Fundamentals and Frontiers of the Josephson Effect (eBook)

Preface 7
General References 14
Acknowledgements 17
Contents 18
Contributors 30
Acronyms 32
Symbols 36
1 Introductory Notes on the Josephson Effect: Main Concepts and Phenomenology 38
1.1 A Brief Historical Survey on the Materials Used for the Realization of Superconducting Junctions 38
1.2 The Coupling Between Macroscopic Quantum Systems and the Equations of the Josephson Effect 40
1.2.1 Josephson Equations in the Tunnel Limit 40
1.2.2 Different Types of Josephson Junctions Other than Tunnel 44
1.3 The Tunneling Hamiltonian and the Scattering Formalism 53
1.3.1 Expression for the Total Current in the Tunneling Hamiltonian Formalism 53
1.3.2 Conductance in a Tunnel Junction 60
1.3.3 From the Tunneling Transfer Hamiltonian to the Scattering Formalism 63
1.3.4 Andreev Reflection 65
1.3.5 Josephson Effect Derived from Quasi-particle Andreev Bound States 66
1.4 Current–Voltage (I–V) Characteristics: From Microscopic Theory to the Resistively Shunted Junction Model 69
1.4.1 I–V: Notes on the Resistively Shunted Junction Model 70
1.5 Temperature Dependence of Ic Rn and of the I–V characteristics 74
1.5.1 Temperature Dependence of Ic in the Tunnel Limit 74
1.5.2 Temperature Dependence of Ic Other than the Tunnel Limit 76
1.6 Magnetic Field Effects 78
1.7 Electrodynamics of the Josephson Junction 86
1.8 Material and Nano Science Open Novel Routes for the Fabrication of Josephson Junctions 87
1.8.1 Low Temperature Josephson Junctions 87
1.8.2 High Temperature Josephson Junctions 88
1.8.3 Hybrid Junctions 89
References 92
2 Josephson Devices as Tests of Quantum Mechanics Towards the Everyday Level 99
2.1 Background 99
2.2 Early History 102
2.3 Consolidation: Work on MQT in the Early 80s 107
2.4 Progress Towards MQC: 1981–1999 108
2.5 The Modern Era: Josephson Qubits 110
2.6 Where Do We Stand? 114
References 115
3 Basic Properties of the Josephson Effect 117
3.1 Introduction 117
3.2 Basic Features and Fundamental Relations 117
3.3 Josephson Effect in Basic Types of Junctions 118
3.4 SNS Junctions 120
3.4.1 Dirty Limit 120
3.4.2 Clean SNS Junctions 122
3.5 Double Barrier SINIS Junctions 123
3.5.1 SINIS Junctions, Clean Limit 124
3.5.2 SINIS Junctions, Dirty Limit 125
3.6 SFS Josephson Junctions 127
3.6.1 Proximity Effect in SF Bilayer 128
3.7 CPR in SFS Junctions 134
3.7.1 -Junctions 136
3.7.2 0-Junctions 138
3.7.3 CPR in Serial SIsFS and SFsFS Junctions 139
References 141
4 Charge Transport in Unconventional Superconductor Junctions 153
4.1 Topological Superconductivity 153
4.1.1 Pair Potential 154
4.1.2 Topological Number and Surface Bound States 157
4.1.3 Tunnel Conductance 161
4.1.4 Josephson Current 164
4.2 Proximity Effect in a Dirty Normal Metal 169
4.2.1 Conductance of a Dirty NS Junction 169
4.2.2 Josephson Effect in a Dirty SNS Junction 173
4.3 Remark: Odd-Frequency Cooper Pair and Majorana Fermion 177
References 180
5 Mesoscopic Features in Nanoscale Superconducting Devices 182
5.1 Introduction 182
5.2 Proximity in Macroscopic Systems 185
5.2.1 Free Energy of the Isolated Superconductor 185
5.2.2 Superconducting Correlations Induced in a Normal Metal by Proximity 186
5.3 Andreev Resonances at Superconductor-Normal Metal Interfaces 189
5.3.1 Andreev Resonances in a Clean N Slab in Proximity with a Superconductor 189
5.3.2 Diffusive N/S Boundary 192
5.3.3 Andreev Reflection Under the Magnetic Field: Magnetoconductance Oscillations in N/S Junctions 194
