1. in collaboration with L. Bulaevskii, J. P. Brison,
Non-uniform (FFLO) states and quantum oscillations in superconductors and superfluid ultracold Fermi gases
A. Buzdin
University of Bordeaux I and Institut Universitaire de France
in collaboration with L. Bulaevskii, J. P. Brison,
M. Houzet, Y. Matsuda, T. Shibaushi, H. Shimahara, D. Denisov, A. Melnikov, A. Samokhvalov
ECRYS-2011, August 15-27, 2011 Cargèse , France

2. Outline
Singlet superconductivity destruction by the magnetic field: - The main mechanisms - Origin of FFLO state Experimental evidences of FFLO state Exactly solvable models of FFLO state Vortices in FFLO state. Role of the crystal structure. 5. Supefluid ultracold Fermi gases with imbalanced state populations: one more candidate for FFLO state?

3. 1. Singlet superconductivity destruction by the magnetic field.

4. (breakdown of Cooper pairs induced by magnetic moment)
Orbital effect (Lorentz force)
Electromagnetic
mechanism
(breakdown of Cooper pairs
by magnetic field
induced by magnetic moment) p B FL FL -p
Paramagnetic effect (singlet pair)
μBH~Δ~Tc
Sz=+1/2
Sz=-1/2
Exchange interaction

5. B Vortex FL -p p Hc2 Normal B Meissner T Vortex lattice in NbSe2 (STM)
Orbital effect B -p FL p
Vortex
x(coherence length)
Flux quantum
l(penetration length）
F0=hc/2e=2.07x10-7Oe・cm2z D
Normal
Meissner T B
Hc2 Tc
Abrikosov Lattice
Vortex lattice in NbSe2
(STM)

6. Superconductivity is destroyed by magnetic field
Orbital effect (Vortices) B -p FL p
Zeeman effect of spin (Pauli paramagnetism)
Maki parameter
Usually the influence of Pauli paramagnetic effect is negligibly small

7. Superconducting order parameter behavior under paramagnetic effect
Standard Ginzburg-Landau functional:
The minimum energy corresponds
to Ψ=const
The coefficients of GL functional are functions of the Zeeman field h= μBH !
Modified Ginzburg-Landau functional ! :
The non-uniform state Ψ~exp(iqr) will correspond to minimum energy and higher transition temperature

8. Ψ~exp(iqr) - Fulde-Ferrell-Larkin-Ovchinnikov state (1964).
Only in pure superconductors and in the rather narrow region.

9. FFLO inventors
Fulde and Ferrell
Larkin and Ovchinnikov

10. The total momentum of the Cooper pair is -(kF -dkF)+ (kF -dkF)=2 dkF

11. Conventional pairing k ( k ,-k ) -k FFLO pairing k q -k+q qky kx E k -k
( k ,-k )
FFLO pairing ky kx E q
q~gmBH/vF
-k+q q k
( k ,-k+q )
pairing between Zeeman split parts of the Fermi surface
Cooper pairs have a single non-vanishing center of mass momentum

12. Pairing of electrons with opposite spins and momenta unfavourable :
But : if
1d SC
2d SC
3d SC 1
At T = 0, Zeeman energy compensation is exact in 1d, partial in 2d and 3d.
0.8
0.6
the upper critical field is increased
Sensivity to the disorder and to the orbital effect:
(clean limit)
0.4
0.2
0.56
0.2
0.4
0.6
0.8 1
0.56
0.2
0.4
0.6
0.8 1

13. FF state uniform ( k ,-k ) ( k ,-k+q ) LO state spatially nonuniform
available for pairing
depaired
( k ,-k+q )
LO state
spatially nonuniform +
The SC order parameter performs one-dimensional spatial modulations along H, forming planar nodes

14. Modified Ginzburg-Landau functional :
May be 1st order transition at 1
0.8
0.6
0.4
0.2
0.56
0.2
0.4
0.6
0.8 1

15. 2. Experimental evidences of FFLO state.
Unusual form of Hc2(T) dependence
Change of the form of the NMR spectrum
Anomalies in altrasound absorbtion
Unusual behaviour of magnetization
Change of anisotropy ….

