JP Ader, A. I. Buzdin A. S. Melnikov

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Vortex states and proximity effect in hybrid
Ferromagnet-Superconductor systems
Vortex states and proximity effect in hybrid
Ferromagnet-Superconductor systems

• J-P Ader, A. I. Buzdin A. S. Melnikov

Condensed Matter Theory Group, CPMOH,
Université Bordeaux I, France Institute for
Physics of Microstructures, RAS, Nizhny
Condensed Matter Theory Group, CPMOH,
Université Bordeaux I, France Institute for
Physics of Microstructures, RAS, Nizhny Novgorod,
Russia

• INSTITUTE for PHYSICS of MICROSTRUCTURESRussian
  Academy of Sciences

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Interplay between magnetism (ferromagnetism) and
superconductivity via

• Orbital effect (Lorentz force)

B
p
FL
FL
-p

• Paramagnetic effect (singlet pair)

µBH?Tc
Sz1/2
Sz-1/2
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Remarkable effects come from the possible shift
of sign of the wave function in the ferromagnet,
allowing the possibility of a  p-coupling 
between the two superconductors (p-phase
difference instead of the usual zero-phase
difference)
 0 phase
? phase  
S/F bilayer
F
S
h-exchange field, Df-diffusion constant
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The Key Idea Exchange effect vs Orbital effect
?(r,?)?(r) exp(i L?)
Commensurability effects
Thin-walled superconducting (S) shell around a
ferromagnetic (F) cylinder
L 0 phase 2Rf lt ?f
L 1 phase 2Rf ?f
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Systems under consideration
Fort Paté (Vauban) Gironde Estuary (1693)
I/F/S/I
I/S/F/I
I/S/F
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The Model and Equations
Linearised Usadel Equations (h gtgtpTc0 dirty
limit ht ltlt 1 Tct ltlt 1)
The boundary conditions
Used simplifications
Thin SC Shell
Absence of Little-Parks effect
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L-Vortex state in thin SC shell around FM cylinder
Solutions
Critical Temperature Tc of FS Structure
The pair-breaking parameter
supercurrent energy
Exchange Effects
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Critical temperature Tc of vortex states
The penetration of Cooper pairs into the FM core
and the phase shift of the pair wave function
due to the exchange interaction can induce vortex
states in the superconducting shell.
The dependence of the critical temperature Tc on
the F core radius Rf for two values of the
vorticity L 0 and L 1 (d0.5?s ss/sf
2.5 ?s/?f 0.265)
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Hybrid FS system requirements
To observe the switching effect we need the FS
systems with a rather large ratio ?f / ?s 3-10
Superconductors with short coherence length lt 10
nm NbSe2 , Nb3Sn
Weak ferromagnets Cu-Ni alloy ?f 10 nm

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Cascades of the transitions
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I/S/F System
Despite the great number of parameters
characterizing the S/F interface, the critical
temperature depends only on two
quantities C and , W being the thickness of
the S metal
The Usadel equation of the F metal is exactly
solved Neglecting the variations of the order
parameter and the Usadel function in the
superconductor, averaging over its thickness
allow us to obtain an analytic expression of the
boundary conditions and, consequently an explicit
formulation of the critical temperature
Depairing parameter corresponding to the I/S/F
system
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Generalized switching effect

• Cumulative effects 1) proximity effect2)
  exterior ferromagnet I/S/F system 3) small
  interface transparency
• 4 ) too large ratio ?f / ?s
• Complete inversion of the hierarchy

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External field Formulation

• The vector potential becomes circulating around
  the cylinder axis
• Neglecting the proximity effect the critical
  temperature is given by (Little and Parks
  experiment (1962))
• Taking into account proximity effect results in
  supplementary terms involving confluent
  hypergeometric U-functions (I/S/F configuration)
  or M-functions (F/S/I)
• U is converging at infinity, whereas M is regular
  at the origin.

Tricky formulation not only cumbersome, but
moreover the limit H ---gt 0 seems very difficult
to reach divergent (and zero) quantities
appear Nevertheless we recover analytically
former results through nice properties of Kummer
functions
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Switching effect and external field

• For H0 (solid lines) the ground state is a
  vortex with L0
• For H0.05F0/(p?2s) (dashed lines) the ground
  state corresponds to L1
• Similar effect between states L1 and L2

• Switchs between states of different vorticities
  are induced by the external field H

2 Parameters
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Generalization of the Little-Park effect
Evolution of critical temperature with the
magnetic flux No F/S interface solid
line Dashed line Point-dashed line The
critical temperature follows decreasing
parabolas maximized at F/S configuration
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Finite size re-entrant effect
I/S/F/I configuration Influence of the thickness
of the F shell on the critical temperature versus
the radius of the S shell. A reentrant behavior
is observed for L1 A non-monotonic dependence is
found for L0 The finite size effects disappear
for high value of the S radius Dashed lines
Solid lines
Parameters
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Last news generalized switching effect II

• Adding a finite size effect by considering
  I/S/F/I results in a gain of an order of
  magnitude ?s/?f 0.1

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Conclusions

• Superconductor-ferromagnet proximity effect can
  create new type of vortex states
• Multiquanta-states are possible in the case of a
  large interface barrier and very large ratio ?f
  /?s
• Multiquanta-states with a more reasonable ratio
• ?f / ?s 10 can be given by finite-size
  effects (S/F/I systems)
• Re-entrant behaviors are also expected from these
  systems
• Interesting interplay with the Little-Parks
  effect is expected
• What about spherical and ellipsoidal systems ?

Severe technical drawback no variable separation
No external field effect
Part of these results have been already
published Samokhvalov, Melnikov and Buzdin,
Phys. Rev. B 76 184519 (2007).