5.4 Scattering Approach to Ballistic Transport in SNS Josephson Junctions 195
5.4.1 Andreev Bound States with Fully Transmitting NS Interfaces 195
5.4.2 Density of Energy States at a Generic SNS Junction 197
5.5 Ballistic and Diffusive SNS Junction Systems 200
5.5.1 Ballistic Short and Long SNS Junctions 201
5.5.2 Diffusive Short and Long SNS Junctions 202
5.6 Semiclassical Approach to Diffusive Systems and Other Signatures of the Mesoscopic Regime 207
5.6.1 Minigap in SNS Diffusive Junctions 209
5.6.2 Low-Temperature Reentrant Behavior of the Resistance in a Diffusive N Wire in Proximity with a Superconductor 211
5.6.3 Resistance Change in a Wire in Contact with a Superconducting Electrode 213
5.7 Mesoscopic Conductance Fluctuations 214
5.7.1 Self Correlations of the Conductance in Magnetic Field 214
5.7.2 Self Correlations of the Conductance in Non Equilibrium 217
5.8 From Few to Single Channel Junctions 226
5.8.1 Shot Noise in Few Channel NS Junctions 227
5.8.2 Single Channel SS Junctions 228
5.8.3 Andreev Qubits and Parity Jumps 232
5.8.4 Transient Dynamics 235
References 238
6 Magnetic Field Effects in Josephson Junctions 243
6.1 Introduction 243
6.2 Static Magnetic Fields 243
6.2.1 Flux Focussing 243
6.2.2 Time-Independent Sine-Gordon Equation 246
6.2.3 Magnetic Interference Patterns 248
6.2.4 Josephson Vortices 255
6.3 Time-Dependent Magnetic Fields 257
6.3.1 Time-Dependent Sine-Gordon Equation 257
6.3.2 Fiske Steps 260
6.3.3 Zero-Field Steps 264
References 265
7 Current–Voltage Characteristics 268
7.1 The Resistively Shunted Junction Model 268
7.1.1 The Noise Term in the RSJ Model, a First Watch at Fluctuations 271
7.2 I–V Curves in the RSJ Model in the Small Capacitance Limit 275
7.3 I–V Curves in the RSJ Model for Finite Capacitance 277
7.3.1 Details of the I–V Curves in the Subgap Region for Finite Capacitance and Nonlinear RSJ Models 278
7.4 Current Biased Tunneling Junction, a More Accurate Description of the Subgap Region for Finite Capacitance 281
7.5 Effects of Thermal Fluctuations 286
7.5.1 Negligible Capacitance 287
7.5.2 Finite Capacitance 290
7.5.3 Large Capacitance 291
7.6 I–V Curves: When They Do Not Match RSJ-Like Predictions 292
7.6.1 Deviations from RSJ, RSJN and TJM Models 292
7.6.2 I–V Curves in Small or Nanoscale Junctions: From the RSJ Model to Phase Diffusion 295
7.6.3 Beyond Classical Smoluchowski Dynamics, from Coulomb Blockade to Quantum Diffusion 298
7.6.4 More on the Amplitude of the Hysteresis 300
7.6.5 Concluding Remarks and a Further Look at Experimental I–V Curves 302
References 302
8 High Critical Temperature Superconductor Josephson Junctions and Other Exotic Structures 308
8.1 Introduction 309
8.2 Complementary Investigations and the Importance of a Structural Feedback 309
8.3 Grain Boundary Junctions 310
8.3.1 Bicrystal Junctions 312
8.3.2 Biepitaxial Junctions 313
8.3.3 Step-Edge Junctions 316
8.4 Locally Affecting Superconductivity, Moving Oxygen in Thin Films and Damaged Junctions 318
8.4.1 Modifying Junctions by Irradiation 320
8.4.2 Electro-Migration Studies 321
8.5 Junctions with an Artificial Barrier 321
8.5.1 Ramp Edge Junctions Realized with Au and Ag Inert Barriers 323
8.5.2 Ramp Edge Junctions Realized with Perosvkite and Layered Materials 325
8.5.3 Trilayer Structures 327
8.6 Interface-Engineered Junctions, a different way of Creating a Barrier 328
8.6.1 Ramp-Edge Junctions for Superconducting Electronics 329
8.7 Junctions with HTS Other Than YBCO 330
8.7.1 La1.85Sr0.15CuO4-Based Trilayer with One-Unit-Cell-Thick Barrier 330
