1

A. BuzdinUniversity of Bordeaux I and Institut Universitaire de France

Non-uniform (FFLO) states and quantum oscillations in superconductors and

superfluid ultracold Fermi gases

ECRYS-2011, August 15-27, 2011 Cargèse , France

in collaboration with L. Bulaevskii, J. P. Brison, M. Houzet, Y. Matsuda, T. Shibaushi, H. Shimahara, D.

Denisov, A. Melnikov, A. Samokhvalov

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?

Outline

1. Singlet superconductivity destruction by the magnetic field.

• Orbital effect (Lorentz force)

B

-pFL

p

FL

• Paramagnetic effect (singlet pair)

Sz=+1/2 Sz=-1/2

μBH~Δ~Tc cTsSI

Electromagneticmechanism

(breakdown of Cooper pairsby magnetic field

induced by magnetic moment)

Exchange interaction

Normal

MeissnerT

B

Hc2

Tc

Abrikosov Lattice

Hc2 0

2 2

Vortex

Orbital effect

(x coherence length)

Flux quantum

(l penetration length ）

F0=hc/2e=2.07x10-7Oe ・ cm2z

B

D

Vortex lattice in NbSe2

(STM)

B

-pFL

p

FL

Pc

orbc

H

H

2

22 1~

F

Superconductivity is destroyed by magnetic field

Orbital effect (Vortices)

Zeeman effect of spin (Pauli paramagnetism)

1

2N H 2

1

2N(0)2

N 1

2(g B )2 N (0) B

Pc g

H

2

2

20

2 2

orbcH

Maki parameter

Usually the influence of Pauli paramagnetic effect is negligibly small

B

-pFL

p

FL

Superconducting order parameter behavior under paramagnetic effect

Standard Ginzburg-Landau functional:

...2222 aF

The minimum energy corresponds to Ψ=const

The coefficients of GL functional are functions of the Zeeman field h= μBH !

Modified Ginzburg-Landau functional ! :

422

24

1

b

maF

The non-uniform state Ψ~exp(iqr) will correspond to minimum energy and higher transition temperature

q

F

q0

242 )( qqqaF

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

FFLO inventors

Fulde and Ferrell

Larkin and Ovchinnikov

E

kkF +dkF

kF -dkF

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

ky

kx

E

Conventional pairing

FFLO pairing

( k ,-k )

( k ,-k+q )

pairing between Zeeman split parts of the Fermi surface

Cooper pairs have a single non-vanishing center of mass momentum

k

-k

-k k

ky

kx

E

q

q~gmBH/vF

-k+q

q

-k+q

k

k

Pairing of electrons with opposite spins and momenta unfavourable :

But :

• the upper critical field is increased

• Sensivity to the disorder and to the orbital effect:

(clean limit)

At T = 0, Zeeman energy compensation is exact in 1d, partial in 2d and 3d.

if

0.2 0.4 0.6 0.8 1cTT /0.56

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

cTT /

/HB1d SC2d SC3d SC

0.56

FF state uniform

(r) (q)exp(iqr)

LO state spatially nonuniform

(r) 0 cos(qr)

~1

q

+

+

+

The SC order parameter performs one-dimensional spatial modulations along H, forming planar nodes

available for pairing

depaired

( k ,-k )

( k ,-k+q )

May be 1st order transition at

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

cTT /

/BB

0.56

...62*222*

2242222

aF

Modified Ginzburg-Landau functional :

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 ….

CSH or D

CSH or D

BEDT-TTF (donor molecule)

BEDT-TTF layer

Cu[N(CN)2]Br layer

15 Å

k-(BEDT-TTF)2Cu(NCS)2 (Tc=10.4K)Organic superconductor

Layered structure Suppression of orbital effects in H parallel to the planes

Talk of Stuart Brownabout 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, 117002 (2008)

Field induced superconductivity (FISC) in an organic compound

S. Uji et al., Nature 410 908 (2001)

L. Balicas et al., PRL 87 067002 (2001)

FISC

Metal

Insulator AF

-(BETS)2FeCl4

c-axis (in-plane) resistivity

Other example:

Temperature [K]

Cri

tica

l fie

ld H

c2 [

Tesl

a]

Jaccarino-Peter effect

Eu-Sn Molybdenum chalcogenide

H. Meul et al, 1984

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

S

S(Eu0.75Sn0.25Mo6S7.2Se0.8)

Strong evidence of inhomogeneous FFLO phasein CeCoIn5

2nd order

2nd

1st

1st

H// ab

H// c

H-T phase diagram of CeCoIn5

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

24

In ferromagnet ( in presence of exchange field) the equation for superconducting order parameter is different

042 aIts solution corresponds to the order parameter which decays with oscillations! Ψ~exp[-(q1 ± iq2 )x]

x

Ψ

Order parameter changes its sign!

