• Generalized Josephson Junctions


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    • Abstract: Junctions with Resistive Channel. G(v) the resistive conductance ... Superconductor. Superconducting Josephson Junction. L. J-1. For a normal junction, the phase ...

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Generalized Josephson Junctions
Outline
1. Junctions with Resistive Channel
2. RCSJ Model
3. 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
G(v)
S-I-S
Josephson Tunneling
Massachusetts Institute of Technology
6.763 2003 Lecture 13
Normal and Superconducting Analogy
Superconductor Superconducting Josephson Junction
LJ-1
For a normal junction, the phase is constantly
Normal metal being driven back to zero so linearize near
zero and add a damping time
for dc drive
for dc drive
and and
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 Model
i
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
Josephson Time Constant Stewart-McCumber Parameter Q2
Massachusetts Institute of Technology
6.763 2003 Lecture 13
Overdamped Junction βc > 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
τRC >> τJ
A. Static Solution:
B. Dynamical Solution
The phase changes
quickly compared to
RC, so the voltage is
just from R and C.
Therefore,
i R
Hysteretic
Massachusetts Institute of Technology
6.763 2003 Lecture 13
Junction with arbitrary βc
A. Static Solution:
B. Dynamical Solution Return Current
Massachusetts Institute of Technology
6.763 2003 Lecture 13
Return Current
Energy 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
where and
Massachusetts Institute of Technology
6.763 2003 Lecture 13
βc = 4
V(t)
A A
i/IC
Β φ(t)
C
V(φ)
/ICR
Β C
V(t) V(t)
(t) φ(t)
V(φ) V(φ)
Massachusetts Institute of Technology
6.763 2003 Lecture 13
βc =0.5
A A
V(t)
B
i/ICC
φ(t)
/ICR
C
V(t)
V(t)
B
φ(t) φ(t)
V(φ) V(φ)
Massachusetts Institute of Technology
6.763 2003 Lecture 13
Pendulum Model for a Josephson Junction
τapp
+
l
Iapp R C
Icsinϕ ϕ
-
mg
• 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
Massachusetts Institute of Technology
6.763 2003 Lecture 13


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