In this case, the behaviour of the system approaches the Josephson effect, with a sinusoidal 30

45 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.8: Resulting modification of the CPR in a weak link as the barrier transparency is adjusted. Γ refers to the coupling strength between the two superconducting regions, with a low Γ value corresponding to strong coupling. From [3] CPR. As the regions are moved closer together, or the barrier between them is made more transparent, they become more strongly coupled, and a phase gradient cannot be easily maintained along the length of the link. The CPR then follows a more distorted shape, correponding to the higher harmonics of equation 2.25 becoming dominant. Figure 2.8. The effect of changing the coupling strength, denoted Γ, is shown in Measurements of the CPR in superconducting junctions were first reported experimentally by Rifkin and Deaver [32] using an RF technique. In their experiment, a ring of superconductor containing a Josephson Junction is inductively coupled to a tank coil circuit. By measurements of the effective inductance, L eff of the coil and Junction loop, the current-phase relation can be numerically reconstructed. The predictions and observations of a non-sinusoidal CPR are of great importance in the fundamental understanding of the Josephson Junction and give us information about the underlying mechanisms of superconductivity. More will be discussed on the topic of the CPR in the section on High temperature superconductors, as in the work presented here and the future experiment proposals, there are some interesting effects to be explored in HTS junction systems. 31

46 CHAPTER 2. JOSEPHSON PHYSICS 2.7 Macroscopic Quantum Tunnelling As the majority of the work described in this report directly concerns the secondary macroscopic effects observable in Josephson systems, in particular the Macroscopic Quantum Tunnelling, it will now be described in detail. As noted in an earlier section, one of the main features of superconductivity is the possibility of describing the Cooper pair condensate in terms of a wavefunction, i.e. all the electrons form a coherent state; their phase modelled by a macroscopic variable. From equation (2.10), the energy landscape of a Junction is modelled by a sinusoidal potential. By considering the decrease in height of the barrier U more carefully as the current is increased, it is interesting to investigate exactly how the phase escapes from the potential well. Indeed it is the escape rate from the well, as a function of applied current, which is available for experimental interrogation. There are several ways in which the superconducting phase can prematurely escape from the confines of a washboard minimum. The energy of the state can be excited by thermal fluctuations, causing the phase state to gain enough extra energy to be thermally activated over the barrier. The second is that if the barrier height (and more importantly the width) becomes small, the wavefunction has a finite probability of extending beyond the barrier, and the phase is able to tunnel through. This phenomenon is known as Macroscopic Quantum Tunnelling (MQT). It should be noted at this point that it is the phase of the entire wavefunction that is described by the tunnelling variable. These two cases of premature escape from the well will now be described in a little more detail. The mathematical treatment has been intensively studied in the cited references, and as such will not be reproduced fully here. However, some of the key aspects and results of the derivations will be highlighted in order for the reader to gain an insight into the fundamental physics of the systems under consideration. A 32

47 CHAPTER 2. JOSEPHSON PHYSICS useful and approachable introduction to the theoretical model is summarised in the 1987 paper by Martinis [33]. Thermal escape The case of thermal activation from a washboard potential was first solved by Kramers [34], in the context of the activation energy barriers associated with chemical reactions. Early theoretical work has been performed on the effect of fluctuations on the Josephson critical current by Ivanchenko [35] and Ambegaokar [30]. In this section we follow the standard notation for treatment of the thermal activation, as developed in these papers. The experimental progress in the field of thermal activation and MQT will be reported in Section 4.3. In the thermal regime the escape is dominated by fluctuations produced by the finite temperature of the system. As such, the escape follows an exponential Boltzmann dependence. The inverse lifetime of the state is known as the escape rate, Γ, and takes the form: ( ω0 ) Γ t = a t exp 2π ( ) U k B T (2.26) Where U is the barrier height previously described by equation 2.14, and ω 0 is the plasma frequency under an applied bias, as defined in equation The escape rate is preceded by a thermal prefactor, a t, derived by Büttiker [36]. a t = 4α ( ) 2 (2.27) 1/2 1 + αqk B T U The numerical factor α is approximately unity although a value of 1.4 has been estimated numerically by Riskin and Voigtlaender [37]. The thermal prefactor, a t is however a small correction and as such the value of α is relatively unimportant. In 33

48 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.9: Illustration of the various regions under consideration in the MQT regime the work presented here, the value of α is taken to be equal to unity. Quantum escape For ease of calculation, the system is first considered with the conditions β << 1 and T = 0. The quantum mechanical case is then treated by considering the wavefunction as localised in one of the wells of the tilted washboard potential. The Schrödinger Equation can then be applied to the system. The three regions under consideration are illustrated in Figure 2.9. In the first region, the solution is identical to the ground state of a harmonic oscillator potential. In the region under the barrier (II), one can apply the WKB approximation to generate the exponential decay of the probability in this region. In region III, the solution is that of an unbound probability wave which decays as it travels to the right. A more thorough treatment can be found, for example, in Likharev [38]. The lifetime of the quantum state can then be determined from the solution of the transmission probabilities between these three regions. τ 1 Q = ω 0 2π ( 864π U hω 0 ) exp ( ) 36 U 5 hω 0 (2.28) 34

49 CHAPTER 2. JOSEPHSON PHYSICS The escape rate can also be inferred from this method: ( ω0 ) [ Γ q = a q exp 7.2 U ( )] 2π hw 0 Q (2.29) Similar to the thermal case, there is also a quantum prefactor, a q associated with the quantum escape rate. ( ) 1/2 864π U a q = (2.30) hw 0 It should be noted that the escape mechanism provided by the quantum tunnelling is independent of temperature. Intermediate escape Between the two regions, there is an intermediate regime [39], [40]. The regime follows the same dependence as the thermal case, but with a modified prefactor: ( ω0 ) Γ i = a i exp 2π ( ) U k B T (2.31) a i = sinh( hω 0/2k B T ) sin( hω 0 /2k B T ) (2.32) Switching histograms Because the switching of the Josephson junction (either thermally or by quantum tunnelling) is a random process, in order to extract useful information it is necessary to compile statistics of repeated switching events, and analyse the probability of a particular switching behaviour. The concept is illustrated in Figure Each time the Josephson junction enters the voltage state, the value of critical current will be slightly different. By compiling the values of critical current, the switching 35

50 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.10: a.) Stochastic switching of a Josephson junction b.) Probability of a switching event as a function of applied current probability as a function of current can be presented as a histogram. It is the shape and spread of the histogram which will be important in the description of the work presented here. The shape of the histogram can be reconstructed from the escape rate. This will be addressed in Section 4.2. The crossover temperature As the temperature, and therefore the thermal fluctuations decrease, the spread of critical currents will become smaller, as the junction is less likely to be excited into the voltage state by a thermal process. The histogram width, σ, therefore narrows. However, quantum fluctuations will persist down to the lowest temperatures, thus the histogram width reaches a constant, finite value in the quantum limit. The value of temperature at which the escape due to thermal fluctuations in the junction equals the escape due to quantum tunnelling is known as the crossover temperature, T*. T = hω 0 2πk B (2.33) This can also be seen from equation The thermal escape rate decreases with temperature due to the exponential Boltzmann term. The escape rate due to tunnelling is, to a first approximation, independent of temperature. Hence, below T*, 36

51 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.11: Diagram illustrating how the width of the switching histogram changes with Temperature the quantum tunnelling will become the dominant mechanism for escape from the potential well. Escape temperature The escape temperature is defined as T esc = U k B ln(2πγ f /ω 0 ) (2.34) and is interpreted as a single average temperature at which the probability of the particle leaving the potential well is above a particular criterion. In practise, one must choose this value, which corresponds to fixing the value of Γ, here denoted Γ f, in order to calculate the escape temperature from a set of data. This selects a particular point on each dataset. The escape rate is an important derived property, as it allows comparison of independent samples, which may have different experimental parameters e.g. capacitance or critical current. By presenting the escape temperature, datasets can be normalised. 37

52 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.12: Effect of dissipation on the escape rate from a cubic potential. The value of α is proportional to the frequency-independent dissipation in the system. From Grabert et al. [4] The effect of dissipation The quantum tunnelling processes will begin to occur at a finite temperature, however, the original model considered the tunnelling process only at T = 0. Caldeira and Leggett theoretically considered a system with dissipation in their 1981 paper [41] in which they derived a method of modelling RSJ behaviour by allowing a friction term into the quantum mechanical description of the junction. This damping is introduced mathematically using a dissipation kernel, which corresponds to a bath of harmonics oscillators effectively acting as the ohmic resistance in the RSJ model. The theory predicts that the effect of a finite temperature on the system is to enhance the tunnelling rate in the presence of an ohmic dissipation, which is described as one where the damping co-efficient is frequency-independent at low frequencies. Grabert et al. have studied the effect of such dissipation on the thermal and quantum escape from various types of potential [4], [39]. One of the results of their calculations of the modified escape rate is shown in Figure They find that if the junction is underdamped, the enhancement of the escape rate is small, however in the presence of moderate damping, the escape rate is enhanced by a factor of T 2. 38

53 CHAPTER 2. JOSEPHSON PHYSICS In the region 0 < T < T, the quantum tunnelling is enhanced by a factor which is calculated numerically. In the region around T* and above, the enhancement has a T 2 dependence. ln Γ Q(T ) Γ Q (0) = s(α)t 2 (2.35) Thus if the junction follows the RSJ behaviour and exhibits moderate damping, one should expect an enhanced escape rate in both the quantum and the thermal regimes. The crossover temperature is modified to become: T = hω ( ) 2 1/2 1 2πk B 2Q 2Q (2.36) Experimentally it is difficult to implement an adjustment for dissipation, as the damping parameter may not be directly related to the measured parameter R N, (the resistance of the junction in the normal state) at low frequencies. 2.8 The qubit One major proposed application of the Josephson Junction in a superconducting circuit as a building block within a quantum bit (qubit) circuit. A qubit is described as a system with two quantum states which can be entangled, such that a wavefunction of the system with a well-defined set of eigenstates is established. The entangled states can be used to hold quantum information, as a tool for performing calculations on states in parallel. The discussions on qubits in this report will be for background interest only, as the work performed in this project concentrates on the fundamental physics of Josephson Junction based devices that give rise to such potential qubit applications. For practical qubit investigations directly stemming from this research, see Section 8.2, which explains some of the proposed applications of the work described here. An excellent review article of current qubit designs and 39

54 CHAPTER 2. JOSEPHSON PHYSICS theory can be found in [42], and a review article summarising the progress of the field of qubit physics can be found in [43]. The Josephson Junction is particularly well suited to addressing the difficult problem of scalable qubit technology. The inclusion of several junctions into a loop gives rise to an inductive component in the system. This causes the washboard potential to be modulated by a parabolic term, creating a double well potential from the conventional response (see Figure 2.13). The MQT is a precursory work to many qubit experiments, as it can help establish the temperature and magnetic conditions in which the quantum effects are dominant. Qubits can be classified into three main types: Charge, phase and flux Qubits. Recently, all three types have been under investigation, in addition to hybrid versions such as the quantronium, which utilises both the charge and phase degrees of freedom [44]. A charge qubit makes use of the electron charge on a small island as the quantum property, and is commonly known as a Cooper pair box. The Japanese group at NEC have published several experimental results on the charge qubit in a Cooper pair box system [45], [46], and were the first group to observe coherent quantum oscillations in their qubit device. Vion et al. have also seen coherence in their mixed charge-phase qubit [44]. However, the charge qubit will not be discussed in detail in this report, as the Josephson devices under consideration in this work are more suited to a flux or phase driven device. The flux/phase qubit A phase qubit can be a single Josephson Junction, and uses the parameter of phase across the junction as the quantum variable. In these devices it is the coherence of the phase on either side the barrier which is the important property. A flux qubit usually consists of one or more Josephson Junctions in a ring, whereby the appropriate variable is the flux in the centre of the ring. The central idea of the flux/phase qubit is that a double well potential can be 40

55 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.13: Schematic of the energy wells in a flux qubit established by suitable biasing of the device. A schematic of such a potential is shown in Figure The coherent superposition of the two states 0 > and 1 > is achieved by allowing the phase variable to tunnel between the two wells. In order to establish the 0 > and 1 > states as distinct, at least two energy levels are required in each well. In the ground state, the phase will thus be localised in one of the wells. Upon the application of a microwave pulse, (or in some experiments, upon tilting the potential) the upper energy level can be populated and the phase will oscillate coherently between the two wells. It is useful to visualise this behaviour in terms of qubit rotations around the Bloch sphere (Figure 2.14). The qubit is prepared by applying a magnetic bias to tilt the double well, and ensure that the phase is well defined in the ground state of one of the wells. This performs a unitary transform on the system, rotating it through 90 from the ground state to one of the these excited states, which corresponds to the phase rotating about the z-axis of teh Bloch sphere. When the pulse is removed, the system undergoes free, coherent oscillations between the two degenerate, excited quantum states. This can be thought of as analogous to the probability density of the wavefunction oscillating between the two wells, or in the Bloch representation as the state rotating about the x-axis and the z-axis in a spiralling fashion. The technique is directly analogous to that of manipulating the spin state of a set of atoms using Nuclear Magnetic Resonance (NMR). The coherent state will persist until the system relaxes, at which 41

56 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.14: Schematic of the Bloch sphere representation of a qubit point it will return to the ground state. The coherent oscillations are known as Rabi oscillations and can be measured experimentally by interrogation of the qubit. An example of such a measurement is shown in Figure Most flux qubit devices are fabricated using e-beam lithography, due to its ease and high speed of batch production. Devices with features as small as 50nm are routinely achieved using this process, although the qubits need not be this small; many enclose an area of several square microns. Most devices have been fabricated using Al as the superconducting material of choice, although Nb is also be a suitable candidate for low T c experiments. In 1999, Mooij et al. showed that by constructing a qubit using 3 Josephson Junctions in a ring, the need for large loop inductance is eliminated, and the loop can be made much smaller [47]. The first experimental evidence of the quantum coherent state in a flux qubit was performed by the Delft group [5]. Their qubit was based on the design aspects highlighted by Mooij. The device was fabricated using a double angle shadow evaporation technique, in which a single mask is used to make two separate depositions forming overlapping areas. These areas define the Josephson junctions. A scanning electron microscope image of the qubit is shown in Figure Subsequent careful measurements of the qubits produced by this group have been documented [48], [6], [49]. In these measurements the sources of decoherence 42

57 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.15: SEM image of a 3 junction flux qubit. From Chiorescu et al. [5] are carefully estimated and eliminated. It is also possible to couple qubits together to use them as gates, or as arrays, both schemes exploit the advantages of quantum parallelism. There are three main proposals for the coupling of qubits: capacitative, inductive, and coupling via a Josephson Junction. If the coupling can be made switchable, or even tunable, the entanglement level of the qubits can be controlled. Entanglement of two qubits using a capacitative coupling technique has been demonstrated [50], and new methods of controlling the interaction between the qubits are under investigation, see for example [51] and [52]. In the flux qubit system it has even been possible to fabricate 4 qubits with controllable inter-qubit couplings [53]. Experimental interrogation of the qubit Qubits are tested using a variety of methods. The work described in this project is an important method of assessing whether a particular Josephson Junction device will be suitable as a potential qubit candidate. The first demonstration, which is of particular relevance here, should be that the system can potentially exhibit Macroscopic Quantum Coherence (MQC), which can be done by ensuring that the device is operating below the thermal-quantum crossover, T*. Level spectroscopy using microwave excitation can then be employed to determine the position of the energy levels within the Josephson potential wells. Once the Josephson junctions have been shown to be suitable, they can then be fabricated in a loop configuration, 43

58 CHAPTER 2. JOSEPHSON PHYSICS Figure 2.16: Rabi oscillations of a phase qubit detected with a DC SQUID readout technique, from Bertet et al. [6]. The two traces show two different measurements methods: Light grey measured using a DC current pulse, dark grey measured using a resonant activation technique. and demonstrations of the coherent oscillations between states can be performed. Biasing of the qubit is achieved by applying a magnetic field such as to ensure a flux of Φ 0 /2 inside the qubit loop. This is usually done by a nearby on-chip bias coil, although it can be done with an external magnetic field, however the need to eliminate magnetic noise usually means that the on-chip biasing method is preferable. There are also some more novel proposals for implementing an intrinsic Φ 0 /2 field inside the qubit, such as the inclusion of a small magnetic dot in the centre of the qubit [54]. Some more fundamental ways of biasing the qubit using the properties of the superconducting materials themselves will be discussed in Section 3.8. The most common way to observe a superposition of states is by the Rabi oscillation method, in which the biased sample is irradiated with microwaves tuned to the frequency of the energy gap corresponding to that bias. Incident microwave radiation initiates a coherent oscillation between the states which can then be measured with a readout technique. The oscillations proceed with a period of τ = h/e J, where E J = ( h/2e)i c0 is the Josephson energy. The oscillations are modulated by an exponential factor caused by dephasing of the qubit. Observing the coherent quantum oscillations is the first step towards performing quantum gate operations. 44

59 CHAPTER 2. JOSEPHSON PHYSICS As the qubit s state is destroyed by the action of a measurement, the Rabi oscillations are measured by preparing the qubit in the same state, and allowing it to evolve. Each measurement of the qubit s state is performed at a different time subsequent to the initial pulse, and thus the oscillation between the states can be seen by reconstructing many of these single-shot measurements. The standard methods of interrogating the state of a qubit consists of either an RF tank circuit, or a DC SQUID, as mentioned in Section 2.5. The tank circuit works via a weak inductive coupling to the flux qubit. The local magnetic conditions of the qubit therefore affect the resonant factor of the coil. In the case of the DC SQUID, the switching voltage is highly sensitive to the flux inside the loop, and thus by placing the qubit inside the SQUID, one can infer the flux state of the qubit. These techniques are well documented in the literature, see for example [42] for a review. The problem with both of these methods is the so called back-action on the qubit. The tank coil is directly coupled to the qubit by a mutual inductance, and therefore crosstalk occurs between the coil drive current and the qubit. In the DC SQUID system, the SQUID remains in the superconducting state most of the time, thus does not affect the qubit as much, however upon switching, the voltage generated across the SQUID perturbs the qubit system. Some novel systems for minimising the back action on the qubit have been proposed (see for example, Petrashov et al [55]), and this area remains currently under intense investigation. Decoherence in qubits Flux and phase based qubits tend to have longer decoherence times compared to charge qubits, due to their insensitivity to charge-based fluctuations (for example from charge moving in the substrate). For the same reasons it is thought that flux qubits will ultimately be more scalable than their charge counterparts, as the larger the distance between entangled qubits, the more prone to charge fluctuations they will be. Even though flux qubits are, by their nature, more tolerant to noise than 45

60 CHAPTER 2. JOSEPHSON PHYSICS charge qubits, however they can still be affected by sources of noise such as electron trapping by defects in the junction regions, or stray magnetic fields. Experiments have been performed on Josephson Junction systems in order to characterise the decoherence due to noise and other sources. An experiment by Berkley et al. [56] probed the microwave levels inside the Josephson washboard potential, and demonstrated that the Josephson system is indeed suitable, if careful design of the devices is employed. 46

61 Chapter 3 The cuprate superconductors This section will focus on properties of several cuprate compounds exhibiting superconductivity. The fundamental differences between these materials and the conventional (Low T c ) superconductors will be highlighted. Three important systems will be discussed: The intrinsic Josephson junction, the narrow track (or 1-Dimensional) superconductor, and the grain boundary junction. The final sections of this chapter are concerned with the potential advantages of using such materials in device fabrication, in particular their advantages when operated as qubit systems. 47

62 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS 3.1 Crystal properties In 1986, the first so called High Temperature superconductor, La 2 x Ba x CuO 4 was discovered, with a T c of 30K. In the following few years, the successful growth of many more superconductors belonging to this family were also reported. The first High Temperature superconductor (HTS) to have a transition temperature above that of the boiling point of liquid nitrogen, 77K, was Y Ba 2 Cu 3 O 7 δ with a T c of 92K. Although there are various other instances of HTS, such as the compound MgB 2, this work will focus on the cuprates. The main feature of the cuprate superconductors is their common crystalline properties. The most common cuprates are Y Ba 2 Cu 3 O 7 δ (YBCO), Bi 2 Sr 2 CCuO 2, (BSCCO) and T l 2 BaCCuO (TBCCO). A schematic of the crystal structure of Y Ba 2 Cu 3 O 7 δ is shown in Figure 3.1. In fully oxygenated YBCO, the crystal structure is orthorhombic. In YBCO with a δ > 0.7 the material is no longer superconducting, and for δ = 1 the crystal structure assumes a tetragonal phase. The cuprates consist of Copper-Oxide planes, in which the supercurrent is localised. This means that the current transport is strongly in-plane. In YBCO, J c values of critical current density of A/cm 2 are typical in the a-b crystallographic direction. Supercurrent can flow between the planes, along the CuO chains, however the maximum current density in the c-axis direction is generally much lower, and the resistivity in the normal state is higher. The anisotropy of such a superconductor is defined by the ratio of the resistivities in the different crystallographic directions. ρ ab /ρ c of 10 is common in YBCO, however in BSCCO and TBCCO the value can be much higher. It is, however, notoriously difficult to fabricate samples from High T c superconductors, and to ensure both reliably and reproducibly. One of the main problems associated with the materials is their tendency to lose oxygen from the surface, resulting in a lowering of the critical temperature and critical current density. The T c of YBCO is a function of the oxygenation. Growing high quality single crystals of 48

63 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.1: Perovskite crystal structure of YBCO High Temperature superconductors is difficult; samples often contain crystal grain boundaries and twinned regions, which may or may not be desirable. Where crystals with a single grain boundary are required, the boundary can often be highly non-ideal (see Section 3.6). The degradation of High T c samples over time is also problematic. 3.2 Symmetry of the order parameter A striking feature of the cuprate superconductors is the large volume of evidence suggesting an unconventional pairing state of the electrons within the condensate. For this reason the materials are often described as exhibiting d-wave superconductivity. This state is characterised by lobes and nodes in the amplitude of the order parameter, Ψ(k x,y ), and a difference in phase of different regions in k-space. Unlike the conventional s-wave case, where Ψ is constant in amplitude in all directions in k-space, in a d-wave superconductor the amplitude and phase of Ψ is intrinsically as- 49

64 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.2: a.) Schematic polar plots showing the properties of electrons in a d-wave superconductor a.) The magnitude and phase of the wavefunction, Ψ; b.) The magnitude of the energy gap, sociated with a real physical property of the lattice, specifically the crystallographic direction. The order parameter therefore changes with the wavevector, k. This is shown schematically in Figure 3.2. The energy gap,, in a d-wave pairing state follows a similar relation, the main feature is that it becomes suppressed at the nodal points, therefore there exist excitations of the superconducting state with particular values of k in the form of quasiparticles even at T = 0K. There are methods of detecting the presence of d-wave superconductivity. Wollman et al. demonstrated experimentally the presence of the d x 2 y 2 symmetry in YBCO by measuring the magnetic diffraction pattern of an YBCO-Nb DC SQUID, with the two tunnel junctions fabricated along different crystallographic directions of the YBCO single crystal [7]. The results were found to be in agreement with a phase shift of π in the order parameter around the SQUID loop, caused by the change in direction of the current within the YBCO. This experiment is of particular relevance to the work in this report. The presence of an intrinsic π shift can be of use in the design of qubit systems, as one can inject current in different directions to induce such a phase shift across the junction, a 50

65 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.3: Measurement of the intrinsic phase shift in a d-wave SQUID a.) Experimental setup b.) Magnetic diffraction patterns. From Wollman et al. [7] 51

