1. Dynamics of intermediate-state flux structures in current-driven mesoscopic type-I superconductors
G. R. Berdiyorov and F. M. Peeters
University of Antwerp, Belgium
A. Hernandez-Nieves and D. Dominguez
Centro Atomico Bariloche, Argentina
CMT, Antwerpen, July 20, 2011

2. Intermediate state of type-I superconductors
The intermediate state in a cylinder.

3. IS of type-I superconductors: Landau model
Bright – normal regions, Dark – superconducting regions
A superconducting Pb disk (diameter 5mm and thickness 1mm) show hysteretic behavior – flux tubes (left) upon flux penetration and lamellae (right) upon flux exit.
R. Prozorov, Phys. Rev. Lett. 98, (2007).
Landau picture: periodically located superconducting and normal domains
L. Landau, Sov. Phys. JETP 7, 371 (1937).
L. Landau, Nature (London) 141, 688 (1938).
How does the flux penetrate and exit the sample?

4. Current induced intermediate state
Current-induced intermediate state in a Pb film of thickness 9 mm and width 95 mm at 4.2 K.
How does the annihilation process takes place between the normal domains that contain flux of opposite polarity?
R.P. Huebener, Magnetic flux structurs in Superconductors (Springer-Verlag, NY, 1979)

5. Model system
L=256(0)
w=128(0)
d=40(0)
=0.4
T=0.8Tc

6. Theoretical approach Time-dependent Ginzburg-Landau equations:
Magnetic inductance:
Boundary conditions:

7. Annihilation of flux tubes
The flux enters the sample in the form of tubes, containing several flux quanta.
Annihilation of tubes containing opposite flux occurs in a discretized form, releasing single-quantum vortices, which results in a measurable traces in the voltage characteristics of the sample.
0-descretized annihilation of normal domains!

8. Annihilation of flux tubes

9. Annihilation of normal stripes
Regardless of the shape of the normal domains annihilation occurs in descretized form!
At larger current values normal stripes are formed, which again annihilates in a discrete form.

10. Effect of edge defects
Surface defects are important in determining the size, as well as the shape of the flux domains.
For the given parameters of the sample, no IS flux structures are observed without the edge defects.
j=0.143j j=0.143j j=0.315j0

11. Complex dynamics of IS flux strutures
The annihilation point oscillates in time.
The size of the newly nucleated flux domains does not depend on this complex annihilation process.

12. Complex dynamics of IS flux strutures
Hall probe measurements (with a probe fixed at ¼ of the length of the channel) show that:
- A train of pulses with two distinct periods are found at small currents (a);
A complex train of large and smaller pulses are observed with increasing the applied current (b), which indicates that both the size and separation of flux tubes are affected;
Purely period-one behavior is found at larger currents (c), followed by fairly flat-topped pulses interspersed with occasional short pulses (d).
What determines the size, shape and the separation of the flux tubes?
S. B. Field and G. Stan, Phys. Rev. Lett. 98, (2007).

13. Complex dynamics of IS flux strutures
Depending on the applied drive the transition from periodic (with single or multiple periods) to non-periodic dynamics of the flux structures are obtained, which is related to the oscillatory behavior of the annihilation point.

14. Complex dynamics of IS flux strutures

15. Complex dynamics of IS flux strutures

16. Complex dynamics of IS flux strutures

17. Conclusions
Depending on the applied current the nucleation of flux domains occurs either in discretized (tubular) or in continuous (laminar) form.
Regardless of the shape of the IS domains the annihilation always occurs in a discretized form through Φ0 -quantized flux quanta.

18. Thickness dependence transition from type-I to type-II bahavior
Transition from type-I regime to type-II behavior can be obtained by decreasing the sample thickness.

19. Effect of sample thickness – transition to type-II regime
Critical thickness for the transition from type-I to type-II regime decreases with decreasing k.

20. 0-descretized annihilation of normal domains!
Conclusions
0-descretized annihilation of normal domains!

21. IS of type-I superconductors: History dependence
Bright – normal regions, Dark – superconducting regions
A superconducting Pb disk (diameter 5mm and thickness 1mm) show hysteretic behavior – flux tubes (left) upon flux penetration and lamellae (right) upon flux exit.
R. Prozorov, Phys. Rev. Lett. 98, (2007).

22. History dependence and effect of sample geometry
Bright – normal regions
Dark – superconducting regions
A superconducting Pb disk (diameter 5mm and thickness 1mm) show hysteretic behavior – flux tubes (left) upon flux penetration and lamellae (right) upon flux exit.
A Pb (hemi)sphere (diameter 4mm ) show no hysteresis with flux tubes dominating the intermediate state.
R. Prozorov, Phys. Rev. Lett. 98, (2007).

23. Effect of sample geometry: ground state configurations
- Variety of possible flux structures in the IS of type-I superconductors are found in the mesoscopic regime.
- Regardless of the sample geometry, the IS consist of flux tubes at small fields, laminar-like structures at larger fields and combination of them at intermediate values of the applied field.

24. Effect of sample geometry: ground state configurations
- In samples with surface parallel to the applied field, normal domains are mostly located parallel to the surface of the sample. In a sphere and a cone normal domains are oriented along the radial direction.

25. Effect of tilted magnetic field
Transition to parallel stripes of normal and superconducting regions can be obtained by applying a small in-plane component to the applied field
Y.V. Sharvin, Sov. Phys. JETP 6, 1031 (1958).

26. Effect of tilted magnetic field
Transition from laminar state to parallel arrangement of normal domains can be obtained by tilting the magnetic field.

27. Effect of tilted magnetic field
Tilting the field does not lead to the transition from the tubular state to laminar state.

28. Complex dynamics of IS flux strutures
The annihilation point oscillates in time.
The size of the newly nucleated flux domains does not depend on this complex annihilation process.

29. IS of type-I superconductors: Landau model
Black – normal regions
White – superconducting regions
A. Hernandez and D. Dominguez, Phys. Rev. B 72, (2005).
Landau picture: periodically located superconducting and normal domains
L. Landau, Sov. Phys. JETP 7, 371 (1937).
L. Landau, Nature (London) 141, 688 (1938).