Applied Superconductivity

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Text of Applied Superconductivity

Applied Superconductivity:Josephson Eect and Superconducting ElectronicsManuscript to the Lectures during WS 2003/2004, WS 2005/2006, WS 2006/2007,WS 2007/2008, WS 2008/2009, and WS 2009/2010Prof. Dr. Rudolf GrossandDr. Achim MarxWalther-Meiner-InstitutBayerische Akademie der WissenschaftenandLehrstuhl fr Technische Physik (E23)Technische Universitt MnchenWalther-Meiner-Strasse 8D-85748 GarchingRudolf.Gross@wmi.badw.de Walther-Meiner-Institut Garching, October 2005ContentsPreface xxiI Foundations of the Josephson Effect 11 Macroscopic Quantum Phenomena 31.1 The Macroscopic Quantum Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Coherent Phenomena in Superconductivity . . . . . . . . . . . . . . . . . . . . 31.1.2 Macroscopic Quantum Currents in Superconductors . . . . . . . . . . . . . . . 121.1.3 The London Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2 Flux Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2.1 Flux and Fluxoid Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2.2 Experimental Proof of Flux Quantization . . . . . . . . . . . . . . . . . . . . . 281.2.3 Additional Topic:Rotating Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.3 Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.3.1 The Josephson Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3.2 Josephson Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 JJs: The Zero Voltage State 432.1 Basic Properties of Lumped Josephson Junctions . . . . . . . . . . . . . . . . . . . . . 442.1.1 The Lumped Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . . . 442.1.2 The Josephson Coupling Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 452.1.3 The Superconducting State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.1.4 The Josephson Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.1.5 Mechanical Analogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.2 Short Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2.1 Quantum Interference Effects Short Josephson Junction in an Applied Mag-netic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50iiiiv R. GROSS AND A. MARX CONTENTS2.2.2 The Fraunhofer Diffraction Pattern . . . . . . . . . . . . . . . . . . . . . . . . . 542.2.3 Determination of the Maximum Josephson Current Density . . . . . . . . . . . 582.2.4 Additional Topic:Direct Imaging of the Supercurrent Distribution . . . . . . . . . . . . . . . . . . 622.2.5 Additional Topic:Short Josephson Junctions: Energy Considerations . . . . . . . . . . . . . . . . 632.2.6 The Motion of Josephson Vortices . . . . . . . . . . . . . . . . . . . . . . . . . 652.3 Long Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.3.1 The Stationary Sine-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . 682.3.2 The Josephson Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.3.3 Junction Types and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 732.3.4 Additional Topic:Josephson Current Density Distribution and Maximum Josephson Current . . . . 792.3.5 The Pendulum Analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 JJs: The Voltage State 893.1 The Basic Equation of the Lumped Josephson Junction . . . . . . . . . . . . . . . . . . 903.1.1 The Normal Current: Junction Resistance . . . . . . . . . . . . . . . . . . . . . 903.1.2 The Displacement Current: Junction Capacitance . . . . . . . . . . . . . . . . . 923.1.3 Characteristic Times and Frequencies . . . . . . . . . . . . . . . . . . . . . . . 933.1.4 The Fluctuation Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.1.5 The Basic Junction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.2 The Resistively and Capacitively Shunted Junction Model . . . . . . . . . . . . . . . . . 973.2.1 Underdamped and Overdamped Josephson Junctions . . . . . . . . . . . . . . . 1003.3 Response to Driving Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.3.1 Response to a dc Current Source . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.3.2 Response to a dc Voltage Source . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.3.3 Response to ac Driving Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.3.4 Photon-Assisted Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.4 Additional Topic:Effect of Thermal Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.4.1 Underdamped Junctions: Reduction of Ic by Premature Switching . . . . . . . . 1173.4.2 Overdamped Junctions: The Ambegaokar-Halperin Theory . . . . . . . . . . . . 1183.5 Secondary Quantum Macroscopic Effects . . . . . . . . . . . . . . . . . . . . . . . . . 1223.5.1 Quantum Consequences of the Small Junction Capacitance . . . . . . . . . . . . 122 Walther-Meiner-InstitutCONTENTS APPLIED SUPERCONDUCTIVITY v3.5.2 Limiting Cases: The Phase and Charge Regime . . . . . . . . . . . . . . . . . . 1253.5.3 Coulomb and Flux Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.5.4 Coherent Charge and Phase States . . . . . . . . . . . . . . . . . . . . . . . . . 1303.5.5 Quantum Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1323.5.6 Macroscopic Quantum Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 1333.6 Voltage State of Extended Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . 1393.6.1 Negligible Screening Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.6.2 The Time Dependent Sine-Gordon Equation . . . . . . . . . . . . . . . . . . . . 1403.6.3 Solutions of the Time Dependent Sine-Gordon Equation . . . . . . . . . . . . . 1413.6.4 Additional Topic:Resonance Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144II Applications of the Josephson Effect 1534 SQUIDs 1574.1 The dc-SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.1.1 The Zero Voltage State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594.1.2 The Voltage State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.1.3 Operation and Performance of dc-SQUIDs . . . . . . . . . . . . . . . . . . . . 1684.1.4 P