66 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS technique which will be discussed further in Section 3.6. The same will occur if the negative lobe of the d-wave order parameter forms one side of the junction and an s-wave region forms the other. There are other methods of introducing a phase shift into a Josephson junction. The use of an thin ferromagnetic layer between the superconducting areas (known as an SFS junction) can also be employed. In this case, the width of the barrier region will define the junction properties. For recent work in this area, see for example [57]. 3.3 Tunnelling in HTS So far the Josephson effect has concentrated on the ideal tunnelling between two ideal, s-wave superconducting regions. In a d-wave superconductor, the tunnelling, and the resulting effects can be rather different. When a Josephson junction is formed from an s-wave superconducting electrode coupled to a d-wave one, the tunnelling is angle dependent, as the magnitude of the order parameter has a spatial dependence, as shown in Figure 3.2. In the 0-0 direction, the tunnelling behaves similarly to the case of an s-s junction. However, when the node of a d-wave system is aligned with the s-wave electrode, the tunnelling behaves more like an NS interface (described in Section 1.4), even at T = 0K, and the dominant transport mechanism across the barrier takes place via Andreev reflection. Quasiparticles with energies greater than 2 can also be injected across the barrier into the s-wave superconductor. The system is therefore similar to the NS interface, and the same characteristic effects are observed in such systems, such as Andreev Bound states and complications from quasiparticle interactions. One can also produce a d-wave-d-wave Josephson tunnelling effect. This is the basic form of the grain boundary junction, which will be described in greater detail in the next section. 52

67 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS 3.4 Intrinsic Josephson Junctions In certain High T c superconductors, Josephson Junctions are formed intrinsically between the superconducting a-b planes of the crystal, i.e. along the c-axis, between the CuO planes. These intrinsic Josephson Junctions (IJJ) can be found in, for example, BSCCO, TBCCO [58], and in de-oxygenated YBCO [59]. IJJs are formed as part of the perovskite crystal structure and are therefore naturally uniformly spaced and homogeneous in composition and width. The use of IJJs helps eliminate some of the inconsistency in the junction fabrication process, and also reduces the number of lithography steps required. The only parameters that need to be tightly controlled are the lateral junction dimensions, in contrast to Niobium and Aluminium film technologies, which both require a highly characterised and controlled process to reproduce the oxide tunnel barrier. Current state of the art junction technology involves Al or Nb trilayer processing, and is used widely in rapid single flux quantum (RSFQ) circuitry [60], whereby junctions and SQUIDs are used to encode information in the form of a quantum of flux, and transfer it around a circuit. Although the technology for these types of junctions is advancing rapidly, they may never be able to be fabricated with as great a precision as the intrinsic junctions available within the HTS materials. Intrinsic Josephson Junctions have already been used for some specific applications; for some recent advancements in the field of BSCCO devices, see for example [61], [62], and [63] Dimensional Superconducting tracks 1D superconducting devices can behave in a similar way to the Josephson Junction, if a region of the small superconducting track becomes normal, forming an SNS-type junction. In this nanowire system, the normal region is known as a phase slip centre (PSC) as the wavefunction can no longer maintain the high phase gradient, and so 53

68 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS advances by 2π at this point. The PSC can take the role of a virtual junction with a normal-region width equal to the coherence length, ζ. The governing thermal and quantum activation processes of the phase slip can be monitored in a similar way to the switching of the Josephson Junction. Phase slip processes always occur in overdamped junction systems, and thus the resistance plays an important part in the dynamics of the system. As the junction resistance is increased, the damping decreases. At sufficiently high resistances the system is so underdamped that the particle within the well occupies an average position high enough for quantum tunnelling to naturally take place between the wells. In this case, even for an untilted washboard potential, all wells become effectively linked. The material can no longer maintain a superconducting state as the phase cannot be localised. The value of resistance at which this occurs is known as the quantum of resistance, R Q, where R Q = h/4e 2. If the normal state resistance of a small track is greater than this value, the superconducting state will never fully manifest. The crossover between the superconducting behaviour and the resistive behaviour is an interesting area to investigate. Thermally Activated Phase slips As the width of the track becomes small, the energy required to turn a section of the track normal decreases, as the difference in Gibbs free energy of the superconducting and normal state below T c is given per unit volume. When this energy becomes comparable to k B T, it will be favourable for these processes to occur. The passing of a small current through the superconductor would under normal circumstances not be sufficient to tilt the washboard potential far enough to cause entry into the voltage state. However, a local thermal fluctuation in addition to this current can incite a small area to become normal, causing a phase slip and a consequent voltage spike. In a standard measurement, the time over which the voltage is recorded is much longer than the duration of the thermal activation events, and so these 54

69 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS voltages are averaged, with the result appearing as if the phase slip were a dissipative ohmic region. The phase slips themselves are spatially located at weak points in the material, where it may require slightly less energy to form the normal region. Current driven phase slips A current passing through the superconductor can cause phase slips to nucleate if the current is such that a relaxation by 2π in the order parameter (as happens during a phase slip event) would lower the energy of the system. In the washboard potential, at a bias current of just higher than I c, the phase can be imagined as a damped particle travelling at its terminal velocity down the washboard potential. As it approaches the points of inflection it will be travelling more slowly and the phase will therefore be advancing more slowly, but as it falls down the steeper parts the phase will suddenly change much more quickly, or slip. For an overview of the physics of 1-dimensional superconductors, see [21]. Current driven phase slips have been investigated in High T c superconductors by Reymond et al. [64] LAMH theory In 1967, Little [65] suggested that the transition from resistive to superconducting behaviour may not be a true phase transition if the sample was of non-infinite dimensions, specifically a 1 dimensional loop. The paper suggested that in this case the persistent current near T c may have a mode of decay through thermal fluctuations. This work was the precursor to a full theory of thermally activated resistive behaviour below T c, initially proposed by Langer and Ambegaokar in 1967 [66], and furthered in 1970, by McCumber and Halperin [67]. Their joint work became known as the LAMH theory of thermally activated phase slips (TAPS). The theory will not be treated here in detail, but it is useful to cite the key result, which is a description of the behaviour of the resistance, R LAMH, below T c 55

70 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS R LAMH (T ) = ( ) ( ) h hω e F/k BT 4e 2 k B T (3.1) where Ω is the attempt frequency, defined as: Ω = L ξ ( ) 1/2 F 1 (3.2) k B T τ GL and F is the barrier height, defined as: F = Where L is the length of the superconducting nanowire, H c ( ) H 2 c Aξ (3.3) 8π is the critical field of the superconductor, and τ GL = [π h/8k(t c T )] is the relaxation time, from the Ginzburg-Landau theory. The exponential dependence of R LAMH upon temperature indicates a strong downturn in the resistance, hence such data are usually plotted on a log scale. There have been many reports in the literature of thermally activated phase slips in conventional superconductors, and many attempts to fit to the theory, however in several cases there is a large deviation from this theory, especially at low resistance, which leads to the possibility of a new regime of phase slippage: The quantum phase slip. Quantum phase slip The quantum tunnelling of phase slips can also produce a resistive term, which adds to the R LAMH defined previously. The theory of quantum phase slips were considered later, by Saito and Murayama, and first observed experimentally by Giordano [68]. Several systems have since been studied, including phase slips in MoGe nanowires [69], [70]. Sharifi et al. [71] found that the LAMH theory fitted to the tails of the resistive transition in Pb nanowires, however upon reducing the cross sectional area further, the resistance deviated substantially from the theory. This cannot be 56

71 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS attributed to quantum phase slips, as no evidence of a crossover from LAMH regime to a finite residual resistance below T c was found. They attribute this behaviour instead to Coulomb blockade effects. Lau et al., [69], see evidence for quantum phase slips in Pb and PbIn wires. Altomare et al. [72] also investigate MQT of Phase slips in a recent experiment. As the phase slip process causes a voltage spike, or switching event, it is possible to gather statistical infomation on the process, in a similar way to the switching measurements of the critical current in a Josephson system. Thus far, there have been no reports of such measurements in the literature, which provides a possible novel use for the MQT measurement system in this relatively unexplored field. Phase slips in High Temperature Superconductors have proved elusive. This is due to the small coherence length in the materials, or the order of ζ = 10nm. Abdelhadi et al. claim to have seen phase slip like behaviour in underdoped YBCO [73]. Unfortunately, there lacks a model to explain the mechanism of current flow in HTS, thus the issue of where the phase slip occurs and its physical origin are still unknown. This will be discussed a little more in Section Grain Boundary Junctions The growth of a film on a substrate where the lattice parameters of substrate and film are well matched can result in the phenomenon of epitaxial growth, whereby the film will grow following the crystal structure of the substrate. A bicrystal is the result of the fusing together of two sections of single crystal substrate, with the crystal lattice of each half orientated in a different direction. When a film is grown upon such a substrate, a natural crystal grain boundary will occur in the film along the line where the two halves of the substrate are fused together. In High T c superconductors, due to the relationship between the order parameter and the crystallographic direction, such a system will exhibit Josephson-like properties. This 57

72 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.4: AFM image of a grain boundary junction. From Mannhart et al. [8] type of weak link is known as a Grain Boundary Junction (GBJ) and is advantageous in that only one lithographical patterning stage is required to define the junction area. An atomic force microscopy image of such a junction is shown in Figure 3.4. GBJs are usually grown on substrates with 22.5 or 12.5 inter-grain angles. It is found that as the grain boundary angle is increased, the weak link behaviour becomes more like a tunnel junction, as reported in [10]. Tunnelling in Grain Boundary Junctions In a perfect 0 45 d-wave GBJ, tunnelling between a node and a lobe of the order parameter, the orientation of which is illustrated in Figure 3.5, will suppress the Josephson supercurrent. However, the existence of Andreev bound states (as described in Section 1.4) can produce another method of tunnelling. It is possible for transfer of a Cooper pair across the barrier to occur by the reflection of a quasiparticle from a positive lobe to a negative one. The supercurrent which flows as a consequence of this tunnelling has double the normal I c (Φ) periodicity. It corresponds to presence of the second harmonic of the CPR, I c2 (Φ), which becomes 58

73 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.5: Order parameter in a 0-45 bicrystal GBJ. From Lindstrom et al. [74] dominant when I c1 (Φ) is suppressed, in cases such as that of the 0 45 GBJ. In a real GBJ, the effects become much more complex. The barrier is no longer perfect, but exhibits faceting and meandering. This affects the local J c of the junction. In GBJs, the magnetic diffraction pattern can vary widely from the ideal Fraunhoferlike modulation, due to the current transport across the junction being non uniform. There are suggestions that this is due to filamentary transport in the barrier region. The effects of different barrier transparencies and interface faceting are explored in the recent work by T. Lindström et al. [74]. Their theoretical modelling and experimental results demonstrate the appearance of second harmonic effects in the magnetic diffraction patterns of Grain Boundary SQUIDs, such as the minimum of I c occurring at zero applied field. 1/f noise in Grain Boundary Junctions In an ideal tunnel junction, the barrier is clean and there are no localised states for electrons to occupy. In High T c junctions, there may be trapping sites available for the electrons. The random hopping of an electron between two states, or the random fluctuation between occupation and non-occupation of a single site generates telegraph noise on the critical current. The effect of a low number of dominant trapping sites is a more Lorentzian-type distribution in the Noise spectrum data. An example of such a distribution can be found in [9], which has been reproduced 59

74 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS in Figure 3.6. When the noise is summed over a large number of states, the results give a characteristic 1/f response, as is derived in [75]. The exact mechanism for the electron hopping is unknown, although there have been several studies of the origin. For example, Rogers et al. look more closely at the behaviour of single two-level fluctuators in their 1984 paper [76]. If there are a low number of trapping sites available, the amount of noise may change considerably as a function of temperature, as certain sites become more energetically favourable. Specifically, a trapping site may become dominant at a particular temperature. This is observed as the tendency of I c to jump between two distinct values, each of which displays the normal spread of critical current values due to thermal activation or MQT. The phenomenon of low-frequency noise in GBJs has been addressed in the literature, however, the phenomenon has not been thoroughly investigated at temperatures lower than 4.2K. In the majority of the publications, the effects are considered only at 77K, due to the main driving factor of the research being the improvement of the HTS SQUID. This factor also means that the noise due to a single GBJ has not been addressed extensively. The 1/f noise can be reduced in a SQUID device by bias reversal scheme, in which the fluctuations are essentially cancelled out by measuring the switching of the SQUID both on the positive and negative cycles of an alternating bias current. However, this method cannot help in the case in a single junction. In the paper by Miklich [77], a single GBJ is considered. The findings of these studies are consistent with a large 1/f noise component in Grain Boundary Systems, which will be of interest when the results of measurements on such a system are considered later in this report. Kawasaki et al. [78] suggest some potential mechanisms for the noise, including oxygen-deficient sites at the surface of the material, and dislocations in the sample forming an array which supresses the order parameter. Single junctions are also considered in [79]. In this paper they find evidence of an 60

75 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.6: Lorentzian modulation of the noise spectrum due to dominant trapping sites. From [9] excess noise component in Low T c Nb-Al 2 O 3 -Nb junctions, showing that even in relatively simple junctions the magnitude and type of noise may be unpredictable. A good review of excess 1/f noise in junctions can be found in [80]. The CPR in Grain Boundary junctions The CPR has also been investigated by Il ichev et al., in samples consisting of a 45 / 0 Grain Boundary, [81], [82]. Their method was similar to that demonstrated by Rifkin et al. They found a deviation from the conventional sinusoidal response that could not be explained by the effects of flux noise, variations of current density in the electrodes, or additional mechanisms mentioned in the literature, all of which can also cause the CPR to appear non-sinusoidal. For a good review of both the theoretical and experimental advancements in this field, the reader is encouraged to consult [3] The temperature dependence of the first and second harmonics has also been investigated. The two behave very differently; the dependence as measured by Il ichev et al (2001) is reproduced in Figure

76 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.7: Temperature dependence of a GBJ critical current (triangles), and the Josephson current harmonic components I 1 (squares) and I 2 (circles). The inset shows the theoretically expected dependence. From Il ichev et al. [10] Smilde et al. take very careful measurements of the angular dependence of the order parameter in their 2005 paper [11]. They find evidence that the symmetry is not that of an ideal d-wave system in their YBCO sample. The paper suggests an admixture of d-wave pairing with a subdominant s-wave component. The presence of this component would indicate that the nodal points of the order parameter never decrease to exactly zero, and thus the energy gap remains finite for all values of k. Figure 3.8 illustrates schematically their experimental setup. The presence of an admixtures could mean that the order parameter does not fall to zero at the nodal positions, and potential problems due to quasiparticles may be reduced. Measuring the CPR of such junctions would give further information about the system. The consequences of an anomalous CPR in High T c superconductors is under intense investigation at the moment due to a potential reduction of the effect of quasiparticle decoherence in the qubit by exploiting the higher harmonics of the effect. This will be mentioned in Section

77 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.8: Schematic of the angle-resolved tunnelling measurement performed by Smilde et al. [11] 3.7 MQT in High Temperature Superconductors Recently work has been undertaken to demonstrate the Macroscopic Quantum Tunnelling in HTS Josephson junctions. It was strongly believed for a time that the quantum regime in HTS would never become fully accessible due to the presence of nodal quasiparticles down to the lowest temperatures, which act to destroy quantum coherence in the system. Recently however, theoretical and experimental evidence would suggest that this is not the case. For example, Fominov et al., [83] calculate the expected decoherence due to nodal quasiparticles in several different systems, and find that it may not be as detrimental to quantum device operation as originally thought. This has sparked a recent renewed interest in the HTS system for qubit applications. Theoretical advancements such as these motivate experimental studies into HTS MQT. The effect has been both calculated and observed in numerous HTS systems. In some cases evidence points towards these HTS systems demonstrating unusual behaviour, and possibly even making them more favourable for applications in the 63

78 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS quantum regime. As the previous experimental work is particularly relevant to the results presented in this thesis, a dedicated review of the experiments performed in this area and a discussion of their relevance will be presented separately in Section The HTS Flux Qubit Many people have succeeded in fabricating LTS structures exhibiting quantum coherence, as the junctions are easy to fabricate and fairly reproducible. However, coherent oscillations in qubits fabricated from HTS have yet to be demonstrated. It is possible to design qubits from HTS materials in a very similar way to LTS, for example by the incorporation of one, or three, Josephson junctions in a loop. However, the fabrication of High T c junctions is usually different, as they may consist of Intrinsic Junctions (c-axis stacks) or Grain Boundary Junctions, or SND ramp junctions. There are some promising candidates for HTS qubits, such as the pi-qubit. By introducing a pi-junction into the superconducting loop, the need for the flux bias can be eliminated. Such a qubit was first proposed by Yamashita et al. [84]. Quite a good overview of d-wave pi qubits, and their equivalence to 5 junction qubits is given in [85] Quantum computing using similar junctions (of an SDS type) has also been investigated theoretically by Ioffe et al. [86]. As of yet there have been no examples in the literature of flux qubits fabricated from intrinsic Josephson junctions in the 3JJ configuration of Mooij [47]. The silent qubit A problem with HTS qubit designs is that the nodal quasiparticles inject noise into the device. Amin et al. propose a silent qubit based on the dominance of the second order CPR harmonic. In their proposal, the external flux coupling into the qubit s 64

79 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS Figure 3.9: From [12] loop does not affect the shape of the potential, nor does the state of the qubit affect the flux. For such a qubit design to be realised, HTS Josephson Junctions are required. In order to fabricate a silent qubit, one requires careful control of the angles and dimensions of the High T c qubits. Understanding and controlling the current-phase relation is therefore an important step towards realising the silent qubit. There are several device proposals for controllable CPR junctions. The work of Il ichev [10] suggests a doubly degenerate ground state exists in the symmetric grain Boundary junction system, and this system warrants further investigation. Amin et al. also investigate the YBCO grain boundary qubit [12]. Their qubit is reproduced in Figure 3.9. The design is such that it is possible to reduce the effect of sources of decoherence on the qubit from the external circuitry to a negligible level. This implies that the only remaining source of decoherence comes from dissipation due to the nodal quasiparticles present in the d-wave order parameter. 65

80 CHAPTER 3. THE CUPRATE SUPERCONDUCTORS A very recent paper has been published by the Twente group [87] in which the group suggest that s/d junctions may be better than both s/s junction (because of their intrinsic π shift and better than d/d junctions because the presence of a full s-wave gap on one side prevents the transport of quasiparticles at low energies, and enhances their quiet mode of operation. They use theoretical data constructed from the experimental work performed on hybrid junctions in [11]. The paper suggests that the presence of an s-wave gap supresses the low-energy quasiparticles, which are most likely to cause decoherence in a qubit structure. 66

81 Chapter 4 Principles of the experiment This chapter will explain the general experimental method for measuring thermal activation and macroscopic quantum tunnelling in Josephson Junction devices. It will provide an overview of past and current research in this field, and a discussion of the insights gained as a result. This will provide a solid groundwork and context for the methods used in the experiments performed here, which will be discussed in the next chapter. 67

82 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT 4.1 The MQT experimental technique The general principle of the technique involves repeated stochastic measurement of the critical current of a Josephson junction. The junction is repeatedly switched, typically several hundred thousand times, and the values of I c are compiled into a histogram, as mentioned in Section 2.7 and illustrated in Figure By performing this measurement at a series of different temperatures, information about the thermal activation and quantum tunnelling regimes of operation can be directly extracted. The block diagram of the apparatus for cooling and measuring the sample in a typical setup is shown in Figure 4.1. It is hoped that this will allow the reader to become familiar with the overall design of the system, as each component section is subsequently discussed in detail in the remainder of this chapter and the next. Figure 4.1: Block diagram of the measurement system 68

83 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT Figure 4.2: The voltage ramp applied to the junction and the resulting switching signal Time of flight technique The commonly used technique for these measurements is known as the time of flight method. A good overview of the technique can be found in the paper by Wallraff et al. [88]. In this method, the current through the junction is ramped from a slightly negative value up to the critical current of the junction. As the value of the applied current passes through zero, a time interval counter is started. The counter is stopped at the point at which the junction enters the voltage state. This point is detected by observing the value of voltage across the junction and comparing this value to a predefined trigger level. With a linear current ramp, the time interval between the two trigger points will be proportional to the effective critical current of the junction. This method is illustrated schematically in Figure 4.2. The method is relatively simple to implement, the results can be accumulated quickly, and the time interval can be measured extremely precisely with commercially available apparatus. It is found that the mean value of the histogram, and the width (standard deviation, 69

84 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT σ) of the histogram change with temperature. This is related back to the theory, as the chance of an early switching event increases with temperature. The true value of I c is therefore not the mean or peak of the histogram, but will typically lie at the high-current end of the distribution. The increased probability shift in mean position is mainly due to the increased probability of early switching events, but may also be slightly affected by any changes in I c0 with respect to temperature. For the small temperature range considered here (20mK-1K) this effect should be negligible. Signal preservation Most previous experiments of this type employ the use of a shielded room. The novelty of the approach taken in this work is that it can be shown that with careful shielding considerations, the use of a shielded room as an essential aspect of quantum coherence experiments is unnecessary. This allows experiments to be performed without large scale upheaval of existing resources. The implementation of the shielding and noise rejection will be discussed in detail in Section 5.3. To help eliminate noise, the signal is generally attenuated down the length of the current lines, and filtered on the voltage lines. This means that in order to calculate the value of current switching, the time interval must be converted. A voltage input into the top of the cryostat will see an impedance, R AT T, which will be the overall impedance of the system. The filter modules should be designed to give an R AT T value of approximately 50Ω, in order to match to the impedance of the current drive. Another factor, I AT T is defined as the ratio of the value of current at the top of the system to that which flows through the junction, i.e. the current attenuation factor. In order to calculate the value of current flowing through the sample, one utilises the relation: 70

85 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT I c = (V pp /2 + V OF F )I AT T R AT T (4.1) Another important experimental variable, the current ramp rate, can also defined in terms of these factors: di/dt = fv pp I AT T R AT T (4.2) The current ramp rate In experiments, the upper bound on the rate at which the current can be ramped must be theoretically less than the plasma frequency of the junction and the escape rate itself, so that the washboard is slowly tilted with respect to the timescales associated with the oscillations of the quantum state. In practise, the rate is limited at a much lower value, as the bandwidth of the signal path is usually limited with low-pass filtering to reduce the absolute noise level. The current ramp rate must however also be high enough to ensure that the mean value at which the junction switches does not lie too far away from the true value of I c, i.e. if the current is ramped too slowly, the early switchings will dominate. In fact, the deviation of the mean from the true critical current is logarithmically dependent on the ramp rate. For these reasons, values of the sweep frequency, f, are usually chosen to be of the order of Hz [38]. The ramp rate of the experiment will change the absolute position of the histogram in the time domain, but when normalised so that the histogram is plotted on a current scale, the ramp rate should not affect its position. Changing the critical current It is useful to apply a magnetic field to change I c0, whilst keeping the other parameters fixed. If a persistent mode superconducting magnet is used, the critical current can be adjusted quite accurately, and it will remain at a stable value without the 71

86 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT need of a constant magnet power supply. Changing the critical current is a good test of whether the system has entered the quantum tunnelling regime, or if it is limited by noise. From equation 2.33, the crossover temperature, T*, is proportional to Ic0. If the system is noise limited, decreasing I c will not result in a change in T*. The noise rejection of the system should be good enough that any noise-induced crossover should fall below the base temperature of the equipment, such that in the absence of a quantum tunnelling regime, the data follow the thermal activation theory down to the lowest experimental temperature. Estimating parameters In order to compare the histograms, the theoretical calculation of the escape rate needs to be converted into histogram representation, as given in equation The fitting is done with parameters which can be adjusted by the user. As the fitting is extremely sensitive to the value of the critical current and ramp rate, an initial parameter range is chosen such that the fit will fall nearby. The parameters are estimated as follows: I c is estimated to an accuracy of ±9.26nA from a reading taken from the scope trace. di/dt is calculated from the settings of the current ramp, as in equation 4.2. R N is estimated from the normal state resistance of the junction as measured at low temperatures, to an accuracy of 0.01% and C is estimated from the hysteresis of the junction, which is limited by the accuracy of R N. The capacitance, C, can be more accurately determined by the observation of Fiske steps (as mentioned in Section 2.5. Once these parameters are input into the fitting program, the results are fairly close to the data, and the final fit can be adjusted manually with the User Interface, the procedure for which is explained in Section 5.5. Even though the measured value of I c is quite accurate, it is different from the true value of I c0. This is the point at which the voltage state is entered in the absence of any mechanisms of dissipation (T=0K) or any quantum tunnelling in the system, i.e. 72

