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1 Suppression of surface barrier in ultrathin superconducting nanostructures by thermal fluctuations W. V. Pogosov Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia W. V. Pogosov, Phys. Rev. B 81, (2010).

2 I. Motivation II. Surface barrier A. Prehistory B. Surface barrier in LLL approximation C. Viscosity D. First passage time III. Summary Outline

3 Motivation No hysteresis for vortex entry/exit Cren et al., PRL (2009)

4 Nishio et al., PRL (2008) - The width of hysteresis was significantly smaller than expected from theory - Temperature was twice lower than in the experiment of Roditchevs group

5 Thermal activation of vortices over the surface barrier?

6 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.

7 Thermoactivation over the surface barrierThermoactivation over the surface barrier vacuum superconductor Petukhov and Chechetkin, ЖЭТФ (1974). Approach: 1. Surface barrier: London theory 2. Viscosity coefficient: in bulk 3. Penetration time: Fokker-Planck equation (kinetics) Thermoactivation is not possible! High-T c cuprates! V. N. Kopylov, A. E. Koshelev, I. F. Schegolev, and T. G.Togonidze, Physica C (1990): pancakes L. Burlachkov, PRB (1993): 3D vortices – thermoactivation is possible H

8 In connection with Roditchevs group experiment Surface barrier disc Model d = 5.5 nm – disc thickness; T = 4.3 K - temperature; H = T – vortex entry/exit = 48 nm R = 149 nm

9 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)Surface barrier profile (ii)Viscosity (iii)Fokker-Planck equation

10 GL energy (dimensionless units) Order parameter expansion Regime of ultimate vortex confinement:

11 Eigen-value equation for the kinetic-energy operator Solution (Kummer function):

12 Energy Optimal path:

15 Vortex motion Magnetic field variation (in time) Electric field Energy dissipation in vortex core Viscosity (Bardeen-Stephen model) Viscosity Supercurrent: An additional vector potential created by the supercurrent: Electrodynamics of the disc: Perturbation theory

16 Electric field (Faraday law) Dissipation rate: Vortex penetration Supercurrent (the leading-order contribution in c 1 ):

17 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:

18 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

19 We now know profile of the surface barrier and viscosity in terms of Landau-level populations First passage time Probability to find a system at t in c 1, provided it was in c 1 at t = 0. The system is homogeneous In time, so we can switch to

20 Backward Fokker-Planck equationBackward Fokker-Planck equation Probability that the system is still within relevant interval Integrate FP equation over c 1

21 Average exit time Integrate over t the equation for G: Boundary conditions: - probability to exit within time dt reflecting absorbing

22 First passage timeFirst passage time abs

23 Final expression (Arrhenius-like)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

24 Summary - An explanation of the recent experimental results on the suppression of magnetic hysteresis in ultrathin superconducting nanostructures is proposed in terms of a thermal activation of vortices over the surface barrier. - Theoretical approach to the problem of vortex thermal activation is suggested. The idea is to incorporate LLL approximation into the kinetic theory (Fokker-Planck equation) – at all steps of the derivation.

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