16. BEDT-TTF (donor molecule)
Organic superconductor
k-(BEDT-TTF)2Cu(NCS)2 (Tc=10.4K)
Layered structure
Suppression of orbital effects in H parallel to the planes
BEDT-TTF layer
Cu[N(CN)2]Br layer
 15 Å C S
H or D
BEDT-TTF (donor molecule)

17. about FFLO in this compound!
Talk of Stuart Brown
about FFLO in this compound!

18. Anomalous in-plane anisotropy of the onset of SC in (TMTSF)2ClO4
S.Yonezawa, S.Kusaba, Y.Maeno, P.Auban-Senzier, C.Pasquier, K.Bechgaard, and D. Jerome, Phys. Rev. Lett. 100, (2008)

19. Field induced superconductivity (FISC) in an organic compound
-(BETS)2FeCl4
Metal
FISC
Insulator AF
c-axis (in-plane) resistivity
S. Uji et al., Nature (2001)
L. Balicas et al., PRL (2001)

20. Jaccarino-Peter effect
Zeeman energy
Exchange energy between conduction electrons in the BETS layers and magnetic ions Fe3+ (S=5/2)
For some reason J > 0 : the paramagnetic effect is suppressed at
Other example:
Eu-Sn Molybdenum chalcogenide
H. Meul et al, 1984
Critical field Hc2 [Tesla]
(Eu0.75Sn0.25Mo6S7.2Se0.8) S S
Temperature [K]

21. Strong evidence of inhomogeneous FFLO phase in CeCoIn5

22. H-T phase diagram of CeCoIn5
2nd
order
1st
H// ab
H// c
Pauli paramagnetically limited superconducting state

23. New high field phase of the flux line lattice in CeCoIn5
This 2nd order phase transition is characterized by a structural transition of the flux line lattice
Ultrasound and NMR results are consistent with the FFLO state which predicts a segmentation of the flux line lattice

24. In the usual case (normal metal):
Proximity effect in a ferromagnet ?
In the usual case (normal metal):
In ferromagnet ( in presence of exchange field) the equation for superconducting order parameter is different
Its solution corresponds to the order parameter which decays with oscillations! Ψ~exp[-(q1 ± iq2 )x] Ψ
Wave-vectors are complex!
They are complex conjugate and
we can have a real Ψ.
Order parameter changes its sign!
Many new effects in
S/F heterostructures! x

25. Df-diffusion constant
Remarkable effects come from the possible shift of sign of the wave function in the ferromagnet, allowing the possibility of a « π-coupling » between the two superconductors (π-phase difference instead of the usual zero-phase difference) F S S S F S
« 0 phase »
«  phase  »
S/F bilayer F S
h-exchange field,
Df-diffusion constant

26. S-F-S Josephson junction in the clean/dirty limit
Damping oscillating dependence of the critical current Ic as the function of the parameter =hdF /vF has been predicted.
(Buzdin, Bulaevskii and Panjukov, JETP Lett. 81)
h- exchange field in the ferromagnet,
dF - its thickness S F S Ic
E(φ)=- Ic (Φ0/2πc) cosφ
J(φ)=Icsinφ 

27. Collaboration with V. Ryazanov group from ISSP, Chernogolovka
Critical current density vs. F-layer thickness (V.A.Oboznov et al., PRL, 2006)
Collaboration with V. Ryazanov group from ISSP, Chernogolovka
Ic=Ic0exp(-dF/F1) |cos (dF /F2) + sin (dF /F2)|
dF>> F1
Spin-flip scattering decreases the
decaying length and increases the
oscillation period.
“0”-state
-state
F2 >F1
Nb-Cu0.47Ni0.53-Nb junctions
“0”-state
-state
I=Icsin
I=Icsin(+ )= - Icsin()

28. Cluster Designs (Ryazanov et al.)
30m
2 x 2
unfrustrated
fully-frustrated
checkerboard-frustrated
6 x 6
fully-frustrated
checkerboard-frustrated