87 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT when the washboard potential reaches its point of inflection. I c0 cannot be measured directly; it must be estimated. In the paper by Wallraff et al. [88], a lower limit is placed on the value of I c0 by recording the maximum switching current observed. One way of producing an accurate estimate of I c0 is to fit theoretical histogram curves to the data, and the resulting fit will be sensitive to the parameter I c0. This method is adopted in the work presented here. A full description of the fitting program used with the data gathered in this project is described in Section Data analysis techniques The techniques described here are those generally employed in this type of experiment. The specific implementation of these methods for the work presented in this report will be described in more detail in the data acquistion section (5.5). The main ways of displaying the data from such experiments are by plotting the escape rates, escape temperatures and switching histograms as a function of sample temperature. These three main methods of presenting the data can also be used as cross-checks, to ensure that all data analysis programs are working correctly. In addition, noise measurements can be performed on the junction samples, to provide extra information on the critical current fluctuations. described in a little more detail. The methods will now be Escape rate It is necessary to test the validity of the data obtained in during the experiment, and to explain any anomalous results, by comparison with the theory described in Section 2.7. This means that the data will need to be manipulated into a form represented by the escape rate Γ, given by equations (2.26) and (2.29) of the system at a particular value of bias current and temperature. The theoretical fits calculated in this report assume that the system is sufficiently underdamped that dissipation 73

88 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT does not affect the escape rate or histograms. The escape rate data is presented in a more useful format by plotting a modified escape rate against current. By calculating ln(w p /2πΓ) 2/3, one can fit the theoretically expected escape rates as straight lines against the data values. Escape temperature As has been shown in Section 2.7, the temperature dependence of the width of distribution can be used as an indicator of a crossover between the thermal and quantum regimes. However, the escape temperature allows the data to be normalised, and therefore allows comparison between samples. The escape temperature plots are calculated from equation (2.34) for each set of switching events. Theoretically reconstructed histograms In addition, as the raw data is most easily presented in the form of a switching histogram, it is useful to reconstruct a theoretical histogram to fit to the acquired dataset. The method of preparing the raw data for analysis follows that described in the early paper by Fulton and Dunkleberger [89]. In the theoretical treatment, the histograms and escape rates are a continuous variable. However the data set for each histogram consists of the discretised probability of a switching event occurring within a current interval di, which is the probability density, given by P (I). The lifetime (or escape rate) of the state, τ 1 (I) is the probability of a switching event occurring per unit time. Thus P (I) can be expressed using the definition: and as P (I) = ( I ) ( di 1 P (u) du 0 dt ) 1 τ 1 (I) (4.3) 74

89 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT 1 I 0 P (u) du = I P (u) du (4.4) P (I) = τ 1 (I) and the escape rate is therefore ( ) 1 di P (u) du (4.5) dt I τ 1 (I) = P (I) ( ) di 1/ dt I P (u) du (4.6) To discretise this, we consider that our data is split into a number of finite current intervals of width I, each represented by a number of counts N k occurring within the k t h interval. The total number of counts, M, can be expressed as M = N(k) (4.7) k=0 And the probability density of a switching event is therefore ( ) I N(k)/ N(k) I k=0 (4.8) Following the definition above, the probability of a switching event occurring above and including the current interval in question is therefore P (k) = k N(k)/ k N(k) (4.9) 0 therefore combining equations (4.6), (4.8) and (4.9) yields τ 1 (k) = ( N(k)/ ) ( I di N(k) N(k)/ I dt 0 k 1 N(k)) (4.10) 0 75

90 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT which reduces to This is equivalent to the form: ( τ 1 (k) = di I N(k)/ dt I ) N(k) k (4.11) τ 1 (k) = di dt ( k 1 I ln j=1 k 1 P (j)/ i=1 P (i) ) (4.12) Which will be used in the data analysis to calculate the escape rate from the histogram data acquired. In order to calculate the theoretical probability histogram, Fulton and Dunkleberger employ interpolation of exponentials between current intervals, and express the result logarithmically. Thus from equation (4.5), we can also construct a probability function, which when plotted will give a theoretical histogram fit: ( p t = di 1 Γ(I) exp dt I ) Γ(I)dI di/dt (4.13) Noise measurements By taking repeated single shot measurements of the value of current at which the junction switches, one can compile a time-domain analysis of the noise on I c. It is then possible to implement a Fast Fourier Transform (FFT) of this to produce a noise spectrum. The data must be taken over a long enough period of time such that the low frequency components of the noise are accessible. The highest frequency available for interrogation is determined by the Nyquist limit of f sampling /2. The sampling must not however be too fast, or the bandwidth of the system will affect the shape of the switching event. 76

91 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT 4.3 Previous experiments The measurement of MQT of the phase in a conventional (s-wave) Josephson junction has been addressed thoroughly in the literature. Fulton and Dunkleberger [89] made current switching measurements on Sn-Sn oxide junctions. The work of Voss and Webb in 1981 [13] furthered the experimental investigation of the the phenomenon. Their work focussed on 1 µm Nb junctions, with an RSJ-like tunnelling characteristic. The histogram data from these early experiments are taken with extreme care. The histograms were taken over several hours of data collection at each temperature, resulting in smooth histograms. The data have been reproduced in Figure 4.4. In the same year, Jackel et al. also performed experiments on a tunnel junction system [90], using Pb and Pb(In) as the junction materials. The thermal to quantum crossover was observed in this system. Caldeira and Leggett [41] also published a dissipative model of the theory of such systems. Martinis et al. [33], furthered these results to include the effect of microwave excitation on the tunnelling, again using Nb-NbO x -Nb junctions. The effect of dissipation on MQT measurements was measured by Washburn in [91], where it is found that the experiment is in good agreement with the theory of an enhanced escape rate above T* for a range of capacitance values. However, not all their data fits the model, which may be a consequence of their assumption that the damping parameter could be calculated from a constant value of R N. In 1988, Cleland et al. also experimentally investigated MQT in the presence of dissipation, specifically in the form of a shunting resistance across the junction [92]. Their fit to the theory described in Section 2.7 was an improvement over the earlier attempts, and their analysis method has been cited in most papers involving MQT calculations. Corato et al. [14] have measured the switching characteristics on a single junction in a loop (RF SQUID configuration), which demonstrates MQT in a more qubit-like 77

92 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT Figure 4.3: Martinis et al. results demonstrating the thermal-quantum crossover (1987) Figure 4.4: Results of MQT in 1 µm Nb junctions, from Voss et al., [13] 78

93 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT Figure 4.5: Results of MQT measurements on an RF SQUID, from Corato et al., [14] system than the single junctions considered so far. Their extremely underdamped devices yielded a quasiparticle resistance R N of 27kΩ, illustrating that the MQT model is valid in this regime. The results are reproduced in Figure 4.5. In 2003, Wallraff et al. demonstrated the effect of applying microwaves to a junction in order to investigate the higher energy levels of the potential well [93]. A microwave frequency is applied to the junction, usually by inserting an antenna-like termination of a microwave line in the vicinity of the sample. Microwaves can couple energy levels within the potential well, causing excited states to be populated. An enhanced tunnelling rate is then observed, due to the smaller potential barrier to these higher energy states. The use of such spectroscopic measurements is a clear demonstration of quantisation within the washboard potential, which is necessary if one wishes to achieve manipulation of a two-level system. Microwave excitation is also a useful mechanism for testing the noise level of the system, as the current distribution of escape events from higher states in the potential well may be intrinsically narrower than those of ground state escape. 79

94 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT Figure 4.6: Diagram illustrating the effect of microwave radiation on tunnelling from the washboard potential. E 1 denotes the ground state, E 2 the excited state 4.4 High T c MQT measurements Although the primary objective of this work is to provide a system capable of characterising low temperature superconducting junctions, it is interesting to consider recent experiments performed on High T c junctions. The system will then be able to be used as a tool for the investigation of new and novel junctions. It was believed since the early reports of the d-wave pairing state in cuprate superconductors that the quantum mechanical properties arising from the macroscopic coherence would be obscured by the nodal quasiparticles. However, recent results have been in contradiction to this statement. The first experimental evidence of MQT in a High T c superconductor was reported by Inomata et al. in 2005 [94]. Their sample consisted of a stack of intrinsic BSCCO junctions, fabricated using a Focussed Ion Beam technique. Each junction in the stack has a slightly different critical current, and it is possible to bias the current through the junction stack such that a single junction can be selected. The group s results suggest a high value of T*, around 1K, prompting further work in this area. Their results are reproduced in Figure

95 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT Figure 4.7: Results from Inomata et al. (2003) demonstrating the first successful Macroscopic Quantum Tunnelling experiments on a HTS sample (BSCCO) More recently, Jin et al. showed that the tunnelling in a stack of N Junctions was enhanced by a factor of N 2 [15]. There have been several theoretical attempts to explain this phenomenon, by considering the interactions between the Josephson junctions in the stack. Savalev et al. describe a long range interaction between adjacent Josephson Junctions in an intrinsic junction stack [95]. Fistul [96] explains the effect by the appearance of a global charge interaction, mediated through an external shunt resistance, which would normally serve to suppress the rate of MQT. An increase in the MQT rate is advantageous for any potential qubit devices, as the crossover into the quantum regime will occur at an elevated temperature, meaning that the devices will be operable over a larger temperature range, and well into the quantum regime, avoiding fluctuations from the intermediate state. As mentioned in Section 3.6, the various harmonics of the CPR can be temperature dependent, and under particular conditions the second harmonic I 2 can become larger than I 1. It is therefore possible, in a junction with an unconventional CPR, to observe an enhanced tunnelling effect below a critical temperature, if I 2 becomes dominant. This has been directly observed in MQT measurements, by Bauch et al. 81

96 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT Figure 4.8: Jin et al. [15] show that the tunnelling rate in a stack of N Josephson Junctions is enhanced by a factor of N 2 Figure 4.9: Results from Bauch et al. (2005) illustrating features in the temperature dependence of the activation processes in a YBCO sample [16]. 82

97 CHAPTER 4. PRINCIPLES OF THE EXPERIMENT [16]. Their paper presented some interesting findings on a YBCO bicrystal sample grown as a twist junction, where the planes are rotated about the a or b directions, rather than the usual c-axis. In their data, there are two distinct regions to the thermal escape in the junction. The enhanced escape from the lower temperature regime is attributed to the presence of a second harmonic of the CPR. The effect of this phenomenon on the width of the distribution is reproduced from their data in Figure 4.9. The group have furthered their work on the Macroscopic quantum properties of High T c junctions to include an experiment demonstrating the existence of well defined energy levels within the potential well of their grain boundary twist junction [97]. These results are of great importance, as they demonstrate the first steps towards the operation of High T c qubits which are not limited by decoherence from their intrinsic quasiparticles. 83

98 Chapter 5 Apparatus design This chapter documents a large proportion of the experimental work performed during this project. It describes the fabrication of the samples to be measured, and the design and building of a low noise setup to perform quantum measurements of Josephson devices, the fabrication of which is also described here. The noise characteristics of the setup are of vital importance in this experiment, as the signature of noise in the data appears very similar to the onset of the thermalquantum crossover. A large section of this chapter therefore describes in detail the sources of noise present in the setup, and discusses ways in which the noise can be characterised and subsequently reduced. The final part of this chapter also explains how several tests were performed in order to ensure that the experimental setup was optimised and describes the final settings and procedures used to gather the results presented in the next chapter. 84

99 CHAPTER 5. APPARATUS DESIGN 5.1 Sample fabrication YBCO Grain Boundary Junctions The samples available consisted of YBCO-LCMO bilayer films on bicrystal substrates. The CMR material LCMO was therefore still present after the junctions had been defined. This may contribute to unwanted magnetic effects in the junction, however the junctions exhibited good critical currents and it was therefore concluded that they were largely unaffected by the presence of this additional magnetic material. A 3-stage lithographical technique was used to pattern the Grain Boundary Junction samples. A diagram illustrating the process is shown in Figure 5.1. The recipe for the lithography steps is given below: Lithography technique Photoresist coating spin on: Shipley positive photoresist 1214 Pre-bake for 5 minutes at 95 C UV expose, 13 seconds at a lamp power of 285W Develop, 20 seconds in Microposit developer Dry samples thoroughly with N 2 gas and leave in a desiccator for 30 minutes Post-bake 10 minutes at 95 C Ar+ Ion-mill, 9 minutes at 0.25A stage ion current. The first step consisted of using an edge-bead mask to define a 0.5mm area around the perimeter of the 5mm x 5mm sample. The YBCO-LCMO was removed by Ion beam milling of the exposed edge for 10 minutes using a stage current of 0.20A. The sample was then etched in a 7% solution of Hydrofluoric acid. The acid etches preferentially along the Grain Boundary, thus revealing its position. Once the Grain Boundary had been revealed, the sample was carefully aligned and patterned using 85

100 CHAPTER 5. APPARATUS DESIGN Figure 5.1: Lithography steps for the preparation of a YBCO Grain Boundary Junction sample. The grey line indicates an invisible GB, the red line an exposed GB. the main mask, to define the junction areas and 4-terminal measurement tracks. The same ion beam milling settings were used for this step. The final step involved a negative resist mask to expose only the contact pads. The sample was ion beam milled for a further 1 minute to clean any remaining residue from the contact pad areas, before quickly transferring it to a thermal evaporation chamber for deposition of a 200nm Ag layer. The Ag layer is required on the contact pad areas, as it is difficult to wire bond to YBCO. Additionally, the presence of this layer helps protect these areas from degradation during unavoidable exposure to atmospheric conditions. The sample is then soaked in acetone and removal of the Ag lift-off layer is achieved by gently wiping the surface of the sample using a cotton bud, whilst observing the sample under a microscope. Sample mounting All samples were mounted in a similar fashion for measurements, involving the adhesion of the chip to a chip carrier, with large contact pads for ease of attachment 86

101 CHAPTER 5. APPARATUS DESIGN Figure 5.2: The chip carrier, installed into the dilution refrigerator. Good thermal contact is ensured by a metallic contact between the sample substrate and the Cu backing plate. to the measurement apparatus. Two different chip carriers were used, for measurements in a continuous flow cryostat system, and in a dilution refrigerator system. The continuous flow chip carrier allowed multiple sets of 4-terminal measurements to be made during a single cooling cycle. The carrier for the dilution refrigerator was a single 4-terminal carrier, specially mounted on a copper plate for optimal thermal contact with the sample. A schematic of the chip carrier is shown in Figure 5.2. The samples are wire bonded to the chip carrier. This ensures reliable connections with low contact resistances. On some samples, it is difficult to achieve a low resistance contact, and the resulting measurement can be limited by the Common-Mode Rejection Ratio of the amplifier. Using the wire bonding method, in combination with a 4-terminal measurement technique, the characteristics of all the Junctions considered here were easily observed independent of the chip carrier wiring. 5.2 Low temperature techniques Samples of Low and High T c superconductors were fabricated, and cooled down to a temperature range within which the quantum properties should be observable. The 87

102 CHAPTER 5. APPARATUS DESIGN MQT and thermal activation phenomena of interest in this work are most easily observed in the temperature range of 0-1K. Most of the samples are first measured in a continuous flow 4 He cryostat in order to check their superconducting properties at 4.2K. This ensured that the samples were behaving correctly, before they were mounted in the dilution refrigeration system. This is a necessary step, as the sample can be measured within a few hours on the continuous flow system, compared to the 2 day changeover time on the dilution refrigerator. The setup of the dilution refrigerator for use in this project will be described in detail in Section 5. An Oxford Instruments Kelvinox 25 dilution refrigerator is used to cool the samples to the low temperatures required for the measurements. The system works by maintaining a concentration gradient between a concentrated and dilute mixture of the two isotopes of Helium, 3 He and 4 He. A mixture of the two gases is circulated round the system. The gases pass through counterflow and discrete heat exchangers, where they begin to condense, through the mixing chamber, and back up towards the still. Due to its higher vapour pressure, the 3 He preferentially evaporates from the still upon gentle heating, and is pumped away. This maintains a concentration gradient between the liquid in the lower half of the mixing chamber and the 3 He rich phase above, promoting 3 He transport across the phase boundary. This dilution of 3 He into the 4 He rich phase causes a cooling. The mixing chamber thus cools to a base temperature, limited by the pumping rate of 3 He. With this system it is possible to achieve a base temperature of 25mK. The dilution refrigerator can be run continuously once the base temperature is achieved, as long as the main Helium bath remains above a minimum level. The system also incorporates a sliding seal, which allows the insert to be removed whilst the main bath remains filled with liquid Helium. This is a useful feature for fixing any problems with the fridge insert which occur during the experimental runs, and for changing samples without the need to fully warm both the fridge and dewar. 88

103 CHAPTER 5. APPARATUS DESIGN Figure 5.3: Schematic of the fridge thermometry Thermometry A fully calibrated RuO thermometer gives the main measure of temperature on the mixing chamber of the dilution fridge. The calibration of this thermometer is accurate down to the base temperature of 25mK. The voltage across the diode thermometer is measured using a SHE bridge. The bridge is designed to allow measurement of the voltage across the thermometer diode with different excitation currents, as the resistance of the diode is highly non-linear with respect to temperature. This value of voltage is also monitored using a Keithley 199 voltmeter, which is connected to the computer via a GPIB interface, in order to record the diode voltage during a measurement. The voltage-temperature conversion is achieved using a previously determined set of measurements [98]. The calibration table for this thermometer is utilised in the custom software in order to convert the voltage reading from the diode, which is discussed more thoroughly in Section 5.5. The temperature of the mixing chamber can be controlled using a heater, which allows stabilisation of the experiment temperature to within ±1mK. 89

104 CHAPTER 5. APPARATUS DESIGN 5.3 Cryostat wiring A large part of this project consisted of the production of a fully shielded wiring system for the dilution refrigerator. A full schematic of the insert wiring for DC signals is shown in Figure 5.5. A photograph of the fridge insert which shows the wiring below 4.2K (inside the IVC) can be found in Figure 5.4. All cables passing down the length of the cryostat probe are shielded to avoid the coupling of interference to the small signal lines. To achieve this, the use of miniature coaxial cables is employed. These are interrupted by 3 stages of attenuation and 2 stages of low pass filtering on the current lines, and 2 low pass filtering stages on the voltage lines. Both types of miniature coaxial cable also act as a lossy transmission lines down the length of the cryostat, increasing the attenuation in addition to the discrete attenuation modules. The attenuation of high frequencies is critical to minimise the amount of noise reaching the junction. The frequency response of the coaxial cables is investigated below. Several coaxial cables are available commercially which serve different and sometimes multiple purposes within the cryostat setup. These will now be described. Coaxial cabling Stainless Steel semi-rigid coax was used for the coaxial feedthroughs between the vacuum can and the main helium bath, and between the bath and the outside of the cryostat. In these cases a good seal was needed to avoid leaks. In the case of the feedthrough into the vacuum can, a good seal is imperative, as the liquid helium can form a superleak along the dielectric of the cable into the vacuum can. A braided, flexible stainless steel miniature coax was used to connect filter stages in the main helium bath, as the filters could easily be attached and their positions adjusted if necessary. The stainless steel inner conductor and braid ensure minimum thermal conduction between the top of the cryostat (T = 300K) and the top of the 90

105 CHAPTER 5. APPARATUS DESIGN Figure 5.4: The fridge insert with wiring (below 4.2K stage) 91

106 CHAPTER 5. APPARATUS DESIGN Figure 5.5: Wiring schematic of the fridge insert 92

107 CHAPTER 5. APPARATUS DESIGN vacuum can (T = 4.2K). By using a miniature coax the heatsinking of the inner conductor is also optimised. For the lowest temperature stages of the fridge wiring, the use of 0.3mm coaxial cable was employed, due to it s low thermal conductivity. The cable was purchased from the Japanese supplier Coax Company, and consisted of a stainless steel inner and outer conductor. In order to maintain compatibility with the fridge wiring, MCX connectors were installed on the end of each piece of this coax. The connectors were attached using a custom designed assembly. The inner conductor is crimped to a small stainless steel tube, which is covered with a PTFE dielectric insulation, and then inserted into a brass connector. One end of the connector is crimped gently to the outer of the coaxial cable, and the other is crimped to the shield of the MCX connector cable. This method is more successful than soldering the cable, as an acid flux is required to solder the stainless steel. The flux enters the small gaps in the connector during the soldering process and is difficult to remove. After continued exposure to water from frost on the cryostat, the connectors can fail due to the damage caused by the residual acid flux. A photograph of the installed crimped MCX connector assemblies can be found in Figure 5.6. The sample box The sample box is designed to house the junction in a low-noise environment, whilst maintaining good thermal contact to the mixing chamber. This is achieved by the entire box being fabricated from a single piece of high purity Cu. A photograph of the sample box can be found in Figure 5.7. The sample chamber is housed in the centre of the design, surrounded by six smaller antechambers. Small coils are installed into these chambers on each of the four DC lines, presenting a final, base temperature filtering stage to the signals entering the box. The remaining two chambers are not used in this project, but are present to allow two extra signal paths to enter the sample space if required at a later stage. 93

108 CHAPTER 5. APPARATUS DESIGN Figure 5.6: Custom designed MCX connector assemblies, installed on each of the 4 DC signal lines Figure 5.7: The Cu sample box 94

109 CHAPTER 5. APPARATUS DESIGN Filter modules The custom made filter modules are mounted along the length of the cryostat probe. The filter components have been selected carefully to ensure correct operation at the relevant temperatures. Metal film resistors are used because of the independence of their resistance on temperature. The resistors are surface-mount metal electrode leadless face 0.125W minimelf resistors. A summary of the construction process for the filter modules and accompanying photographs of the modules can be found in Appendix 9.2. The component values used are given in the overview of the current circuit, shown in Figure Thermal properties of the wiring It is important to maintain the correct temperature at each stage down the length of the dilution refrigeration insert. At each temperature stage, heat sinking and thermal isolation issues must be addressed. The stainless steel coaxial adapters located on the top and bottom of each mini-circuits filter are thermally anchored to the still region (0.7K) and metallic contact between the 0.3mm coaxial cables near to the mixing chamber are isolated from being in contact with the mixing chamber using GE varnished paper. The SMA connectors need specific attention, as the connector screw is used to secure the filters in place, leading to a potentially loose outer (ground) connection. Silver paint is therefore applied to these connectors to form a good electrical and thermal link between the two parts of the connector should they become loose or undergo thermal contraction. The main problem with using coaxial lines in a dilution fridge environment arises from the inadequate heat sinking of the inner conductor, which can transfer heat down from the higher temperature stages. To address this problem, silver paint is also used inside the Copper sample box to ensure contact between the enamelled 95

110 CHAPTER 5. APPARATUS DESIGN inner conductor and the sample box. In this region the shielding is provided by the surrounding copper of the sample box. This heat sinking process is repeated for each of the four DC signal lines. Electrical response of the wiring In order to test the microwave properties of the insert wiring, a HP8722C network analyser (50Mhz-40GHz) was connected in series with the various filter modules, attenuative coaxial lines and discrete attenuators. The response of the individual components can be found in Figures 5.8 to All sections of the wiring help to eliminate microwave frequencies. The largest response comes from the low pass filters and the small coaxial cables. The attenuators (Figure 5.8) do not contribute much to the overall filtering. The total response of the cabling (Figures 5.11 and 5.12) attenuates frequencies in the range 2-20GHz to less than the minimum resolution available with the network analyser (-90dB). In the intermediate frequency range (0.5-2GHz) the signal strength rolls off smoothly between -50dB and -80dB. The rise times of the signal lines were also investigated. The current line response to a step input was a 10% - 90% rise in 0.32µs. The voltage line response to the same input was 2.6µs. The rise time is limited by the bandwidth, W of the system. From these measurements it is concluded that W 1MHz, as calculated. The attenuator model A circuit diagram of the attenuator network and filtering on the current and voltage lines is shown in Figure The system consists of 3 attenuative sections at 4.2K, a filter stage at 0.7K and a filter stage at mixing chamber temperature on the current lines. On the voltage lines there is a filter stage at 0.7K and a filter stage at 4.2K. All 4 DC lines also pass through small coil filters housed inside the antechambers of 96

111 CHAPTER 5. APPARATUS DESIGN Figure 5.8: Microwave response of -20dB (blue line), -10dB (yellow line) and -6dB (red line) attenuators compared with a microwave short (top line). Figure 5.9: Microwave response of low pass filter (blue line) and coil filter (pink line) compared with a microwave short (top line). 97

