Generalized Josephson JunctionsJunctions with Resistive ChannelTunneling between two superconductorsNormal and Superconducting AnalogyICRn ProductCapacitance of a Josephson JunctionGeneralized Josephson JunctionRCSJ ModelDC Current drive in the RSCJ ModelOverdamped Junction bc << 1Overdamped Junction bc << 1Underdamped Junction bc >> 1Junction with arbitrary bcReturn CurrentDynamical Analysisbc = 4bc =0.5Pendulum Model for a Josephson JunctionPendulum Model for a vortexMassachusetts Institute of Technology 6.763 2003 Lecture 13Generalized Josephson JunctionsOutline1. Junctions with Resistive Channel2. RCSJ Model3. DC Current Drive• Overdamped and Underdamped Junctions• Return Current• Dynamical Analysis4. Pendulum ModelOctober 16, 2003Massachusetts Institute of Technology 6.763 2003 Lecture 13Junctions with Resistive ChannelG(v) the resistive conductanceMassachusetts Institute of Technology 6.763 2003 Lecture 13Tunneling between two superconductorsGiaever TunnelingJosephson TunnelingS-I-SG(v)Massachusetts Institute of Technology 6.763 2003 Lecture 13Normal and Superconducting AnalogySuperconductorSuperconducting Josephson JunctionLJ-1For a normal junction, the phase is constantly being driven back to zero so linearize near zero and add a damping timeNormal metalfor dc drivefor dc driveandandMassachusetts Institute of Technology 6.763 2003 Lecture 13ICRnProductThe condition is equivalent to Experimentally, For Nb at 2K,Massachusetts Institute of Technology 6.763 2003 Lecture 13Capacitance of a Josephson JunctionMassachusetts Institute of Technology 6.763 2003 Lecture 13Generalized Josephson JunctionandTherefore,Massachusetts Institute of Technology 6.763 2003 Lecture 13RCSJ ModeliandTherefore,Massachusetts Institute of Technology 6.763 2003 Lecture 13DC Current drive in the RSCJ ModelandTherefore, The equation of motion can be rewritten aswhereStewart-McCumber Parameter Q2Josephson Time ConstantMassachusetts Institute of Technology 6.763 2003 Lecture 13Overdamped Junction βc<< 1τJ>> τRCA. Static Solution: B. Dynamical Solution for i > IcThis is periodic with periodMassachusetts Institute of Technology 6.763 2003 Lecture 13Overdamped Junction βc<< 1v(t)/IcRt<v>/(IcR)i/IcThe time averaged voltage isUse the voltage-phase relation,Therefore,Non-hystereticMassachusetts Institute of Technology 6.763 2003 Lecture 13Underdamped Junction βc>> 1τRC>> τJA. Static Solution: B. Dynamical SolutionThe phase changes quickly compared to RC, so the voltage is just from R and C.Therefore,<v(t)> i RHystereticMassachusetts Institute of Technology 6.763 2003 Lecture 13Junction with arbitrary βcA. Static Solution: Return CurrentB. Dynamical SolutionMassachusetts Institute of Technology 6.763 2003 Lecture 13Return CurrentEnergy Loss per cycle = Energy supplied by sourcwhere V= IR and τ = Φ0 / (2 π I R), thereforeSo thatMassachusetts Institute of Technology 6.763 2003 Lecture 13Dynamical AnalysisandwhereMassachusetts Institute of Technology 6.763 2003 Lecture 13βc= 4M(t)V(φ)<V>/ICRi/ICΒΒAACCV(t)φ(t)V(φ)V(t)V(t)φ(t)V(φ)Massachusetts Institute of Technology 6.763 2003 Lecture 13βc=0.5V(t)φ(t)V(φ)φ(t)V(φ)i/IC<V>/ICRCCBAAV(t)φ(t)V(t)BMassachusetts Institute of Technology 6.763 2003 Lecture 13Pendulum Model for a Josephson JunctionτappmgϕlR-+IcsinϕCIapp• Single junction (RCSJ model) pendulum (damped)• Coupled junctions – can support non-linear excitations (breathers and moving vortices)Massachusetts Institute of Technology 6.763 2003 Lecture 13Pendulum Model for a