Search for inphase Josephson vortex solutions in BSCCO type junctions: Modeling and Numerical simula

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Search for in-phase Josephson vortex solutions in
BSCCO type junctions Modeling and Numerical
simulations
N.F. Pedersen
Oersted -DTU, Technical University of Denmark
DK-2800 Lyngby
Work done in collaboration with Søren Madsen
Oersted -DTU, Technical University of Denmark,
DK-2800 Lyngby
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Stacked Josephson junctions
High Tc Josephson junction
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Equations for the Josephson stack (coupled
sine-Gordon equations)
(NxN )
(sine-Gordon in each junction)
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Normalisations
Bias current ? is normalised to the critical
current Io

• x is normalised to the Josephson penetration
  length

time is normalised to the inverse of the single
junction plasma frequency
?c is the McCumber parameter,
the magnetic thickness d
the coupling parameter. s
Normalised coupling parameter, s s/d
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Solutions for N-stack (N Coupled sine-Gordon
equations)

Two types of solutions, 1) non-linear
fluxons (numerical solution) and 2) linear
plasma oscillations (analytical). Each type of
solutions have N different layer configurations
with N different velocities. Some geometric
similarities between 1) and 2) for same mode
number i. i1....N
1) Fluxon solutions (2p kinks)
2) Plasma oscillation solutions
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Plasma oscillations Dispersion relations, N
50, m 1 ...50
In-phase
50 electromagnetic velocities
Electromagnetic modes Fiske modes
Anti-phase
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In-phase fluxon solutions, i.e. fluxons in the
different layers move in-phase
We look for
Three known mechanisms

• Intrinsic phase locking
• Magnetic field induced phase locking
• Cavity induced phase locking

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1) Intrinsic in-phase locking - motion in a stack
in-phase
anti-phase
v
0
Illustration by G. Filatrella,2004
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N3. Anti-phase and in-phase plasmon and fluxon
modes (3rd mode not shown)
Slt0, C- lt C
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The 5 plasma modes in a 5 stack
Mode 3
Mode 4
Mode 2
Mode 5 inphase
Mode 1antiphase
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5 stack, mode 4,
1,2
5 4 3 2 1
4,5
Fluxon trajectories
?
1,2
3
4,5
?
Analytic solution for plasma waves
Numerical solution for fluxons
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2) Magnetic field induced in-phase locking
Magnetic field breaks the symmetry and creates a
unidirectional flow of fluxons. The in-phase
solution becomes an in-phase motion, also called
square lattice. The anti-phase solution
requires the largest distance between fluxons and
becomes triangular lattice.
BSCCO The flux flow voltage or the flux flow
resistance as a function of the magnetic field is
usually measured. Typically it shows oscillations
with a periodicity corresponding to one flux
quantum per layer (high fields) or one half flux
quantum per layer (low fields) as a function of
magnetic field.
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Oscillation of flux flow resistance in IJJs
S. Ooi, et al., PRL 89, 247002(2002)
Where s is the distance between superconducting
layers, 1.5 nm in BSCCO-2212 w is the junction
width perpendicular to magnetic fields.
Please be referred to 4ER02, 4ER05
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Important parameters L junction length 2.5 -20 S
coupling param. 0.01 -0.3 N number of junctions
1-10 ? bias current 0,05 0,15 a damping
L20, N 10
L4, N 10
L20, N 10
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Square lattice N10, L4, H10
Triangular lattice N10, L4, H9
? Fluxon position
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Now the simplest case N2
Vertical lines G np/L, n0,1,2.. (Fiske steps)
N2, L2.5, ?0.5, S -0.4, a 0.1
Resistance
Voltage
Magnetic field ?
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Definitions
N2, L2.5, ?0.5, S-0.4, a 0.1
i.e. number of fluxons
Boundary conditions
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N2, L2.5, ?0.5, S-0.4,a 0.1
The different fluxon configurations (a-h)
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N2, L2.5, ?0.5, S-0.4,a 0.1
(00)?
(11)?
(11)?
(22)?
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N2, L2.5, ?0.5, S-0.4,a 0.1
(22)?
(32)?
(33)?
Fiske modes
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Changing parameters L5.0
N2, L 5.0, ?0.5, S -0.4, a 0.1
a(00)? b(22)? c (33)? d(44)? e(44)?
f(55)? g Fiske modes
Note more square and triangular modes as length
becomes bigger
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Changing parameters ?0.25
N2, L2.5, ?0.25, S -0.4, a 0.1
Note still in-phase modes as bias becomes
smaller
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Changing parameters L10.0
N2, L10.0, ?0.25, S -0.4, a 0.1
Apparently no in-phase modes as length becomes
big and bias still small
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Changing parameters ?0.5, L10.0
N2, L10.0, ?0.5, S -0.4, a 0.1
In-phase modes reappear as bias becomes big and
length still big
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Changing parameters N5, ?0.5
N5, L 2.5, ?0.5, S -0.4, a 0.1
In-phase modes remain as number of junctions
increase
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3) Cavity induced in-phase locking
Each stack voltage pulse injects a charge into
the cavity. Cavity current builds up. At
equilibrium cavity inject current back into
stack, and locks stack to cavity resonance.
Non-linear stack-cavity interaction near cavity
resonance
Cavity resonance corresponding to junction
in-phase velocity
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Equations for a Josephson stack in a cavity
Boundary condition coupling between stack and
cavity
O v(1/(NCoL))/ ?j Cavity resonnce frequency
normalised to the plasma frequency
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N3 Fluxons locked to the resonance at low bias