112 CHAPTER 5. APPARATUS DESIGN Figure 5.10: Microwave response of a section of co-axial cable (blue line) compared with a microwave short (top line). the sample box. The final current through the Josephson Junction is stabilised by passing through a 1KΩ resistor on the I + lead inside the sample box itself. As described in Section 4.1, the important parameters from the attenuation network are the input impedance to the voltage signal, and the overall attenuation of the current from the top of the system to the sample at the bottom. These values were found from a spreadsheet calculation [99] to be I AT T = and R AT T = Ω. The attenuator model is useful for calculating noise, which will be addressed in Section The measurement system A system has been devised to test the main aspects of the junctions without the need for any complicated electronics or shielding, which can be added step by step to build towards the final system. This involves the use of a commercially available 30M Hz digital arbitrary function and waveform generator, the Stanford instrument DS345. The function generator provides the current ramp which passes down through the current line I +, through the junction, and then to ground. By controlling the amplitude of the signal, the user can bias the junction to switch into the voltage 98

113 CHAPTER 5. APPARATUS DESIGN Figure 5.11: Microwave response of the current leads (blue and pink lines) from the top of the cryostat to the sample box including all inline filtering, compared with a microwave short (top line). Figure 5.12: Microwave response of the voltage leads (blue and pink lines) from the top of the cryostat to the sample box including all inline filtering, compared with a microwave short (top line). 99

114 CHAPTER 5. APPARATUS DESIGN Figure 5.13: The current circuit for the attenuator network 100

115 CHAPTER 5. APPARATUS DESIGN Figure 5.14: Block diagram of the measurement system state on each cycle of the waveform. Both the block diagram of apparatus used and the waveform diagram for the ramp and junction switching have been discussed previously, and can be found in Figures 4.1 and 4.2, however the block diagram of the apparatus is reproduced in Figure 5.14 for ease of reference. The current ramp To maximise the Signal to Noise Ratio (SNR), 36dB of attenuation is included down the length of the filter lines, such that the signal coming from the DS345 can be maximised whilst still supplying a small current to the junction at the bottom, and thus attenuating any interference by the same amount. The maximum voltage provided by the DS345 is V P P = 10V. Using the attenuation factors calculated in Section 5.3, the maximum current which can be passed through the junction is therefore given by I MAX = V P P 2 I AT T R IN (5.1) 101

116 CHAPTER 5. APPARATUS DESIGN Figure 5.15: Final circuit diagram for the low noise preamplifier. The initial 6231 amplifiers were later replaced with the superior 8620 package. Thus with V P P = 10V, I MAX = 92.6µA. This gives a sensible range over which critical current values can be investigated. It is not advisable, however, to run the system at this maximum level, as the resistors in the primary attenuator modules are only rated to a maximum power dissipation level of 0.125W. If the first attenuator appears to the signal to have an input impedance of 50Ω to ground, then the current through this attenuator will be 0.1A at maximum V IN. Therefore the power dissipated by the first resistor in the chain will be approximately 0.41W. As the resistors are cooled in Helium gas, their power dissipation will be better than quoted. However, based on these calculations it is advisable to avoid running the system above about V P P = 6V for too long. The purpose of the shielded electronics box, mounted on the top of the fridge, is to pass the current ramp signal into the 4 terminal wiring, and to retrieve the voltage signal from the junction so that the switching events may be observed electronically. 102

117 CHAPTER 5. APPARATUS DESIGN The voltage preamplifier The circuit involves the use of Analog Design AD8620 dual op-amp in a low power package. This is important as the entire circuit will be run from battery power. The AD8620 is an ultra-low current noise package, with 5fA/ Hz at 1KHz per amplifier. The current noise is given in the amplifier datasheet as 5fA/ Hz at 1KHz. It is difficult to measure the current noise in order to verify this value, however the amplifier can be checked against the datasheet values by measuring the voltage noise. The voltage noise per Hz of the AD8620 amplifiers was measured using a Stanford Research Systems SRS870 Dual-Channel Spectrum Analyser at a range of frequencies up to 100KHz. A typical value of the noise power spectral density at 56Hz was measured to be 800nV/ Hz. When divided by the gain of the amplifier and the addition of noise in quadrature from the 2 amplifers on the differential input stage, the input voltage noise per amplifier is 5.7nV/ Hz. The datasheet value quotes 6nV/ Hz, so the amplifiers were behaving as expected. The preamplifier is constructed on a daughterboard, which enabled the circuit to be plugged in and out of the mainboard for replacement with a preamplifier more suitable for any given measurement. For example, if a low voltage noise circuit was needed, a second daughterboard module could be constructed using the Linear Technologies LT6231 amplifier, which is supplied in the same chip package as the AD8620, and has a voltage noise of V n = 1.1nV/ Hz. A circuit diagram of the daughterboard used in this experiment is shown in Figure 5.15, and a photograph of the circuitboard is shown in Figure The preamplifier box also contained a ± 5V voltage regulator circuit, so that the amplifier could be run using two 9V batteries located inside the box itself, to minimise the risk of mains interference before pre-amplification, where the voltage signal levels are very small. 103

118 CHAPTER 5. APPARATUS DESIGN Figure 5.16: Photograph of the low noise preamplifier illustrating the daughterboard concept Noise considerations In order to understand the sources of noise, it is useful to refer back to the current circuit in Figure In addition to gaining a quantitative measure of the measurement error due to the wiring and circuitry, the results from the noise analysis will be a useful aid in deciding how best to improve the system in future. A spreadsheet was used to model the noise sources down the length of the cryostat wiring. It should be noted that these sources will produce different amounts of noise due to the differing component values and temperatures along the length of the fridge insert. The attenuator network is designed such that the larger series resistances are at the highest temperatures, and as such produce the largest Johnson noise, which will be attenuated by the subsequent stages. Each stage was treated as a Thevenin equivalent circuit and then added to the next stage as a single voltage noise source and source impedance. The noise from each section is calculated as follows: V n = 4k B T R 1 + (R 1 R 2 ) (5.2) V s = V s R 2 R s + R 1 + R 2 + V n (5.3) 104

119 CHAPTER 5. APPARATUS DESIGN R s = R 1 + (R 2 (R 1 + R 2 )) (5.4) Where V s is the voltage noise source from all the previous stages, V n is the voltage noise from the next stage to be added, and R s is the combination of the previous source resistance and the next resistance stage to be added. R 1 gives the values of the two series resistors and R 2 the resistor to ground in each attenuator. The input voltage noise for the first stage was measured as the voltage noise on the current ramp, and gave a value of less than 50µV. The result of the calculation gives two voltage noise terms from the two current lines at the junction, which can be converted into a current noise by applying Ohm s law to the final V s and R s in the chain (nearest to the junction). In addition, the input current noise of the preamplifier consisting of AD8620 amplifiers will also be injected directly into the voltage lines, which are filtered but not attenuated. The input voltage noise of the preamplifier will therefore also appear as a voltage noise across the 50Ω input impedance, adding a two further current noise terms. In addition to the noise sources described above, there are some additional potential sources of noise. There will be a mains component due to the non-isolated equipment being run from a standard rack setup, for example the oscilloscope. The four main noise sources are added in quadrature (as they are non-correlated), to give the total current noise at the junction: I N T otal = ( ) V 2 ( ) s+ V 2 s + + I 2 R s+ R s IN+ + I2 IN (5.5) The final current noise value was therefore calculated to be 0.66nA at the junction over the system bandwidth of 1MHz. The noise was also modelled using a SPICE circuit simulator. The current noise 105

120 CHAPTER 5. APPARATUS DESIGN Figure 5.17: Current noise at the junction from the attenuator network calculated using SPICE simulation software (total current noise is shown as the pink curve). is shown as the pink curve in Figure The curve is peaked around 1MHz, which indicates the point at which the bandwidth limiting 2.2nF capacitors begin to dominate the path to ground over the resistive parts of the voltage lines. The calculation yields I n (RMS) = 0.45nA at the junction at 1MHz, and I n (RMS) = 0.14nA at the junction at DC. By observing the histogram widths in previous measurements, [13], it can be seen that the smallest distribution has a critical current of approximately 1.6µA and 2σ of the distribution falls within 0.05µA. To resolve 100 points on such a histogram requires a noise level on the histogram mean of below 0.5nA, and thus it is established that the noise level calculated using these methods should be acceptable. The cryostat wiring has been designed to have a single grounding point, to avoid forming ground loops, which can pick up interference. The ground point was defined as the top-plate of the cryostat, which corresponds to the entire metallic body of the cryostat being grounded. The shielding in this experiment relied upon the sensitive 106

121 CHAPTER 5. APPARATUS DESIGN signal paths being well isolated and the ground connections being well defined, by good electrical contact to the body of the cryostat. Though the signal paths were well shielded from the diagnostic wiring, some electrical noise from the heaters and thermometry was unavoidable, as they are earthed via the Oxford Instruments temperature controller. One method of further reducing this source of noise would be to unplug the thermometry wiring once the fridge has stabilised, and run the heater power supply from a battery and potentiometer setup. Although this was not implemented in the system described here, it is a useful improvement which could be applied to future versions of the measurement system. Future measurement scheme The measurement system described so far uses commercially available equipment. A potential replacement was investigated, which involves a custom designed circuit. Although in the work presented here, the main measurement scheme has been used to take the experimental data, work has been performed towards a superior system to replace the commercially available instruments, and is a potential Future measurement scheme. A prototype version of the future measurement circuit has been built. It will therefore be described here as part of the main apparatus design for implementation into the system at a future point. A schematic circuit diagram of the proposed (and prototyped) system can be found in The entire circuit is mounted on a motherboard, again with the preamplifier as a daughterboard, so it could be replaced, as mentioned previously. The current ramp is created by charging a capacitor using a constant current source. The OP77 in combination with the INA121 creates a steady current source for this purpose. The second op-amp of the pair creates a current ramp from the voltage across the capacitor. The buffering action of the op-amp circuit should prevent any loading by the source. The voltage across the sample is read using an additional INA121 precision instrumentation amplifier with an adjustable gain setting. 107

122 CHAPTER 5. APPARATUS DESIGN Figure 5.18: The proposed circuit for future MQT measurements 108

123 CHAPTER 5. APPARATUS DESIGN Optical isolation Once the current value reaches the threshold set by the resistor network defining the current reference threshold, the output of the top comparator will switch. Upon the sample voltage reaching a similar threshold, the bottom comparator will switch. The TTL signal outputs from these two ultra-fast LT1016 comparators each pass through a small circuit to convert the signal to an optical format, and then leave the setup via fibre-optical cables. The resulting signals are converted back to voltages before being fed into the SR620 Universal Time Interval Counter ports as the START and STOP signals. The optical components allow the system to be fully isolated from the measurement apparatus. The signals are output from the circuit using HFBR-X41X ST optical output modules with discrete transmitter (TX) and receiver (RX) units. Two TX units are mounted onto the chassis of the shielded box and connected to the circuit motherboard. The TX modules have a TTL input stage. The rise time of the TX modules is t r = 1.30ns [100]. Given a current ramp rate of approximately di/dt = 0.1As 1, and assuming that the input circuitry triggers half-way up the rising edge, this yields a current offset error of approximately di/dt t r /2 = 6nA. However, this should only cause a systematic error (offest) in the time interval, rather than a jitter, and can thus be taken into account when compiling data. An auxiliary circuit is used to receive the optical signals and convert them back to TTL voltages. The receiver circuit is housed in a separate, shielded box so it can be positioned close to the measurement apparatus. This allows the long lengths of cable between the top of the cryostat and the measurement apparatus to be optical, rather than noise-prone electrical cables. Testing the circuit Preliminary tests were performed to ensure that the electronics would switch correctly on the rising edge of the critical current of a real junction, cooled using a 109

124 CHAPTER 5. APPARATUS DESIGN Figure 5.19: Testing the auxiliary circuit. Top trace shows comparator 1 output; Bottom trace shows comparator 2 output continuous flow cryostat. The current ramp was observed to sweep through the critical current of the junction. The setpoint was adjusted so that signal from the junction switching comparator T2 occurred just as the critical current appeared. To visualise the switching, the outputs of comparators 1 and 2 were monitored using a digital oscilloscope. The resulting waveform can be found in Figure Comparison with the existing scheme The differences between the two schemes have a slight impact on how the noise in the system is treated. These differences will now be addressed. As the current signal is generated internally to the circuit, no signal generator is needed to supply the waveform. This also reduces possible sample heating, as the charging capacitor is reset by the CMOS analog switch once the sample has switched into the voltage state. In the future measurement scheme, the current noise on the ramp itself will be different, and will be related to the amplifier output noise, although in both cases it will subsequently be attenuated down the length of the cryostat. 110

125 CHAPTER 5. APPARATUS DESIGN At each relevant point in the circuit grounding is achieved by connection to the single-track grounding rail to eliminate any possibility of ground loops being formed. Thus in the future measurement scheme the grounding all takes place within a single metal box housing. In addition, the entire circuit is to be operated from the same ±9V battery power source, to eliminate mains noise and crosstalk between power supplies. The interference from external sources may interact differently with the two setups. This is dependent on the local environment, and would be measured in each case. The fidelity of the voltage signal is ensured by using fast comparators to reduce the rise time of the logic output. The operational amplifiers act to intrinsically buffer the input and output stages, so that a changing load impedance should not present too many problems. In the main measurement scheme, the comparator response is limited by the SR620 input comparators. It is expected that from a lowest noise point of view, the future measurement scheme would be a superior design. However, at this stage of the system design, the discretecomponent measurement scheme using commercially available equipment has been implemented for the majority of the junction measurements. The following discussion of setup and the results obtained will therefore refer to that of the main measurement scheme. 5.5 Computer Interfacing Due to the large volume of data necessary for the histogram measurements, data collection needed to be automated to a large extent. A series of programs were developed to communicate with the experimental apparatus. The programs have been written using the Python scripting language [101], which interacts with two other programs, an API to the GPIB interface, and an instance of the Open Source mathematical plotting package, Scilab [102]. It was decided to allocate the task of 111

126 CHAPTER 5. APPARATUS DESIGN Figure 5.20: Block diagram of the MQT Control Centre software plotting the data to a program such as Scilab, as opposed to handling the plotting fully in Python, due to its speed and ability to handle large amount of data efficiently. The system consisted of a main program, a hardware interface, a graphical user interface, a data acquisition program, and several plotting programs. A block diagram clarifying how these components interact can be found in Figure The general algorithms and purpose of the various parts of the program are discussed in the following section, but no specific code will be reproduced. Functions shown in the block diagram are italicised in the text. Hardware Interface Both the Stanford Time Interval counter and a Keithley 199 voltage meter were linked to the experimental computer using a GPIB bus, which has a standard protocol. A National Instruments GPIB-USB convertor was used to physically link the instruments to the PC. The convertor was supplied with a driver which allowed the user to supply commands to the bus simply by writing to the device. 112

127 CHAPTER 5. APPARATUS DESIGN The SHE bridge voltage was monitored on its analogue output by a Keithley 199 DVM over the GPIB. The commands were used to read the voltage from the Keithley back to the program, at the start and the end of taking each histogram, and the average was calculated, which was used as the temperature of the data point. This system also allows a measure of error to be calculated. To initialise the instruments and set-up the system to take a measurement, the following process was used: START Check GPIB and instruments are responding Initialise GPIB addresses (Stanford SR620 (Time counter) and Keithley 199 DVM) Initiated file write process and generate output file Set Stanford default values: No. of samples / source / mode etc. END Graphical User Interface A user interface has been specifically designed and implemented for the data acquisition and for real time data analysis. The interface is entitled MQT CONTROL CENTRE, and a screenshot is shown in Figure The program allows the user to grab histograms, either one at a time, or repetitively, directly from the SR560 and add them to a current list of histograms. The program then allows the user to perform manipulation and analysis of these raw data files. Data handling The Main Program links the Graphical User interface to the data acquisition and temperature calibration sub-programs. When the user selects filenames, the main 113

128 CHAPTER 5. APPARATUS DESIGN Figure 5.21: The User interface for the MQT Control Centre software program handles the passing of the relevant filenames and calibrated temperature values to the sub-programs. The function Device Query is the main data acquisition routine, which comprises of several components, as can be seen in Figure The Get Single Histogram function talks directly to the GPIB driver which interrogates the SR620, and returns the data in the form of a pre-binned histogram. Get Single Histogram writes the raw data to a file in the form of time interval values and associated counts per histogram bin. The datafiles are kept in a separate folder so that the raw data will always remain accessible after any further processing. The filename is recorded as fridgedata.* where * is the voltage returned from the SHE bridge reading, so that the files can be listed in order of increasing temperature. The value of sigma, calculated by the Stanford 560, is also written to the file. The program runs repeatedly, taking histograms with a user-specified number of points. The number of points is usually set to 1E5, as this produces a clean histogram in approximately 120 seconds. This routine runs as the user warms or cools the mixing chamber accordingly. 114

129 CHAPTER 5. APPARATUS DESIGN START Take histogram mean Centre histogram and autoscale Measure scales, set no. of samples, write to file Take a sweep, write to file END Mean drift and noise analysis The function Get Mean Drift writes the value of time interval at which single switching events occur to a file. The program allows the user to input the number of points, which, in combination with the frequency defines for how long the measurement will run. The sample number (number of switchings per histogram or point) is set to 1 by the program, i.e. the Stanford SR560 does not compile a histogram, it simply relays every individual point to the acquisition system. The program then plots the individual values of critical current against time using the calculated current ramp rate. This is useful to monitor any unusual correlations of the switching current, as there should be no correlation between consecutive switching events. The noise data is also analysed using a Fast Fourier Transform, and the result can also be plotted using the same program. Fits to the Fourier Spectrum can give useful information about non-random switching, which would deviate from the expected white noise. The program plots the values of current, rather than the default time interval, by calculating the current ramp rate from the experimental input parameters, which can be seen in Figure The program can also add a fit line if necessary, such as a 1/f line, as a guide to the eye. 115

130 CHAPTER 5. APPARATUS DESIGN Figure 5.22: Simulated current switching histograms in Niobium junction, I c = 10.0µA Simulating the mean drift The dependence of the mean position with respect to temperature is difficult to produce analytically for a given junction, however it can be investigated by simulating the data. A junction with a T c of 9.12K was modelled using Scilab, with a similar implementation as is performed in the fitting procedure. The plots are shown in Figures 5.22 and In the histograms, each successive plot shows an increment of 5mK, starting at 5mK. The histograms widen with temperature and decrease in height. A plot of this drift is shown in Figure 5.23, which was calculated by taking the mean position of theoretically generated histogram plots. The simulated mean drift shows a non-linear dependence upon temperature. The true mean drift tends to show variations from this form, which can be used as an estimation of error on the T esc plots. Temperature Calibration The Temperature Calibration routine takes the filename under which the histograms are saved, and passes this value to a lookup routine which returns the corresponding 116

131 CHAPTER 5. APPARATUS DESIGN Figure 5.23: Simulated drift on mean in Niobium junction, I c = 10.0µA calibrated temperature value. The program works by selecting the two voltage points above and below the measured value, and performs a linear interpolation of the corresponding temperature values. An error in sigma can be caused by variation in temperature during a measurement. To help keep track of this error, two measurements of the temperature are taken, one at the start of each reading, and another one at the end. It is the average of these two values which corresponds to the average temperature of the sample during the histogram acquisition. Plotting programs Scilab was also used to perform some of the preliminary data analysis, including calculating the escape rates, the escape temperatures and the theoretically predicted values of both these variables for the current values at which the data were taken. 117

132 CHAPTER 5. APPARATUS DESIGN Figure 5.24: Fitting histograms using the MQT control centre software Histograms The Plot Histograms function reads in the data files, and sends the appropriate data to Scilab, plotting the selected histograms, as a function of current, on the same axes. An example of a graphical output from the Plot Histograms function can be found in Figure This function also accesses Temperature Calibration in order to append the legend with the temperature corresponding to each histogram. The histogram plotting function also plots theoretical histogram fits to the data, by calculating the theoretically expected normalised probability, p t, of an escape event at each recorded value of current. The implementation of the theoretical fit follows the theory discussed in Section 4.2. The routine allows the user to change the fitting parameters, and see the effect on the histograms in real time. Escape Rates The Plot Escape Rates function also reads in the data from the raw files and converts the time intervals to current values, and the filename voltage to temperature. It then uses a Scilab implementation of the escape rate reconstruction equations, (see 118

133 CHAPTER 5. APPARATUS DESIGN Figure 5.25: Example escape rates from data values and theoretical calculation Section 4.2), to calculate the experimental escape rates from each set of data points. The function also calculates the expected (theoretical) escape rates for the same current values. The theoretical escape rates are calculated from equations 2.26 and 2.29 using the following user-adjustable parameters: Current Ramp rate, I Critical current (T=0), I c0 Junction capacitance, C Junction resistance, R N Using these values it is possible to calculate the theoretical thermal escape rate, Γ th, the theoretical quantum escape rate Γ q, and the theoretical Intermediate escape rate, Γ i, as described in Section 4.2. The user adjusts the parameters via use of the sliding bars in the GUI (Figure 5.21), and the graphical output adjusts in real-time. A screenshot of the function output is shown in Figure

134 CHAPTER 5. APPARATUS DESIGN Histogram width The Plot Sigma function reads in the value of sigma as calculated and returned from the Stanford instrument, and plots this against temperature, using the separate Temperature Calibration routine. The value of sigma is read in directly from the datafile, as calculated by the SR620 Time Interval Counter. Escape temperature The Plot Escape Temp function calculates the escape temperature from each of the selected filenames, and plots this variable against the bath temperature on a logarithmic scale. The escape temperature is represented by a single point from each histogram, and as such requires a selection criterion. The datapoint is selected by a fixing parameter in the code, which is usually set to be between Γ = 10.8 and Γ = 11.2 following the method of Martinis et al. [33]. Testing the modelling software A theoretically generated dataset was fitted using the model [103]. The parameters of the junction were cited as I c = 1.3µA, di/dt = 0.1, and Q = 40, giving a resistance value of R N = 812Ω. The results can be seen in Figure 5.26, 5.27 and From these fits it is concluded that the main aspects of the fitting program, namely the histogram, escape rate and escape temperature fitting are all working well. 5.6 System testing This section describes some of the tests that have been performed on the system in order to establish that the data acquisition is working well. Several procedures 120

135 CHAPTER 5. APPARATUS DESIGN Figure 5.26: Testing the histogram modelling using theoretically generated data Figure 5.27: Testing the escape rate model using theoretically generated data Figure 5.28: Testing the escape temperature model using theoretically generated data 121

136 CHAPTER 5. APPARATUS DESIGN Figure 5.29: Example of a graphical histogram output from the SR620 for optimising the system and testing the noise level are described, and the settings procedures for acquiring measurements once the apparatus has been cooled are documented. Optimising the histogram acquisition The principles of use of compiled histograms and the time-of-flight technique has already been mentioned in Section 4.1, following the early work of Voss [13], Fulton [89] and Jackel [90]. In this setup discussed here, the histograms are compiled using the SR620 Time Interval Counter. SR620 can be found in Figure An example of a histogram output from the To determine the intrinsic noise limit of the experiment imposed by the use of the Universal Time Interval counter and ramp generator, the counter was connected to the ramp generator alone and histogram measurements accumulated. It was possible to adjust several settings of the apparatus, including the polarity of the two trigger signals, the trigger levels, the pre-amplifier gain and filtering. The trigger levels are best set to rapidly rising or falling edges, to mimic the junction switching. If it is necessary to trigger on a rising ramp, the level should be set to the point where the 122

137 CHAPTER 5. APPARATUS DESIGN Figure 5.30: The effects of changing the trigger level on the width of the histogram voltage crosses zero, as this is where the error on slope jitter is minimal. Figure 5.30 shows the result of a test on a sharp rising edge, as would be seen in a junction switching event. The histogram is independent of the trigger level over the majority of the rising edge. Components can also be introduced one by one into the system, including the preamplifier and the attenuator network, a dummy sample and an artificial Josephson Junction. The attenuation can be adjusted using a dummy attenuator network constructed to mimic the internal wiring of the fridge, which is illustrated in Figure During these tests, histogram width is monitored, to see if it is affected by the addition of any components, or the adjustment of any settings. The drifting mean To check that the equipment had stabilised, the Stanford was connected to its own internal test signal, and the position of the mean was monitored over a period of several hours, to test this piece of equipment independently. It was found that the Stanford equipment needed to be left to settle for a minimum of a few hours before taking a large set of data. The same result was found when observing the 123

