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Microscopic structure and properties of superconductivity on the density wave background

Microscopic structure and properties of superconductivity on the density wave background. P. D. Grigoriev. L. D. Landau Institute for Theoretical Physics, Chernogolovka, Russia. Superconductivity and charge/spin-density wave:

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Microscopic structure and properties of superconductivity on the density wave background

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  1. Microscopic structure and properties of superconductivity on the density wave background P. D. Grigoriev L. D. Landau Institute for Theoretical Physics, Chernogolovka, Russia Superconductivity and charge/spin-density wave: 1). How can these two phenomena coexist? What is the microscopic structure of such phase? 2). How do the properties of SC change on the DW background? The results obtained explain many properties in layered organic DW superconductors: high Hc2, unconventional order, high Tc, upward curvature of Hc2z(T), triplet pairing on SDW background, etc. Publications:1). L.P. Gor'kov, P.D. Grigoriev, Europhys. Lett. 71, 425 (2005). 2). L.P. Gor'kov, P.D. Grigoriev, Phys. Rev. B 75, 020507 (2007). 3). P.D. Grigoriev, Phys. Rev. B 77, 224508 (2008). 4). P.D. Grigoriev, in preparation.

  2. 7 CDW / SDW band structure E Electron Hamiltonian in the mean field approximation:  2 The order parameter is a number for CDW, and a spin operator for SDW: kx Energy band diagrams Empty states E Energy spectrum in the CDW /SDW state 2  Perfect nesting condition: The energy gap in DW state prevents from SC ky

  3. CDW superconductors 3 Review paper: A.M. Gabovich, A.I. Voitenko, J.F. Annett and M. Ausloos, Supercond. Sci. Technol. 14, R1-R27 (2001)

  4. 3a SDW superconductors Review paper: A.M. Gabovich, A.I. Voitenko, J.F. Annett and M. Ausloos, Supercond. Sci. Technol. 14, R1-R27 (2001)

  5. 4b Coexistence of CDW and superconductivity in NbSe3 Phase diagram of NbSe3 Fermi surface Phys. Rev. B 64, 235119 (2001) S. Yasuzuka et al., J. Phys. Soc. Jpn. 74, 1782 (1982)

  6. 4a Coexistence of CDW and superconductivity in sulfur Phase diagram of sulfur Fermi surface Observed maximum atomic displacement in S-IV and S-V as a function of pressure and temperature, shown as open diamond symbols. The temperature of the superconducting transition Tc from Ref. [E. Gregoryanz et al., Phys. Rev. B 65, 064504 (2002)] is shown by yellow triangles. The temperature is given on a logarithmic scale. O. Degtyareva et al., PRL 99, 155505 (2007)

  7. 4 Experimental phase diagrams in organic metals (TMTSF)2PF6:T.Vuleticet al., Eur. Phys. J. B 25, 319 (2002) -(BEDT-TTF)2KHg(SCN)4:D. Andres et al., Phys. Rev. B 72, 174513 (2005) External pressure damps SDW, but SC appears before SDW is completely destroyed. ! There is a pressure region where SC coexists with SDW or with CDW

  8. 4 Quasi-1D metals and Peierls instability Electron dispersion in quasi-1D metals (tight-binding approximation) ky antinesting term Nesting condition: kx Fermi surface External pressure increases the antinesting term t’y and damps the DW. Nesting vector QN What is the structure of coexisting SC and DW? (TMTSF)2PF6

  9. 29b Macroscopic coexistence of superconductivity or normal metal with DW insulator SC This model explains the anomalous increase of Hc2 and its upward curvature only if the domain size dS <<SC. The nonuniform DW structure costs energy 0>> SC , and the soliton structure is more favorable, where the energy loss 0 is compensated by the gain ~t’b of the kinetic energy in the soliton band. I. J. Lee et al, PRL 88, 207002 (2002) dS E  soliton band 2 ky

  10. 29 Two mechanisms of microscopic coexistence of superconductivity or normal metal with DW [ L.P. Gor'kov, P.D. Grigoriev, Phys. Rev. B 75, 020507 (2007) ] 1. Ungapped pockets of FS. Empty band E The antinesting dispersion 2  ungapped pockets Pockets appear when ky 2. Soliton phase (non-uniform). E The SDW order parameter depends on the coordinate along the 1D chains:  soliton band 2 or ky

  11. P1 Procedure of the theoretical analysis Step 1: Describe the DW in the mean field approximation. a). Calculation of the quasi-particle energy spectrum and Green functions as function of pressure (imperfect nesting). b). Renormalization of the e-e coupling by the DW critical fluctuations. Step 2: Describe superconductivity with the new quasi-particle spectrum and new e-e interaction potential. a). Estimate the SC transition temperature with new quasi-particle energy spectrum and new e-e interaction potential. b). Consider the influence of the spin-structure of SDW on SC. c). Calculate the upper critical field Hc2 for SC on the CDW and SDW background. This procedure allows to investigate the superconducting properties on the DW background and to explain many experimental observations !

