Fundamentals and Frontiers of the Josephson Effect (Springer Series in Materials Science (286))

Preface
	General References
Acknowledgements
Contents
Contributors
Acronyms
	Symbols
1 Introductory Notes on the Josephson Effect: Main Concepts and Phenomenology
	1.1 A Brief Historical Survey on the Materials Used  for the Realization of Superconducting Junctions
	1.2 The Coupling Between Macroscopic Quantum Systems and the Equations of the Josephson Effect
		1.2.1 Josephson Equations in the Tunnel Limit
		1.2.2 Different Types of Josephson Junctions Other than Tunnel
	1.3 The Tunneling Hamiltonian and the Scattering Formalism
		1.3.1 Expression for the Total Current in the Tunneling Hamiltonian Formalism
		1.3.2 Conductance in a Tunnel Junction
		1.3.3 From the Tunneling Transfer Hamiltonian to the Scattering Formalism
		1.3.4 Andreev Reflection
		1.3.5 Josephson Effect Derived from Quasi-particle Andreev Bound States
	1.4 Current–Voltage (I–V) Characteristics: From Microscopic Theory to the Resistively Shunted Junction Model
		1.4.1 I–V: Notes on the Resistively Shunted Junction Model
	1.5 Temperature Dependence of Ic Rn and of the I–V characteristics
		1.5.1 Temperature Dependence of Ic in the Tunnel Limit
		1.5.2 Temperature Dependence of Ic Other than the Tunnel Limit
	1.6 Magnetic Field Effects
	1.7 Electrodynamics of the Josephson Junction
	1.8 Material and Nano Science Open Novel Routes  for the Fabrication of Josephson Junctions
		1.8.1 Low Temperature Josephson Junctions
		1.8.2 High Temperature Josephson Junctions
		1.8.3 Hybrid Junctions
	References
2 Josephson Devices as€Tests of€Quantum Mechanics Towards the€Everyday Level
	2.1 Background
	2.2 Early History
	2.3 Consolidation: Work on€MQT in€the€Early 80s
	2.4 Progress Towards MQC: 1981–1999
	2.5 The Modern Era: Josephson Qubits
	2.6 Where Do We Stand?
	References
3 Basic Properties of the Josephson Effect
	3.1 Introduction
	3.2 Basic Features and Fundamental Relations
	3.3 Josephson Effect in Basic Types of Junctions
	3.4 SNS Junctions
		3.4.1 Dirty Limit
		3.4.2 Clean SNS Junctions
	3.5 Double Barrier SINIS Junctions
		3.5.1 SINIS Junctions, Clean Limit
		3.5.2 SINIS Junctions, Dirty Limit
	3.6 SFS Josephson Junctions
		3.6.1 Proximity Effect in SF Bilayer
	3.7 CPR in SFS Junctions
		3.7.1 -Junctions
		3.7.2 0-Junctions
		3.7.3 CPR in Serial SIsFS and SFsFS Junctions
	References
4 Charge Transport in Unconventional Superconductor Junctions
	4.1 Topological Superconductivity
		4.1.1 Pair Potential
		4.1.2 Topological Number and Surface Bound States
		4.1.3 Tunnel Conductance
		4.1.4 Josephson Current
	4.2 Proximity Effect in a Dirty Normal Metal
		4.2.1 Conductance of a Dirty NS Junction
		4.2.2 Josephson Effect in a Dirty SNS Junction
	4.3 Remark: Odd-Frequency Cooper Pair  and Majorana Fermion
	References
5 Mesoscopic Features in Nanoscale Superconducting Devices
	5.1 Introduction
	5.2 Proximity in Macroscopic Systems
		5.2.1 Free Energy of the Isolated Superconductor
		5.2.2 Superconducting Correlations Induced in a Normal Metal by Proximity
	5.3 Andreev Resonances at Superconductor-Normal Metal Interfaces
		5.3.1 Andreev Resonances in a Clean N Slab in Proximity with a Superconductor
		5.3.2 Diffusive N/S Boundary
		5.3.3 Andreev Reflection Under the Magnetic Field: Magnetoconductance Oscillations in N/S Junctions