138 CHAPTER 5. APPARATUS DESIGN Figure 5.31: Circuit diagram showing the dummy attenuator network for noise testing Figure 5.32: The drift on the histogram mean and sigma at the start of the data acquisition switching histograms directly from the junction. The drifting mean is most probably due to component warming. The results of observing the mean over a course of approximately 30 minutes can be seen in Figure After leaving the equipment to stabilise for several hours, the drift on the mean can be substantially reduced (Figure 5.33). The width of the histogram as a function of number of samples per histogram was also investigated, as shown in Figure A good measure of the histogram width can be obtained even for relatively low numbers of samples. 124

139 CHAPTER 5. APPARATUS DESIGN Figure 5.33: The drift on the histogram mean and sigma several hours into the data acquisition Figure 5.34: Demonstration that sigma is accurate even for low numbers of samples. The x-axis shows the natural logarithm of the number of samples. 125

140 CHAPTER 5. APPARATUS DESIGN RF rejection testing Tests were performed to understand the response of the junction to a deliberate source of microwave interference. A 3W RF source operating at 439M hz was introduced into the vicinity of the cryostat. A dipole transmitting antenna was used to produce a wide, double lobed RF field pattern. The radiation intensity was simultaneously measured using a HP8590A spectrum analyser, with the input connected to a receiving antenna of a similar dimension to the transmitter. With the RF source turned off, the response of the analyser gave several peaks, above the noise. The analyser noise limit could be reduced to -72dBm. Above this level, environmental RF interference was observed at particular frequencies: 962MHz at -70dBm, 398MHz at -63dBm, 30MHz at -56dBm. With the RF source turned on, and located at 3 different distances from the cryostat, the readings were as follows: RF level (dbm) Effect on histogram Change in time interval of 600ns Change in time interval of 518ns No detectable change in histogram This suggests that the value of I c will not be changed significantly by any interference of this frequency below -50dBm. The allowable upper limit of environmental RF interference can therefore be concluded to be -50dBm. All background interference observed via pickup of the receiving antenna in the lab environment were below this threshold level, and thus they are not expected to harm the junction switching characteristics. It was noted that the most susceptible area of the apparatus to the RF interference was the BNC cable located between the preamplifier inside the shielded box at the top of the cryostat and the SR570 amplifier. Altering the position of this cable 126

141 CHAPTER 5. APPARATUS DESIGN resulted in differing amounts of RF coupling into the system, and the maximum effect on the histogram was observed when the source was placed close to the cable. This again provides strong evidence for the use of an optically isolated system, such as that described in the future measurement system, Section 5.4. A Josephson Junction simulator A Josephson Junction simulator was used [104] to test the response of the system without the need to cool the dilution refrigerator. The circuit diagram is shown in Figure A typical IV response of the simulator is shown in Figure The critical current of the junction could be changed in a range of 1 100µA, and also the noise level of the current-voltage characteristic, to mimic a finite temperature situation. Switching measurements were compiled using the artificial junction, in order to test that the setup was working properly. An example of the histograms obtained can be seen in Figure In this measurement, the noise level on the switching event was changed to mimic an increasing temperature. The histogram widths can be seen to increase accordingly. 127

142 CHAPTER 5. APPARATUS DESIGN Figure 5.35: Circuit diagram of the Josephson junction simulator 128

143 CHAPTER 5. APPARATUS DESIGN Figure 5.36: IV characteristics of the artificial Josephson Junction at different values of I c Figure 5.37: Current switching histograms of the artificial Josephson Junction 129

144 Chapter 6 Results Several different junction structures were investigated with the measurement system, each having various tests performed as described above. Junctions of both Low temperature superconductor (Niobium) and High Temperature Superconductor (YBCO) from various sources were investigated. In this section, the procedures for measuring the junctions will be explained, and the results will be presented and analysed. 130

145 CHAPTER 6. RESULTS 6.1 Measurement procedures Samples are cooled using a continuous flow cryostat to check their Josephson properties at 4.2K. Samples are then cooled using the dilution refrigerator setup down to 40mK. The IV characteristic is clearly visible using the monitor scope. I c becomes much sharper, due to both the lower temperature and the improved shielding of the fridge system. An example of such a characteristic can be seen in Figure 6.1. The scope is a useful technique of visually checking the state of the junction. By careful monitoring of the critical current, it is possible to observe effects which may be obscuring the true thermal or quantum switching distribution, as the measurement occurs quickly, whereas a compiled histogram can average out the signature of noise or drift. Such effects are easy to distinguish once the user is familiar with the true distribution shape of the switching events observed on the scope. Figure 6.1: Monitoring switching events using an oscilloscope, sample IPHT, I c = 9.63µA. Horizontal scale = 3.7µA/div, Vertical scale = 1mV/div A series of preliminary histograms is taken at base temperature, similar to those shown in Figure 6.2 to establish that the noise level is acceptable and that the datasets are reproducible. The temperature of the mixing chamber is adjusted by 131

146 CHAPTER 6. RESULTS Figure 6.2: An example of a series of switching histograms taken at different temperatures allowing the dilution unit to cool at maximum power (highest rate of circulation) whilst applying a variable amount of heat to the mixing chamber. Datasets of temperature dependent histograms are taken as the fridge warmed and cooled. Controlling the temperature When compiling escape rates or histograms as a function of temperature, it is easier to allow the temperature to slowly drift and take continuous measurements than to try and stabilise the temperature. With a slow drift, the error on temperature is low, and as temperature readings are taken at the beginning and end of each measurement, the error is recorded as part of the measurement. However, this error does not take into account the thermal lag between the temperature reading on the thermometer and the temperature of the sample substrate, or the fact that the electron temperature may differ from the substrate temperature. The implication of these effects will be discussed further in Section 6.5. For the majority of the measurements, the error in temperature was kept to within ±0.002K. The error in sigma due to a significant warming or cooling rate can be estimated 132

147 CHAPTER 6. RESULTS by σ = I(T final ) I(T initial ). For a typical histogram this would be an error of approximately 2-5ns. Adjusting the magnetic field For junctions with a high critical current, and indeed for all future measurements, a reliable method of suppressing I c was necessary. A superconducting magnet assembly was installed around the vacuum can containing the sample. The magnet was measured with a Hall probe for calibration at a current of 8.88mA, which produced a change in field of 0.24mT. The magnet was therefore able to produce fields of 0.027T/A. To check that this is adequate for the junction, the active junction cross section is estimated to be 2λd, where d is the width of the junction. Given a penetration depth in Niobium of 39nm and a junction width of 40µm yields a cross sectional area of m 2 To ensure a quantum of flux in this area, a field density of T m 2 is needed, therefore the necessary magnetic field is 0.64mT. This was easily achievable with the setup. The magnet was connected up to a power supply enabling operation in persistent current mode. The critical current of the junction sample could be easily suppressed using the magnet. The calibration was found to be 14mA per Φ 0 for a typical Niobium tunnel junction. This was determined by observing the magnitude of I c on the oscilloscope screen as the current was changed. The maximum value of I c was periodic, as expected, with a periodicity of 14mA. It was therefore easy to access a wide range of critical currents using this system. A second method of changing the critical current, I c is possible with some junctions, by switching in and out the preamplifier. This allows the junction to settle into a particular magnetic state. This behaviour is most probably caused by rearrangement 133

148 CHAPTER 6. RESULTS of magnetic flux in the vicinity of the junction, thereby affecting the local magnetic field. It is equivalent to the application of a fixed magnetic field, and was utilised before the addition of the magnet to change the value of the critical current. This result is not unsurprising as it gives a direct connection to the voltage leads of the sample. Taking data The DS345 ramp generator is adjusted such that the waveform used is a sawtooth shape of a non-inverted polarity (sharp falling edge). The ramp generator had a variable input voltage V P P and voltage offset V Off. These two parameters should be noted down for each experiment, as they are used to calculate the ramp rate of the system. The time interval is measured using the SR620 as the difference between when the start signal exceeded a trigger level T RIGA and the stop signal exceeded a trigger level T RIGB. It is important to keep the maximum amplitude, frequency and offset of the signal constant throughout the experiment, as changing the first two of these variables will alter the current ramp rate, and changing the offset will affect the mean position of the histogram. The critical current of a junction can also be highly sensitive to any changes in electrical conditions during the course of a dataset, and so it is necessary to avoid adjusting or unplugging nearby electrical equipment, for example the experimental computer, or the oscilloscope. Observing I c on the scope is a useful method of monitoring any changes to the setup. Although the inclusion of an oscilloscope is detrimental to the overall grounding scheme, it is indispensable as a visual monitor of the switching events. Occasionally the Stanford SR620 Time Interval Counter would shift trigger levels by 10mV. This was due to the physical potentiometers on the front panel of the instrument. As the shift in trigger level changes the mean position and can render datasets useless, it must be carefully monitored. In addition to any such unavoidable changes in the mean, there will be also be a steady drift on the mean when the apparatus is first 134

149 CHAPTER 6. RESULTS used due to component warming and stabilising. It is therefore necessary to leave the apparatus to stabilise for at least 2 hours before it is possible to obtain accurate histograms. The SR620 is then altered such that the triggering occurs on the falling edge of the current ramp. The advantage of this method is the reduction of noise on the start signal. The disadvantage is that the time spent in the voltage state is increased. Other parameters which need to be adjusted are the number of samples (on the SR620 ) and the frequency (on the DS345 ). All the results presented in this section were taken between 1KHz and 5KHz. To ensure that the fluctuation in the value of I c are not obscuring the measurements, the SR620 should be set to take the highest number of datapoint without a discernable change in the value of sigma upon changing the number of datapoints. Fitting to the data The user can choose between any of the functions discussed in Section 5.5 to yield the escape rates, escape temperatures, current switching histograms, mean histogram value versus temperature (here referred to as as mean drift), and noise on the critical current. During the experiment, these raw (preprocessed) time interval data from the experiment are saved in a file and then processed by Scilab. This processed data can be fitted to theoretically produced lines, during the experiment, by using the MQT software. Before fitting, the user is therefore required to input the values of V P P, V OF F SET and the frequency of the current ramp, to allow the data processing to take place. The fitting parameters are adjusted using the sliding bars, as illustrated in Figure 5.21 until the fit is good. The main adjustable fitting parameters are critical current, capacitance and resistance. The sliders allow the data to be reprocessed in real time, so it is relatively quick to get a good manual fit. Each time the slider is moved, the data complete with fit are replotted, as the fit is easiest to assess from the 135

150 CHAPTER 6. RESULTS Figure 6.3: Example of an escape temperature plot taken from the grain boundary sample P395, illustrating the fit to the theoretical thermal and quantum regimes. Figure 6.4: Example of a sigma plot 136

151 CHAPTER 6. RESULTS straight line of the escape rate. An example of an escape temperature plot is given in 6.3. The quantum tunnelling limit is shown as the purple line. The crossover temperature occurs at the intersection between the two lines. Although it is possible to implement auto-fitting routines for such data, the initial parameters for critical current are vastly different for each measurement, and thus an initial good guess is required on behalf of the user. A manual method of sliding bars proved more useful than an autofit, as it was quick and allowed the user to see what effect changing the parameters had on the fitting. There were also sets of data for which the theory did not fully apply due to various non-ideal junction properties, and an autofit may not have worked in many of these cases. Whilst the data is being recorded, the user can plot the value of sigma, the width of the histogram, against ln(t ). This gives a quick visual idea of how well the measurement is proceeding. The sigma plot is not a good measure of the junction properties, as if there is non-random noise, the histogram will be artificially widened and sigma will be raised. It is however not easy to distinguish if this is the case from the sigma plot alone. In order to present the data in a normalised fashion, the escape rates must be calculated and fitted to the theoretically expected escape rates. An example of an unprocessed sigma plot generated from raw data is shown in Figure 6.4. In addition, as discussed previously, the user can take a mean drift measurement, either to observe the noise on the critical current in more detail, or to see if the mean value of the histogram is drifting unusually. 6.2 Preliminary studies Two samples were tested, NBSJ and NBARR. Each sample consisted of two devices. NBSJ contained 2 single independent Josephson Junctions, and NBARR contained two independent arrays of Josephson Junctions. The devices were all accessible by 137

152 CHAPTER 6. RESULTS Figure 6.5: IV characteristic of NBARR junction array, 4.2K 4-Terminal measurement leads. The two single junctions SJ1 and SJ2 measured 2µm x 2µm and 10µm x 10µm respectively. The two arrays on the second sample, AR1 and AR2 consisted of 25 junctions each, with AR1 junctions measuring 2µm x 2µm and AR2 junctions measuring 10µm x 10µm. The array structure is an important junction variant, as in the case of BSCCO the junction stacks behave in a similar fashion. The junction was therefore tested in order to establish if it was possible to switch on a single branch of a multi-branch IV characteristic. The junction array exhibited a structure consisting of steps in the IV curve. The steps are an indication of the junctions switching in turn, each with its own critical current value. A plot of the IV curve is shown in Figure 6.5. The individual critical current branches themselves are not visible due to the type of current sweep used, with the same maximum value of current each time. The junction therefore returns to the superconducting state along just one of the branches. Upon cooling in the dilution refrigerator, the junction characteristics were observed to have less thermal rounding and it was thus concluded that the fridge wiring and junction connections were working well at the base temperature. These preliminary studies helped to establish how best to operate the system, and 138

153 CHAPTER 6. RESULTS Figure 6.6: IPHT junction IV characteristic measured at 4.2K in a continuous flow cryostat. The critical current is suppressed due to poor filtering of the system that it is indeed compatible with multi-junction samples. 6.3 Low T c Junctions Commercial Junction The primary Low T c junction to be measured was IPHTB1, a Niobium sample obtained from IPHT [106]. This junction had a well defined critical current of 14µA. The sample was measured using both a continuous flow cryostat and a dip probe dewar system. The IV characteristic did not show a well defined critical current in either case. A feature at the gap voltage was however quite well pronounced, which gave evidence toward the good quality of the junction. It is quite common for the real value of I c to undergo both thermal rounding and supression by external interference in the minimally shielded continuous flow cryostat. There was also a sub-gap current with a finite slope. The shape of this region of the IV characteristic 139

154 CHAPTER 6. RESULTS Figure 6.7: Escape rate, sample IPHT, I c0 = 9.63uA Figure 6.8: Escape temperatures, sample IPHT, I c0 = 9.63uA persisted down to low temperatures, suggesting a mechanism of dissipation other than the quasiparticle resistance, which tends to zero as T 0. After the measurement of a second junction in the continuous flow cryostat system, it was discovered that a similar IV characteristic was seen, demonstrating that the measurement system was indeed thermally or noise rounding the data. On taking the same measurement using the dilution fridge setup, this problem was eliminated. I c was adjusted between 3 different values, and full data sweeps across a temperature range of 40mK-1K were acquired for each. The values were 9.63µA, 1.85µA, and 1.11µA respectively. The value of I c0 was adjusted to 9.63µA. Fitting of the data from this sample can be seen in Figures 6.9 and 6.8. It can be seen from this data that the escape rate does 140

155 CHAPTER 6. RESULTS Figure 6.9: Histograms, sample IPHT, I c0 = 9.63uA not fit the thermal regime, even at high temperatures, as the crossover occurs well above the quantum limit. The data moves smoothly between the thermal activation regime and the temperature independent regime. However, the data does not show a typical crossover. The deviation is over an extremely large temperature range, as opposed to a neat thermal-quantum crossover similar to those seen in the literature. This is indicative of excess noise in the system, and some possible causes for this are discussed in Section 6.5. Comparison Junction A Niobium junction (PTB) from the NPL group [107] was acquired mainly for use as a calibration sample, as a few measurements had already been performed on this sample The junction was fairly large, with an area of 40µm 2 and a nominal I c of approximately 120µA. An IV characteristic of the junction was measured, again using the continuous flow cryostat system. The result is shown in Figure The IV characteristic illustrates a large hysteresis, and a well defined gap feature, suggesting that the junction is tunnel-like. Examples of current switching histograms can be found in Figure 6.11, and show that the distributions are somewhat wider than the theoretical predictions at all temperatures. This is also apparent from the escape rate plots, shown in Figure 141

156 CHAPTER 6. RESULTS Figure 6.10: IV characteristic, sample PTB, 4.2K Figure 6.11: Current switching histograms, sample PTB, I c = 43.9µA 142

157 CHAPTER 6. RESULTS Figure 6.12: Escape rates, sample PTB I c = 43.9µA 6.12, where the discrepancy manifests as an overall reduction in the escape rate gradient when compared to the theoretical curves. The sigma plot, given in Figure 6.13 demonstrated a crossover, however when viewed in light of the histograms and escape rates, may be attributed again to noise-induced escape. The crossover does not however seem to be as wide as in the IPHT sample, and thus the noise level does appear to have been improved, by following the technique for optimising the histograms, as documented in section 5.6. By taking measurements on several junctions, the system noise level was able to be improved with each successive run, due to an increasing familiarity with the apparatus and the potential sources of noise. By comparing the escape temperature plot (Figure 6.14) with the mean drift (Figure 6.15) a correlation can be seen. Changes on the mean directly affect the escape temperature plot. This is intuitive, as if the mean position of the histograms change, the value of current at the fixed point will deviate from the expected value. This appears as an error on the value of T esc. There are several reasons for the temperature dependence of the histogram mean. The first reason is trivial; the early switching events are more likely to occur if the 143

158 CHAPTER 6. RESULTS Figure 6.13: Sigma plot, sample PTB, I c = 43.9µA Figure 6.14: Escape temperatures, sample PTB, I c = 43.9µA 144

159 CHAPTER 6. RESULTS Figure 6.15: Mean drift, sample PTB, I c = 43.9µA temperature is higher, and thus the mean position of the histogram moves to the right, indicated by a decrease in the mean drift plot with respect to temperature, as described in Section 6.1, and illustrated in Figure The second is that the fitting parameter I c0 itself may change due to the dependence of critical current upon temperature, as given by the Ambegaokar-Baratoff relation, described in Section 2.1. The equation is reproduced below: I c R N = π ( ) 2e tanh 2kT (6.1) If the sample temperature is well below T c, and the junction is in the dirty limit, the dependence upon temperature is negligible. However, the temperature dependence can become important if the junction is in the clean limit. The third is that the value of I c can change if there are changes in the local magnetic conditions, for example flux moving in the vicinity of the barrier, or fluctuations in the applied field itself. All the above effects manifest in the mean drift data. The first is accounted for by the theory when fitting histograms, however the latter two are not. 145

160 CHAPTER 6. RESULTS It can be concluded that the T esc data is only truly valid if the shape of the mean histogram plot can be explained fully. The mean drift is therefore a good way of estimating the error in the fit of T esc. 146

161 CHAPTER 6. RESULTS Low T c SFS precursor Sample CAM03 was a Nb-NbO x -Nb tunnel junction, which acts as a precursor to the SFS devices produced by the Cambridge group [108]. The junction will be used as a standard with which to compare the SFS structures produced subsequently. The SFS junctions are fabricated to the same specification as this tunnel junction; the insulating barrier replaced with a ferromagnetic-insulator bilayer of variable width. These junctions are proposed as potential qubit candidates, and as such, it is necessary to test the junction behaviour at low temperatures. Preliminary studies on the junction properties are therefore of great importance. Figure 6.16: Fiske steps observes on the IV curve, sample CAM03, Junction 2 The junction sample CAM03 had a nominal critical current of 100µA, as measured in the continuous flow cryostat system. A field of 3.05mT was applied to the junction in order to suppress the critical current to within the measurement range. critical current was reduced to 50µA at this field. The A Fiske step structure was observed on the IV characteristic of the junction, as shown in Figure The capacitance fitting parameter was estimated using the 147

162 CHAPTER 6. RESULTS Figure 6.17: Multivalued I c observed in CAM03 sample. The white vertical lines have been added as a guide to the eye, indicating the positions of I c voltage value of the first Fiske step in equations 2.23 and 2.24 to be F. However, the inductance per unit length is difficult to determine, so this estimate is only a guide. The capacitance of a tunnel junction is also commonly estimated to be approximately 20fF per µm 2, thus giving a value of F. The capacitance was therefore likely to lie between these two values, and indeed it was found that the higher of the two values gave a reasonable fit to the data. Duty cycle tests The sample was tested at different values of ramp rate and duty cycle (percentage of overall cycle time spent in the voltage state). The tests were done at 4.2K, and the temperature was allowed to stabilise over an extensive period. The sample was tested at various duty cycles, and the corresponding fits are shown in plots 6.18, 6.19, and In the first case the duty cycle was minimised and the fit at 4.11K is good. Upon increasing the duty cycle, the value of critical current was slightly higher, but there was no other adverse effect upon the fit. Lowering the duty cycle again showed only a small change in the fitting parameter I c0. These 148

163 CHAPTER 6. RESULTS Figure 6.18: Escape rates, CAM03 sample, duty cycle 1.3%. measurements establish two key points: The fitting of the data at 4.2K is accurate, and the fit is insensitive to the ramp rate. Upon successful fitting at 4.2K, the sample temperature could be changed to observe the dependence of T esc upon T. See Figure 6.29 for the escape rate fitting. The curves fit at higher temperatures, but at the two lowest temperatures the escape curves appear to become non-linear. This is attributed to noise limiting the measurement of the macroscopic quantum tunnelling. The width of the histogram appears to follow the correct form, a straight line dependence when plotted on a log temperature scale, with a crossover to a temperature independent behaviour. However, upon plotting the Escape Temperature (Figure 6.23), it can be seen that there are deviations from the thermal activation, and an elevated crossover, above that expected for quantum tunnelling. It can also be seen by comparing the escape temperature plot with the mean drift plot, that there is a correlation between the two. This is to be expected, as the escape temperature is calculated from a point on the escape rate curve, the position of which depends on the mean position of the histogram. 149

164 CHAPTER 6. RESULTS Figure 6.19: Escape rates, CAM03 sample, duty cycle 35.8% Figure 6.20: Escape rates, CAM03 sample, duty cycle 2.6% 150

165 CHAPTER 6. RESULTS Figure 6.21: Current switching histogram, CAM03 sample, duty cycle 1.3%. The two anomalous points arise due to the autoranging of the SR620 and should be disregarded. Measurements performed on the junction at a temperature of 1K demonstrated that at particular values of magnetic field the critical current was noted to take two distinct value. This resembled that of a two-level fluctuator, as described in Section 3.6. A scope trace illustrating the observed effects is shown in Figure In this image, the trace intensity has been maximised to clarify the rising edge of the onset of the voltage state. The two-state system was a function of applied field, and it was possible to adjust the current difference between the two states by changing the field. By choosing the value appropriately, I c could be made single valued. It was found that the IV characteristic could be changed by thermally cycling the junction about T c. The hysteresis was reduced and the multi-valued I c was no longer present. However upon changing the critical current using the magnetic field, the Fiske step structure and multivalued I c was again observed, thus the effect was reproducible. A possible explanation for this effect will now be discussed. 151

166 CHAPTER 6. RESULTS Figure 6.22: Sigma plot, sample CAM03, I c = 67.7µA. A clean crossover can clearly be seen at around 300mK, which is of the correct order of magnitude for the estimated I c0 and C values Critical current fluctuations Unlike a two-level fluctuator, the two values of I c were not independent of each other. By adjusting the maximum current below the level of the higher critical current, it was found that the lower value was no longer excited. This gave evidence towards the junction system existing in an oscillatory state. This is not surprising; the Josephson Junction is a complex non-linear system. A possible physical feedback mechanism could be one in which the local magnetic field generated by a particular value of critical current causes its own self-suppression. Similar effects have been seen in junctions before. In a paper by Cronemeyer, [109], negative tails are observed on the IV characteristics. These are regions where the junction jumps from a higher voltage state to a lower one as the current is increased, corresponding to a negative dynamic resistance. The features are attributed to a chaotic behaviour in the junction. Whilst there is not the scope in this project to explore these effects, it is a possibility that in the junction region two or more values of critical current may become self-sustaining. For the junction devices proposed, 152

