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
1
Outline
1. Singlet superconductivity destruction by the magnetic
field:
- The main mechanisms
- Origin of FFLO state.
2. Experimental evidences of FFLO state.
3. Exactly solvable models of FFLO state.
4. Vortices in FFLO state. Role of the crystal structure.
5. Supefluid ultracold Fermi gases with imbalanced state
populations: one more candidate for FFLO state?
1. Singlet superconductivity destruction by the
magnetic field.
• Orbital effect (Lorentz force)
p
FL
B
-p
FL
Electromagnetic
mechanism
(breakdown of Cooper pairs
by magnetic field
induced by magnetic moment)
• Paramagnetic effect (singlet pair)
μBH~Δ~Tc
Sz=+1/2
Sz=-1/2
 
 
I S  s  Tc
Exchange interaction
Orbital effect
p
B
FL
FL
-p
Hc2
Hc2 
Vortex
Flux quantum
0
2 2
Normal

coherence
B
length)
D
Abrikosov Lattice
Meissner
T
Vortex lattice in NbSe2
(STM)
Tc
lpenetration length）
0hc/2e=2.07x10-7Oe・cm2z
Superconductivity is destroyed by magnetic field
Orbital effect (Vortices)
p
FL
B
H
orb
c2
FL
-p
0

2 2
Zeeman effect of spin (Pauli paramagnetism)
1
1
 N H 2  N(0)D2
2
2
1
 N  (g B ) 2 N (0)
2
H
P
c2

2D
g B

Maki parameter

H corb
2 P2
H c2
~
D
F
 1
Usually the influence of Pauli paramagnetic effect is negligibly small
Superconducting order parameter
behavior under paramagnetic effect
Standard Ginzburg-Landau
functional:
1
b 4
2
2
F a 
  
4m
2
The minimum energy corresponds
to Ψ=const
The coefficients of GL functional are functions of the Zeeman field h= μBH !
Modified Ginzburg-Landau functional ! :
2
F  a          ...
2
2
2
The non-uniform state Ψ~exp(iqr) will correspond to
minimum energy and higher transition temperature
F
F  (a  q  q ) q
2
q0
4
q
Ψ~exp(iqr) - Fulde-Ferrell-Larkin-Ovchinnikov state (1964).
Only in pure superconductors and in the rather narrow region.
2
FFLO inventors
Fulde and Ferrell
Larkin and Ovchinnikov
E
kF -dkF
k
kF +dkF
The total momentum of the Cooper pair is
-(kF -dkF)+ (kF -dkF)=2 dkF
Conventional pairing
E
k
-k
kx
ky
FFLO pairing
k
( k ,-k )
-k
E
k
q
-k+q
q
q~gBH/vF
ky
k
( k ,-k+q )
kx
-k+q
pairing between Zeeman split parts of the Fermi surface
Cooper pairs have a single non-vanishing center of mass momentum
Pairing of electrons with opposite spins and momenta unfavourable :
But :
if
At T = 0, Zeeman energy compensation
is exact in 1d, partial in 2d and 3d.
1d SC
2d SC
3d SC
 B H / D1
0.8
0.6
• the upper critical field is increased
• Sensivity to the disorder and to the orbital
effect:
0.4
0.2
0.2
0.56
0.4
0.6
0.8
1
T / Tc
(clean limit)
0.56
0.2
0.4
0.6
0.8
1
T / Tc
FF state
D(r)  D(q)exp( iq  r)
uniform
( k ,-k )
available for pairing
depaired

( k ,-k+q )
LO state
D(r)  D 0 cos(q  r)
spatially nonuniform
+


+
~
1
q
+

The SC order parameter performs one-dimensional spatial modulations
along H, forming planar nodes
Modified Ginzburg-Landau functional :
2
F  a                
2