8.7.2 Electron Doped HTS 331
8.8 Intrinsic Stacked Junctions 331
8.9 HTS Junctions and Wires on the Meso/nano Scale 333
8.9.1 GB Junctions Realized with Ultra-Thin Films and Superlattices 333
8.9.2 HTS Nanostructures and Nanowires 334
8.9.3 Submicron Josephson Junctions, Energy Scales and Mesoscopic Effects 336
8.10 General Criteria on I–V Curves and the Estimation of Junction Parameters 338
8.10.1 The Shape of I–V Curves 339
8.10.2 From I–V Curves and Their Modelling to Junction Parameters 340
8.10.3 Capacitance and Related Electromagnetic Properties of Junction Interfaces 341
8.11 Dependence of the Josephson Current on the Temperature 343
8.12 Notes on the Magnetic Properties of HTS Junctions 347
8.12.1 Dependence of the Critical Current and I–V Characteristics on the Magnetic Field 347
8.12.2 Spontaneous Magnetization with Random Orientation 350
8.13 Fractional Shapiro Steps: Time-Dependent Effects 352
8.14 Other Exotic Structures: Josephson Junctions Based on Interface Superconductors 352
References 355
9 Pairing Symmetry Effects 371
9.1 Dependence of Josephson Critical Currents on Junction Geometry 372
9.2 Quantum Interference of Josephson Currents 375
9.3 Spontaneous Josephson Currents 381
References 391
10 Intrinsic Josephson Junctions in High Temperature Superconductors 399
10.1 Introduction 399
10.2 Fabrication Methods and Materials 405
10.3 Basic Properties 409
10.3.1 Resistivity and Out-of-Plane Critical Current Density 409
10.3.2 Current Voltage Characteristics 411
10.3.3 Interlayer Tunneling Spectroscopy 419
10.3.4 Modelling of One-Dimensional Stacks: Coupling by Charge Fluctuations 424
10.4 Josephson Plasma Oscillations and Collective Fluxon Dynamics 430
10.4.1 Coupled Sine-Gordon Equations 430
10.4.2 Static Josephson Fluxons Lattices 433
10.4.3 Collective Josephson Plasma Oscillations 438
10.4.4 Fluxon Dynamics 443
10.5 Generation of THz Radiation with Intrinsic Junction Stacks 453
References 471
11 Phase Dynamics and Macroscopic Quantum Tunneling 487
11.1 Escape Out of a Metastable State 488
11.1.1 Theoretical Background, Effects of Dissipation and the Underdamped Limit 488
11.1.2 The First Experiments 492
11.1.3 The Effect of the Magnetic Field on SCD 496
11.1.4 Notes on Resonant Activation and Quantized Energy Level 497
11.1.5 The Master Equation for Phase Dynamics 499
11.1.6 The Retrapping Current 500
11.1.7 Thermal Activation and Macroscopic Quantum Tunneling in SQUIDs and Annular Junctions 501
11.2 Moderately Damped Regime 503
11.3 Thermal Activation, Macroscopic Quantum Tunneling and Phase Diffusion in Unconventional Josephson Junctions 511
11.3.1 HTS Josephson Junctions 512
11.3.2 In the `Far' Low Critical Current Regime in LTS and HTS JJs 515
11.3.3 Phase Dynamics Diagram: Influence of Dissipation 520
11.3.4 Ferromagnetic Junctions 522
11.3.5 SCDs in Junction with Graphene Barriers 531
11.4 SCDs in Junctions with High Values of Jc 531
11.4.1 SCDs in Nanowires 536
References 537
12 High Frequency Properties of Josephson Junctions 545
12.1 Simple Voltage Source Model 546
12.2 Finite Dimension Effect in Tunneling Junctions 549
12.3 Current Source Model 551
12.4 Resonant Activation 556
References 559
13 Josephson Effect in Graphene and 3D Topological Insulators 561
13.1 Introduction 561
13.2 Superconductor - Graphene - Superconductor Junctions 562
13.3 Superconductor - Topological Insulator - Superconductor Junctions 568
13.4 Fabrication of Superconducting Hybrid Devices 571
13.5 Effective Area of a Planar Josephson Junction 575
