1. Observation of a possible Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in CeCoIn 5 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. CeCoIn 5 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).

2. T c < 200 mK P ~ 25 kbar Superconductors, T c up to 2.3 K at ambient pressure Ce 2 CoIn 8, Ce 2 RhIn 8 under pressure

4. 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. CeCoIn 5 combines both of these properties

5. 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 H P for s-wave BCS singlet superconductor is For CeCoIn 5 this formula gives H P = 4.2 T, if we use weak coupling BSC value for  0 = 1.76 T c and g = 2. Problem: experimental values: H c2 = 5 T for H || [001] and 12 T for H ||[110]!!!  theoretical estimate of 4.2 T is unphysical since H P can not be less then experimental value H c2. Solution: g  2, strong coupling.

6. Superconductivity is suppressed with respect to theoretical prediction of H c2 without PL  CeCoIn 5 is Pauli limited.

7. R. Movshovich et al., PRL 86, 5152 (2001)   T 3 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.

8. Symmetry of the order parameter of CeCoIn 5 Pauli limiting   Specific heat Thermal conductivity  NQR  + = d-wave

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

14. First order nature of the superconducting phase transition is reflected in a step in thermal conductivity at T c. C. Capan et al., submitted to PRB. H||[100]

15. for CeCoIn 5 : Experimentally, for H || [001]: H c2 = 5 T and H c2 0 =13.2 T & GG gives H P = 5.8 T, α = 3.6, T 0 =.35T c 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

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

19. CeCoIn 5, C/T (J/mol K 2 ) A. Bianchi et al., Phys. Rev. Lett. 91, 257001 (2003)H. Adachi and R. Ikeda, Phys. Rev. B 68, 186510 (2003)

21. Conclusions: CeCoIn 5 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 H c2, 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.

22. C  exp (-  /T)   exp (-  /T) C  T 2 T in impurity dominated region,   universal limit. T 3, clean limit Specific heat C and thermal conductivity  can help to determine the symmetry of the superconducting order parameter.

23. R. Movshovich et al., PRL 86, 5152 (2001) C el  aT + bT 2 at low temperature  lines of nodes in the energy gap