2
2

        
*2
2
2
4
  d 
* 2
6
2
2
 ...
May be 1st order transition at
1
B B / D
0.8
0.6
0.4
0.2
0.56
0.2
0.4
0.6
0.8
1
T / Tc
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 ….
Organic superconductor
Layered structure
k-(BEDT-TTF)2Cu(NCS)2 (Tc=10.4K)
Suppression of orbital effects in H parallel to the
planes
Cu[N(CN)2]Br layer
 15 Å
BEDT-TTF layer
H or D
S
C
H or D
S
C
BEDT-TTF (donor molecule)
Talk of Stuart Brown
about FFLO in this compound!
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, 117002 (2008)
18
Field induced superconductivity (FISC) in an organic
compound
l-(BETS)2FeCl4
Metal
FISC
Insulator AF
c-axis (in-plane)
resistivity
S. Uji et al., Nature 410 908 (2001)
L. Balicas et al., PRL 87 067002 (2001)
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
Eu-Sn Molybdenum
chalcogenide
(Eu0.75Sn0.25Mo6S7.2Se0.8)
H. Meul et al, 1984
Critical field Hc2 [Tesla]
Other
example:
S
S
Temperature [K]
Strong evidence of inhomogeneous FFLO phase
in CeCoIn5
H-T phase diagram of CeCoIn5
1st
H// ab
2nd
order
1st
H// c
2nd
Pauli paramagnetically limited superconducting state
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
a 
Proximity effect in a ferromagnet ?
In the usual case (normal
1metal):
2
qx
4m
   0, and solutionfor T  Tc is   e
, where q  4ma
In ferromagnet ( in presence of exchange field) the
equation for superconducting order parameter is different
a       0
2
Ψ
4
Its solution corresponds to the order parameter which
decays with oscillations!
Ψ~exp[-(q1 ± iq2 )x]
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
24
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)
D
D
S
F
S
S
D
D
S
D
S
« 0 phase »
«  phase »
F
F
S/F bilayer
D
 f  D f / h  (1  10)nm
h-exchange field,
25
Df-diffusion constant
S-F-S Josephson junction in the clean/dirty
limit
S
F
S
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
Ic
E(φ)=- Ic (Φ0/2πc) cosφ
J(φ)=Icsinφ

26
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
“0”-state
Spin-flip scattering decreases the
decaying length and increases the
oscillation period.
-state
F2 >F1
0
“0”-state
I=Icsin
-state
Nb-Cu0.47Ni0.53-Nb junctions
I=Icsin(+ )= - Icsin()
27
Cluster Designs (Ryazanov et al.)
30m
2x2
unfrustrated
fully-frustrated
checkerboard-frustrated
6x6
fully-frustrated
checkerboard-frustrated
28
Scanning SQUID Microscope images
(Ryazanov et al., Nature Physics, 2008))
Ic
T
T = 1.7K
T = 2.75K
T
T = 4.2K
29
FFLO State in Neutron Star
Color superconductivity
R.Casalbuoni and G.Nardulli
Rev. Mod. Phys. (2004)
Bose-Einstein-Condensate
Vortices
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)
3. Exactly solvable models of FFLO state.
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
 B H / D1
0.8
0.6
D( x)  D0 sn( x /  , k )
0.4
0.2
0.56
at T  0
D(x)
B H 
2

T / Tc
D
Magnetic moment
x
Spin-Peierls transitions - e.g. CuGeO3
Cu
O
Ge
un  (1)n D( x  na)
Chains direction
TSP=14.2 K
In 2D superconductors
( k ,-k )
Y.Matsuda and H.Shimahara
J.Phys. Soc. Jpn (2007)
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
4. Vortices in FFLO state. Role of the crystal
structure.
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