	5.4 Scattering Approach to Ballistic Transport in SNS Josephson Junctions
		5.4.1 Andreev Bound States with Fully Transmitting NS Interfaces
		5.4.2 Density of Energy States at a Generic SNS Junction
	5.5 Ballistic and Diffusive SNS Junction Systems
		5.5.1 Ballistic Short and Long SNS Junctions
		5.5.2 Diffusive Short and Long SNS Junctions
	5.6 Semiclassical Approach to Diffusive Systems and Other Signatures of the Mesoscopic Regime
		5.6.1  Minigap in SNS Diffusive Junctions
		5.6.2 Low-Temperature Reentrant Behavior of the Resistance in a Diffusive N Wire in Proximity  with a Superconductor
		5.6.3 Resistance Change in a Wire in Contact  with a Superconducting Electrode
	5.7 Mesoscopic Conductance Fluctuations
		5.7.1 Self Correlations of the Conductance in Magnetic Field
		5.7.2 Self Correlations of the Conductance  in Non Equilibrium
	5.8 From Few to Single Channel Junctions
		5.8.1 Shot Noise in Few Channel NS Junctions
		5.8.2 Single Channel SS Junctions
		5.8.3 Andreev Qubits and Parity Jumps
		5.8.4 Transient Dynamics
	References
6 Magnetic Field Effects in Josephson Junctions
	6.1 Introduction
	6.2 Static Magnetic Fields
		6.2.1 Flux Focussing
		6.2.2 Time-Independent Sine-Gordon Equation
		6.2.3 Magnetic Interference Patterns
		6.2.4 Josephson Vortices
	6.3 Time-Dependent Magnetic Fields
		6.3.1 Time-Dependent Sine-Gordon Equation
		6.3.2 Fiske Steps
		6.3.3 Zero-Field Steps
	References
7 Current–Voltage Characteristics
	7.1 The Resistively Shunted Junction Model
		7.1.1 The Noise Term in the RSJ Model, a First Watch at Fluctuations
	7.2 I–V Curves in the RSJ Model in the Small Capacitance Limit
	7.3 I–V Curves in the RSJ Model for Finite Capacitance
		7.3.1 Details of the I–V Curves in the Subgap Region for Finite Capacitance and Nonlinear RSJ Models
	7.4 Current Biased Tunneling Junction, a More Accurate Description of the Subgap Region  for Finite Capacitance
	7.5 Effects of Thermal Fluctuations
		7.5.1 Negligible Capacitance
		7.5.2 Finite Capacitance
		7.5.3 Large Capacitance
	7.6 I–V Curves: When They Do Not Match RSJ-Like Predictions
		7.6.1 Deviations from RSJ, RSJN and TJM Models
		7.6.2 I–V Curves in Small or Nanoscale Junctions:  From the RSJ Model to Phase Diffusion
		7.6.3 Beyond Classical Smoluchowski Dynamics, from Coulomb Blockade to Quantum Diffusion
		7.6.4 More on the Amplitude of the Hysteresis
		7.6.5 Concluding Remarks and a Further Look at Experimental I–V Curves
	References
8 High Critical Temperature Superconductor Josephson Junctions and Other Exotic Structures
	8.1 Introduction
	8.2 Complementary Investigations and the Importance  of a Structural Feedback
	8.3 Grain Boundary Junctions
		8.3.1 Bicrystal Junctions
		8.3.2 Biepitaxial Junctions
		8.3.3 Step-Edge Junctions
	8.4 Locally Affecting Superconductivity, Moving Oxygen in Thin Films and Damaged Junctions
		8.4.1 Modifying Junctions by Irradiation
		8.4.2 Electro-Migration Studies
	8.5 Junctions with an Artificial Barrier
		8.5.1 Ramp Edge Junctions Realized with Au and Ag Inert Barriers
		8.5.2 Ramp Edge Junctions Realized with Perosvkite and Layered Materials
		8.5.3 Trilayer Structures
	8.6 Interface-Engineered Junctions, a different way of Creating a Barrier
		8.6.1 Ramp-Edge Junctions for Superconducting Electronics
	8.7 Junctions with HTS Other Than YBCO