29. Scanning SQUID Microscope images (Ryazanov et al
Scanning SQUID Microscope images (Ryazanov et al., Nature Physics, 2008)) Ic T T
T = 1.7K
T = 2.75K
T = 4.2K

30. FFLO State in Neutron Star Bose-Einstein-Condensate
Color superconductivity
Vortices
R.Casalbuoni and G.Nardulli
Rev. Mod. Phys. (2004)
Glitches
Supefluid ultracold Fermi gases
with imbalanced state populations:
one more candidate for FFLO state?
Massachusetts Institute of Technology:
M.W. Zwierlein, A. Schirotzek, C. H. Schunck, W.Ketterle (2006)
Rice University, Houston:
Guthrie B. Partridge, Wenhui Li, Ramsey I. Kamar, Yean-an Liao, Randall G. Hulet (2006)

31. 3. Exactly solvable models of FFLO state.

32. FFLO phase in the case of pure paramagnetic interaction and BCS limit
Exact solution for the 1D and quasi-1D superconductors ! (Buzdin , Tugushev 1983)
The FFLO phase is the soliton lattice,
first proposed by Brazovskii, Gordyunin and Kirova in 1980 for polyacetylene.
1d SC 1
0.8
0.6
0.4
0.2
0.56
Magnetic moment x

34. Spin-Peierls transitions - e.g. CuGeO3
Chains direction
Importance de la mesure de susceptibilité et de son caractère isotrope.
Période du soliton : L =1/delta k_SP
Neutrons dans la phase Iseulement depuis qu’ont peut atteindre la zone T (avant, il fallait se contenter des composés dopés où Hc est plus petit, mais la physique plus complexe)
TSP=14.2 K

35. In 2D superconductors ( k ,-k ) Y.Matsuda and H.Shimahara
　　　　　J.Phys. Soc. Jpn (2007)

36. In 3D superconductors
The transition to the FFLO state is 1st order. The sequence of phases is similar to 2D case. Houzet et al. 1999; Mora et al. 2002

37. 4. Vortices in FFLO state. Role of the crystal structure.

38. FFLO phase in the case of paramagnetic and orbital effect (3D BCS limit) – upper critical field
Note : The system with elliptic Fermi surface can be tranformed by scaling transformation to ihe isotropic one.
Sure the direction of the magnetic field will be changed.
Lowest m=0 Landau level solution, Gruenberg and Gunter, 1966
FFLO exists for Maki parameter α>1.8.
For Maki parameter α>9 the highest
Landau level solutions are realized
– Buzdin and Brison, 1996.

39. FFLO phase in 2D superconductors in the tilted magnetic field - upper critical field
Highest Landau level solutions are realized –
Bulaevskii, 1974; Buzdin and Brison, 1996; Houzet and Buzdin, 2000. q B

40. Exotic vortex lattice structures in tilted magnetic field
Generalized Ginzburg-Landau functional
Near the tricritical point, the characteristic length is
Microscopic derivation of the Ginzburg-Landau functional :
Instability toward FFLO state
Instability toward 1st order transition
Next orders are important :
Validity:
large scale for spatial variation of  : vicinity of T *
small orbital effect, introduced with
we neglect diamagnetic screening currents (high- limit)

41. 2nd order phase transition at
 higher Landau levels
Near the transition: minimization of the free energy with solutions in the form
gauge
 Parametrizes all vortex lattice structures at a given Landau level N
is the unit cell
All of them are decribed in the subset :

42. Rectang. Triangular square Rectangular Triangular lattice n=2 n=1 n=0
Analysis of phase diagram :
cascade of 2nd and 1st order transitions between S and N phases
1st order transitions within the S phase
exotic vortex lattice structures 3
n=2
Landau levels
Magnetic field
Rectang. 2
n=1
n=0 1
Triangular
square
Tricritical point
Rectangular
Triangular lattice -2 -1
Temperature __
1st order transition
2nd order transition
2nd order transition in the paramagnetic limit
At Landau levels n > 0, we find structures with several points of vanishment of the order parameter in the unit cell and with different winding numbers w =  1,  2 …

43. Order parameter distribution for the asymmetric and square lattices with
Landau level n=1.
The dark zones correspond to the maximum of the order parameter and the white
zones to its minimum.