  12. D2 DoS in the open-pocket scenario (DW-SC separation in the momentum space) 1 The density of states (DoS) in the density wave (DW) state with open pockets remains large in DW: Empty band E 2  () 0 ungapped pockets of size   ky 0 0  Due to the small open pockets at the Fermi level, the DoS is the same, as in the metallic phase. Hence, the superconducting transition temperature is not exponentially smaller in the DW state! Renormalization of the effective e-e interaction in the Cooper channel by critical DW fluctuations can make TcSC even higher than without DW [ P.D. Grigoriev, “Properties of superconductivity coexisting with a density wave with small ungapped FS parts”, Phys. Rev. B 77, 224508 (2008). ]

  13. Suppression of spin-singlet SC by SDW backgroundappears in both models in agreement with experiments[ L.P. Gor'kov, P.D. Grigoriev, Phys. Rev. B 75, 020507 (2007) ] 2 Critical magnetic field Hc exceeds ~5 times the paramagnetic limit: Knight shift does not change as temperature decreases: absorption I.J. Lee, P. M. Chaikin, M. J. Naughton, PRB 65, 18050(R)(2002) I.J. Lee et al., PRB 68, 092510 (2003) ! Critical magnetic field and the Knight shift in (TMTSF)2PF6 in the superconductivity-SDW coexistence phase confirm the triplet paring. The absence of gap nodes suggests px symmetry of order parameter.

  14. 13 Equations for SC instability in SDW phase If we introduce the diagonal and non-diagonal Cooper bubbles: L L R R L R L fLR = fRL+ fLR R L R R L R L R R L L R L R fRL = fLR + fRL L R L L R L R the self-consistency equations for superconductivity rewrite: SDW spin structure The spin-singlet superconducting order parameter anticommutes with SDW order parameter: which results in the SC equation: and Tc is exponentially smaller than without SDW.

  15. 15 Triplet superconductivity in SDW or CDW. The self-consistency equations for superconductivity: The triplet superconducting order parameter is Using the commutation identity for triplet pairing with we obtain the SC equation on SDW background: Infrared singularities cancel each other as for singlet SC on SDW. one obtains For Infrared singularities do not cancel. while for one has

  16. Why the spin structure of SDW background suppresses the spin-singlet superconductivity (illustration) Spin-dependent scattering:the sign of the scattered electron wave function depends on its spin orientation. spin-triplet SC pair -QN singlet SC pair after scattering by SDW Direct SC singlet pairing The two-electron wave function acquires “” sign after scattering by SDW if the electron spins in this pair look in opposite directions. QN Fermi surface Nesting vector QN This affects only the infrared divergence in the Cooper logarithm. The ultraviolet divergence remains unchanged. 16

  17. 27a Electron dispersion in the ungapped FS pockets on the DW background is strongly changed 3 Small ungapped pockets on a FS sheet, which get formed when the antinesting term in the electron dispersion exceeds CDW energy gap. The quasi-particle dispersion in these small pockets is where

  18. 13 3 Result for Hc2z on uniform DW background [ P.D. Grigoriev, Phys. Rev. B 77, 224508 (2008). ] where is the size of the new FS pockets. the constant C1 depends on electron dispersion. For some dispersion For tight-binding dispersion with only two harmonics In all cases, since the size of new FS Hence,Hc2 diverges as PPc1 : which agrees well with experiment.

  19. 26 Critical magnetic field in the coexistence phase CDW + superconductivity: -(BEDT-TTF)2KHg(SCN)4:D. Andres et al., Phys. Rev. B 72, 174513 (2005) (TMTSF)2PF6:J. Lee, P. M. Chaikin and M. J. Naughton, PRL 88, 207002 (2002) ! The critical magnetic field Hc2 has very unusual temperature and pressure dependence.

  20. 29 Two mechanisms of microscopic coexistence of superconductivity or normal metal with DW 1. Ungapped pockets of FS lead to SC with unusual properties. Empty band E 2 [ P.D. Grigoriev, PRB 77, 224508 (2008) ]  The antinesting dispersion ungapped pockets ky 2. Soliton phase (non-uniform). E The SDW order parameter depends on the coordinate along the 1D chains:  soliton band 2 or ky

  21. 35 Energy of soliton phase in Q1D case Soliton phase linear energy: Schematic picture of energy bands E where nis the soliton wall linear density,  Boundaries E_ of the soliton level band 2 is the soliton wall energy per chain, ky is the width of center allowed band (appearing due to periodic domain walls) and The soliton level band is only half-filled and the system gains the energy (the second term in A) which can be greater than the soliton wall energy cost gives the soliton wall interaction energy. ! Then the soliton phase is the thermodynamically stable state. [S.A. Brazovskii, L.P. Gor'kov, A.G. Lebed', Sov. Phys. JETP 56, 683 (1982)]

