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Heavy Fermion Superconductivity: Competition and Cooperation of Spin

Heavy Fermion Superconductivity: Competition and Cooperation of Spin Fluctuations and Valence Fluctuations. K. Miyake KISOKO , Osaka University. KISOKO = Graduate School of Engineering Science. Outline of the talk. % Fundamental concepts of superconductivity of heavy fermion

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Heavy Fermion Superconductivity: Competition and Cooperation of Spin

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  1. Heavy Fermion Superconductivity: Competition and Cooperation of Spin Fluctuations and Valence Fluctuations K. MiyakeKISOKO, Osaka University KISOKO = Graduate School of Engineering Science

  2. Outline of the talk % Fundamental concepts of superconductivity of heavy fermion metals (mainly Ce-based compounds): Brief introduction of experiments and theoretical wisdom % Two kinds of SC mechanism for heavy fermion metals, spin fluctuations and valence fluctuations: Experiments and theoretical attempts. % Effect of magnetic field on valence transition % Signatures of valence transition or crossover in Fermi surface change of CeRhIn5 % Outlook for the future & New universality class of QCP associated with critical end point of valence transition in heavy fermions & Connection to high-Tc cuprates

  3. Current collaborators S. Watanabe (Osaka Univ. ⇒ Kyushu Institute of Technology), A. Tsuruta (Osaka Univ. ), J. Flouquet (CEA / Grenoble, ESPCI) Former collaborators Y. Onishi (NEC), O. Narikiyo (Kyushu Univ.), H. Maebashi (ISSP, Univ. Tokyo) , A. T. Holmes (Univ. Birmingham), D. Jaccard (Univ. Geneva), M. Imada (Univ. Tokyo) T. Sugibayashi (Ehime Univ.),

  4. Fundamental concepts of superconductivity of heavy fermion metals (mainly Ce-based compounds): experiments and theoretical wisdom

  5. 1979: year of paradigm change of superconductivity Report of superconductivity in CeCu2Si2 which is barely magnetic material Steglich et al: PRL 43 (1979) 1892 zero resistance C/T ∝ m*(N/V)1/3 Meissner effect ρ heavy electron 103 times larger than usual metals T(K) T(K) Long silence till1984 and “Strum und Drang” of research developments after then

  6. BCS superconductivity is very fragile against magnetic impurities La1-xGdx Tc T(K) Ferromagnetic state Gd (%) Matthais et al : PRL (1958)

  7. (Osaka Univ. 2002)

  8. General clue for pairing interaction in heavy fermions -k k’ -k’ k % Pairing interaction among quasi-particles (weight of quasi-particle in electron) % Coupling constant for Cooper pairingl* In order that Tc is high enough to be observable, Pairing and heaviness of quasi-particles should be the same origin: magnetic fluctuations, quadruple fluctuations, etc.

  9. -p p’ p -p’ Electronic (spin-fluctuation or charge-fluctuation) mechanism cf. Kohn-Luttinger： PRL 15 (1965) 524 pairing interaction finite Tc“in principle” even for purely repulsive interaction Pairing interaction in triplet channel Theory for superfluid 3He Nakajima: Prog. Theor. Phys. 50 (1973) 1101. Anderson & Brinkman: Phys. Rev. Lett. 30 (1973) 1108 Para-magnon = ferromagnetic spin fluctuations Spin triplet P-wave pairing

  10. Ferromagnetic spin-fluctuation mechanism was so successful for understanding the existence of 3He-A phase (ABM phase) Anderson-Brinkman: Phys. Rev. Lett. 30 (1973) 1108. Kuroda: Prog. Theor. Phys. 51 (1974) 1269 . Analogy between heavy fermion SC and 3He was stressed till mid ’80‘, early stage of research of heavy fermion SC The same process is valid also for antiferromagnetic spin fluctuations KM, Schmitt-Rink, Varma: PRB 34 (1986) 6554. Pairing interaction in singlet channel Scalapino, Loh, Hirsch: PRB 34 (1986) 8190. AF fluctuations promote “d-wave” General expression of pairing interaction in RPA spin triplet spin singlet

  11. Since mid ‘90, SC’s appeared near pressure induced AF-QCP CePd2Si2 Grosche et al: Physica B 223/224 (1995) 50 CeIn3 Non-Fermi liquid behavior 3d AF-QCP moderate enhancement Mathur et al: Nature 394 (1998) 39 AF spin fluctuations should play an important role : Recent Dogma

