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 RoditchevвЂ™s group No hysteresis for vortex entry/exit ! Cren et al., PRL (2009) вЂў Experiment of HasegawaвЂ™s group - The width of hysteresis was significantly smaller than expected from theory - Temperature was twice lower than in the experiment of RoditchevвЂ™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. вЂў 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 вЂ“ thermoactivation is possible State of the art: In low-Tc not possible, in high-Tc possible Surface barrier In connection with RoditchevвЂ™s group experiment Model disc d = 5.5 nm вЂ“ disc thickness; T = 4.3 K - temperature; H = 0.235 T вЂ“ vortex entry/exit = 48 nm R = 149 nm General idea ! - London model is definitely not applicable, moreover vortex coordinate is not a вЂњgoodвЂќ independent variable - We will use, at all steps of the derivation, Landau level populations as вЂњgoodвЂќ variables instead of the вЂњbadвЂќ vortex coordinate (i) (ii) (iii) Surface barrier profile Viscosity Fokker-Planck equation вЂў GL energy (dimensionless units) Landau-level representation of the order parameter (F.M. Peeters with coworkers) Regime of вЂњultimate vortex confinementвЂќ: вЂў Eigen-value equation for the kinetic-energy operator Solution (Kummer function): вЂў Energy Optimal path: Viscosity вЂў 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: вЂў Electric field (Faraday law) Dissipation rate: Vortex penetration Supercurrent (the leading-order contribution in c1): вЂў 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 вЂў We now know profile of the surface barrier and viscosity in terms of Landau-level populations Probability to find a system at t in c1вЂ™, provided it was in c1 at t = 0. The system is homogeneous In time, so we can switch to вЂў Backward Fokker-Planck equation Probability that the system is still within relevant interval Integrate FP equation over c1 вЂў Average exit time - probability to exit within time dt вЂў Integrate over t the equation for G: reflecting Boundary conditions: absorbing вЂў First passage time a s b вЂў 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 RoditchevвЂ™s and HasegawaвЂ™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) вЂ“ at all steps of the derivation.