13.6 Planar Josephson Junctions with TI Barriers 578
References 582
14 Physics and Applications of NanoSQUIDs 586
14.1 Introduction 586
14.2 Superconducting “Weak-Link” Response and the Josephson Effects 588
14.2.1 Josephson Junctions for NanoSQUIDs 589
14.2.2 Josephson Tunnel Barrier 589
14.2.3 Trilayer Junctions 590
14.2.4 Normal Metal Barriers 591
14.2.5 Dayem Bridge Weak Links 592
14.2.6 Focussed Ion Beam Milling 594
14.2.7 Electron Beam Lithography (EBL) 595
14.2.8 Niche Fabrication Developments 596
14.2.9 Comparison of Tunnel Junctions and Other Weak Links 598
14.3 Practical NanoSQUID Realisations 598
14.3.1 Grenoble Group 598
14.3.2 CSIRO SQUIDs 599
14.3.3 NPL SQUIDs 599
14.3.4 Other Nanosuperconducting Structures 600
14.3.5 Single Josephson Tunnel Junction 601
14.3.6 3D NanoSQUIDs 601
14.3.7 State-of-the-Art 603
14.4 Nanoscale Leads to Improved Energy Sensitivity 603
14.4.1 How Reproducible is NanoSQUID Fabrication? 604
14.4.2 Further Miniaturization? 605
14.5 Applications of NanoSQUIDs 606
14.5.1 Nano Electro-Mechanical Systems (NEMS) 606
14.6 Superconducting Qubits—At the Nanoscale? 608
14.7 High Frequency Readout of SQUIDs 609
14.8 New Materials 611
14.9 Summary and Outlook 612
Bibliography 612
15 Josephson Junctions for Metrology Applications 617
15.1 Introduction 617
15.2 Overview of Voltage Metrology and Applications 618
15.3 Voltage Quantization 620
15.4 Programmable DC Voltage Standards 624
15.5 Intrinsic AC Voltage Standards and Arbitrary Waveform Synthesis 627
15.6 Temperature Metrology with a Quantum Voltage Noise Source 631
References 634
16 Josephson Junctions for Digital Applications 640
16.1 Introduction 640
16.2 Digital Circuits 642
16.2.1 Rapid Single Flux Quantum Logic 642
16.3 Energy-Efficient Single Flux Quantum Circuits 645
16.4 DC Biased Energy-Efficient Circuits 645
16.5 AC Biased Energy-Efficient Circuits 647
16.6 Adiabatic Flux Quantum Parametron Logic 649
16.6.1 Introduction 649
16.6.2 Operation Principle of Adiabatic Quantum Flux Parametron (AQFP) Logic 650
16.6.3 Energy Efficiency of an AQFP Logic Gate 653
16.6.4 AQFP Logic Circuits 656
16.7 Memory for Cryogenic Supercomputer 661
16.7.1 Introduction 661
16.7.2 SQUID Memory 661
16.7.3 Abrikosov Vortex Memory 669
16.7.4 Cryotron Memory 672
16.7.5 CMOS Memory 675
16.7.6 Memory Proposals Using Hybrid Superconductor/Ferromagnet Structures 677
16.7.7 Novel Room-Temperature Memory Proposals Considered for Cryogenic Applications 691
16.7.8 Conclusion and Outlook 695
16.8 Fabrication of Low-Critical-Temperature Josephson Junctions and Integrated Circuits 695
16.8.1 Introduction 695
16.8.2 Circuit Elements of Superconducting Digital Circuits 696
16.8.3 Josephson Junctions 697
16.8.4 Fabrication Process 698
16.8.5 Nb/AlOx/Nb Josephson Junction Fabrication 699
16.8.6 Planarization 701
16.8.7 Device Structure for Digital Circuits 704
16.8.8 Ic Controllability 706
16.8.9 Device Yield 707
16.8.10 Evolution of Digital Circuit Fabrication 713
16.8.11 Application to Other Superconducting Devices 715
References 718
17 Quantum Bits with Josephson Junctions 731
17.1 Introduction 731
17.1.1 What Is a Qubit? 731
17.1.2 Why Josephson-Junction Qubits? 733
17.1.3 Outline 734
17.2 Quantizing Electrical Circuits 734
17.3 The Three Basic Josephson-Junction Qubits 738
17.3.1 Charge Qubit 739
17.3.2 Flux Qubit 740
17.3.3 Phase Qubit 742
17.4 Further Josephson-Junction Qubits 742
17.4.1 The Transmon Qubit 743
17.4.2 Other Qubit Refinements 744
17.5 Quantum Computing with Josephson-Junction Qubits 746
