1

A. Buzdin University of Bordeaux I and Institut Universitaire de France

Revealing the FFLO phase by the in-plane critical field anisotropy in layered

superconductors

in collaboration with M. Croitoru

ECRYS-2014, August 11-23, 2014 Cargèse , France

1. Singlet superconductivity destruction by the magnetic field: - The main mechanisms - Origin of FFLO state. 2. Experimental evidences of FFLO state. 3. Quasi-2D superconductors: in-plane anisotropy of the critical field due to FFLO modulation. 4. Quasi-1D superconductors

Outline

• Orbital effect (Lorentz force)

B

-p FL

p

FL

• Paramagnetic effect (singlet pair)

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

μBH~Δ~Tc ( ) cTsSI ≈⋅

Electromagnetic mechanism

(breakdown of Cooper pairs by magnetic field

induced by magnetic moment)

Exchange interaction

1. Singlet superconductivity destruction by the magnetic field.

Pc

orbc

HH

2

22≡α 1~ <<∆

Fεα

Superconductivity is destroyed by magnetic field

Orbital effect (Vortices)

Zeeman effect of spin (Pauli paramagnetism)

12

χN H 2 =12

N(0)∆2

χN =12

(gµ B )2 N (0) B

Pc g

Hµ

∆=

22

20

2 2πξΦ

=orbcH

Maki parameter

Usually the influence of Pauli paramagnetic effect is negligibly small

B

-p FL

p

FL

μBH~Δ~Tc

Superconducting order parameter behavior under paramagnetic effect

Standard Ginzburg-Landau functional:

...2222 +Ψ∇+Ψ∇−Ψ= ηγaF

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

241

Ψ+Ψ∇+Ψ=b

maF

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

q

F

q0

242 )( qqqaF Ψ+−= ηγ

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

P. Fulde, R. A. Ferrell, Phys.Rev. 135, A550 (1964) A. I. Larkin, Yu. N. Ovchinnikov, JETP 47, 1138 (1964)

E

k kF +δkF

kF -δkF

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

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~gµBH/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 1 cTT /0.56

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

cTT /

∆/HBµ1d SC 2d SC 3d SC

0.56

•Anisotropies of the Fermi surface and the gap function can stabilize the FFLO state. Stability of FFLO state crucially depends on the dimensionality of the system.

11

Requirements for the formation of the FFLO sate

• Strongly type-II superconductors with very large Ginzburg-Landau parameter

• Large Maki parameter, such that the upper critical field can easily approach the Pauli paramagnetic limit:

pairing • Very clean, , since the FFLO state is sensitive to the presence of impurities.

1D SC 2D SC 3D SC

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

HH

2

22≡α 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 and orbital effect

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.

θ B

3. 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 ….

15

Layered organic superconductor

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

C S H or D

BEDT-TTF (donor molecule)

17

Possible FFLO state in Heavy Fermion superconductors

This 2nd order phase transition is characterized by a structural transition of the flux line lattice (FLL).

Ultrasound and NMR results are consistent with the FFLO state which predicts a segmentation of the flux line lattice

Y. Matsuda and H. Shimahara, J. Phys. Sos. Japan 76, 051005 (2007)

Pauli paramagnetically limited superconducting state

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

λ-(BETS)2FeCl4

c-axis (in-plane) resistivity

Other example:

Temperature [K]

Crit

ical

fiel

d H

c2 [T

esla

]

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)

20

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)

21

Model system –quasi 2 metal

We assume that the corrugation of the Fermi surface is small

22

Eilenberger equation for layered SC

Here:

Taking into account that the system is near the second-order phase transition, the linearized Eilenberger equation describing layered superconducting systems acquires the form

The order parameter is defined self-consistently as (s-wave)

23

Choice of the solution The solution of the Eilenberger equation can be chosen without loss of generality as a Bloch function

At the same time, the order parameter can be expanded as

This expansion takes into account the possibility for the formation of the pairing state with finite center-of-mass momentum:

• The direction of the FFLO modulation vector is fixed by the symmetry of the Fermi surface.

• |q| is taken that maximizes Hc2 calculated in the Pauli paramagnetic limit.

M. Croitoru, M. Houzet, and A. I. Buzdin, Phys. Rev. Lett. 108, 207005 (2012).

25

Solution

Here:

Making use of the self-consistency relation we finally obtain

General case:

Resonance:

Out of resonance:

26

Orbital correction (out of resonance) Normalized orbital correction of the superconducting onset temperature as a function of in-plane magnetic field H for several angles between magnetic field and the FFLO modulation vector q.

Absolute value of the wave vector q of the FFLO phase (dashed lines) and of the wave vectors Q (solid lines) versus the reduced temperature calculated for several values of Fermi velocity .

28

Orbital correction (resonance)

Resonance condition:

29

Resonance condition • Vector potential of the parallel magnetic field results in a modulation of the interlayer coupling;

Resonance condition:

Conducting layer

• The period of this modulation may interfere with the in-plane FFLO modulation leading to the anomalies in the critical field behavior;

Conducting layer

30

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

31

Intrinsic Josephson vortex pinning in the FFLO phase for parallel field orientation

S. Uji et al., Phys.Rev.Lett. 97, 157001 (2006) L. Bulaevskii et al., Phys.Rev.Lett. 90, 067003 (2006)

Normalized superconducting transition temperature, as a function of the angle between the directions of the applied magnetic field and the vector q for several values of .

32

In-plane anisotropy of the onset of superconductivity

Resonance case Non-resonance case

Croitoru, Buzdin, PRB, 2012

34

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)

Normalized correction of the superconducting onset temperature as a function of in-plane magnetic field H for several angles between magnetic field and the FFLO modulation vector q.

35

Orbital correction (d-wave)

Orbital correction FFLO modulation vector

Normalized superconducting transition temperature, as a function of the angle between the directions of the applied magnetic field and the vector q for several values of .

36

In-plane anisotropy of the onset of superconductivity (d-wave)

37

Phase diagram with FFLO • Normalized correction of the superconducting onset temperature as a function of (i) the reduced temperature, (ii) of the magnetic field direction; • Polar plots of .

0.0 0.2 0.4 0.6 0.8 1.0-2.0

-1.5

-1.0

-0.5

0.0

α = 0°

α = 20°

α = 70°

α = 45°

∆TcP

/TcP

T2 c0

/t2 c0

T/Tc0

mx/my = 1, 10 η = 5.5

0 45 90 135 180-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

η = 5.5

T/Tc = 0.05

T/Tc = 0.1

T/Tc = 0.2

T/Tc = 0.4

T/Tc = 0.6T/Tc = 0.8

my/mx = 1

∆TcP

/TcP

T2 c0

/t2 c0

angle (°)

0.6

0.8

1.00

20

40

60

80

100

120

140

160180200

220

240

260

280

300

320

340

0.6

0.8

1.0

Croitoru, Houzet, Buzdin, PRL, 2012

Quasi 1D superconductors

Croitoru, Buzdin, PRB, 2014

41

Conclusions

• There are strong experimental evidences of the existence of the the FFLO state in organic layered superconductors.

• The interplay between FFLO modulation and orbital effect in

layered superconductor in tilted field results in the very special oscillatory-like angular dependence of the critical field.

• The careful studies of the in-plane critical field anisotropy

could provide the information about the direction and the absolute value of the FFLO wave-vector.

42

System of coupled equations

Substituting one gets system of coupled equations

where

Assumptions: We keep term up to:

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.

θ B

4. Vortices in FFLO state.

45

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.

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-κ 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

ζ 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

fiel

d

0

1

2

3

Tricritical point

n=1

n=0

n=2

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 …