  12. Another SC mechanism for heavy fermions, enhanced valence fluctuations: Experiments

  13. Two-types of P-induced heavy fermion SC near AF quantum critical point (QCP) CeCu2Si2: Thomas et al, J. Phys. Condens. matter 8 (1996) L51 CeCu2Si2: Steglich et al, PRL 43 (1979) 1892 CePd2Si2: Grosche et al, Physica B 223/224 (1996) 50 CeIn3: Mathur et al, Nature 394 (1998) 39 Strong-coupling theory near magnetic QCP Moriya et al: JPSJ 52 (1900) 2905 Monthoux &Lonzarich PRB 63 (2001) 054529 CeCu2Si2: Bellarbi et al, PRB 30 (1984)1182 CeCu2Ge2: Jaccard et al, IRAPT’97; Physica B 259/261 (1996) 1 CeRhIn5: Heeger et al, PRL 84 (2000) 4986 Suggesting new SC mechanism of repulsive origin

  14. QCP of Valence transition in Cu Ge Ce D. Jaccard et al, Physica B 259-261 (1999) 1 Rapid decrease of Kadowaki & Woods: SSC 58 (1986) 507 Rapid change of nf (valence of Ce) M. Rice & K. Ueda, PRB 34 (1986) 6420 cf. Gutzwiller arguments K.M. et al, PhysicaB 259-261 (1999) 676

  15. T-linear resistivity (T>Tc) in CeCu2Ge2 near P~Pv Jaccard et al: Physica B 230-232 (1997) 297

  16. Pressure scale is shifted such that Pc=0 CeCu2(Si1-xGex)2 x=0 (Thomas et al ’96) x=0 (Holmes et al ’03) x=1 (Jaccard et al ’97) x=0.1 (Yuan et al ’03) Two distinct Tc domes ! Enhancement of r0 at P=4GPa r(T) - r0 ∝ Ta r(T) - r0 ∝ T at P=4GPa H. Q. Yuan et al, Science 302 (2003) 2104 SCES’02 Krakow

  17. Pressure Signature of critical valence fluctuations observed in CeCu2(Ge,Si)2 around the critical pressure Pv PV 1) Enhancement of Tc (SC) 2) Enhancement of r0 3) T-linear reesistivity r(T) ~ T 4) Shoulder of g=C/T (at T=Tc) Holmes, Jaccard, KM: PRB 69 (2004) 024508 Two separate domes of Tc and 2) & 3) H. Q. Yuan et al, Science 302 (2003) 2104

  18. Fujiwara, Kobayashi et al: JPSJ 77 (2008) 123711 (1/T1T) ∝ A ∝γ2 NQR relaxation rates Line nodes: T3 – law at all pressures

  19. K. Fujiwara: SCES2011 suggesting sharp valence crossover 6

  20. Another SC mechanism for heavy fermions, enhanced valence fluctuations: Theoretical attempts.

  21. (Kondo regime)

  22. r～a : lattice const. (if kF~p/a) Real-space picture of pairing potential G(0)(q) Strong on-site repulsion short-range attraction d-wave pairing

  23. e +n U ~m f c fc super- conductivity Extended periodic Anderson model (PAM) with f-c Coulomb repulsion Ufc PAM pressure % Superconductivity with d-wave symmetry in the valence crossover region % Phase diagram at T=0 K in U-ef slave boson MF slave-boson mean-field & fluctuations 1st-order valence transition Mixed Valence QCP 1st-order valence transition Kondo crossover paramagnetic metal Onishi & KM: JPSJ 69 (2000) 3955 Watanabe et al.: JPSJ 75 (2006) 043710 1d DMRG calculations

  24. Intuitive picture for enhancement of residual resistivity Charge distribution around impurity at far from P~PV cond-electron (a) f-electron impurity Charge distribution around impurity at around P~PV cond-electron (b) f-electron xV impurity Effect of impurity remains as short-ranged r0 : not enhanced Effect of impurity extends to over long-range region xV diverging as P → PV r0 : highly enhanced

  25. Enhancement of r0 due to critical valence fluctuations as many-body effect on impurity potential p+k/2 q p-k/2 KM, Maebashi: JPSJ 71 (2002) 1007 In the forward scattering limit (k ~ 0), Ward-Pitaevskii identity cf. Betbeder-Matibet & Nozieres: Ann. Phys. 37 (1966) 17 for single component Fermi liquid Renormalized impurity potential Residual resistivity divergent at criticality ( ) : higher order corrections do not change the result. (cf. Rutherford scattering)

