Observation of a possible Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in CeCoIn5  

Roman Movshovich

Andrea Bianchi Los Alamos National Laboratory, MST-10

Cigdem Capan

Filip Ronning

Pascoal Pagliuso

John Sarrao

• Fulde-Ferrell-Larkin-Ovchinnikov inhomogeneous superconductivity - competition between superconductivity and Pauli paramagnetism.

CeCoIn5 meets all the requirements:

• Very clean heavy-fermion superconductor, most likely d-wave

• First order phase transition, phase diagram strong Pauli limiting

• Low temperature anomaly in specific heat second superconducting phase. FFLO?A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003), R. Movshovich et al., Nature 427, 802 (2004).

Tc < 200 mKP ~ 25 kbar

Superconductors,Tc up to 2.3 K atambient pressure

Ce2CoIn8,Ce2RhIn8

under pressure

FFLO state (III) appears if certain conditions are satisfied.

From Gruenberg and Gunther, Phys. Rev. Lett. 16, 996 (1966)

(1) clean superconductor

(2) Pauli limited

(3) PL is strong enough compared to orbital limiting: Maki parameter is large enough. GG: > 1.8

Good candidates:• low dimensional sc (organics)• heavy fermion sc: weak orbital limiting.

CeCoIn5 combines both of these properties

Pauli limiting

PL is due to the competition between Zeeman energy of electrons’s spins in the normal state and the superconducting condensation energy. PL is mostly pronounced for the singlet superconductivity, with S = 0, since superconducting electrons in a pair with opposite spins can not take advantage of the Zeeman energy.Pauli limiting will have effect of suppressing superconductivity and the superconducting critical field.

PL field HP for s-wave BCS singlet superconductor is B

P gH

22

0

For CeCoIn5 this formula gives HP = 4.2 T, if we use weak coupling BSC value for 0 = 1.76 Tc and g = 2.

Problem: experimental values: Hc2 = 5 T for H || [001] and 12 T for H ||[110]!!! theoretical estimate of 4.2 T is unphysical since HP can not be less then experimental value Hc2.Solution: g 2, strong coupling.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

h = Hc2

/(Tc·-(dH

c2/dT

IT

c

))

l = 0, clean limit

h

t = T/Tc

0.0 0.5 1.0 1.5 2.0 2.5 3.00

10

20

30

40

50

CeCoIn5 – upper critical field for H II c

dHc2

/dT = -66.7 (kG/K)

Tc = 2.27 K

E. Helfand and N.R. Werthamer Phys.Rev. 147 313 (1967)

Hc2

(kG

)

T (K)

Superconductivity is suppressed with respect to theoretical prediction of Hc2 without PL CeCoIn5 is Pauli limited.

0.1 1 10

0.01

0.1

1

10

CeCoIn5

(W

/Km

)

T (K)

0.0 0.2 0.4 0.6 0.8 1.00

1

2

/T

(W

/K2 m

)

T2 (10-2K2)

R. Movshovich et al., PRL 86, 5152 (2001)

T3 at low temperature lines of nodes in the energy gap in clean limit,Impurity band width is less than 30 mK very clean material.Order of magnitude rise in /T qp mean free path of few m.

Symmetry of the order parameter of CeCoIn5  

Pauli limiting

Specific heat

Thermal conductivity  NQR

+ = d-wave  

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.3

0.6

0.9

1.2

1.5

H || c

T (K)

CeCoIn5 – second order – first order

4.8

4.9

4.54.6

4.7

4.75

CP (

J/m

ol K

)

A. Bianchi et al., PRL 89, 137002 (2002)

H

T

Pha se Bo und a ry

M ag n e to ca lo r ic e ffe c tfo r firs t o rd e r p h ase tran s itio n

d T /d H = T /C (-d M /d T )S H H

H ig h e n tro p y p h ase

L o w e n tro p y p h ase

A. Bianchi et al., PRL 89, 137002 (2002)

A. Bianchi et al., PRL 89, 137002 (2002)

A. Bianchi et al., PRL 89, 137002 (2002)

First order nature of the superconducting phase transition is reflected in a step in thermal conductivity at Tc.

C. Capan et al., submitted to PRB.

H||[100]

H||[100]

for CeCoIn5:

Experimentally, for H || [001]: Hc2 = 5 T and Hc20=13.2 T

& GG gives HP = 5.8 T, α = 3.6, T0=.35Tc

L . W . G ru e n b erg an d L . G u n th er, P R L 9 9 6 (1 9 6 6 )1 6 ,

B

p gH

22

0p

c

H

H 202 cc

c TdT

dHH 2

20 7.0

Conditions for formation of the FFLO state:

(1) clean superconductor

(2) Pauli limited

(3) PL is strong wrt orbital limiting: Maki parameter is large enough. GG: > 1.8

0.0 0.4 0.8 1.2 1.6 2.0 2.40

1

2

3

4

0.2 0.3 0.4

1.0

1.2 12 T 11.4 T 11.2 T 11 T 10.6 T 10 T 8 T 6 T

(C-C

Sch

)/T

(J/

mol

K2 )

(a)

CeCoIn5, H || [110]

T (K)

C

/T (

J/m

ol K

2 )

T2

A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Tc

T2

CeCoIn5, H || [110]

(C-C

Sch

)/T

(J/

mol

K2 )

T (K)

10.77 T 11 T 10.51 T

A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003)

0.0 0.2 0.4 0.6 0.8 1.0 1.29.8

10.0

10.2

10.4

10.6

10.8

11.0

11.2

11.4

T2Tc

T (K)

H (

T)

0

0.5000

0.7500

1.000

1.250

1.500

2.000

2.500

3.050

3.800

5.000

CeCoIn5, C/T (J/mol K2)

A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003) H. Adachi and R. Ikeda, Phys. Rev. B 68, 186510 (2003)

Conclusions:

CeCoIn5 is a clean Type II strongly Pauli limited superconductor, as seen from (1) the phase diagram and (2) the change of the superconducting transition to first order at high magnetic fields close to the superconducting critical field Hc2, as predicted by K. Maki in 1960’s.

The second phase transition within the superconducting state in the high field-low temperature part of the phase diagram is consistent with the formation of the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state predicted in 1960’s.

Needs: • Theoretical support on the detailed predictions of various properties of the FFLO state to compare with experiments.• Experiments that probe directly the microscopic structure of the FFLO state.

F S

F S

C exp (-/T)

exp (-/T)

C T2

T in impurity dominated region, universal limit.

T3, clean limit

Specific heat C and thermal conductivity can help to determine thesymmetry of the superconducting order parameter.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0 1 2 30

1

2

CeCoIn5

(C

-CS

ch)/

T (

J/m

ol K

2 )

T (K)

In NQ Schottky

T (K)

C/T

(J/

mol

K2 )

R. Movshovich et al., PRL 86, 5152 (2001)

Cel aT + bT2 at low temperature lines of nodes in the energy gap