		8.7.1 La1.85Sr0.15CuO4-Based Trilayer with One-Unit-Cell-Thick Barrier
		8.7.2 Electron Doped HTS
	8.8 Intrinsic Stacked Junctions
	8.9 HTS Junctions and Wires on the Meso/nano Scale
		8.9.1 GB Junctions Realized with Ultra-Thin Films and Superlattices
		8.9.2 HTS Nanostructures and Nanowires
		8.9.3 Submicron Josephson Junctions, Energy Scales and Mesoscopic Effects
	8.10 General Criteria on I–V Curves and the Estimation of Junction Parameters
		8.10.1 The Shape of I–V Curves
		8.10.2 From I–V Curves and Their Modelling to Junction Parameters
		8.10.3 Capacitance and Related Electromagnetic Properties of Junction Interfaces
	8.11 Dependence of the Josephson Current  on the Temperature
	8.12 Notes on the Magnetic Properties of HTS Junctions
		8.12.1 Dependence of the Critical Current and I–V Characteristics on the Magnetic Field
		8.12.2 Spontaneous Magnetization with Random Orientation
	8.13 Fractional Shapiro Steps: Time-Dependent Effects
	8.14 Other Exotic Structures: Josephson Junctions Based on Interface Superconductors
	References
9 Pairing Symmetry Effects
	9.1 Dependence of Josephson Critical Currents  on Junction Geometry
	9.2 Quantum Interference of Josephson Currents
	9.3 Spontaneous Josephson Currents
	References
10 Intrinsic Josephson Junctions in High Temperature Superconductors
	10.1 Introduction
	10.2 Fabrication Methods and Materials
	10.3 Basic Properties
		10.3.1 Resistivity and Out-of-Plane Critical Current Density
		10.3.2 Current Voltage Characteristics
		10.3.3 Interlayer Tunneling Spectroscopy
		10.3.4 Modelling of One-Dimensional Stacks:  Coupling by Charge Fluctuations
	10.4 Josephson Plasma Oscillations and Collective Fluxon Dynamics
		10.4.1 Coupled Sine-Gordon Equations
		10.4.2 Static Josephson Fluxons Lattices
		10.4.3 Collective Josephson Plasma Oscillations
		10.4.4 Fluxon Dynamics
	10.5 Generation of THz Radiation with Intrinsic Junction Stacks
	References
11 Phase Dynamics and Macroscopic Quantum Tunneling
	11.1 Escape Out of a Metastable State
		11.1.1 Theoretical Background, Effects of Dissipation and the Underdamped Limit
		11.1.2 The First Experiments
		11.1.3 The Effect of the Magnetic Field on SCD
		11.1.4 Notes on Resonant Activation and Quantized Energy Level
		11.1.5 The Master Equation for Phase Dynamics
		11.1.6 The Retrapping Current
		11.1.7 Thermal Activation and Macroscopic Quantum Tunneling in SQUIDs and Annular Junctions
	11.2 Moderately Damped Regime
	11.3 Thermal Activation, Macroscopic Quantum Tunneling and Phase Diffusion in Unconventional Josephson Junctions
		11.3.1 HTS Josephson Junctions
		11.3.2 In the `Far\' Low Critical Current Regime in LTS and HTS JJs
		11.3.3 Phase Dynamics Diagram: Influence of Dissipation
		11.3.4 Ferromagnetic Junctions
		11.3.5 SCDs in Junction with Graphene Barriers
	11.4 SCDs in Junctions with High Values of Jc
		11.4.1 SCDs in Nanowires
	References
12 High Frequency Properties  of Josephson Junctions
	12.1 Simple Voltage Source Model
	12.2 Finite Dimension Effect in Tunneling Junctions
	12.3 Current Source Model
	12.4 Resonant Activation
	References
13 Josephson Effect in Graphene and 3D Topological Insulators
	13.1 Introduction
	13.2 Superconductor - Graphene - Superconductor Junctions
	13.3 Superconductor - Topological Insulator - Superconductor Junctions
	13.4 Fabrication of Superconducting Hybrid Devices