Wave-vectors are complex!They are complex conjugate and

we can have a real Ψ.

Proximity effect in a ferromagnet ?

In the usual case (normal metal):

4maq where, is TTfor solution and ,04

1c

2 qxem

a

Many new effects in

S/F heterostructures!

25

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)

SF

S

S F

« phase  » « 0 phase »

S F S

S/F bilayer nmhD ff )101(/

h-exchange field,Df-diffusion constant

26

S-F-S Josephson junction in the clean/dirty limit

S SF

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 thicknessIc

E(φ)=- Ic (Φ0/2πc) cosφ

J(φ)=Icsinφ

27

Critical current density vs. F-layer thickness (V.A.Oboznov et al., PRL, 2006)Collaboration with V. Ryazanov group from ISSP, Chernogolovka

-stateI=Icsin

I=Icsin(+ )= - Icsin()

Nb-Cu0.47Ni0.53-Nb junctions

Ic=Ic0exp(-dF/F1) |cos (dF /F2) + sin (dF /F2)|

dF>> F1

“0”-state

F2 >F1

“0”-state-state

0

Spin-flip scattering decreases thedecaying length and increases the oscillation period.

28

Cluster Designs (Ryazanov et al.)

2 x 2

6 x 6

fully-frustrated checkerboard-frustrated

fully-frustrated unfrustrated checkerboard-frustrated

30m

29

T

T = 1.7K T = 4.2KT = 2.75K

Scanning SQUID Microscope images(Ryazanov et al., Nature Physics, 2008))

T

Ic

FFLO State in Neutron Star

R.Casalbuoni and G.Nardulli Rev. Mod. Phys. (2004)

Glitches

Color superconductivity 　 Bose-Einstein-Condensate

Vortices

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.

0.2

0.4

0.6

0.8

1

cTT /

/HB1d SC

0.56

),/()( 0 kxsnx

20 HTat B

x)(x

Magnetic moment

Spin-Peierls transitions - e.g. CuGeO3

TSP=14.2 K

Cu

O Ge

Chains direction

)()1( naxu nn

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

Lowest m=0 Landau level solution, Gruenberg and Gunter, 1966

Pc

orbc

H

H

2

22 FFLO exists for Maki parameter α>1.8.

For Maki parameter α>9 the highest Landau level solutions are realized – Buzdin and Brison, 1996.

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.

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

Exotic vortex lattice structures in tilted magnetic field

Near the tricritical point, the characteristic length is

Microscopic derivation of the Ginzburg-Landau functional :

Instability toward FFLO state

Instability toward 1st order transition

Validity:

• large scale for spatial variation of : vicinity of T *

small orbital effect, introduced with

• we neglect diamagnetic screening currents (high- limit)

Generalized Ginzburg-Landau functional

Next orders are important :

• 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 :

__ 1st order transition__

2nd order transition

__ 2nd order transition in the paramagnetic limit

Temperature

-2 -1 0

Mag

netic

fie

ld

0

1

2

3

Tricritical point

n=1

n=0

n=2Landau levels

Triangular lattice

squareTriangular

Rectang.

Rectangular

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

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

Quasi-2D heavy fermion

(Tc=2.3K)

Strong antiferromagnetic

fluctuation

d-wave symmetry

Tetragonal symmetry

CeIn3

CoIn2

CeIn3

CeCoIn5

z

x yisotropic part

Modified Ginzburg-Landau functional:

jjj iAc

e

2

z

=/2xy-plane modulation

=0.5 arccos(z/(-

z))

~

=0z-axis modulation

=2z

~

~

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.

),~( z

Magnetic field along z axis

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.

z

=2z

qz>0, n>0

qz=max, n=0

qz=0, n=max

z=

x

=x

~

~

~

H||z

Magnetic field along z axis

z

=2z

qz>0, n>0

qz=max, n=0

qz=0, n=max

z=

x

=x

~

~

~

z

=/2xy-plane modulation

=0.5 arccos(z/(-

z))

~

=0z-axis modulation

=2z

~

~

z

z

intermediate

xy

xy

intermediate

Magnetic field along x axis

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.

z

qx=max, n=0

qx>0, n>0

qx0, n=max

=-x

~

=3z-2

x

~

=-3z

~ ~

H||x

Magnetic field along x axis

z

qx=max, n=0

qx>0, n>0

qx0, n=max

=-x

~

=3z-2

x

~

=-3z

~ ~

z

=/2xy-plane modulation

=0.5 arccos(z/(-

z))

~

=0z-axis modulation

=2z

~

~

z

z

intermediate

xy

xy

intermediate

Small angle neutron scattering from the vortex lattice for H //c

A.D.Bianch et al.Science (2007)

FFLO? Neutron form factor

Neutron form factor seems to be consisitent with FFLO state

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

Supefluid core

Normal gas

rf transitions76 MHz

hyperfine states

Rotating supefluid ultracold Fermi gases in a trap

MIT: Ketterle et al (2005)

Fermion condensate

Laser beams

Coils generating magnetic field

Images of vortex lattices

Vortex as a test for superfluidity

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.

