Generalized Josephson Junctions - MITweb.mit.edu/6.763/www/FT03/Lectures/Lecture13.pdfMassachusetts...
Transcript of Generalized Josephson Junctions - MITweb.mit.edu/6.763/www/FT03/Lectures/Lecture13.pdfMassachusetts...
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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
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
Junctions with Resistive Channel
G(v) the resistive conductance
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
Tunneling between two superconductors
Giaever Tunneling
Josephson Tunneling
S-I-S G(v)
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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
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
ICRn Product
The condition is equivalent to
Experimentally, For Nb at 2K,
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
Capacitance of a Josephson Junction
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
Generalized Josephson Junction
and
Therefore,
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
RCSJ Modeli
and
Therefore,
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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
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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
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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
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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
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
Junction with arbitrary βc
A. Static Solution:
Return CurrentB. Dynamical Solution
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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
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
Dynamical Analysis
andwhere
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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)
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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
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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)
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Massachusetts Institute of Technology 6.763 2003 Lecture 13
Pendulum Model for a vortex