  26. Self-energy due to critical valence fluctuations 1st version A. T. Holmes, D. Jaccard, KM: Phys. Rev B 69 (2004) 024508 CeCu2Si2 Umklapp scattering 0<q<3kF/2 Peak structure of effective mass at P=Pv

  27. Variety of valence transition Location of critical point (Pvc,Tvc) in P-T plane depends on the details of materials parameters Assumption: there exist compounds such that Tvc~0 or Tvc << EF* Ce critical point Typical example of 1st order valence transition Ce:g-a transition Z. Fisk et al: J. Appl. Phys. 55 (1984) 1921

  28. Effect of magnetic field on valence transition and fluctuations causing a metamagnetic behavior S. Watanabe, A. Tsuruta, KM, J. Flouquet: PRL 100 (2008) 236401 & JPSJ 78 (2009) 104706.

  29. Drastic effect of magnetic field on valence transition Expected Phase Diagram in P-T-B space Drymiotis et al: J. Phys.: Condens. Matter 17 (2005) L77 How about in the case Tc<0 (i.e., in the crossover regime) ? Tc B B

  30. Mixed valence Kondo at h =0.01 QCP at h =0.02 Metamagnetic jump appears in crossover regime cf. A. J. Millis, A. J. Schofield, G. G. Lonzarich & S. A. Grigera, PRL 88 (2002) 217204 Field induced VQCP Reduction of TK Metamagnetism Field-induced VQCP 3 dimension Ufc ef S. Watanabe et al, PRL 100 (2008) 236401

  31. CeIrIn5 1st order V-T V-QCP Ufc Crossover of valence ef (P) In-NQR nQ starts to change at P ~ 2.1GPa 1st-order transition at h~42T Yashima & Kitaoka (2010) S. Kawasaki, et al: PRL 96(2006)147001 T. Takeuchi, et al: JPSJ 70(2001) 877 E. C. Palm, et al: Physica B329-333(2003) 587

  32. Promising candidate for magnetic field induced QCP-VT CeCu6 1st order V-T V-QCP Ufc Crossover of valence Fine tuning possible by changing H and P ef (P) (P) Raymond & Jaccard J. Low Temp. Phys. 120 (2000) 107 Jaccard et al: Physica B 259-261 (1999) 1

  33. Magneto resistivity of CeCu6 under pressure J // b, H // c ● ● ● % Signature of H and P induced QCP-VT ● ● ● ● % Measurements at higher H and P expected to exhibit much sharper structure Y. Hirose et al (Onuki group): J. Phys. Soc. Jpn. Suppl. (2012) accepted for publication

  34. Signatures of valence transition or crossover in Fermi surface change of CeRhIn5 S. Watanabe & K.M. JPSJ 79 (2010) 033707

  35. NK Sato Group (Nagoya) Knebel et al: JPSJ 77 (2008) 144704

  36. Pc dHvA result in CeRhIn5 H. Shishido, R. Settai, H. Harima & Y. Onuki, J. Phys. Soc. Jpn. 74 (2005) 1103 “Small” Fermi surfaces similar to those of LaRhIn5 for P<Pc drastic change at Pc=2.35 GPa “Large” Fermi surfaces similar to those of CeCoIn5 for P>Pc What is the nature of transition? Knebel et al: JPSJ 77 (2008) 144704 cf. Park et al: Nature 440 (2006) 65

  37. Transport anomalies in CeRhIn5 Pc e T. Muramatsu et al., JPSJ 70 (2001) 3362 e = 1 G. Knebel et al., JPSJ 77 (2008) 114704 “residual resistivity” has a peak at P=Pc Pc P (GPa) T. Park et al., Nature 456 (2008) 366 Signature of sharp valence crossover T-linear resistivity emerges most prominently near P=Pc

  38. scaling under pressure G. Knebel et al., JPSJ 77 (2008) 114704 at H=15 T Cyclotron mass of b2 branch by dHvA at H=12~17 T Shishido et al., JPSJ 74 (2005) 1103 Pc m* scales with Mass enhancement near P=Pc not from AF QCP but from band effect cf. K.M. et al., Solid State Commun. 71 (1989) 1149

  39. c f c c f f S. Watanabe, A. Tsuruta, K. Miyake & J. Flouquet, JPSJ 78 (2009) 104706 Extended periodic Anderson model 2D-like b2 branch square lattice (cf. half filling n = 1) filling: E(k) k first-order transition Mixed Valence Kondo