17.5.1 Fulfilling the DiVincenzo Criteria 746
17.5.2 Adiabatic Quantum Computing and Quantum Annealing 750
17.5.3 Quantum Simulation 751
17.5.4 Quantum Error Correction 752
17.6 Quantum Optics and Atomic Physics with Josephson-Junction Qubits 754
17.6.1 New Prospects for Textbook Quantum Optics 755
17.6.2 New Coupling Strengths 756
17.6.3 New Selection Rules 757
17.6.4 New Atom Sizes 758
References 760
18 Quantum Superconducting Networks: From Josephson to QED Arrays 770
18.1 Introduction 770
18.2 Josephson Junction Arrays 772
18.2.1 Model of a Josephson Junction Array in the Quantum Regime 773
18.2.2 The Zero-Field Phase Diagram 777
18.3 Circuit-QED Arrays 780
18.3.1 The Model Hamiltonian of a Cavity Array 782
18.3.2 Effective Models 786
18.3.3 Open System Dynamics 787
18.4 Concluding Remarks: Fron Josephson to Circuit-QED Arrays 789
References 790
19 Josephson Effects in Superfluid Helium 792
19.1 Introduction 792
19.2 Superfluid Weak Links 793
19.2.1 Josephson Equations for Quantum Fluids 793
19.2.2 Relevant Coupling Dimensions 794
19.3 Experimental Apparatus, Techniques, and Superfluid Hydrodynamics 796
19.3.1 Superfluid Weak Link Aperture Arrays 796
19.3.2 Description of Physical Cell 796
19.3.3 Superfluid Hydrodynamics 797
19.4 Josephson Dynamics in Superfluid 3He 800
19.4.1 Early Work 800
19.4.2 Superfluid 3He Josephson Oscillation 801
19.4.3 Superfluid 3He Plasma Mode 803
19.4.4 Superfluid 3He Current-Phase Relation 805
19.4.5 Superfluid 3He ? State 807
19.4.6 Superfluid 3He Shapiro Effect 807
19.4.7 Superfluid 3He Fiske Effect 808
19.5 Josephson Dynamics in Superfluid 4He 809
19.5.1 Superfluid 4He Josephson Oscillation 809
19.5.2 Superfluid 4He Current-Phase Relation 813
19.5.3 Superfluid 4He Junction Size Effect and Phase Coherence 816
19.5.4 Superfluid 4He Chemical Potential ``Battery'' 817
19.5.5 Superfluid 4He Plasma Mode Bifurcation 818
19.6 Superfluid Helium Quantum Interference Devices 820
19.6.1 Principle of Quantum Interference in Superfluids 820
19.6.2 Sensitivity to ``Rotation Flux'' Instead of Magnetic Flux 821
19.6.3 Superfluid ``Gyrometers'' 825
19.6.4 Superfluid Quantum Interference Grating 827
19.6.5 Further Progress 829
19.7 Conclusion 833
References 834
20 Weak Link for Ultracold Bosonic Gases 838
20.1 Introduction 838
20.2 Two Linearly Coupled Interacting Bose-Einstein Condensates 840
20.3 The Quantum Hamiltonian in Schwinger Collective Spin Representation 843
20.4 Weak Link Quantum Dynamics as Rotation and Shear of Collective Spin 844
20.4.1 The Most Classical Collective Spin State 844
20.4.2 Generalized Bloch Sphere and Husimi Representation 846
20.4.3 Rotation and Shear of Collective Spin 847
20.5 The Classical Mean Field Hamiltonian 850
20.6 Phase Portrait of the Classical Hamiltonian 852
20.7 The Analog Mechanical System—Momentum Shortened Pendulum 853
20.8 Experimental Realization of a Bosonic Weak Link 855
20.8.1 Spatial Weak Link: The Atomic Double-Well System 856
20.8.2 Internal Weak Link: The Atomic Two-State System 857
20.8.3 Overview of the Experimental Sequence 857
20.8.4 Control of Initial State 859
20.8.5 Detection of Imbalance and Relative Phase 861
20.9 Classical Dynamics of Macroscopic Quantum Systems 863
20.9.1 The First Observation of Weak Link Dynamics in Bose Einstein Condensates 863
20.9.2 From the Rabi to the Josephson Regime 866
20.9.3 The Phase Portrait of an Atomic Weak Link 868
20.10 Application to Thermometry—Fluctuations are the Signal 869
References 872
Appendix A 875
Index 876