167 CHAPTER 6. RESULTS Figure 6.23: Escape temperature, sample CAM03, I c = 67.7µA. The red line shows the theoretically expected thermal escape, the purple line shows the expected escape due to quantum tunneling. Figure 6.24: Mean Drift, sample CAM03, I c = 67.7µA 153

168 CHAPTER 6. RESULTS Figure 6.25: Fluctuations of I c as a function of time, sample CAM03, 1K, demonstrating telegraph noise on the critical current. There are at least three distinct stable values, with fluctuations between two of the levels appearing to dominate. Figure 6.26: Frequency analysis of fluctuations with 1/f fit, sample CAM03, 1K 154

169 CHAPTER 6. RESULTS Figure 6.27: Fitted histograms, CAM03 sample, I c0 = 58.8µA showing temperature effects such as the qubit, this behaviour would be detrimental, as the critical current must be well defined and the coherent phase states must be free from mechanisms of noise-induced escape over the potential barrier. This feedback mechanism could be occurring on several different time and current scales. This could explain the apparent regions of stability at particular bias currents. Although this phenomenon directly affects the junction measurements, it is difficult to characterise the behaviour fully. It is suggested that junctions exhibiting this behaviour should be regarded with caution, even with an ideal apparatus setup. Noise measurements were taken on this sample at 1K. It should be noted that these results are a measure of current noise, as they were recorded using the technique described in Section 5.5. The results can be seen in Figures 6.25, which shows the time domain fluctuations of the critical current, and 6.26, which shows the Fourier spectrum of the noise. A 1/f line is included in the plot as a guide to the eye. The fluctuations look very similar to telegraph noise, which indicates the presence of fluctuating occupation sites within the barrier, as mentioned in Section 3.6. The good fit to a 1/f dependence again provides evidence for trapping sites in the barrier. A second data set was taken on the sample after adjusting the critical current to 155

170 CHAPTER 6. RESULTS 24.0µA. The results can be seen in Figures 6.28, 6.29, 6.30 and Low T c junction conclusions It is concluded from performing tests on the Low T c junctions that the system is working reasonably well, and all the relevant data are able to be obtained. The fitting looks good on the CAM03 sample at 4.2K, however when the temperature is lowered the data deviate from the theoretical fits. The origin of this discrepancy is as of yet unknown, but it is likely to be a combination of the drifting mean, a residual noise in the system of unknown origin (including noise from the junction itself) and self heating of the junction. Because these effects are difficult to isolate and test individually, further improvements to both the system and the theoretical model need to be made in order to obtain a better fit to the data. In particular, the theoretical model has assumed no effects due to the finite damping present in the system. The effect of dissipation, and quantum corrections, as discussed in Section 2.7 can influence the escape rates and crossover temperature, and will need to be modelled more thoroughly. Secondly, many effects have been seen in junctions which can obscure the true switching distributions, and one should be careful when selecting junctions for use in such experiments. It is found however that monitoring the mean position of the histogram is an excellent indicator of data quality, and is easy to fit to a theoretical curve, from which the validity of the other fits can be established. It should also be noted that further measurements on the commercial Low T c junctions would be of great benefit, as the measurement system is improved further. 156

171 CHAPTER 6. RESULTS Figure 6.28: Current switching histograms, CAM03 sample, I c0 = 24.0µA Figure 6.29: Escape rates, CAM03 sample, I c = 24.0µA. 157

172 CHAPTER 6. RESULTS Figure 6.30: Escape temperatures, CAM03 sample, I c0 = 24.0µA Figure 6.31: Mean drift with fit, CAM03 sample, I c0 = 24.0µA 158

173 CHAPTER 6. RESULTS 6.4 High T c junctions Grain Boundary Junctions A bicrystal [100] tilt Grain Boundary YBCO film was patterned (sample P395) using the standard lithographical techniques and mask described in Section 5.1. The resulting film consisted of 3 GBJ regions, with widths of 3µm, 5µm and 10µm. Figure 6.32 shows the IV characteristic of the 3µm junction, as measured using the continuous flow cryostat setup mentioned previously. All 3 junctions show RSJ-like properties, with the two smaller ones having well defined critical currents. However, due to the thermal and noise rounding properties of this measurement system, the hysteresis of the junction samples is not directly observable until the samples are cooled using the dilution refrigerator. Figure 6.32: IV characteristic, sample P395 JJ1, 10K The smallest junction had a width of 3µm. In the low noise measurement apparatus the IV curve obtained from this junction became much clearer, and demonstrated a hysteresis of approximately 0.2I c, showing that the system was appropriately damped for the switching current measurements. It should however be noted 159

174 CHAPTER 6. RESULTS Figure 6.33: Escape temperature, sample P395 JJ1, I c0 = 25.4µA that evaluation of the damping paramter using Equation 2.20 yields a value for the McCumber parameter of β = This is only just in the underdamped regime, and thus one should be careful in drawing conclusions from fitting to the underdamped model of the junction. At a temperature of approximately 75mK, and with zero applied field, the value of I c was 25.4µA. Several temperature sweeps were taken, and typical datasets are shown in Figures 6.33, 6.34 and It can be seen from the mean drift plot that the datasets were quite noisy. This is due to the drift on the mean changing over a timescale similar to that over which the histograms were compiled. If the mean value changes rapidly during a measurement, then sigma will artificially be higher. One can conjecture that, given a long period fluctuation, there will be times when the mean value is changing rapidly, and times when it is not. observed here. This could produce noise on the value of sigma similar to that This is, however, a difficult problem to rectify. Taking more points per histogram would obscure the true switching distribution, and taking fewer points per histogram 160

175 CHAPTER 6. RESULTS Figure 6.34: Current switching histograms, sample P395 JJ1, I c0 = 25.4µA. It should be noted that the histograms look significantly more Gaussian than theoretically expected. Figure 6.35: Mean drift, sample P395 JJ1, I c0 = 25.4µA 161

176 CHAPTER 6. RESULTS gives a less accurate calculated value of sigma due to the random nature of the switching events. By adjusting the frequency of sweep and number of samples, (both of which essentially reduce to altering the measurement time) one can empirically reduce this noise contribution. However, it was found to be impossible to eliminate entirely, and a possible reason for this will be discussed. It was noted by monitoring the switching events on the scope screen whilst data was being compiled that the mean value of I c was changing with respect to time. Though the motion was apparently random, it was dominated by long period fluctuations. It is possible that these fluctuations could be caused by 1/f noise in the barrier region. Several attempts were made to isolate the source of noise, and it was concluded that the residual noise may be intrinsic to the junction itself. The measurements were repeated at different times of day to ensure that the noise was not similar to the occasional non-random fluctuations in I c observed in the Niobium samples. It should be noted that this noise was of a different character from that seen in the Niobium samples, which was almost certainly due to external sources, and that conversely no noise of the character seen in this sample was observed in the Niobium samples. The application of a magnetic field was required in order to investigate the noise and switching at different critical currents. With the magnet current set to a value of 160mA, the value of I c was 31.4µA (corresponding to a field of 4.32mT ), This value was higher than the zero-field I c. It is possible that there may have been trapped flux in the barrier region, which had been supressing I c under zero applied field. The critical current was recorded as a function of field, and the result can be found in Figure The critical current varied smoothly with applied field, with a minimum very close to zero field. This could have been due to the presence of the LCMO layer on top of the YBCO film meaning that flux would become trapped in the junction region upon cooling below T c. 162

177 CHAPTER 6. RESULTS Figure 6.36: Escape rates, Sample P395, JJ1, I c0 = 35.4µA The sigma plot, escape temperature plot and mean drift plot for I c = 31.4µA are shown in Figures 6.36, 6.37 and 6.38 respectively. The escape temperature can be seen to decrease steadily, with a slightly improved fit to the theoretical thermal escape over the initial value of I c. The histograms do not fit even though the escape temperature fit appears to be better than in the Low T c junctions. With both values of I c, the sigma plot was observed to exhibit unusual feature. Sigma remained constant above 900mK, and upon further heating, began to decrease. This can be seen in the sigma plot shown in Figure All parameters, including the warming rate, were kept constant, and the mixing chamber was warmed using the usual heating technique, To eliminate the possibility that the heater was affecting the results, the dataset was taken again upon removal of the mixture from the dilution unit, a process which causes warming of the mixing chamber without the need for applying heat. The same result was observed in this case. A possible explanation of this effect is that the junction has entered the phase diffusion branch. Phase diffusion in High T c superconductors was first observed by Kivioja et al. [17]. For an overview of phase diffusion in Low T c SQUIDs as a 163

178 CHAPTER 6. RESULTS Figure 6.37: Escape temperature, Sample P395, JJ1, I c0 = 35.4µA Figure 6.38: Mean drift, Sample P395, JJ1, I c0 = 35.4µA 164

179 CHAPTER 6. RESULTS Figure 6.39: Evidence for phase diffusion behaviour in sigma plot, Sample P395, JJ1, I c0 = 35.4µA function of magnetic field, see also [110]. A similar result has recently been seen in BSCCO junctions at low temperatures ( 1K) by Li et al. [111], at the same temperature, and by Krasnov et al. [112] at higher temperatures, close to T c. In Figure 6.36, an upturn on the escape rate characteristics at high currents can be seen. Similar shaped features on the histograms and escape rates were also observed in the BSCCO studies. In the results of Li et al. they use the signature of the phase diffusion regime to calculate the resistance parameter of the phase diffusion branch, R 0. This allows them to restrict their fit to no free parameters, unlike the previous results obtained by Inomata [94]. In the phase diffusion regime they fit to the histograms by allowing I c and T to be free parameters. It is concluded that the effect should be investigated further in the junction measured here, to try and restrict the free parameters in a similar way. This will allow better fitting of the data in the future. It is also interesting that in this experiment, the effect has only been observed in the High T c sample P395, even though several junctions were measured at these temperature. The observation of such a regime may even prove useful in future experiments, as demonstrated in the paper by Warburton et al. [113]. In this report, the high resistance of the junction, which allowed 165

180 CHAPTER 6. RESULTS Figure 6.40: Regimes of escape from the potential well. From Kivioja et al. [17] the phase diffusion regime to manifest, was also indicative of a possible mechanism by which junctions may become isolated from the environment. There are several reasons why this may not be the case. To enter the phase diffusion regime, the Josephson energy of the junction must be large compared to k B T, in order for retrapping to become significant. An appropriate limit would be when E J < 10k B T. In this junction, E J = , and 10k B T = , suggesting that the junction is well into the running phase regime. A schematic of the different regimes of operation has been reproduced in Figure The junction under consideration here would be well above the top left hand corner of the plot (running state). The effect was only seen at the higher value of I c, which is inconsistent with the phase diffusion regime becoming more favourable as I c is decreased. The cause of the effect cannot therefore be attributed to the phase diffusion regime, as it was not observed in the sample with the lower critical current. The cause of the decrease in σ as a function of temperature in this dataset therefore remains unknown. The effect is possibly due to the self-heating in the junction region, or thermal resistance between the junction and the sample box, becoming temperature dependent, and is 166

181 CHAPTER 6. RESULTS another example of how the data can be rather misleading. The same fluctuations on I c were observed at this higher value of critical current, and appeared to be independent of applied field. In this track the noise was investigated a little further by recording the noise spectrum. Telegraph noise was observed on the position of I c at a base temperature of 40mK, which can be seen in Figure The analysis of the noise spectrum looks like (1/f) 2 noise, as shown in Figure The purple line shows the (1/f 2 ) fit, included as a guide to the eye. The origin of the (1/f 2 ) noise is unknown. There are many possible mechanisms of excess noise generated in a High T c sample. High temperature junction conclusions The appearance of this effect is however consistant with self-heating in the junction, as the junction electron temperature may be much higher than the mixing chamber if the junction remains in the voltage state for a substantial amount of time. From the High T c samples it is concluded that there is likely to be a self-heating effect due to the junction spending too much time in the voltage state. There is also an excess noise component present in the measurements. The combination of these two factors allow the data to be fitted over a small temperature range, or a larger range by manually adjusting T. It is possible to fit the histograms over a small region, but not over the entire dataset. One should be cautious when making assumptions based on the BCS theory when dealing with High T c samples. The assumption that I c0 is temperature independent may not hold in this case. Further data and analysis are required from these samples, and from similar Grain Boundary Junctions for verification. It is noted that this particular problem has not been addressed much in the literature. The intrinsic noise of junctions, especially the 1/f noise, has been extensively studied in SQUIDs, however it has not been considered at length in the case of High 167

182 CHAPTER 6. RESULTS Figure 6.41: Fitted histograms, sample P395, JJ2, I c0 = 4.26µA Figure 6.42: Fitted escape temperature, sample P395, JJ2, I c0 = 4.26µA 168

183 CHAPTER 6. RESULTS Figure 6.43: Time Domain noise, sample P395, JJ2 50mK Figure 6.44: Frequency Domain noise, sample P395, JJ2 50mK. The purple line illustrates the (1/f 2 ) fit. 169

184 CHAPTER 6. RESULTS Figure 6.45: Magnetic characteristic, sample P395 JJ2, 50mK. The right hand side of the trace is potentially increasing toward the first maximum of the Fraunhofer pattern. T c qubits. It is therefore concluded from this study that it is necessary to perform careful noise measurements on all potential qubit devices based on HTS schemes. It should be noted that in the symmetric GBJs discussed here, the intrinsic noise observed appears to be obscuring the observation of a crossover to quantum behaviour. This may be an important consideration when designing qubits using GBJ systems. In theoretical studies such as [83] it is suggested that there are fundamental differences between the quantum properties of different types of GBJ. This may be the case, but if excess noise dominates all such junctions, these differences may not be able to be exploited in real GBJs with non-ideal interfaces. The characterisation of the noise is important in High T c junctions, due to their potential use as qubits in future measurements. In this measurement, the long period 1/f noise cannot be avoided. However, in the case of quantum bits, the operating frequency will be much higher, and thus these low-frequency noise effects may not cause as much of a problem as they do in Macroscopic Quantum Tunnelling experiments. In addition, the decoherence time for current qubit device proposals is of the order of ns-µs, which is much higher than the frequency range considered here. These low frequency noise effects may be more detrimental to the qubit state 170

185 CHAPTER 6. RESULTS as new ways of increasing the decoherence time are discovered. 6.5 Difficulties with the measurements There are two major factors to be considered when taking the measurements: Junction self heating effects, and the effect of noise. These issues will now be addressed. Self-heating The self heating of the junction manifests in two forms, both of which involve Joule heating. The first is due to the resistance of the leads in contact with the mixing chamber, which will artificially raise the temperature near the junction. The second form of heating comes from the power dissipated due to the junction remaining in the voltage state for a finite period of time after switching. It is possible to minimise the amount of time spent in the voltage state, by reducing the amplitude of the current ramp. In hysteretic junctions, a current offset must also be employed, as I r occurs at a much lower value than I c, and thus the junction will spend unnecessary time in the voltage state during the negative half of the current ramp. Self-heating was investigated quantitatively, using two of the sample investigated. The histogram width and position are dependent only upon the temperature and the current ramp rate, thus by adjusting the frequency and maximum voltage, it was possible to keep the current ramp rate constant. The temperature of the mixing chamber was then recorded as a function of the time spent in the voltage state. It was found that the recorded temperature did indeed increase, suggesting a selfheating effect pushing the junction into the thermal regime of operation when the junction was allowed to remain in the voltage state for a significant amount of time. In the first sample, the IPHT junction, the least time spent in the voltage state was measured to be 2us, which was 0.6% of the total cycle time. The power transferred to the junction in this minimum case was 2mV 40µs = 80nW. In the second 171

186 CHAPTER 6. RESULTS Figure 6.46: Schematic of thermal resistances at low temperature stage sample, the CAM03 sample, the power transferred to the junction in the minimum case was 2mV 210µ s = 420nW. The results can be found in Figures 6.47 and The errors arise from the accuracy of the mixing chamber temperature readout. It should be noted that this is the effect of sample self heating as read by the mixing chamber thermometer. There is a thermal resistance between the position of the thermometer and the junction itself. The largest part of this thermal resistance is expected to be the unknown interface resistances, labelled R T HMC S and R T HS J in the diagram. Interface resistances become dominant at low temperatures. The substrate is adhered to the chip carrier using silver paint to try and minimise the thermal resistance of the layer, as a metallic contact is optimal at low temperatures. Thermal boundary resistances between solids at temperatures above 0.1K are considered at length in [114]. A raised fitting temperature may also manifest if the electrons which constitute the quasiparticle fraction of the transport current come out of thermal equilibrium with the lattice. This effect has been investigated in the case of the SQUID [115], but it is difficult to characterise. The results of Figure 6.50 seem somewhat reminiscent of this 172

187 CHAPTER 6. RESULTS Figure 6.47: Self heating effects of the junction in the voltage state, sample IPHT effect. This effect may be more apparent in the High T c case, where understanding effects due to the quasiparticles is of great importance. In a dataset where the noise level appears to be low (histograms are the correct shape and the drift on the mean is low), but the thermal offsets are high, the thermal lag in the system can be estimated as a function of temperature. This is possible because the value of I c itself does not change very much, so the percentage of the duty cycle spent in the voltage state should be fairly constant. The data can then be investigated by allowing the temperature to become a fitting parameter. In the absence of all other noise, this temperature should yield the true temperature of the electrons in the junction. This has been illustrated in Figure 6.49 where the temperature parameter for each histogram has been adjusted until the data fits the theoretical curves. The error between the theoretical temperature and the measured temperature is then plotted as a function of measured temperature in Figure It can be seen that the measured temperature deviates from the fitted temperature in a systematic way, suggesting a constant temperature difference which is a function of the cooling power of the fridge. 173

188 CHAPTER 6. RESULTS Figure 6.48: Self heating effects of the junction in the voltage state, sample CAM03 Figure 6.49: Fitted histograms with variable temperature parameter, sample CAM03 174

189 CHAPTER 6. RESULTS Figure 6.50: Fitted Temperature value vs. measured value, sample CAM03 Noise on I c It was noted that occasionally the value of I c would fluctuate between two distinct values. This was observable on both the oscilloscope as two regions around which the switching events would alternately cluster, and on the histogram data as a double peaked histogram. This noise was attributed to external sources, as it appeared for durations of several minutes, and was not present whilst running the apparatus during building quiet times. The source of this noise was most likely switch mode power supplies, or pump power supplies injecting noise into the local building ground. However, it was easy to avoid this noise by running the majority of the experiment during quiet times. Fortunately, there are several different ways of recognising the presence of noise on the signals and resulting datasets. The widening of the histogram can be seen both from the histogram data itself, and from the escape rate data, as illustrated on the lower temperature escape rates shown in Figure 6.29, where the escape rates have characteristic curves where they should appear as straight lines. Such a phenomenon is easy to see in an escape rate plot, whereas it may be difficult to spot in the original histogram. A similar finding is discussed in detail in [89]. 175

190 Chapter 7 Additional work This chapter will discuss some of the novel techniques used in the production of High Temperature superconducting samples. Although not all of the samples were successful, the methods used to fabricate and subsequently pattern them have been optimised and are therefore included in this report due to their general applicability to HTS samples. 176

191 CHAPTER 7. ADDITIONAL WORK 7.1 Fabrication of Single Crystal BSCCO whiskers In an attempt to obtain single crystal BSCCO whisker samples, a method of crystal growth was adopted following a documentation of successful BSCCO growth by Nagao et al. [116]. Some recent publications have focussed on the optimisation of growth of BSCCO, e.g. [117], and these were studied carefully in order to try to understand the way in which the whiskers grow. However, even though great care was taken to follow the recipes in the literature, the following method was attempted several times, but was unsuccessful. Precursor powders of Bi 2 O 3, SrCO 3, CaCO 3, CuO and T eo 2 were weighed out into the correct molar ratios to form the composite Bi2212, and ground with a mortar and pestle until the mixture was homogeneous. The powder was transferred to an alumina crucible and calcined in an oven at 775C for 16 hours. The mixing and calcining process was repeated a further 2 times. The powder was then pressed into pellets of varying thicknesses between 1 and 3mm. The pellets were homogeneous, and of a dark grey colour. The pellets were transferred to a tube furnace where the material was exposed to 880C for 15 minutes to promote the melt phase, and then 870C for 5000 minutes for crystal growth to occur. The growth of whiskers was indeed observed on the pellets after this treatment. The whiskers were harvested by carefully moving the pellet across a white ceramic plate, so that they could then be selected under a microscope. Several batches of whiskers were grown in this way, and numerous attempts to make contacts to them were undertaken. None of the whiskers exhibited superconductivity, and in one particular case, showed strong evidence of semiconducting behaviour. Although the whiskers grown during this project were not suitable for measurements, they allowed development of a robust method of defining 4-terminal contacts, and a method of isolating stacks of High-T c Josephson Junctions. These methods will now be discussed. 177

192 CHAPTER 7. ADDITIONAL WORK Figure 7.1: Schematic of the FIB system operation 7.2 The Focussed Ion Beam (FIB) Microscope Principles and operation The equipment used for the fabrication and subsequent modification of devices in modern device research environments consists of a Focussed Ion Beam system (FIB). In this work the system used was an FEI Strata DualBeam 235 SEM/FIB running the xp v2.29 software suite. The equipment consists of an electron column to provide SEM imaging capabilities, and an Ion column mounted at 52 o to the electron column for imaging and patterning applications. Additionally, the setup contains a Gas Injection System (GIS) for gas enhanced etching, and localised metal and insulator deposition. For more information regarding the operation of the system the reader is encouraged to consult [118]. SEM and FIB imaging The system contains an integrated SEM column, used for imaging. The FIB column can also be used for imaging. To achieve this, the beam is scanned quickly over the entire viewable area, defined by the magnification and sample position, and the secondary electron emission is monitored. Standard High Resolution (SHR) and Ultra High Resolution (UHR) modes of operation are available for the SEM. In 178

193 CHAPTER 7. ADDITIONAL WORK addition, the machine is supplied with an Energy Dispersive X-Ray (EDX) module for sample material analysis. The SEM function is used as the primary method of sample imaging, as this avoids excessive exposure of the sample to the Ion beam, which is detrimental to the sample. Milling Although the ion beam can be used for imaging the sample, the main function it serves is as a tool for device patterning. The ion beam is focussed into a spot of minimum size approximately 7-8nm. The pattern areas are defined using the FEI xp software using a graphical interface. With this method it is possible to draw rectangles, circles, lines and custom polygons as an overlay to a captured SEM or FIB image of the area of interest. The beam is then raster-scanned over the defined patterns, at a rate determined by the user, which results in precision removal of material in the patterned area. By selecting the Enhanced Ion Etch mode an additional gas flow of XeF over the area can provide enhanced rates of etching. Metal Deposition A useful feature of the DB 235 system is the ability to deposit Platinum tracks upon the sample. The deposition process involves raster scanning the Ion beam across the pre-defined area, in the same way as during the milling, whilst exposing the sample surface to a continuous vapour stream of an organo-metallic compound containing Platinum. In the regions where the beam passes through the gas, the molecules are decomposed and the resulting charged Platinum ions are attracted to the grounded substrate. The image shown in Figure 7.2 illustrates the main the features of the system, SEM imaging of a sample, FIB milling of small slots into the BSCCO sample itself, and deposition of Platinum to form sample contact leads. 179

194 CHAPTER 7. ADDITIONAL WORK Figure 7.2: SEM image of a BSCCO device illustrating Platinum deposition (horizontal strips) and precision FIB milling (dark trenches) 7.3 Contacts to BSCCO whiskers Various methods of contact to the BSCCO whiskers were considered throughout the course of this project. It is possible to place rough 4 terminal contacts to larger whiskers using Electrodag adhesion Ag paint. This method is quick and simple, and involves no lithography step. However, it is only possible to place the contacts to within an accuracy of approximately 1mm, and so with smaller whisker samples this method is not feasible. Several other contact methods were tried, including the use of Indium metal pressed down onto the sample, and photolithographically defined Au tracks, patterned from a layer of Au deposited on top of the whiskers by RF sputtering. A further method of contact using the Focussed Ion Beam microscope was also investigated. Contact methods to whiskers have been addressed in the literature (see for example [119]), however it was found that the main method discussed in this report (the use of Silver paint contacts) did not provide suitable results in the case of small whiskers such as the sub-1mm whiskers grown for this project. The best way of attaching contacts to whiskers was found to be using the FIB Plat- 180