  40. Slave-boson mean field theory G. Kotliar & A. E. Ruckenstein, PRL 57 (1996) 1362 : probability for empty, singly-, & doubly-occupied states l , l’ , dl: Lagrange multipliers Q = (p,p): AF-ordered vector : mas renormalization factor 7 equations are solved self-consistently

  41. Mixed Valence nf Ufc Kondo 0.0 0.5 1.5 1.0 ef Ufc=0.5 1.177 (QCP) ms AF-paramagnetic boundary almost coincides with valence transition & valence crossover line Suppression of AF order by valence fluctuations ef in CeRhIn5 H=0 Ground state phase diagram n = 0.9 At quantum critical end point (QCP) of first-order valence transition, valence fluctuation diverges: t = 1, V = 0.2, U=∞ Ufc=1.0 cv Ufc 0.5 0.0 ef ef

  42. S. Watanabe et al., JPSJ 79 (2010) 033707 Drastic change of Fermi surface h=0.005 t = 1, V = 0.2, U=∞, Ufc=0.5, n = 0.9, kF :kF for conduction band ek at nc=0.8 “Small” Fermi surface changes to large Fermi surface at AF to paramagnetic transition discontinously

  43. E E S. Watanabe et al., JPSJ 79 (2010) 033707 Mass enhancement h=0.005 t = 1, V = 0.2, U=∞, Ufc=0.5, n = 0.9, As ef increases toward , Zsincreases Gap between original lower hybridized band & the folded band increases f-dominant flat part of the folded band approaches Fermi level m Mass enhancement by band effect This explains scaling G. Knebel et al., JPSJ 77 (2008) 114704

  44. Pc Comparison with dHvA measurement Larger D(m) for ef > missing b2 branch for P >Pc h makes D(m) small CeCoIn5 For ef=-0.4, D(m)=0.84 is about 10 times larger than Dc(m)=0.092 at nc=0.8 At P = 0 g = 50 mJ/molK2 in CeRhIn5 is about 10 times larger than g = 5.7 mJ/molK2 in LaRhIn5 H. Shishido et al., JPSJ 74 (2005) 1103

  45. Effect of hybridization strength on P-T phase diagram (C) (B) (A) V β-YbAlB4 etc. CeCu2(Si,Ge)2 CeRhIn5 under H CeCoIn5, CeIrIn5 (C’) S. Watanabe & KM: J. Phys. Condens. Matter, 23 (2011) 094217.

  46. New universality class of QCP: Critical end point of valence transition in heavy fermions S. Watanabe, KM: PRL 105 (2010) 186403

  47. Unconventional criticality in b-YbAlB4 r ~ T c ~ T -0.5 T(K) S. Nakatsuji et al., Nature Phys. 4 (2008) 603 Y. Matsumoto et al., arXiv:0908.1242 Uniform magnetic susceptibility is enhanced as c ~ T -0.5 even though the system in not close to the FM phase -logT Enhanced Wilson ratio

  48. Unconventional criticality Self Consistent Renormalization (SCR) theory for spin fluctuations: T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springre-Verlag, Berlin, 1985) T. Moriya & K. Ueda, Rep. Prog. Phys. 66 (2003) 1299 J. A. Hertz, PRB 14 (1976) 1165 A. J. Millis, PRB 48 (1993) 7183 RG study: r C/T c0 cQ1/T1T Fermi liquid T 2 constant 3D AF T 3/2 c-T 1/2c-T1/4T -3/2 C.W.T -3/4 3D F T 5/3 -lnT T -4/3 C.W.T -4/3 2D AF T -lnTc -lnT/T C.W. -lnT/T 2D F T4/3T -1/3 -1/(T lnT ) C.W. -1/(T lnT )3/2 YbRh2Si2 T -lnT T -0.6 T -0.5 b-YbAlB4 T 1.5T-lnT T -0.5 exp. desired

  49. c f Quantum criticality of VQCP Periodic Anderson model K. M, J. Phys.:Condens. Matter 19 (2007) 125201 S. Watanabe & K. M., Phys. Status Solidi B 247 (2010) 490 large U small U fc fc valence T T T crossover critical end point 1st order valence transition VQCP P P P Ce b-YbAlB4 Most of Ce, Yb compounds Construction of mode-coupling theory for valence fluctuations starting from HPAM cf. Hubbard model SCR theory for spin fluctuations Local correlation effect by U

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