H corb
2
2
P
H c2
FFLO exists for Maki parameter α>1.8.
For Maki parameter α>9 the highest
Landau level solutions are realized
– Buzdin and Brison, 1996.
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.
B
q
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
Next orders are important :
Validity:
• large scale for spatial variation of D :
vicinity of T *
small orbital effect, introduced with
• we neglect diamagnetic screening currents (high-k limit)
Instability toward 1st
order transition
• 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 :
• cascade of 2nd and 1st order transitions
between S and N phases
• 1st order transitions within the S phase
• exotic vortex lattice structures
3
Magnetic field
Analysis of phase diagram :
n=2
2
n=1
n=0
1
Tricritical point
0
-2
-1
0
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 …
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.
43
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(φ - φ
n
φn- φn+1=2πxHs/Φ0 + πn
s-interlayer distance, x-coordinate
along q
n+1)
44
CeCoIn5
Quasi-2D heavy fermion
CeIn3
(Tc=2.3K)
Strong antiferromagnetic
fluctuation
CoIn2
z
d-wave symmetry
CeIn3
Tetragonal symmetry
Modified Ginzburg-Landau functional:
isotropic part
x
j j 
2e
iA j
c
y
No orbital effect
~

q=0
z-axis modulation
q=/2
xy-plane modulation
z
~
q=0.5 arccos(z/(-z))
~
=2z
Modulation
(~,  z )
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.
Magnetic field along z axis
~
 z=x
H||z
qz=0, n=max
~
=x
qz=max, n=0
z
qz>0, n>0
~
=2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.
Magnetic field along z axis
~
 z=x
xy
qz=0, n=max
~
=x
qz=max, n=0
z
z
intermediate
q >0, n>0
z
~
=2z
~

xy
q=0
z-axis modulation
q=/2
xy-plane modulation
z
~
intermediate
q=0.5
arccos(z/(-z))
~
=2z
z
Magnetic field along x axis
~
~
=-3z

qx=max, n=0
z
~
=-x
qx 0, n=max
H||x
qx>0, n>0
~
=3z-2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.
Magnetic field along x axis
~
~
=-3z

xy
qx=max, n=0
z
z
~
=-x
qx 0, n=max
qx>0, n>0
intermediate
~

xy
~
=3z-2x
q=0
z-axis modulation
q=/2
xy-plane modulation
z
intermediate~
q=0.5 arccos(z/(-z))
~
=2z
z
Small angle neutron scattering from the vortex lattice for H //c
FFLO?
Neutron form factor seems to be
consisitent with FFLO state
A.D.Bianch et al.Science (2007)
Neutron form factor
 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.
5. 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)
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
Rotating supefluid ultracold Fermi gases in a
trap
Coils generating magnetic field
Fermion
condensate
Vortex as a test for
superfluidity
Laser beams
MIT: Ketterle et al (2005)
Images of vortex
lattices
Questions:
1. What is the effect of confinement (finite system size) on FFLO
states?
2. Effect of rotation on FFLO states in a trap (effect of magnetic
field on FFLO state in a small superconducting sample).
3. Possible quantum oscillation effects.
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
vs
1
Ф/Ф
0
L=-1
L=0
-ΔTc/Tc0
L=1
o
c
0 
e
Ф/Ф
A.Bezryadin, A.I. Buzdin, B. Pannetier (1994)
Examples of quantum oscillation effects.
Little-Parks effect.
Simply-connected systems
Mesoscopic samples
dimensions ~ several coherence lengths
O.Buisson et al (1990)
R.Benoist, W.Zwerger (1997)
V.A.Schweigert, F.M.Peeters
(1998)
H.T.Jadallah, J.Rubinstein,
Sternberg (1999)
H
R~ξ
Tc (H) oscillations
Origin of Tc oscillations:
Transitions between the states
with different vorticity L
iLq


D | D r | e
L - Vorticity
(orbital momentum)
Multiquantum vortices
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
 Hz
a 
0 k02
Model: Modified Ginzburg-Landau functional (2D)
 2
2 2