	13.5 Effective Area of a Planar Josephson Junction
	13.6 Planar Josephson Junctions with TI Barriers
	References
14 Physics and€Applications of€NanoSQUIDs
	14.1 Introduction
	14.2 Superconducting “Weak-Link” Response and the Josephson Effects
		14.2.1 Josephson Junctions for€NanoSQUIDs
		14.2.2 Josephson Tunnel Barrier
		14.2.3 Trilayer Junctions
		14.2.4 Normal Metal Barriers
		14.2.5 Dayem Bridge Weak Links
		14.2.6 Focussed Ion Beam Milling
		14.2.7 Electron Beam Lithography (EBL)
		14.2.8 Niche Fabrication Developments
		14.2.9 Comparison of€Tunnel Junctions and€Other Weak Links
	14.3 Practical NanoSQUID Realisations
		14.3.1 Grenoble Group
		14.3.2 CSIRO SQUIDs
		14.3.3 NPL SQUIDs
		14.3.4 Other Nanosuperconducting Structures
		14.3.5 Single Josephson Tunnel Junction
		14.3.6 3D NanoSQUIDs
		14.3.7 State-of-the-Art
	14.4 Nanoscale Leads to€Improved Energy Sensitivity
		14.4.1 How Reproducible is NanoSQUID Fabrication?
		14.4.2 Further Miniaturization?
	14.5 Applications of€NanoSQUIDs
		14.5.1 Nano Electro-Mechanical Systems (NEMS)
	14.6 Superconducting Qubits—At the Nanoscale?
	14.7 High Frequency Readout of€SQUIDs
	14.8 New Materials
	14.9 Summary and€Outlook
	Bibliography
15 Josephson Junctions for€Metrology Applications
	15.1 Introduction
	15.2 Overview of€Voltage Metrology and€Applications
	15.3 Voltage Quantization
	15.4 Programmable DC Voltage Standards
	15.5 Intrinsic AC Voltage Standards and€Arbitrary Waveform Synthesis
	15.6 Temperature Metrology with€a€Quantum Voltage Noise Source
	References
16 Josephson Junctions for€Digital Applications
	16.1 Introduction
	16.2 Digital Circuits
		16.2.1 Rapid Single Flux Quantum Logic
	16.3 Energy-Efficient Single Flux Quantum Circuits
	16.4 DC Biased Energy-Efficient Circuits
	16.5 AC Biased Energy-Efficient Circuits
	16.6 Adiabatic Flux Quantum Parametron Logic
		16.6.1 Introduction
		16.6.2 Operation Principle of€Adiabatic Quantum Flux Parametron (AQFP) Logic
		16.6.3 Energy Efficiency of€an€AQFP Logic Gate
		16.6.4 AQFP Logic Circuits
	16.7 Memory for€Cryogenic Supercomputer
		16.7.1 Introduction
		16.7.2 SQUID Memory
		16.7.3 Abrikosov Vortex Memory
		16.7.4 Cryotron Memory
		16.7.5 CMOS Memory
		16.7.6 Memory Proposals Using Hybrid Superconductor/Ferromagnet Structures
		16.7.7 Novel Room-Temperature Memory Proposals Considered for€Cryogenic Applications
		16.7.8 Conclusion and€Outlook
	16.8 Fabrication of€Low-Critical-Temperature Josephson Junctions and€Integrated Circuits
		16.8.1 Introduction
		16.8.2 Circuit Elements of€Superconducting Digital Circuits
		16.8.3 Josephson Junctions
		16.8.4 Fabrication Process
		16.8.5 Nb/AlOx/Nb Josephson Junction Fabrication
		16.8.6 Planarization
		16.8.7 Device Structure for€Digital Circuits
		16.8.8 Ic Controllability
		16.8.9 Device Yield
		16.8.10 Evolution of€Digital Circuit Fabrication
		16.8.11 Application to€Other Superconducting Devices
	References
17 Quantum Bits with Josephson Junctions
	17.1 Introduction
		17.1.1 What Is a Qubit?
		17.1.2 Why Josephson-Junction Qubits?
		17.1.3 Outline
	17.2 Quantizing Electrical Circuits
	17.3 The Three Basic Josephson-Junction Qubits
		17.3.1 Charge Qubit
		17.3.2 Flux Qubit
		17.3.3 Phase Qubit
	17.4 Further Josephson-Junction Qubits
		17.4.1 The Transmon Qubit
		17.4.2 Other Qubit Refinements
	17.5 Quantum Computing with Josephson-Junction Qubits