195 CHAPTER 7. ADDITIONAL WORK Figure 7.3: Resistance vs. Temperature curve from the 4-terminal measurement of a non-superconducting BSCCO whisker inum deposition. Figure 7.3 shows a 4 terminal measurement on a semiconducting sample of BSCCO, and demonstrates how accurately the characteristic can be determined when the contact resistances are kept low. In this work only semiconducting BSCCO was grown, however the technique remains applicable for any small single crystal superconductors. 7.4 Manipulation of BSCCO using the FIB There have been many reports of lithographical techniques to pattern BSCCO single crystals, see for example [120]. However, it is difficult to perform c-axis transport measurements with conventional lithography on single crystal mesas due to their tendency to grow as flat platelets. 3-Dimensional patterning of BSCCO using a FIB microscope was first reported in the literature in 1999 by Kim et al. [121]. In their paper, whiskers are placed on a substrate and the FIB is used at a high beam angle to mill an undercut structure into the material. A stack of c-axis Josephson junctions can be defined using this process. A schematic of the process can be found in Figure 7.4. See Figure 7.6 for an SEM image of such a prototype device fabricated 181

196 CHAPTER 7. ADDITIONAL WORK Figure 7.4: Schematic of the isolation of a stack of Josephson junctions in a layered superconductor using an undercutting technique during this project. Transport Range of Ions in Matter (TRIM) simulations [122] were performed for the Ar+ Ion milling process and for the Ga+ FIB milling process. A typical example of the graphical output can be seen in Figure 7.5. The red dots show primary Ga Ion implantation, the other colours represent secondary ion collisions, i.e. lattice damage events. It can be seen from this data that the damage region extends approximately 300nm into the material, which is even higher than the values reported in the literature of around 30nm. This favours the use of protective masks when using the FIB to produce narrow tracks. The lateral damage is unavoidable, but by depositing an inert layer on top of the YBCO, for example a thin layer of Strontium Tintanate or Silicon, the damage from the incident beam can be reduced. 7.5 Phase slips in Narrow tracks A system for investigating the 1D properties of High T c superconducting nanowires as described has been developed and several studies have been performed upon these systems. There will now follow a short discussion of some of the findings and possible explanations of the interesting results obtained. The tracks of YBCO 182

197 CHAPTER 7. ADDITIONAL WORK Figure 7.5: TRIM simulation of 30keV Ga Ions incident on a BSCCO sample Figure 7.6: Undercut in a BSCCO whisker demonstrating 3D patterning with the FIB 183

198 CHAPTER 7. ADDITIONAL WORK Figure 7.7: A narrow track of YBCO produced by FIB milling are first patterned using a standard lithographical technique, down to a minimum width of 5µm. The sample containing the tracks is placed in the focussed ion beam microscope and the tracks are trimmed down to a range of smaller sizes. The investigation of these tracks then consists of measuring the resistance as a function of temperature and track thickness. For the publication of the work see reference [123], which is also included at the end of this report. Tracks produced by FIB milling The FIB can be used to pattern thin films of the HTS material YBCO into narrow tracks. The fabrication of individual bridges has an extremely low throughput and results in a varied dose of Ga+ ions into the YBCO tracks, which can cause unpredictable results during measurement. An example of a track produced using this method is shown in Figure 7.7. Narrow tracks can display evidence of phase slip behaviour, as described in Section 3.5. Tracks of several different widths produced using the FIB narrowing technique were measured in a continuous flow cryostat setup. The results can be seen in Figure 7.8. The data shows a series of resistance versus temperature curves taken from the tracks. As the track width is decreased, the room temperature resistance of the track 184

199 CHAPTER 7. ADDITIONAL WORK Figure 7.8: Evidence for phase slips in YBCO narrow tracks increases. As this value approaches the quantum of resistance, R Q = 6.46kΩ, the track crosses the superconductor-insulator transition, as shown in track 3, and the R- T characteristic shows no clear T c. The track with the most interesting characteristic is track 4 in the plot, which displays a finite and exponentially decreasing R(T ) below T c. The data fits to the LAMH theory (Figure 7.9) with a single parameter, the track width. It is found that this fitting parameter must be made much smaller than the physical size of the junction. It is suggested that this may be due to filamentary transport within the microbridge, in which the active superconducting elements are actually restricted to filaments of small cross sectional area threading the microbridge. Tracks produced by chemical etching A more gentle etching process or ion beam milling technique is preferable to the FIB method. To avoid all chance of poisoning, a controllable wet process was established so that YBCO would be removed rather than poisoned. The lithographically patterned tracks were visible under an optical microscope and the progress of the wet etching could be monitored in stages by observation of the 185

200 CHAPTER 7. ADDITIONAL WORK Figure 7.9: Track 4, 500nm YBCO, resistance below T c. LAMH theory The solid line is a fit to the sample, until the track was just under 1µm wide. At this point the resolution limit of the microscope is reached, and the track is no longer visible. Subsequent monitoring of the track must be done by measuring its 4-terminal resistance. The process of monitoring the track consisted of using silver paint to attach wire bonds to the YBCO surface, which allowed the sample to be bonded up to a chip carrier. The photoresist remained in place, however the silver paint solvent was able to dissolve the resist and therefore the silver made a good contact to YBCO even when the resist was left in place. It may be better for the sample that this resist remains as a protective layer to prevent oxygen loss from the surface. The bonded sample is connected to a 4-terminal Keithley DVM, and clamped into position. Using a pipette, a small drop of nitric acid is introduced to the centre of the sample. The measured 4-Terminal resistance is then monitored as the YBCO underetches, beneath the photoresist. It is found that the resistance does appear to increase exponentially as expected, values of 0.1 Ω/s are typical at the beginning of the etch rising to 10 Ω/s for an R value of several hundred Ohms. continues to increase even with the very dilute acid. The rate 186

201 CHAPTER 7. ADDITIONAL WORK Figure 7.10: Underetching in Nb thin film microbridges Figure 7.11: Results of in-situ monitoring of resistance during the Ion beam milling process 187

202 CHAPTER 7. ADDITIONAL WORK Tracks produced by ion beam milling A third method of narrowing the tracks was also investigated. After the chemical etch, the photoresist was removed and the tracks were thinned further using an ion beam milling process. The resistance was monitored as a function of milling time (shown in Figure 7.11), in a similar way to that performed during the chemical etching. The resistance can be seen to increase controllably following a power law, which is expected, as R 1/w, where w is the track width. Discussion of narrow track behaviour Figures 7.12, 7.14 and 7.13 show the results from three different tracks produced by chemical underetching and milling of YBCO. The behaviour of each track is different, which highlights the problem of irreproducibility with this method. However, there are general features seen in all the tracks produced. The samples exhibit one of two types of behaviour: i.) The track displays a downturn in the data at low temperatures, and a more linear region near T c. The downturn can have a positive slope right up to T c, as shown in the Figure 7.14, or it can flatten out (Figure 7.12). ii.) A single linear region extends to the lowest temperatures, and the downturn is not seen. In this case the data is substantially more noisy (Figure 7.13). In case i.) the IV curves remain linear on all current scales tested (1µA, 2µA, 5µA, 10µA, 20µA), which rules out the possibility that a single grain boundary could be dominating the track, as if this were the case, the sample would exhibit a Josephson-like IV characteristic. It also demonstrates that a semiconducting contact resistance is unlikely to be the cause of the finite residual resistance. However, in the samples demonstrating behaviour described in ii.), the IV curves are non-linear. This behaviour is therefore attributed to a possible semiconducting region in the track, which would also explain the data becoming more noisy, as it is difficult to make an 188

203 CHAPTER 7. ADDITIONAL WORK Figure 7.12: YB784b Resistance Temperature characteristic detail. The track has been subject to chemical etching. The inset shows the full characteristic. Figure 7.13: YB724b track Resistance Temperature characteristic detail. Track has been subject to chemical etching and ion beam milling. The inset shows the full characteristic. 189

204 CHAPTER 7. ADDITIONAL WORK Figure 7.14: YB721b Resistance Temperature characteristic detail. The track has been subject to chemical etching. The inset shows the full characteristic. accurate measurement of a semiconducting sample. The semiconducting behaviour is unlikely to be due to a contact resistance problem as the contacts themselves have not been disturbed between successive etches (the acid etch is confined to the centre of the sample). The effects can possibly be explained by the presence of multiple Grain Boundary Junctions threading throughout the YBCO bridge, which dominate the transport of current below T c. The substrates upon which the chemically etched films have been grown are MgO and therefore the lattice mismatch with YBCO is quite high. This is consistent with the poorer results being obtained from films grown on MgO when compared to those on STO substrates. Another possibility is that the presence of such grain boundaries may not produce the effects as such, but when combined with a harsh chemical process which preferentially etches along such boundaries, the sample may degrade in an unknown fashion. It is difficult to draw conclusions from these samples, and the effects warrant further investigation in order to establish the origin of the finite resistance below T c. A sys- 190

205 CHAPTER 7. ADDITIONAL WORK tematic way of narrowing the tracks is currently under investigation, and measurements of the tracks in a cryostat at temperatures below 4.2K are being undertaken in order to try to observe the system entering the regime of quantum phase slip. Several different superconducting systems are under consideration in an attempt to reproduce some of the results in the literature using narrow tracks of conventional, LTS materials. It should be mentioned that both the chemical undercutting technique and the ion beam milling technique are suitable for controllably increasing the resistance of a narrow track. How small a track can be made is still unknown, as the samples investigated in this work appear to be limited by the intrinsic film quality rather than the lithographic preparation method. 191

206 Chapter 8 Conclusions and future prospects The work described here will hopefully form the basis of a longer term study into quantum effects in devices. Here I will summarise the main findings of this report, and provide an overview of the future experiments to be performed in this area. 192

207 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS 8.1 Implications of this work The measurement system The main implication of the work presented here is that the Birmingham group are now in possession of a fully functional system for Macroscopic Quantum Tunnelling measurements. There is strong evidence that the system is working well, with only a few minor changes required. The changes should make fitting the data acquired from LTS junctions a routine procedure. Once this is established, the system can then be satisfactorily used for the investigation of novel HTS junctions. The expertise developed during the process of building the system will be invaluable to the group s future progress in the area of quantum device physics and the investigation of qubit systems. In particular, the experience gained from the design, construction and testing of low-noise quantum measurement system will help the group s future work in construction of a system to probe the properties of the qubit itself. A full custom suite of software has also been designed and implemented, which allows control of the apparatus and allows relevant data analysis techniques to be performed in real time. This can help the user to identify problems with the measurement by fitting the histograms they are acquired. Further understanding of non-ideal junction behaviour In all junctions measured, there was some evidence of intrinsic noise. In the Niobium junctions, a fluctuating I c was observed with a strong 1/f frequency component, giving evidence to there being two-level fluctuators in the barrier. In the High T c junction, the noise behaved in a 1/f 2 manner. It will be of interest to investigate the grain boundary junctions once the system has been optimised, to confirm that the noise was indeed originating from the junction itself. The noise in junctions may be worse for noise than that of 0 45 grain boundaries due to the absence of Andreev bound states which aid in the tunnelling, and again 193

208 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS this should be verified experimentally. It is concluded that the current system is adequate for measuring Low T c junctions, as long as the experiment is performed carefully and the system settings are optimised. The CAM03 Niobium junction demonstrated some unusual behaviour, specifically the appearance of multi-valued critical current under the application of a magnetic field. This will be useful in the development of the process for these particular junctions, which will be the precursor to the SFS pi-qubit candidates. However, in the Niobium Junctions investigated so far, the noise level has been lower than in the HTS junction. This already gives strong support for further investigation of the SFS devices. This work also gives evidence towards some of the problems which can be encountered when dealing with Junction technology, which is especially relevant in the case of the High T c superconductors. The work described here has been invaluable in understanding the non-ideal properties of Junctions subject to this type of measurement. In particular, the excess noise often described in the literature has been observed and several potential mechanisms for the origin of this noise have been described. In order to fully understand the behaviour of High T c junctions, we must understand that even a junction which may appear ideal can exhibit some unusual features as a function of current, magnetic field and temperature. The work on Grain boundary junctions is also particularly relevant to understanding current transport in YBCO narrow tracks, which may well consist of grain boundaries on a sub-micrometre scale, if the lattice mismatch between the YBCO film and the substrate is considerable. The work has also provided new insights into fabrication methods for High T c junctions of YBCO and BSCCO, which will allow the investigation of MQT in new and novel junction systems. 194

209 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS 8.2 Future proposals As explained in the previous chapters, Macroscopic Quantum Tunnelling is a fundamental demonstration of the existence of a coherent quantum state in a superconducting device, and there are several advantages to using High Temperature Superconductors as the basis system in such devices. There are many different systems involving High Temperature superconductors which can yield interesting physics and applications. The next section will explain some of the ways in which the group will move forward in this area of research. Improving the measurement system The system designed and built for this project will be useful in future measurements, However, there are several changes which can be made to the system in order to improve the overall performance. These changes should allow the noise levels to be reduced further, and more experimental parameters investigated. The future measurement system described in Section 5.4 will be completed and installed. The design for this system is sound, and a printed circuit board housing the system is currently in the final stages of design. Once the auxiliary system is installed, it will hopefully replace the main system, reducing problems such as the intrinsic timing error of the current ramp. It is hoped that the dynamic aspects of the Josephson Junctions will be able to be characterised using a microwave setup. The installation of a microwave line is currently in progress, including a -20dB attenuator at 4.2K, and a -10dB attenuator at 1K. It may be necessary to install a resistance thermometer to monitor the onchip temperature. This will give a more accurate estimate of whether or not the junction is self-heating, and if the junction temperature is affected by the microwave irradiation. 195

210 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS Improving methods of data capture In the work of Voss et al. [13], data points were taken using a sweep rate of 10-20Hz. This meant that each measurement was stable over the course of approximately 5 hours. It may be necessary in our system to follow this procedure. In future experiments it will be necessary to allow the dilution refrigerator to settle at a particular temperature, and check that each histogram fits the model for that temperature as soon as they are acquired. If the histograms do not fit, steps must be taken to minimise the width before continuing to take data. Histograms taken using the return critical current, I r, can also be used to give information about the state of the junction, in particular dissipation process within the junction itself. These recapture histograms may prove useful to investigate in the future. A measurement system for monitoring IV curves is also in the process of implementation, which means that IV curves can be captured and analysed in real time using the computer software. This is an invaluable tool, as anomalous results in the histogram or escape rate data can be observed as changes in the IV characteristic, for example the onset of the phase diffusion regime, or the appearance of multi-valued critical currents. At the moment the system in place allows the user to monitor such events, but it is not possible to record them electronically. Work towards qubit measurements The work suggests that several different junction designs may serve as potential qubit candidates. Specifically, the High T c junctions may be useful due to the ability to exploit the higher harmonics of the CPR, which may help the quantum states remain coherent for longer. It is relatively simple to extend the system described in this report to encompass qubit measurements, and work towards this is currently underway. 196

211 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS By applying microwaves to the junction, the higher energy levels within the potential well can be excited, as described in Section 4.3. Subsequent to this report, wiring has been installed to perform these measurements. Once the microwave properties of the junctions have been established, the next stage of the experimental design will involve producing a setup for the measurement of Rabi oscillations in junction systems, as explained in Section 2.8. It is possible to use the existing setup to perform qubit measurements and manipulations. The method involves the installation of 2 additional RF lines, one to provide microwave pulses with which to bias the qubit, and one to supply a fast current pulse to a measurement SQUID. Using the existing sample box setup, there are 2 free positions for these lines. 2 DC lines are required, one to measure the voltage across the SQUID, and one to provide a current to an on-chip coil for flux biasing. The fridge setup is therefore already well equipped for this measurement. This method of performing qubit manipulations in conjunction with the measurement of switching histograms is described in [124]. 8.3 A wider perspective The work presented here has established a solid groundwork for several experiments which are both underway and proposed for the future. To close the report, some of these experiments are described to give the reader a flavour of the progress of the group in the field of quantum device physics. Experimental production of Low T c qubits The flux qubits can initially be fabricated from Low T c superconductors, consisting of either Al-AlOx-Al or Nb-AlOx-Nb Josephson junctions as the basic building blocks. This is achieved using a fully in-situ double-angle e-beam shadow evaporation technique. In the first instance, Nb junctions will be fabricated on a large 197

212 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS scale, to check the properties of the oxide barrier, then scaled down to conventional junction sizes. Nb will be used primarily due to the T c value, which is accessible using a continuous flow cryostat system. If the Niobium junctions are successful, Aluminium devices will be fabricated. Qubit technology using Aluminium is more advanced than that of Niobium, and better quality junctions can be fabricated more easily. Fabrication of Hybrid Junctions The work performed on Grain Boundary Junctions illustrates that the measurement system is suitable for the characterisation of HTS samples. The next stage of the project involves the fabrication of HTS π-junctions involving YBCO and Nb. As explained in Section 3.8, these junctions take advantage of the two different types of superconductor, in order to produce a built-in π phase shift of the order parameter across the barrier between the d-wave and s-wave regions. qubit biasing is intrinsic to the materials used. This means that the These devices will hopefully utilise the dominant second harmonic of the CPR. By modifying the CPR of the device, the amount of decoherence can be minimised. This is the idea behind the operation of the silent qubit, as first proposed by Amin in 2005 [12]. In order to fabricate a silent qubit, a reproducible, clean interface tunnel junction must be established. By placing a thin layer of normal metal between the YBCO and Nb, the YBCO is protected from deoxygenation caused by the Nb layer, whilst still allowing the two superconductors to remain in contact via the proximity effect from YBCO into the Au. By introducing a thin layer of Au interface layer is made thin such that the proximity effect allows the layer to become superconducting. The combination of components can be controlled by producing ramp junctions in a variety of different directions on the same sample, and by careful geometric considerations. The angle of ramp is, for example, an important consideration. In addition, testing if these junctions are suitable as qubit candidates will be possible 198

213 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS using the MQT technique developed. Qubits in cavities One of the measurements to be performed on qubit devices involves the installation of a qubit inside a resonant cavity. The joint microwave properties of the cavity-qubit system will be investigated. The energy levels of a cavity can be well isolated from the environment if the cavity has a high Q-value, as the cavity resonant modes will be well-defined. By placing a qubit inside the cavity, the energy levels of the qubit system interact with those of the cavity and act as a perturbation, whilst the qubit remains well isolated from environmental sources of decoherence. This approach helps prevent decoherence because the high Q factor of the cavity corresponds to only a very narrow band of long lived excitations, reducing the noise level from all other frequencies. The energy spectrum of the joint qubit-cavity system can then be explored, and the qubit state inferred. The qubit designs for use in this work will initially be the Al or Nb systems, although other types of qubits can be employed if they prove to be successful. Work on Phase slip centres in High-T c superconductors If the ongoing work into High T c phase slips is successful, these systems may also be a potential qubit candidate. The reader is advised to consult the recent paper by Mooij concerning phase slip qubits [125]. In addition, there are several systems which have been recently proposed as containing Intrinsic Josephson Junctions, such as the Strontium Ruthenate (Sr 2 RuO 4 ) eutectic system, containing metallic Ruthenium inclusions. It is believed that there may be a possible mechanism of superconductivity involving the interfaces between the two materials, resulting in intrinsic Josephson effects on the scale of the inclusions ( 1µm), [126], [127]. The investigation of phase slip-like behaviour and the 199

214 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS onset of resistance in these unusual systems maintains a constant interest in this field. 8.4 Conclusion The work presented has provided several advancements into the field of Quantum device measurements for the Birmingham Condensed Matter Group. A system for measuring the switching distributions and analysing the Thermal Activation and Macroscopic Quantum Tunnelling in a range of Josephson Junctions has been designed, implemented and tested. Several junctions have been measured using the system, and it has been found that the system is indeed capable of accurately measuring the switching distributions down to a temperature of 50mK. User-friendly software has been written to control the experiment and allow fitting of the data as it is acquired. The measurement system is still under development, and now that several studies have been successfully carried out, further work can be undertaken to improve the noise rejection, overall resolution and reliability of data from the measurements. The measurement system can be expanded to perform microwave excitation measurements, quantum coherence (Rabi oscillation) measurements and further Macroscopic Quantum Tunnelling measurements in High-Temperature Superconducting systems. Meanwhile, work has been performed towards the fabrication of new and novel junction systems to measure, including single crystal samples of BSCCO, YBCO Grain Boundary Junctions, YBCO nanobridges, and hybrid SND junction devices. Several effects have been observed in the novel junction systems during the testing, which are interesting in their own right, however they also serve to warn against making assumption that even conventional LTS junctions will follow an ideal behaviour. The careful documentation of these effects will help ensure minimise such non-ideal effects in future fabrication of junctions for quantum measurements. 200

215 CHAPTER 8. CONCLUSIONS AND FUTURE PROSPECTS Finally, the ability to demonstrate that various Josepshon junctions can be biased into the quantum regime will be a valuable tool for the Birmingham group in the testing of future qubit devices. 201

216 Bibliography [1] B. Chesca, D. Doenitz, T. Dahm, R. P. Huebener, D. Koelle, R. Kleiner, Ariano, H. J. H. Smilde, and H. Hilgenkamp, Observation of Andreev bound states in Y Ba 2 Cu 3 O 7 x /Au/Nb ramp-type Josephson junctions, Phys. Rev. B, vol. 73, no. 1, [2] W. C. Stewart, Current-voltage characteristics of Josephson junctions, Appl. Phys. Lett., vol. 12, no. 8, [3] A. A. Golubov, M. Y. Kupriyanov, and E. Il ichev, The current-phase relation in Josephson Junctions, Rev. Mod. Phys., vol. 76, no. 2, [4] H. Grabert, P. Olschowski, and U. Weiss, Quantum decay rates for dissipative systems at finite temperatures, Phys. Rev. B., vol. 36, no. 4, [5] I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Coherent Quantum Dynamics of a Superconducting Flux Qubit, Science, vol. 299, no , [6] P. Bertet, I. Chiorescu, K. Semba, C. J. P. M. Harmans, and J. E. Mooij, Detection of a persistent-current qubit by resonant activation, Phys. Rev. B., vol. 70, no. 10, [7] D. A. Wollman, D. J. V. Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett, Experimental Determination of the Superconducting Pairing State in YBCO from the Phase Coherence of YBCO-Pb dc SQUIDs, Phys. Rev. Lett., vol. 71, no. 13,

217 BIBLIOGRAPHY [8] J. Mannhart, H. Kilgenkamp, B. Mayer, C. Gerber, J. R. Kirtley, K. A. Moler, and M. Sigrist, Generation of Magnetix Flux by Single Grain Boundaries of Y 1 Ba 2 Cu 3 O 7 δ, Phys. Rev. Lett., vol. 77, no. 13, [9] R. Gross and B. Mayer, Transport processes and noise in Y 1 Ba 2 Cu 3 O 7 δ Grain Boundary Junctions, Physica C., vol. 180, no , [10] E. Il ichev, M. Grajcar, R. Hlubina, R. P. J. Ijselsteijn, H. E. Hoenig, H.-G. Meyer, A. Golubov, M. H. S. Amin, A. M. Zagoskin, A. N. Omelyanchouk, and M. Y. Kupriyanov, Degenerate Ground State in a Mesoscopic Y Ba 2 Cu 3 O 7 x Grain Bounday Josephson Junction, Phys. Rev. Lett., vol. 86, no. 23, [11] H. J. H. Smilde, A. A. Golubov, G. Rijnders, J. M. Dekkers, S. Harkema, D. H. A. Blank, H. Rogalla, and H. Hilgenkamp, Admixtures to d-wave Gap Symmetry in Unteinned YBa 2 Cu 3 O 7 Superconducting Films Measured by Angle-Resolved Electron Tunneling, Phys. Rev. Lett., vol. 95, no , [12] M. H. S. Amin, S. A. Yu, T. Lindstrom, S. A. Charlebois, T. Claeson, and A. Y. Tzalenchuk, Silent Phase qubit based on d-wave Josephson Junctions, Phys. Rev. B., vol. 71, no. 6, [13] R. F. Voss and R. A. Webb, Macroscopic Quantum Tunneling in 1µm Nb Josephson Junctions, Phys. Rev. Lett., vol. 47, no. 4, [14] V. Corato, S. Rombetto, C. Granata, E. Esposito, L. Longobardi, M. Russo, R. Russo, B. Ruggiero, and P. Silvestrini, Quantum behaviour of underdamped Josephson devices, Phys. Rev. B., vol. 70, no , [15] X. Y. Jin, J. Lisenfeld, Y. Koval, A. Lukashenko, A. V. Ustinov, and P. Mller, Enhanced Macroscopic Quantum Tunelling in Bi 2 Sr 2 CaCu 2 O 8+δ Intrinsic Josephson Junction stacks, Phys. Rev. Lett., vol. 96, no , [16] T. Bauch, F. Lombardi, F. Tafuri, A. Barone, G. Rotoli, P. Delsing, and T. Claeson, Macroscopic Quantum Tunnelling in d-wave Y Ba 2 Cu 3 O 7 δ 203

