Is Sr2RuO4 a triplet superconducor? ---analysis of specific heat under fields---

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1 Is Sr2RuO4 a triplet superconducor? ---analysis of specific heat under fields--- Kazu. Machida Okayama Univ collaborators M. Ichioka, T. Mizushima, H. Adachi, N. Nakai, Y. Tsutsumi

2 Outline Phenomena related to Pauli paramagnetism in various superconductors: 1) Sr2RuO4 (214) 2) CeCoIn5 (115) 3) TmNi2B2C (boro-carbide) 4) URu2Si2 (122) 5) UPd2Al3 (123) cf imbalanced Fermi superfluids FFLO state Theoretical framework; quasi-classical Eilenberger

3 Is Sr2RuO4 a triplet superconductor? Supporting evidence for triplet pairing Knight shift (KS) experiments---> no change for both c-axis and ab-plane cf There must be a field direction which shows change of KS when applied parallel to d-vector. How to understand anomalous (H) behaviors for H//ab; specific heat---> bulk property

4 Sr 2 RuO 4 K. Deguchi, et al, JPSJ 75 (2004) 1313

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6 Sr2RuO4

7 Sr 2 RuO 4 triplet pairing? strong suppression 2nd order K. Deguchi, et al, JPSJ 71 (02)2839.

8 Sr2RuO4 Tenya et al 0.01 H c2 0 0 M eq 0 (emu/g) K 0.41 K 0.60 K 0.80 K 1.0 K H k Sr 2 RuO 4 H // [100] H (koe)

9 Sr2RuO4 Tenya et al * [100] 1 [100] 2 [100] [100] [001] 2 [001] 2 [001] T (K) indicating that Pauli effect is important.

10 2

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12 Gor kov equation (Green s function G F Hamiltonian F G Bogoliubov-de Gennes Equation eigen-value equation Superconductivity-version of Schrodinger eq. E eigen-energy u v wave functions : label of eigen-state center of mass coordinate Quasi-classical approximation >> 1/k F Eilenberger equation (quasi-classical Green s function relative momentum perpendicular to the Fermi surface on the Fermi surface Relative momentum space k Fermi surface

13 Quasiclassical Eilenberger theory Free energy with paramagnetic effect Zeeman effect Normal state susceptibility Average flux density Internal field distribution

14 Self-consistent equation Pairing potential Vector potential paramagnetic parameter B(r)/B 0 Total internal field Paramagnetic contribution Diamagnetic (super-current) contribution Paramagnetic magnetization Normal state magnetization

15 Results of calculation Magnetic field dependence T = 0.1 Tc Amplitude of pair potential spatial average s-wave paring Paramagnetic effect A:very small B: C Large d-wave pairing DOS at E=0 ~Low temp. specific heat Paramagnetic susceptibility if GL parameter is large) B Suppression of superconductivity at high field Similar effect both for s-wave and d-wave pairing) B Total magnetization Paramagnetic Diamegnetic Dotted lines Paramagnetic moment in the normal state

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20 Heavy Fermion superconductors URu 2 Si 2, Sr 2 RuO 4, CeCoIn 5 Pauli paramagnetic effect on vortex lattice Zeeman effect -----> up spin and down spin population imbalance > Fulde-Ferrell-Larkin-Ovchinikov (FFLO)

21 Phase diagram in H vs T CeCoIn 5

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23 URu 2 Si 2 another Pauli-limited superconductor H// c H// a point nodes at two poles on Fermi sphere, but we need paramagnetic effect with rather large μ value

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25 vortex core structures ordinary Abrikosov vortex paramagnetic vortex case small μ-case large μ-case enhanced paramagnetic moment accumulated at core majority spin component accommodated exclusively at core because of -phase shift physics

26 CeCoIn 5 prime candidate for FFLO Eskildsen et al Science in press

27 Comparison between quasi-classical theory and exp.

28 Conclusion and perspectives rich vortex physics associated with Pauli paramagnetic effects described microscopically by quasi-classical Eilenberger framework TmNi2B2C Pauli effect arised from sf exchange int. between 4f-localized moment and s-electrons URu2Si2 unconventional pairing with point nodes substaintial Pauli effect -->possible first order at Hc2 CeCoIn5 Pauli paramagnetic effect is important for high field phase in both H-directions, but several mysteries remain associated with non-fermi liquid phenomena. Sr2RuO4 It seems that Pauli effect may be important, in particular in understanding of the in-plane properties; C/T etc It comes from either (A) The pairing symmetry is singlet, (B) or triplet where d-vector locked in plane.

29 References 1) 2) PRL in press 3) arxiv: