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Universal Scaling Relations in Molecular Superconductors

Universal Scaling Relations in Molecular Superconductors. Francis Pratt ISIS Facility, Rutherford Appleton Laboratory, Oxfordshire, UK Stephen Blundell Clarendon Laboratory, University of Oxford, UK. Main collaborations: Oxford, St. Andrews, Birmingham, Zurich groups

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Universal Scaling Relations in Molecular Superconductors

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  1. Universal Scaling Relations in Molecular Superconductors Francis Pratt ISIS Facility, Rutherford Appleton Laboratory, Oxfordshire, UK Stephen Blundell Clarendon Laboratory, University of Oxford, UK

  2. Main collaborations: Oxford, St. Andrews, Birmingham, Zurich groups mSR experiments at PSI and ISIS Tohoku University Samples University of Queensland Theory

  3. Outline of Talk • Universal scaling relations: Uemura and Homes laws • Measurement of superfluid strength rs using mSR • Uemura-style scaling Tc : rs • Homes-style scaling rs : s0(Tc)Tc • Possible interpretations and challenges for theory PRL 94, 097006 (2005)

  4. Universal Scaling Relations • Search for simple ‘laws’ within the class of molecular superconductors • Find aspects independent of the fine details • Guidance for developing theories of sc • Stringent test for ‘good’ theories

  5. Interested here in scaling between: • ‘Strength’ of superconductor against B: rs(superfluid strength = c2/l2) • Energy scale of pairing interactions: Tc (sc transition temperature) • Strength of normal state interactions s0 (normal state conductivity)

  6. Superfluid Strength London Equation for the supercurrent induced by a magnetic field: m0 js = - (rs / c2) A rs referred to as Superfluid ‘Density’,‘Stiffness’ or ‘Strength’ in terms of a superfluid plasma frequency: rs = wp,s2 Tc andrs= c2/l2 are the primary experimental observables of a superconductor

  7. ‘Uemura Plot’ for Cuprates and Other Superconductors • Tc s (mSR relaxation rate) • Equivalently: Tc  ns/m* Tc  rs (superfluid strength) Tc  1/l2 (l is penetration depth)

  8. Interpretation of Uemura Scaling Universal treatment combining 2D and 3D systems: TB  TF (2D/3D) Tc ~ 0.1 TB 2D Bose Einstein Condensation for which Tc  ns/mb Y.J. Uemura, Physica C282-287,194 (1997) Also superfluid phase stiffness interpretation for which Tc  rs  ns/mb Emery and Kivelson, Nature 374,434 (1995)

  9. Nature 430, 639 (2004) ‘Homes Plot’ for Cuprates and Other Superconductors Superfluid strength rs scales with the product of conductivity and Tc

  10. Interpretation of Homes Scaling Sum-rule for optical response: i.e. area of superfluid peak = area lost to gap Small scattering rate G: e0rs = (2/p)s0 (p/2) G e0rs= s0G Large scattering rate G: e0rs = (2/p)s02D/ e0rs=(2/p) (kB/ )  s0 Tc Where 2D =  kB Tc Drude form s(w) = s0/(1+(w/G)2)

  11. Measuring Properties of Type II Superconductors H < Hc1 : Meissner state Surface measurement: l Abrikosov Vortex Lattice Hc1 < H < Hc2 : Vortex state Bulk measurement: l, x saddles RMS Width: Brms or s Lineshape: b = (Bave - Bpk) / Brms (skewness) cores minima

  12. Muon Spin Rotation Spectrum

  13. Extracting rs : Field Dependence Example of TMTSF2ClO4 :

  14. Extracting rs : Orientational Averaging Example: 3D powder average for lc >> la,lb Example: Average for Ha (needle axis)

  15. Extracting rs : Flux phases Example: measuring 2D melting/decoupling in k-ET2Cu(NCS)2

  16. What about Uemura-style Scaling in Molecular Superconductors? n and m* don’t vary much across the family of molecular superconductors: they should all occupy the same region on the Uemura plot

  17. Take a Closer Look at the Molecular Family BETS2GaCl4 provides a dramatic example:

  18. Resistive Transition in k-BETS2GaCl4 Tanaka et al, J. Mater Chem 10,245 (2000) GaBr4 GaCl4 k-BETS2GaBr4 FLP + Ali Bangura

  19. rs across the range of Molecular Superconductors

  20. rs across the range of Molecular Superconductors

  21. Uemura Plot for the Molecular Superconductors Molecular systems have their own empirical scaling law: Tc follows 1/l3 rather than 1/l2 ⇒ Tc (ns/mb) 3/2

  22. For boson dispersion law: CS 1 / mB Tc for BEC in d dimensions is: Casas et al, Phys. Lett. 245, 55 (1998) Departure from Linear Uemura Scaling (1) BEC view: effect of dimensionality and boson dispersion law on Tc For 2D quadratic case (d=2, s=2):Tc nB / mB (Uemura) But linear dispersion is expected for normal conditions (Schrieffer 1964): Tc nB / mB (d=1) Tc nB1/2 / mB (d=2) Tc nB1/3 / mB (d=3) None of these show Tc  (nB/mB) 3/2

  23. Departure from Linear Uemura Scaling (2) Czart and Robaszkiewicz PRB 64,104511 (2001) Penson and Kolb PRB 33,1663 (1986) Fermion hopping versus Cooper pair hopping : the Penson-Kolb model

