Massachusetts Institute of Technology 6.763 2003 Lecture 13

Generalized Josephson Junctions

Outline

1. Junctions with Resistive Channel2. RCSJ Model3. DC Current Drive

• Overdamped and Underdamped Junctions• Return Current• Dynamical Analysis

4. Pendulum Model

October 16, 2003

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Junctions with Resistive Channel

G(v) the resistive conductance

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Tunneling between two superconductors

Giaever Tunneling

Josephson Tunneling

S-I-S G(v)

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Normal and Superconducting AnalogySuperconductor Superconducting Josephson Junction

LJ-1

For a normal junction, the phase is constantly being driven back to zero so linearize near zero and add a damping time

Normal metal

for dc drivefor dc drive

andand

Massachusetts Institute of Technology 6.763 2003 Lecture 13

ICRn Product

The condition is equivalent to

Experimentally, For Nb at 2K,

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Capacitance of a Josephson Junction

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Generalized Josephson Junction

and

Therefore,

Massachusetts Institute of Technology 6.763 2003 Lecture 13

RCSJ Modeli

and

Therefore,

Massachusetts Institute of Technology 6.763 2003 Lecture 13

DC Current drive in the RSCJ Model

and

Therefore,

The equation of motion can be rewritten as

where

Stewart-McCumber Parameter Q2Josephson Time Constant

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Overdamped Junction βc << 1τJ >> τRC

A. Static Solution:

B. Dynamical Solution for i > Ic

This is periodic with period

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Overdamped Junction βc << 1

v(t)/IcR

t

<v>/(IcR)

i/Ic

The time averaged voltage is

Use the voltage-phase relation,

Therefore,

Non-hysteretic

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Underdamped Junction βc >> 1τRC >> τJ

A. Static Solution:

B. Dynamical Solution

The phase changes quickly compared to RC, so the voltage is just from R and C.Therefore,

<v(t)> i R

Hysteretic

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Junction with arbitrary βc

A. Static Solution:

Return CurrentB. Dynamical Solution

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Return CurrentEnergy Loss per cycle = Energy supplied by sourc

where V= IR and τ = Φ0 / (2 π I R), therefore

So that

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Dynamical Analysis

andwhere

Massachusetts Institute of Technology 6.763 2003 Lecture 13

βc = 4

(t)

V(φ)

<V>/ICR

i/ICΒ

Β

A A

C

C

V(t)

φ(t)

V(φ)

V(t)V(t)

φ(t)

V(φ)

Massachusetts Institute of Technology 6.763 2003 Lecture 13

βc =0.5

V(t)

φ(t)

V(φ)

φ(t)

V(φ)

i/IC

<V>/ICR

C

C

B

AA V(t)

φ(t)

V(t)B

Massachusetts Institute of Technology 6.763 2003 Lecture 13

Pendulum Model for a Josephson Junction

τapp

mg

ϕ

l

R

-

+

Icsinϕ

CIapp

• 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 13

Pendulum Model for a vortex