		17.5.1 Fulfilling the DiVincenzo Criteria
		17.5.2 Adiabatic Quantum Computing and Quantum Annealing
		17.5.3 Quantum Simulation
		17.5.4 Quantum Error Correction
	17.6 Quantum Optics and Atomic Physics  with Josephson-Junction Qubits
		17.6.1 New Prospects for Textbook Quantum Optics
		17.6.2 New Coupling Strengths
		17.6.3 New Selection Rules
		17.6.4 New Atom Sizes
	References
18 Quantum Superconducting Networks: From Josephson to QED Arrays
	18.1 Introduction
	18.2 Josephson Junction Arrays
		18.2.1 Model of a Josephson Junction Array in the Quantum Regime
		18.2.2 The Zero-Field Phase Diagram
	18.3 Circuit-QED Arrays
		18.3.1 The Model Hamiltonian of a Cavity Array
		18.3.2 Effective Models
		18.3.3 Open System Dynamics
	18.4 Concluding Remarks: Fron Josephson to Circuit-QED Arrays
	References
19 Josephson Effects in Superfluid Helium
	19.1 Introduction
	19.2 Superfluid Weak Links
		19.2.1 Josephson Equations for Quantum Fluids
		19.2.2 Relevant Coupling Dimensions
	19.3 Experimental Apparatus, Techniques,  and Superfluid Hydrodynamics
		19.3.1 Superfluid Weak Link Aperture Arrays
		19.3.2 Description of Physical Cell
		19.3.3 Superfluid Hydrodynamics
	19.4 Josephson Dynamics in Superfluid 3He
		19.4.1 Early Work
		19.4.2 Superfluid 3He Josephson Oscillation
		19.4.3 Superfluid 3He Plasma Mode
		19.4.4 Superfluid 3He Current-Phase Relation
		19.4.5 Superfluid 3He π State
		19.4.6 Superfluid 3He Shapiro Effect
		19.4.7 Superfluid 3He Fiske Effect
	19.5 Josephson Dynamics in Superfluid 4He
		19.5.1 Superfluid 4He Josephson Oscillation
		19.5.2 Superfluid 4He Current-Phase Relation
		19.5.3 Superfluid 4He Junction Size Effect and Phase Coherence
		19.5.4 Superfluid 4He Chemical Potential ``Battery\'\'
		19.5.5 Superfluid 4He Plasma Mode Bifurcation
	19.6 Superfluid Helium Quantum Interference Devices
		19.6.1 Principle of Quantum Interference in Superfluids
		19.6.2 Sensitivity to ``Rotation Flux\'\' Instead of Magnetic Flux
		19.6.3 Superfluid ``Gyrometers\'\'
		19.6.4 Superfluid Quantum Interference Grating
		19.6.5 Further Progress
	19.7 Conclusion
	References
20 Weak Link for Ultracold Bosonic Gases
	20.1 Introduction
	20.2 Two Linearly Coupled Interacting Bose-Einstein Condensates
	20.3 The Quantum Hamiltonian in Schwinger Collective Spin Representation
	20.4 Weak Link Quantum Dynamics as Rotation and Shear of Collective Spin
		20.4.1 The Most Classical Collective Spin State
		20.4.2 Generalized Bloch Sphere and Husimi Representation
		20.4.3 Rotation and Shear of Collective Spin
	20.5 The Classical Mean Field Hamiltonian
	20.6 Phase Portrait of the Classical Hamiltonian
	20.7 The Analog Mechanical System—Momentum Shortened Pendulum
	20.8 Experimental Realization of a Bosonic Weak Link
		20.8.1 Spatial Weak Link: The Atomic Double-Well System
		20.8.2 Internal Weak Link: The Atomic Two-State System
		20.8.3 Overview of the Experimental Sequence
		20.8.4 Control of Initial State
		20.8.5 Detection of Imbalance and Relative Phase
	20.9 Classical Dynamics of Macroscopic Quantum Systems
		20.9.1 The First Observation of Weak Link Dynamics in Bose Einstein Condensates
		20.9.2 From the Rabi to the Josephson Regime
		20.9.3 The Phase Portrait of an Atomic Weak Link
	20.10 Application to Thermometry—Fluctuations  are the Signal
	References
Appendix A
Index