Multiply-connected systemsSuperconducting thin-wall cylinder

Multiquantum vortices

Superconductor with a columnar defect or hole

A.Bezryadin, A.I. Buzdin, B. Pannetier (1994)

Little-Parks effect. Switching between the vortex states.

Ф/Фo

-1 1

0

L=-1 L=0 L=1

-ΔTc/Tc0

Ф/Фo

Tc (H) oscillations

sv

e

c 0

Examples of quantum oscillation effects.

Tc (H) oscillations

Mesoscopic samples dimensions ~ several coherence lengths

Simply-connected systemsO.Buisson et al (1990)R.Benoist, W.Zwerger (1997)V.A.Schweigert, F.M.Peeters (1998)H.T.Jadallah, J.Rubinstein, Sternberg (1999)

H

Origin of Tc oscillations: Transitions between the states

with different vorticity L

Examples of quantum oscillation effects.

Little-Parks effect.

iLer || L - Vorticity (orbital momentum)

R~ξ

Multiquantum vortices

FFLO states and Tc(H) oscillations in infinite 2D superconductors

A.I. Buzdin, M.L. Kulic (1984)

Examples of quantum oscillation effects.

Hz

H|| ~ Hp

200k

H za

Model: Modified Ginzburg-Landau functional (2D)

Trapping potential

dxdyDDrVaF )||||||))((( 2222

2)( 22rMrV

H

T

S

N

FFLO

Range of validity: vicinity of tricritical

point

FFLO instability

Confinement mechanisms:

1. Zero trapping potential. Boundary condition at the system edge

2. nonzero trapping potential

],[2 riMD

cAieD

2

0Dn

)( 0cTTa

FFLO states in a 2D mesoscopic superconducting disk

Hz

r

z H|| ~ Hp

R~ξ

Interplay between the system size, magnetic length, and FFLO length scale

Perpendicular magnetic field component Hz = 0

Eigenvalue problem:

Wave number of FFLO instability

The critical temperature:

H - T Phase diagram: Hz = 0

T

H

L=1

L=2L=0

FFLO state

H ↑ → L↓

),(2 0 THk

Tilted magnetic field: Hz ≠ 0

Eigenvalue problem:

0

2

z

a

HR

Field induced superconductivity

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

r

z H|| ~ Hp

dxdyDDF )||||( 2242

221

T

S

N

FFLO

Beyond the range of FFLO

instability

21 We focus on the limit

Condition of the first vortex entry:

0)0()1( LFLF

iLe(H*,T*)

||H

Condition of the first vortex entry:

0ln432

ln 322

2

m

m

m

R

R

R

),max( 21 m

0

2

RH z

21

2

42

m

Limiting cases:

12 1

lnR

1m

R

32

2

R

12

ln1

R

RH z

341

RH z

Change in the scaling law

12

||* HH

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)

0)(2 20

24 v

20 k40ka

60

20 2/ kMv

FFLO length scale

Temperature shift

(Trapping frequency)2

Fourier transform:

qde qqi

2

rk

0 Dimensionless coordinate

242

2

0 2qqq

v

xq 1224 412)( xqqqU

2xq e

0

1

v

q

)(qU

0L

00 v

FFLO states in a 2D system in a parabolic trapping potential (no rotation)

Phase diagram

021 v

0v

1

Condensate wave function

40

0

022

0)4cos( vrkerkrk

10 rk

rk0

Suppression of wave function oscillations by the increase in the trapping frequency

12

4/10

v

=Number of observable oscillations

FFLO states in a rotating 2D gas in a parabolic trapping potential.

0)(2 20

24 LLL fvfDfD

2

2 1

a

LD

0

)()(n

nLnL ucf

nLanLnLnL uLLnuquD )1||2(222

Expansion in eigenfunctions of the problem without trap:

nm

mnmnnLnL ccvcqq )2( 42

0

30 duuvv nLmLnm

20 k40ka

60

20 2/ kMv

202 kMa

FFLO length scale

Temperature shift

(Trapping frequency)2

rotation frequency

iLL erfr )()(

rk

0 Dimensionless coordinate

2200

0)12(4)12)(4(max

LLvLv aaaa

L

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?