218 BIBLIOGRAPHY Josephson Junctions, Phys. Rev. Lett., vol. 94, no. 8, [17] J. M. Kivioja, T. E. Nieminen, J. Claudon, O. Buisson, F. W. Hekking, and J. P. Pekola, Observation of Transition from Escape Dynamics to Underdamped Phase Diffusion in a Josephson Junction, Phys. Rev. Lett., vol. 94, no. 24, [18] C. E. Gough, M. S. Colclough, E. M. Forgan, R. G. Gordan, M. Keene, C. M. Muirhead, A. I. M. Rae, N. Thomas, J. S. Abell, and S. Sutton, Flux quantisation in a High-T c superconductor, Nature, vol. 326, no. 855, [19] J. Ireland, R. Chakalov, M. S. Colclough, and C. M. Muirhead, Spin-polarized injection into YBaCu 3 O 7 grain boundary junctions, Appl. Phys. Lett., vol. 83, no. 4, [20] A. V. Ustinov and V. K. Kaplunenko, Rapid single-flux quantum logic using π shifters, Journal of Applied Physics, vol. 94, no. 8, [21] M. Tinkham, Introduction to superconductivity. Dover Publications, Inc., [22] J. R. Waldram, Superconductivity of metals and cuprates. IoP publishing, [23] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev., vol. 108, no. 1175, [24] B. D. Josephson, Possible new effects in superconductive tunneling, Phys. Lett., vol. 1, no. 7, [25] V. Ambegaokar and A. Baratoff, Tunneling between superconductors, Phys. Rev. Lett., vol. 10, no. 11, [26] S. T. Ruggiero and D. A. Rudman, Superconducting Devices. Academic Press, Inc., [27] R. P. Feynmann, R. B. Leighton, and M. Sands, Lectures on Physics, Volume III. Addison Wesley Publishing,

219 BIBLIOGRAPHY [28] K. K. Likharev, Superconducting weak links, Rev. Mod. Phys., vol. 51, no. 101, [29] D. E. McCumber, Effect of ac Impedence on dc Voltage-Current Characteristics of Superconductor Weak-Link Junctions, J. Appl. Phys, vol. 39, no. 7, [30] V. Ambegaokar and B. I. Halperin, Tunneling between superconductors, Phys. Rev. Lett., vol. 22, no. 25, [31] D. D. Coon and M. D. Fiske, Josephson ac and Step Structure in the Supercurrent Tunneling Characteristic, Phys. Rev., vol. 138, no. 2A, [32] R. Rifkin and B. S. Deaver, Current-phase relation and phase-dependent conductance of superconducting point contacts from rf impedence measurements, Phys. Rev. B, vol. 13, no. 9, [33] J. Martinis, M. H. devoret, and J. Clarke, Experimental tests for the quantum behaviour of a macroscopic degree of freedom: The phase difference across a Josephson Junction, Phys. Rev. B, vol. 35, no. 10, [34] H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, vol. 7, no. 4, [35] Y. H. Ivanchenko and L. A. Zil berman, Destruction of Josephson current by fluctuations, Soviet JETP, vol. 8, no. 4, [36] M. Büttiker, E. P. Harris, and R. Landauer, Thermal activation in extremely underdamped Josephson-junction circuits, Phys. Rev. B, vol. 28, no. 3, [37] H. Risken and K. Voigtlaender, Eigenvalues and eigenfunctions of the Fokker- Planck equation for the extremely underdamped Brownian motion in a doublewell potential, J. Stat. Phys, vol. 41, no. 5-6, [38] K. K. Likharev, Dynamics of Josephson Junction and circuits. Gordon and Breach Science Publishers,

220 BIBLIOGRAPHY [39] H. Grabert and U. Weiss, Quantum Tunneling in Dissipative Systems at Finite Temperatures, Phys. Rev. Lett., vol. 52, no. 25, [40] I. Affleck, Quantum-Statistical Metastability, Phys. Rev. Lett., vol. 46, no. 6, [41] A. O. Caldeira and A. J. Leggett, Influence of Dissipation on Quantum Tunneling in Macroscopic Systems, Phys. Rev. Lett., vol. 46, no. 4, [42] Y. Makhlin, G. Schön, and A. Shnirman, Quantum-state engineering with Josephson-junction devices, Rev. Mod. Phys., vol. 73, no. 2, [43] G. Wendin and V. S. Shumeiko, Quantum bits with Josephson junctions, Low Temp. Phys., vol. 33, no. 9, [44] D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. H. Devoret, Manipulating the Quantum State of an Electrical Circuit, Science, vol. 296, no. 5569, [45] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Coherent control of macroscopic quantum states in a single-cooper-pair box, Nature, vol. 398, no , [46] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Quantum coherence in a single- Cooper-pair box: experiments in the frequency and time domains, Physica B., vol. 280, no , [47] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H. van der Wal, and S. Lloyd, Josephson Persistent-Current Qubit, Science, vol. 285, no. 13, [48] I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Coherent dynamics of a flux qubit coupled to a harmonic oscillator, Nature, vol. 431, no ,

221 BIBLIOGRAPHY [49] P. Bertet, I. Chiorescu, G. Burkard, K. Semba, C. J. P. M. Harmans, D. P. DiVincenzo, and J. E. Mooij, Dephasing of a Superconducting Qubit Induced by Photon Noise, Phys. Rev. Lett., vol. 95, no. 25, [50] R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, Simultaneous State Measurement of Coupled Josephson Phase Qubits, Science, vol. 307, no. 1299, [51] C. Granata, B. Ruggiero, M. Russo, A. Vettoliere, V. Corato, and P. Silvestrini, Josephson devices for controllable flux qubit and interqubit coupling, Appl. Phys. Lett., vol. 87, no. 17, [52] A. O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura, S. Lloyd, and J. S. Tsai, Quantum Coherent Tunable Coupling of Superconducting Qubits, Science, vol. 316, no. 5825, [53] M. Grajcar, A. Izmalkov, S. H. W. van der Ploeg, S. Linzen, T. Plecenik, T. Wagner, U. Hüubner, E. Il ichev, H. G. Meyer, A. Y. Smirnov, P. J. Love, A. M. den Brink, M. H. S. Amin, S. Uchaikin, and A. M. Zagoskin, Four- Qubit Device with Mixed Couplings, Phys. Rev. Lett., vol. 96, no. 4, [54] D. S. Golubovic, W. V. Pogosov, M. Morelle, and V. V. Moshchalkov, Magnetic Phase Shifter for Superconducting Qubits, Phys. Rev. Lett., vol. 92, no. 17, [55] V. T. Petrashov, K. G. Chua, K. M. Marshall, R. S. Shaikhaidarov, and J. T. Nicholls, Andreev Probe of Persistent Current States in Superconducting Quantum Circuits, Phys. Rev. Lett., vol. 95, no. 14, [56] A. J. Berkeley, H. Xu, M. A. Gubrud, R. C. Ramos, J. R. Anderson, C. J. Lobb, and F. C. Wellstood, Decoherence in a Josephson-junction qubit, Phys. Rev. B., vol. 68, no. 6,

222 BIBLIOGRAPHY [57] J. W. A. Robinson, S. Piano, G. Burnell, c. Bell, and M. Blamire, Zero to π transition in superconductor-ferromagnet-superconductor junctions, Phys. Rev. B, vol. 76, no. 9, [58] P. A. Warburton, A. R. Kuzhakhmetov, C. Bell, G. Burnell, M. G. Blamire, H. Wu, C. R. M. Grovenor, and H. Schneidewind, Sub-Micron Thin Film Intrinsic Josephson Junctions, IEEE Trans. Appl. Supercond., vol. 13, no. 2, [59] M. Rapp, A. Murk, R. Semerad, and W. Prusseit, c-axis Conductivity and Intrinsic Josephson Effects in Y Ba 2 Cu 3 O 7 δ, Phys. Rev. Lett., vol. 77, no. 5, [60] I. Kuwosawa, H. Nakagawa, M. Aoyagi, M. Maezawa, Y. Kameda, and T. Nanya, A basic circuit for asynchronous superconductive logic using RSFQ gates, Supercond. Sci. Technol., vol. 8, [61] H. B. Wang, P. H. Wu, J. Chen, K. Maeda, and T. Yamashita, Threedimensional arrays of BiSrCaCuO-2212 intrinsic Josephson junctions and zerocrossing Shapiro steps at 760 Ghz, Appl. Phys. Lett., vol. 80, no. 9, [62] H. Ishida, K. Okanoue, K. Hamasaki, H. Shimakage, and Z. Wang, Selfplanarizing process for the fabrication of Bi 2 Sr 2 CaCu 2 O 2 stacks, Appl. Phys. Lett., vol. 86, no. 12, [63] I. E. Batov, X. Y. Jin, S. V. Shitov, Y. Koval, P. Mller, and A. V. Ustinov, Detection of 0.5THz radiation from intrinsic Bi 2 Sr 2 CaCu 2 O 8+δ Junctions, Appl. Phys. Lett., vol. 88, no , Josephson [64] S. Reymond, L. Antognazza, M. Decroux, E. Koller, and O. Fischer, Currentinduced highly dissipative domains in high-t c thin films, Phys. Rev. B, vol. 66, no. 1, [65] A. Bezryadin, C. N. Lau, and M. Tinkham, Decay of Persistent Currents in Small Superconductors, Phys. Rev., vol. 156, no. 2,

223 BIBLIOGRAPHY [66] J. S. Langer and V. Ambegaokar, Intrinsic Resistive Transition in Narrow Superconducting Channels, Phys. Rev., vol. 164, no. 2, [67] D. E. McCumber and B. I. Halperin, Time Scale of Intrinsic Resistive Fluctuations in Thin Superconducting Wires, Phys. Rev. B., vol. 1, no. 3, [68] N. Giordano, Evidence for Macroscopic Quantum Tunneling in One- Dimensional Superconductors, Phys. Rev. Lett., vol. 61, no. 18, [69] C. N. Lau, N. Markovic, M. Bockrath, A. Bazryadin, and M. Tinkham, Quantum Phase Slips in Superconducting nanowires, Phys. Rev. Lett, vol. 87, no. 21, [70] A. Bezryadin, C. N. Lau, and M. Tinkham, Quantum supression of superconductivity in ultrathin nanowires, Nature, vol. 404, no , [71] F. Sharifi, A. V. Herzog, and R. C. Dynes, Crossover from Two to One Dimension in In Situ Grown Wires of Pb, Phys. Rev. B, vol. 66, no. 1, [72] F. Altomare, A. M. Chang, M. R. Melloch, Y. Hong, and C. W. Tu, Evidence for Macroscopic Quantum Tunneling of Phase Slips in Long One-Dimensional Superconducting Al Wires, Phys. Rev. Lett, vol. 97, no. 1, [73] M. M. Abdelhadi and J. A. Jung, Phase-slip-like resistivity in underdoped Y Ba 2 Cu 3 O 7, Phys. Rev. B., vol. 67, no. 5, [74] T. Lindstrom, J. Johansson, T. Bauch, E. Stepantsov, and F. Lombardi, Josephson dynamics of bicrystal d-wave YBa 2 Cu 3 O 7 δ dc-squids, Phys. Rev. B, vol. 74, no. 1, [75] P. D. P. Dutta and P. M. Horn, Energy Scales for Noise Processes in Metals, Phys. Rev. Lett., vol. 43, no. 9, [76] C. T. Rogers and R. A. Buhrman, Composition of 1/f Noise in Metal- Insulator-Metal Tunnel Junctions, Phys. Rev. Lett., vol. 53, no. 13,

224 BIBLIOGRAPHY [77] A. H. Miklich, J. Clarke, M. S. Colclough, and K. Char, Flicker (1/f) Noise in biepitaxial grain boundary junctions of YBa 2 Cu 3 O 7 x, Appl. Phys. Lett., vol. 60, no. 15, [78] M. Kawasaki, P. Chaudhari, and A. Gupta, 1/f Noise in Y 1 Ba 2 Cu 3 O 7 δ Superconducting Bicrystal Grain Boundary Junctions, Phys. Rev. Lett., vol. 68, no. 7, [79] B. Savo, F. C. Wellstood, and J. Clarke, Low-frequency excess noise in Nb- Al 2 O 3 -Nb Josephson tunnel junctions, Appl. Phys. Lett., vol. 50, no. 24, [80] P. Dutta and P. Horn, Low-frequency fluctuations in solids: 1/f noise, Rev. Mod. Phys., vol. 53, no. 3, [81] E. Il ichev, V. Zarosarenko, R. P. J. Ijsselsteijn, V. Schultze, H. G. Meyer, and H. Hoenig, Nonsinusoidal Current-Phase Relationship of Grain Boundary Josephson Junctions in High-T c Superconductors, Phys. Rev. Lett., vol. 81, no. 4, [82] E. Il ichev, V. Zakosarenko, R. P. J. Ijsselsteijn, H. E. Hoenig, V. Schultze, and H. G. Meyer, Anomolous periodicity of the current-phase relationship of grain-boundary Josephson junctions in High-T c superconductors, Phys. Rev. B, vol. 60, no. 5, [83] Y. V. Fominov, A. A. Golubov, and M. Y. Kupriyanov, Decoherence due to Nodal Quasiparticles in d-wave Qubits, JETP Lett., vol. 77, no. 10, [84] T. Yamashita, S. Takahashi, and S. Maekawa, Superconducting pi qubit with three Josephson Junctions, Appl. Phys. Lett., vol. 88, no. 13, [85] G. Blatter, V. B. Geshkenbein, and L. B. Ioffe, Design aspects of superconducting-phase quantum bits, Phys. Rev. B., vol. 63, no. 17, [86] L. Ioffe, V. B. Geashkenbein, M. V. Feigel man, A. L. Fauchere, and G. Blatter, Environmentally decoupled sds-wave Josephson Junctions for quantum 210

225 BIBLIOGRAPHY computing, Lett. to Nature, vol. 398, [87] S. Kawabata, A. A. Golubov, Ariando, C. J. M. Verwijs, H. Hilgenkamp, and J. R. Kirtley, Macroscopic quantum tunneling and quasiparticle-tunneling blockade effect in s-wave/d-wave hybrid junctions, Phys. Rev. B., vol. 76, no. 6, [88] A. Wallraff, A. Lukashenko, C. Coqui, A. Kemp, T. Duty, and A. V. Ustinov, Switching current measurements of large area Josephson tunnel junctions, Rev. Sci. Instr., vol. 74, no. 8, [89] T. A. Fulton and L. N. Dunkleberger, Lifetime of the zero-voltage state in Josephson tunnel junctions, Phys. Rev. B., vol. 9, no. 11, [90] L. D. Jackel, J. P. Gordon, E. L. Hu, R. E. Howard, and L. A. Fetter, Decay of the Zero-Voltage State in Small-Area, High Current-Density Josephson Junctions, Phys. Rev. Lett., vol. 47, no. 9, [91] S. Washburn, R. A. Webb, R. F. Voss, and S. M. Faris, Effects of dissipation and Temperature on Macroscopic Quantum Tunneling, Phys. Rev. Lett., vol. 54, no. 25, [92] A. N. Cleland, J. M. Martinis, and J. Clarke, Measurement of the effect of moderate dissipation on macroscopic quantum tunneling, Phys. Rev. B, vol. 37, no. 10, [93] A. Wallraff, T. Duty, A. Lukashenko, and A. V. Ustinov, Multiphoton Transitions between Energy Levels in a Current-Biased Josephson Tunnel Junction, Phys. Rev. Lett., vol. 90, no. 3, [94] K. Inomata, S. Sato, K. Nakajima, A. Tanaka, Y. Takano, H. B. Wang, M. Nagao, H. Hatano, and S. Kawabata, Macroscopic Quantum Tunneling in a d- wave High-T c Bi 2 Sr 2 CaCu 2 O 8+δ Superconductor, Phys. Rev. Lett., vol. 95, no. 10,

226 BIBLIOGRAPHY [95] S. Savel ev, A. L. Rakhmanov, and F. Nori, Quantum Terahertz Electrodynamics and Macroscopic Quantum Tunneling in Layered Superconductors, Phys. Rev. Lett., vol. 98, no. 7, [96] M. V. Fistul, Macroscopic quantum tunneling in globally couple series arrays of Josephson junctions, Phys. Rev. B., vol. 75, no. 1, [97] T. Bauch, T. Lindström, F. Tafuri, G. Rotoli, P. Delsing, T. Claeson, and F. Lombardi, Quantum Dynamics of a d-wave Josephson Junction, Science, vol. 311, no. 5757, [98] Calibration data provided by E. M. Forgan. [99] Spreadsheet calculations performed by C. M. Muirhead. [100] Agilent Technologies Fiber Optic components technical data booklet, HFBR series, Copyright 1999 Agilent Technologies, Inc. [101] Copyright Python Software Foundation. [102] INRIA, Scilab. Copyright INRIA. [103] Theoretical histogram data provided by T. Lindstrom. [104] Josephson Junction simulator circuit provided courtesey of M. S. Colclough. [105] Sample provided courtesy of E. Tarte. [106] Institute for Physical High Technology, D Jena, Germany. [107] Sample provided courtesy of A. Tzalenchuk, National Physical Laboratory, Teddington, Middlesex, UK, TW11 0LW. [108] Sample provided courtesy of M. Blamire, Device Materials group, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK. [109] D. C. Cronemeyer, C. C. Chi, A. Davidson, and N. F. Pederson, Chaos, noise, and tails on the I-V curve steps of rf-driven Josephson junctions, Phys. Rev. B., vol. 31, no. 5,

227 BIBLIOGRAPHY [110] J. Männik, S. Li, W. Qui, W. Chen, V. Patel, S. Han, and J. E. Lukens, Observation of Transition from Escape Dynamics to Underdamped Phase Diffusion in a Josephson Junction, Phys. Rev. B., vol. 71, no. 22, [111] S.-X. Li, W. Qui, S. Han, Y. F. Wei, X. B. Zhu, C. Z. Gu, S. P. zhao, and H. B. Wang, Observation of Macroscopic Quantum Tunneling in a Single Bi 2 Sr 2 CaCu 2 O 8 δ, Phys. Rev. Lett., vol. 99, no. 3, [112] V. M. Krasnov, T. Bauch, and P. Delsing, Probing the intrinsic Josephson coupling potential in Bi 2 Sr 2 CaCu 2 O 8+δ superconductors by thermal activation, Phys. Rev. B, vol. 72, no. 1, [113] P. A. Warburton, A. R. Kuzhakhmetov, G. Burnell, M. G. Blamire, and H. Schneidewind, Decoupling of a current-biased intrinsic Josephson junction from its environment, Phys. Rev. B., vol. 67, no. 18, [114] E. T. Swartz and R. O. Pohl, Thermal boundary resistance, Rev. Mod. Phys., vol. 61, no. 3, [115] F. C. Wellstood, C. Urbina, and J. Clarke, Low-frequency noise in dc superconducting quantum interference devices below 1 K, Appl. Phys. Lett., vol. 50, no. 12, [116] M. Nagao, M. Sato, H. Maeda, S. Kim, and T. Yamashita, Growth and superconducting properties of Bi 2 Sr 2 CaCu 2 O 8+δ single-crystal whiskers using tellurium-doped precursors, Appl. Phys. Lett., vol. 79, no. 16, [117] A. B. Kulakov, I. G. Naumenko, S. A. Zver kov, A. V. Kosenko, S. S. Khasanov, I. K. Bdikin, G. A. Emelchenko, M. Fehlmann, L. J. Gauckler, G. Yang, and J. S. Abell, The primary crystallisation field and growth of Bi-2212 crystals in Platinum and Gold crucibles, Journal of Crystal Growth, vol. 231, pp , [118] Strata xp DualBeam 235 Workstation - User s Guide - FEI Company,

228 BIBLIOGRAPHY [119] S. Aukkaravittayapun, K. A. Benedicts, I. G. Gorlova, P. J. King, Y. I. Latyshev, C. Staddon, and S. G. Zybtsev, Low-resistance electrical contacts to single crystal BSCCO whiskers, Supercond. Sci. Technol., vol. 8, pp , [120] A. Yurgens, D. Winkler, N. V. Zavaritsky, and T. Claeson, Strong Temperature dependence of the c-axis gap parameter of Bi 2 Sr 2 CaCu 2 O 8+δ intrinsic Josephson Junctions, Phys. Rev. B., vol. 53, no. 14, [121] S. J. Kim, Y. I. Latyshev, and T. Yamashita, Submicron stacked-junction fabrication from Bi 2 Sr 2 CaCu 2 O 8+δ whiskers by focused-ion-beam etching., Appl. Phys. Lett., vol. 74, no. 8, [122] Calculations performed using TRIM 2000 software, [123] P. Mikheenko, X. Deng, S. Gildert, M. S. Colclough, and C. M. Muirhead, Phase Slip Centres in sub-micron Y Ba 2 Cu 3 O 7 δ bridges, Phys. Rev. B, vol. 72, no , [124] T. Wirth, Coupled Josephson Phase Qubits. PhD thesis, Physikalisches Institut III, Friedrich-Alexander-Universität Erlangen-Nürnberg, [125] J. E. Mooij and C. J. P. M. Harmans, Phase-slip flux qubits, New Journal of Physics, vol. 7, no. 219, [126] H. Yaguchi, M. Wada, T. Akima, Y. Maeno, and T. Ishiguro, Interface superconductivity in the eutectic Sr 2 RuO 4 Ru: 3-K phase of Sr 2 RuO 4, Phys. Rev. B, vol. 67, no. 21, [127] J. Hooper, Z. Q. Mao, K. D. Nelson, Y. Liu, M. Wada, and Y. Maeno, Anomolous Josehson network in the Ru Sr 2 RuO 4 eutectic system, Phys. Rev. B, vol. 70, no. 1,

229 Appendices 215

230 Appendix A - Assembly of MCX filter modules Circuit construction Cut a piece of veroboard into strips containing 7 holes Using a file, carefully reduce the veroboard to half its original thickness Place 4 link pins in alternate holes and glue in place with GE varnish When dry, trim link pins to half their original height Place 3 components between the link pins, sticking them down with a little GE varnish Solder components to link pins in desired configuration To create a linking connection use a small piece of enamelled wire between component ends To create a ground line, solder a trailing non-enamelled wire to the relevant link pin, with the other end to be soldered to the case later. Solder an assembled male MCX connector central pin to one of the end link pins of the circuit board Figure 8.1: Assembly of the custom filter circuits

231 Filter assembly Cut a CuNi tube of diameter 3.5mm into 30mm sections, and mark at 5mm and 25mm. File a window carefully between the two marks using a small file Cut a piece of heatshrink in half along its length, the same size as the circuit s veroboard Place one half of the heatshrink inside the tube Slide in the circuitboard from the side, making sure the heathrink remains aligned Insert a second assembled male MCX connector into the open end of the tubing Solder the central MCX connector to the link pin through the tubing window Trim the trailing ground wire and solder to the outer casing (edge of window) Take the second half of the heatshrink and cut into 2 sections Place these over the exposed circuit board on either side of the ground wire Take a second piece of the CuNi tubing and cut or file down the length to open it up Place this around the filter to close off the window Crimp very lightly at each side near the connectors to secure the entire module Figure 8.2: Assembly of the custom filter modules