  24. Correlation between Normal State Conductivity and Superconducting Properties:Search for Homes-style scaling

  25. Parameter values for the molecular superconductors Label Material Tcls0(Tc) (K) (mm) (103 S cm-1) 1 k-BETS2GaCl4 0.16(2) 7 2.3(1) 7 250(25) 13 2 TMTSF2ClO4 1.1(1) 8 1.27(3) 8 39(6) (a-axis) 14 3 a-ET2NH4Hg(SCN)4 1.1(1) 9 1.1(1) 9 36(6) 15 4 b-ET2IBr2 2.2(1) 9 0.90(3) 9 26(2) 16 5 l-BETS2GaCl4 5.5(1) 7 0.72(2) 7 11(1) 17 6 k-ET2Cu(NCS)2 9.2(2) 9,10 0.54(2) 9,10 6(1) 18 7 K3C60 18.9(1) 11 0.48(2) 11 2.9(9) 19 8 Rb3C60 29.3(1) 12 0.42(2) 12 2.5(6) 20 Values for Tc and l are derived simultaneously from muon spin rotation studies. s0(Tc) is the normal state conductivity in the most highly conducting direction. The conductivity is derived from reported multi-contact resistance measurements in the case of the organics, single-domain STM measurements for K3C60 and far-infrared reflectivity for Rb3C60. lab High s

  26. ‘Homes Plot’ comparing Molecular and Other Superconductors • Key: • k-BETS2GaCl4 • TMTSF2ClO4 • a-ET2NH4Hg(SCN)4 • b-ET2IBr2 • l-BETS2GaCl4 • k-ET2Cu(NCS)2 • K3C60 • Rb3C60 Homes scaling law rs  s0Tc doesn’t apply to molecular superconductors

  27. 2D 1D 2D 2D 2D 2D 3D 3D 1D, 2D & 3D systems SC properties correlate with highest s direction Closer look at Superconducting Parameters vs Conductivity • Key: • k-BETS2GaCl4 • TMTSF2ClO4 • a-ET2NH4Hg(SCN)4 • b-ET2IBr2 • l-BETS2GaCl4 • k-ET2Cu(NCS)2 • K3C60 • Rb3C60 s0- 1.05 s0- 0.77 s0+ 0.75 Note the completely opposite rs - s0 scaling between molecular and cuprate superconductors

  28. Superconducting Parameters vs Conductivity Low-conductivity direction across a range of materials Dordevic et al PRB 65,134511 (2002) Low s direction: rs s0m m = +0.85 (all systems) High s direction: m = +0.75 (cuprates) m = -0.77 (molecular)

  29. ‘Uemura Plot’ on Log-Log Scale including Cuprates etc For molecular superconductors Tc rs3/2 (compared with Tc rs for underdoped cuprates) • Key: • k-BETS2GaCl4 • TMTSF2ClO4 • a-ET2NH4Hg(SCN)4 • b-ET2IBr2 • l-BETS2GaCl4 • k-ET2Cu(NCS)2 • K3C60 • Rb3C60

  30. Departure from Homes Scaling in Molecular Superconductors • Key: • k-BETS2GaCl4 • TMTSF2ClO4 • a-ET2NH4Hg(SCN)4 • b-ET2IBr2 • l-BETS2GaCl4 • k-ET2Cu(NCS)2 • K3C60 • Rb3C60 Assuming G>2D :e0rs=(2/p) (kB/ )  s0 Tc Effective gap parameter h Weak coupling BCS strong coupling → h > 3.53 forh < 3.53 either m*s > m*n or ns < nn

  31. Measured Optical Properties Clearly for Fullerenes G>2D For organic metals it seems that G<2D Degiorgi et al PRB 49, 7012 (1994) Kornelsen et al PRB 44,5235 (1991) Drichko et al Europhys. Lett. 59, 774 (2002)

  32. Effective Frequency Width Ge versus Tc Take Ge = e0rs/s0 : if G<2D then Ge = G(Tc) if G>2D then Ge = (2/p) 2D/  • Key: • k-BETS2GaCl4 • TMTSF2ClO4 • a-ET2NH4Hg(SCN)4 • b-ET2IBr2 • l-BETS2GaCl4 • k-ET2Cu(NCS)2 • K3C60 • Rb3C60 Ge  Tcn with n=1.58 is consistent with the typical resistivity vs T laws in range n=1.5 to n=2

  33. Is there a single controlling parameter? k-ET2X Phase Diagram Tc decreases as s0(Tc) increases ← increasing U/W Caulfield et al JPCM 6, 2911 (1994) • The simplicity of the scaling suggests a single dominant control parameter • U/W is a likely candidate for molecular systems, which are generally rather close to a Mott insulator phase • Pressure can be used to tune U/W • Increasing pressure decreases U/W, increases s0 and decreases Tc and rs , following the trends expected from the scaling curves

  34. Dynamical Mean-Field Theory Loss of quasiparticle spectral weight is expected as the Mott-Hubbard transition is approached

  35. Superfluid Strength vs U/W Merino and McKenzie PRB61, 7996 (2000) Powell and McKenzie PRL94, 047004 (2005) DMFT RVB rsZ Experimental picture Feldbacher et al, PRL93, 136405 (2004) DMFT

  36. Summary and Conclusion • New ‘Universal Scaling’ properties revealed for molecular superconductors by combining mSR studies with transport data • Modified form of Uemura scaling found • No Homes scaling seen: both rs and Tc decrease with increasing normal state conductivity (contrast to other superconductors such as cuprates) • The scaling properties provide a challenge for theory • Studies of low Tc systems are important, k-phase provides widest Tc range

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