2
F   (( a  V (r )) |  |   | D |  | D  | )dxdy
a   (T  Tc0 )
T
N
Trapping
potential
FFLO instability
Range of validity:
vicinity of tricritical
point
S

 
D    2iM [, r ] 


D    2ieA  c
Confinement mechanisms:
1. Zero trapping potential.
Boundary condition at the
system edge

nD  0
FFLO
H
2.
nonzero trapping potential

V (r )  M 2r 2 2
FFLO states in a 2D mesoscopic superconducting disk
Hz
z
q
H|| ~ Hp
r
R~ξ
Interplay between the system size, magnetic length,
and FFLO length scale
Perpendicular magnetic field component Hz = 0
The critical temperature:
Eigenvalue problem:
Wave number of FFLO
instability
H - T Phase diagram: Hz = 0
T
 2  k0 ( H , T )
H ↑ → L↓
L=1
L=2
FFLO state
L=0
H
Tilted magnetic field: Hz ≠ 0
Field induced
superconductivity
Eigenvalue problem:
a 
 R2 H z
0
Tilted magnetic field: Hz ≠ 0
Transitions with large jumps in vorticity
Vortex solutions beyond the range of FFLO
instability. Critical field of the vortex entry.
Hz
z
q
H|| ~ Hp
r
 2 4 2 2
F   ( | D |  2 | D  | )dxdy
2
1
We focus on the limit1   2
T
  eiLq
N
S
(H*,T*)
Condition of the first vortex entry:
Beyond the
range of FFLO
instability
F ( L  1)  F ( L  0)  0
FFL
O
H||
Condition of the first vortex entry:
R
 m2  2
R
3
ln     2  3  4 ln     0
m
2 R 
m

m  max(1, 2 )
H z R 2

0
 24
 2 2
 m1
Limiting cases:
2  1

2  1
R
 1
m
H *  H||
  ln
R
1
R
   
 2 
Hz 
23
1
R
ln
R 2 1
1
Hz   
R
43
Change in the scaling
law
FFLO states in a trapping potential
Interplay between the rotation effect, confinement,
and FFLO instability
FFLO states in a 2D system in a parabolic trapping potential (no
rotation)
4
2
2
D   2D   (  v0  )  0
k0   2
FFLO length scale
  a k
4
0
6 (Trapping frequency)2
0


  k0r

  e  d q

iq 
Temperature shift
v0  M / 2k
2
Fourier transform:


L0
Dimensionless
coordinate
U (q )

U (q)  q 4  2q 2  1  4 x 2
 q  e
2
2
 v0  2   q 4  2q 2   
q
q 1  x
 lx 2

q
1
l
v0
v0  0
q
FFLO states in a 2D system in a parabolic trapping potential (no
rotation)
Phase diagram
Condensate wave function

1
k0r  1

k r

cos(k0 r   4)e
k0 r
  1  2 v0
2 2
0
v0

v0 4
Suppression of wave function oscillations by the increase in
the trapping frequency
k0 r
2
1/ 4
0
v

1

=Number of observable oscillations
FFLO states in a rotating 2D gas in a parabolic trapping potential.

(r )  f L (r )eiLq
D4 f L  2D2 f L  (  v0  2 ) f L  0

1     L



D 
     a  

      

2
2
Expansion in eigenfunctions of the problem without
trap:

f L (  )   cnunL (  )
n 0
2
 D2unL  qnL
unL  2a (2n | L | L 1)unL
2
4
(2qnL
 qnL
)cn   vnmcm  cn

m
vnm  v0  umL unL  3d
0
k0   2
FFLO length scale
  a k04
Temperature shift
v0  M 2 / 2k06
a  2M  k02


  k0r
(Trapping frequency)2
rotation frequency
Dimensionless
coordinate
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:
  max (4a  v0 a )( 2 L  1)  v0 L a  4a2 (2 L  1)2
L0
rotation induced
superfluid phase


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 uperconductorferromagnet heterostructures. The -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?
73