Today Lecture 16 The Josephson effect magnetic field

Today Lecture 16: The Josephson effect --- magnetic field effects in extended junctions Discussion of the Josephson effect in five parts: 1. Theory and phenomena 2. RSJ model 3. Magnetic field effects in extended junctions 4. Fluctuations and quantum tunneling 5. Beyond tunnel junctions (SNS, microbridges, SFS, …) Next time Lecture 17: The Josephson effect --- fluctuations and quantum tunneling

Model junction as a Josephson junction in parallel with a resistor and capacitance I S “mass” “potential” “damping” Josephson dynamics: “phase particle” moving in a tilted washboard potential I < Ic: static solution I=0 = constant V=0 Plasma oscillations: I=I I > Ic: dynamic solution c evolves in time V > 0 I voltage oscillates at the Josephson frequency (Resistively-Shunted Junction) model RSJ U

(neglect inertia) V Period t non-linear non-sunusoidal Time-averaged voltage: I IS IN

Competition between damping and inertial terms – can get hysteresis in I-V curves RSJ QP tunneling

Return current IV Particle barely gets over barrier Must solve by computer non-hysteric Define hysteresis parameter: hysteric 0 1 I I I 1 100 0. 1 high-damping I 1000 low-damping

Shape of the I-V characteristic depends on the damping “Mc. Cumber parameter” Low damping (large R): c > 1 High damping (small R): c < 1 I-V is single-valued I-V is hysteretic I I IS IN V IS V t For I>Ic, the junction switches abruptly to finite voltage and the supercurrent is sinusoidal and averages to zero IS t For I>Ic, the supercurrent is nonsinusoidal has a finite average and averages to zero Practical devices (e. g. dc SQUIDS) – shunt with external resistors to remove hysteresis for IV characteristics

Josephson Effect in extended junctions Local relation --- tunneling is highly-directional so the supercurrent depends on the phase difference at each location across the junction The supercurrent depends on the local CPR and the local gauge-invariant phase difference across the junction: “local current-phase relation” Phase coherent --- phases at each location are related interference 1 J 2 Critical current: w = junction width t = barrier thickness Small-junction limit --- ignore self-field effects (no screening of field by tunneling currents)

Josephson Interferometry: response to a magnetic field magnetic thickness of barrier m = b +2 barrier thickness m b x 0 (y) y B Phase coherence magnetic field induces a phase variation: Uniform magnetic field and small junction limit Uniform junction linear phase variation Fourier transform Fraunhofer diffraction pattern Single-slit optical interference

Josephson Interferometry: what it can tell you Critical current variation Gap anisotropy Domains Charge traps Magnetic field variations Currentphase relation Non-sinusoidal terms -junctions Exotic excitations e. g. Majorana fermions Order parameter symmetry Unconventional superconductivity Flux focusing Self-field from tunneling current Trapped vortices Magnetic particles

t y x J B x y z z Integrate phase around path: wz w Currents: (1) junction tunneling current – along x (2) screening currents – along y but only within λ of interface ⟹ negligible contribution

magnetic thickness of junction Phase dynamics Josephson equations: specific capacitance “Sine-Gordon” equation wave equation for the junction phase w/ damping and non-linearity junction acts like a resonant cavity with phase modes

Special cases: (1) Static magnetic field Josephson penetration depth – limits where bias currents flow in a junction MKS: 3 2 1 Assume:

Small (no screening effects) phase at the center Supercurrent: rectangular junction Set by external current to maximize Ic minimize the Josephson coupling energy

ϕ ϕ ϕ y phase y current y

Interference pattern: Fraunhofer pattern - as in single-slit interference -4 -3 -2 -1 0 1 2 3 4 100 nm 10 μm Magnetic coupling area: d x ω Modifications: (1) shape (2) non-uniform current flow (3) self-field effects → size (not small)

Shape: I(H) does not scale with area proportionally because cross-sections have different areas I z y d z Define section critical current density: maximum current if all parts in phase (zero field) y

But in a field, current reduced as before so: diffraction pattern will be a test of barrier uniformity Rectangular junction (width w) → Circular junction (radius r) →

Self-field limiting Current through junction & films create magnetic fields in the junction ⇒ macroscopic phase effects (2) Equation: Example 1 1 -D No self field effects – current is uniform

Exact solutions (Owen & Scalapino) 1 2 3 4 saturates ℓ fixed 0 1 2 3 4 5

Example 2 0 x ℓ Magnetic field behavior: – I I I 0 ℓ 0. 5 I I I

Compare to Type II SC Similarities Differences (1) vortices in JJ (1) ID chain – not a 2 D lattice (3) structure of the Josephson vortices I I 0 ξ