44. Intrinsic vortex pinning in LOFF phase for parallel field orientation
Δn= Δ0cos(qr+αn)exp(iφn(r))
t – transfer integral
Josephson coupling between layers is modulated
Fn,n+1=[-I0cos(αn_- αn+1)+I2cos(qr) cos(αn_+ αn+1)] cos(φn- φn+1)
φn- φn+1=2πxHs/Φ0 + πn
s-interlayer distance, x-coordinate along q

45. Modified Ginzburg-Landau functional: x y
CeCoIn5
CeIn3
CoIn2
Quasi-2D heavy fermion (Tc=2.3K)
Strong antiferromagnetic fluctuation
d-wave symmetry
Tetragonal symmetry z
Modified Ginzburg-Landau functional:
isotropic part x y

46. No orbital effect
Modulation diagram in the case of the absence of the orbital effect (pure paramagnetic limit).
Areas with different patterns correspond to different orientation of the wave-vector modulation. The phase diagram does not depend on the εx value.

47. Magnetic field along z axis
H||z
Modulation diagram in the case when the magnetic field applied along z axis. There are 3 areas on the diagram corresponding to 3 types of the solution for modulation vector qz and Landau level n. Modulation direction is always parallel to the applied field and εx here is treated as a constant.

48. Magnetic field along z axisxy z
intermediate xy z
intermediate

49. Magnetic field along x axis
H||x
Modulation diagram (ἕ, εz) in the case when the magnetic field is applied along x axis. There are three areas on the diagram corresponding to different types of the solution for modulation vector qx and Landau level n. Modulation direction is always parallel to the applied field. The choice of the intersection point is determined by the coefficient εx.

50. Magnetic field along x axisxy z
intermediate xy z
intermediate

51. Neutron form factor seems to be consisitent with FFLO state
Small angle neutron scattering from the vortex lattice for H //c
Neutron form factor
FFLO?
Neutron form factor seems to be consisitent with FFLO state
A.D.Bianch et al.Science (2007)

52. The crystal structure effects influence on the FFLO state is very important.
The FFLO states with higher Landau level solutions could naturally exist in real 3D compounds (without any restrictions to the value of Maki parameter).
Wave vector of FFLO modulation along the magnetic field could be zero.
In the presence of the orbital effect the system tries in some way to reproduce optimal directions of the FFLO modulation by varying the Landau level index n and wave-vector of the modulation along the field.

53. 5. Supefluid ultracold Fermi gases with imbalanced state populations: one more candidate for FFLO state?

54. Experimental result: phase separation
Massachusetts Institute of Technology:
M.W. Zwierlein, A. Schirotzek, C. H. Schunck, W.Ketterle (2006)
Rice University, Houston:
Guthrie B. Partridge, Wenhui Li, Ramsey I. Kamar, Yean-an Liao, Randall G. Hulet (2006)
Experimental system: Fermionic 6Li atoms cooled in magnetic and optical traps (mixture of the two lowest hyperfine states with different populations)
Experimental result: phase separation
rf transitions
76 MHz
hyperfine states
Supefluid core
Normal gas

55. Rotating supefluid ultracold Fermi gases in a trap
Coils generating magnetic field
Fermion condensate
Vortex as a test for superfluidity
Laser beams
Images of vortex lattices
MIT: Ketterle et al (2005)

56. Questions:
What is the effect of confinement (finite system size) on FFLO states?
Effect of rotation on FFLO states in a trap (effect of magnetic field on FFLO state in a small superconducting sample).
Possible quantum oscillation effects.

