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Thermal suppression of surface
barrier in ultrasmall
superconducting structures
W. V. Pogosov
Institute for Theoretical and Applied Electrodynamics,
Russian Academy of Sciences, Moscow, Russia
W. V. Pogosov, Phys. Rev. B 81, 184517 (2010)
Outline
I. Motivation
II. Surface barrier
A. Prehistory
B. Surface barrier in LLL approximation
C. Viscosity
D. First passage time
III. Geometry-induced fluctuations
IV. Conclusions
Motivation
Experiment of Roditchev’s group
No hysteresis for vortex entry/exit !
Cren et al., PRL (2009)
• Experiment of Hasegawa’s group
- The width of hysteresis was significantly smaller than expected from theory
- Temperature was twice lower than in the experiment of Roditchev’s group
Nishio et al., PRL (2008).
Thermal activation of vortices
over the surface barrier?
Prehistory
- Bean and Livingston, PRL (1964).
- Instability of Meissner state with respect to perturbations of the order parameter
and magnetic field
essentially the same result.
- Full GL treatment in mesoscopic superconductors: F. M. Peeters and coworkers.
• Thermoactivation over the surface barrier
Petukhov and Chechetkin, JETP (1974).
vacuum
superconductor
Approach:
1. Surface barrier: London theory
2. Viscosity coefficient: in bulk
3. Penetration time: Fokker-Planck equation (kinetics)
Thermoactivation is not possible!
H
High-Tc cuprates!
Burlachkov et al., PRB (1994): 2D pancake vortices and 3D vortices
– thermoactivation is possible
State of the art: In low-Tc not possible, in high-Tc possible
Surface barrier
In connection with Roditchev’s group experiment
Model
disc
d = 5.5 nm – disc thickness;
T = 4.3 K - temperature;
H = 0.235 T – vortex entry/exit
= 48 nm
R = 149 nm
General idea !
-
London model is definitely not applicable,
moreover vortex coordinate is not a “good”
independent variable
-
We will use, at all steps of the derivation,
Landau level populations as “good” variables
instead of the “bad” vortex coordinate
(i)
(ii)
(iii)
Surface barrier profile
Viscosity
Fokker-Planck equation
• GL energy (dimensionless units)
Landau-level representation of the order parameter (F.M. Peeters with coworkers)
Regime of “ultimate vortex confinement”:
• Eigen-value equation for the kinetic-energy operator
Solution (Kummer function):
• Energy
Optimal path:
Viscosity
• Vortex motion
Magnetic field variation
(in time)
Electric field
Energy
dissipation in vortex core Viscosity
(Bardeen-Stephen model)
Electrodynamics of the disc: Perturbation theory
Supercurrent:
An additional vector potential created by the supercurrent:
• Electric field (Faraday law)
Dissipation rate:
Vortex penetration
Supercurrent (the leading-order contribution in c1):
• Additional vector potential (after integrating
kernel on angle)
where E and K are complete elliptic integrals of the first and second kind
Electric field:
Dissipation rate:
Thus, we have introduced and estimated a
viscosity associated with the projections of the
order parameter on Landau levels
Demagnetization effects were included into consideration
First passage time
• We now know profile of the surface barrier and
viscosity in terms of Landau-level populations
Probability to find a system
at t in c1’, provided it was in
c1 at t = 0.
The system is homogeneous
In time, so we can switch to
• Backward Fokker-Planck equation
Probability that the system is still within relevant interval
Integrate FP equation over c1
• Average exit time
- probability to exit within time dt
• Integrate over t the equation for G:
reflecting
Boundary conditions:
absorbing
• First passage time
a
s
b
• Final expression (Arrhenius-like)
Reasons for short exit/entry time:
1) Very small thickness (low barrier + strong demagnetization)
2) Very small lateral dimensions (suppression of the
order parameter + weak diamagnetic response)
3) Temperature is not very low
Conclusions
- We proposed an explanation of the recent experimental
results by Roditchev’s and Hasegawa’s groups in terms of
a thermal activation of vortices over the surface barrier.
- We suggested a new theoretical approach to the problem
of vortex thermal activation, suitable for nanosized
superconductors. The idea is to incorporate LLL
approximation into the kinetic theory (Fokker-Planck
equation) – at all steps of the derivation.
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