• fluxons exist in all three junctions
• In-phase dynamics
• C voltage
• High cavity current (50 times larger)
• - 45 phase relation

• b) ? 0.35

Illustration by G. Filatrella,2004
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Time pictures. Anti-phase O 0.60 (a) and O
1.2 (b).
Time pictures for a 3 stack with a cavity.
Parameters of the simulations are n3, L5,
a0.05, S0.1, Q100. c 0.0001, ?0.15, O 0.60
(a) and O 1.2 (b). The solid line denotes the
current in the cavity (left axis) while the thick
and thin dotted lines denote the middle and
top/bottom junctions, respectively (right axis).
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Not so strong coupling to the cavity.
Cavity current increase at resonance, but not
enough to fully force fluxons inphase
N3, L4, a0.1, S-0.01,Q100, O 0.7, and c
0.004
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Strong coupling to the cavity. Cavity current is
large and increases at resonance. Fluxons are
fully forced to in-phase
N7, L4, a0.1, S-0.01,Q100, O 0.6, and c
0.1
F4 f6 0, antiphase
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Conclusions
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F4 f6 0, i.e. antiphase
As cavity frequency is approached fluxons are
forced in-phase
N7, L4, a0.1, S-0.01,Q100, O 0.6, and c
0.1
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N7, L4, a0.1, S-0.01,Q100, O 0.6, and c
0.1
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N7, L4, a0.1, S-0.01,Q100, O 0.6, and c
0.1
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N3, L4, a0.1, S-0.01,Q100, O 0.7, and c
0.004
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N3, L4, a0.1, S-0.01,Q100, O 0.7, and c
0.004
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N3, L4, a0.1, S-0.01,Q100, O 0.7, and c
0.004
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1 junction stack, Flux flow branch
N1 .. The special case of no vortex lattice
ordering. AV Ustinov and NF Pedersen (Phys. Rev.
B 2005, in print)
Note both Fo and Fo/2 periods
Fiske step spacing
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Oscillation of flux flow resistance in IJJs
S. Ooi, et al., PRL 89, 247002(2002)
Where s is the distance between superconducting
layers, 1.5 nm in BSCCO-2212 w is the junction
width perpendicular to magnetic fields.
Please be referred to 4ER02, 4ER05
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The different plasma and fluxon
modesN1,2,3,4,5,6,7
2
4
3
5
6
7
N
Antiphase modes?
Mode2
Plasma prediction
Actual fluxon
Mode3
Mode4
Mode5
Yellow some problems
fluxon
antifluxon
Mode6
Inphase modes
M McCumber switch
Mode7
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Dispersion relations, N 50, m 1 ...50