57. Multiquantum vortices
Examples of quantum oscillation effects.
Little-Parks effect. Switching between the vortex states.
Multiply-connected systems
Superconducting thin-wall cylinder
Superconductor with a
columnar defect or hole
Tc (H) oscillations
Multiquantum vortices -1 1
Ф/Фo
A.Bezryadin, A.I. Buzdin, B. Pannetier (1994)
L=-1
L=0
L=1
-ΔTc/Tc0
Ф/Фo

58. H Examples of quantum oscillation effects. Little-Parks effect. R~ξ
O.Buisson et al (1990)
R.Benoist, W.Zwerger (1997)
V.A.Schweigert, F.M.Peeters (1998)
H.T.Jadallah, J.Rubinstein,
Sternberg (1999)
Simply-connected systems
Mesoscopic samples
dimensions ~ several coherence lengths H
R~ξ
Origin of Tc oscillations:
Transitions between the states
with different vorticity L
Tc (H) oscillations
L - Vorticity
(orbital momentum)
Multiquantum vortices

59. Examples of quantum oscillation effects.
FFLO states and Tc(H) oscillations in infinite 2D superconductors
A.I. Buzdin, M.L. Kulic (1984) Hz
H|| ~ Hp

60. Range of validity: vicinity of tricritical point
Model: Modified Ginzburg-Landau functional (2D)
Trapping potential
FFLO instability T
Range of validity: vicinity of tricritical point N
Confinement mechanisms:
1. Zero trapping potential. Boundary condition at the system edge
nonzero trapping potential S H
FFLO

61. FFLO states in a 2D mesoscopic superconducting diskHz
H|| ~ Hp
R~ξ
Interplay between the system size, magnetic length, and FFLO length scale

62. Perpendicular magnetic field component Hz = 0
The critical temperature:
Eigenvalue problem:
Wave number of FFLO instability

63. H - T Phase diagram: Hz = 0 T
H ↑ → L↓
L=1
L=2
FFLO state
L=0 H

64. Tilted magnetic field: Hz ≠ 0
Field induced superconductivity
Eigenvalue problem:

65. Tilted magnetic field: Hz ≠ 0
Transitions with large jumps in vorticity

66. Vortex solutions beyond the range of FFLO instability
Vortex solutions beyond the range of FFLO instability. Critical field of the vortex entry. Hz
H|| ~ Hp
We focus on the limit T N S
(H*,T*)
Condition of the first vortex entry:
Beyond the range of FFLO instability
FFLO

67. Condition of the first vortex entry:
Limiting cases:
Change in the scaling law

68. FFLO states in a trapping potential
Interplay between the rotation effect, confinement, and FFLO instability

69. FFLO states in a 2D system in a parabolic trapping potential (no rotation)
FFLO length scale
Fourier transform:
Temperature shift
(Trapping frequency)2
Dimensionless coordinate q

70. Condensate wave function
FFLO states in a 2D system in a parabolic trapping potential (no rotation)
Phase diagram
Condensate wave function
Suppression of wave function oscillations by the increase in the trapping frequency
=Number of observable oscillations

71. FFLO states in a rotating 2D gas in a parabolic trapping potential.
FFLO length scale
Temperature shift
Expansion in eigenfunctions of the problem without trap:
(Trapping frequency)2
rotation frequency
Dimensionless coordinate

72. FFLO states in a rotating 2D gas in a parabolic trapping potential.
Suppression of quantum oscillations by the increase in the trapping frequency.
First-order perturbation theory:
rotation induced superfluid phase

73. Conclusions
There are strong experimental evidences of the existence of the the FFLO state in organic layered superconductors and in heavy fermion superconductor CeCoIn5
FFLO –type modulation of the superconducting order parameter plays an important role in uperconductor-ferromagnet heterostructures. The p-junction realization in S/F/S structures is quite a general phenomenon.
The interplay between FFLO modulation and orbital effect results in new type of the vortex structures, non-monotonic critical field behavior in layered superconductor in tilted field.
Special behavior of fluctuations near the FFLO transition – vanishing stiffness.
FFLO phase in ultracold Fermi gases with imbalanced state populations?