  22. 36 Region of soliton phase in Q1D metals for various electron dispersions To determine the phase diagram one has to compare the energies of uniform DW phase, soliton phase and normal metal phase. Fortight- binding model with only two harmonics in the dispersion all critical values 2t’y=0 coincide and the soliton phase has zero region. For step-like dispersion E the soliton phase has very large region ky [ L.P. Gor'kov, P.D. Grigoriev, "Soliton phase near antiferromagnetic quantum critical point in Q1D conductors", Europhysics Letters 71, 425 (2005)].

  23. 37 Energy of soliton phase (intermediate general case dispersion) For the intermediate electron dispersion the interval of soliton phase can be about 10% of pC in agreement with experiment in (TMTSF)2PF6 . The SDW–SP transition at pC1 is of the second kind while the SP–Metal transition at pCis of the first kind also in agreement with experiment. [ L.P. Gor'kov, P.D. Grigoriev, Europhysics Letters 71, 425 (2005)] The domain phase observed in (TMTSF)2PF6 may be the soliton phase.

  24. 38 Superconductivity in the soliton phase (suppression of spin-singlet SC by SDW background) The Green functions in the soliton phase are 4x4 matrices: L L R R L R L fLR = fRL+ fLR R L R R L R L Self-consistency Gor’kov equations for superconductivity in soliton phase: R R L L R L R fRL = fLR + fRL L R L L R L R The sign “-” leads to the cancellation of diagonal and non-diagonal Cooper blocks in the SC equations for singlet superconductivity in the soliton band,which means the suppression of spin-singlet SC by the DW background.This cancellation doesn’t happen for singlet SC in CDW soliton, or for triplet SC in the SDW soliton phase.

  25. 40 Calculation of SC upper critical field on the soliton phase background We use again the Ginzburg-Landau approximation: Upper critical field where The electron dispersion :

  26. 41 Width of soliton band in Q1D metals From the soliton phase linear energy where the soliton wall linear density and one obtains the width of the soliton band: In the tight-binding model with only two harmonics near the transition at P = Pc1(where2t’b=0) and

  27. Upper critical field in SC state on soliton-phase background. 42 Result: close to Tc and the constant C1s depends on the electron dispersion. where For tight binding dispersion The width of the soliton band and Hc2 diverges as PPc1 : which agrees well with experiment.

  28. Upward curvature of Hc2z(T) insulator SC Solitons create a layered structure, which is described by the Lawrence-Doniach model of 1D Josephson lattice. Upper critical field in this Josephson lattice is s where ds=s is the interlayer distance. This model was generalize for finite width of SC layers in [G. Deutcher and O. Entin-Wohlman, Phys. Rev. B 17, 1249, (1978) ]. The divergence of upper critical field is cut off by Hc2 in a superconducting slab:

  29. Upper critical field Hc2z in-(BEDT-TTF)2KHg(SCN)4 CDW + superconductivity: TcSC<TcDW 100 times, and the energy of SC state is 4 orders less than DW energy. Hence, no strong influence of SC on DW is possible (as adjusting of the size of DW domains with magnetic field), an the macroscopic domains cannot explain this Hc2z behavior -(BEDT-TTF)2KHg(SCN)4:D. Andres et al., Phys. Rev. B 72, 174513 (2005)

  30. 44 Origin of hysteresis. Phase diagram The observed hysteresis in resistance at temperature change can be explained in both scenarios. For open-pocket scenario of DW1 hysteresys is due the shift of the DW wave vector at P>Pc1 In the soliton scenario of DW1 the hysteresys is due the sliding of soliton walls.

  31. Conclusions • There are, at least, 2 possible structures of a DW1 state, where superconductivity coexists microscopically with density wave. • The SC properties of such state are investigated for both structures: • 1). The DoS on the Fermi level in DW1 is rather high, giving possibility of SC. • 2). The SDW background suppressed the spin-singlet SC coupling, • leaving the triplet SC transition temperature almost without change. • 3). The upper critical field increases at critical pressure Pc1, where SC first appears, and shows unusual temperature (upward curvature) and pressure dependence. • III. The results agree with experiment in organic metals (TMTSF)2PF6 and -(BEDT-TTF)2KHg(SCN)4, explaining many unusual properties. Publications:1). L.P. Gor'kov, P.D. Grigoriev, "Soliton phase near antiferromagnetic quantum critical point in Q1D conductors", Europhys. Lett. 71, 425 (2005). 2). L.P. Gor'kov, P.D. Grigoriev, " Nature of superconducting state in the new phase in (TMTSF)2PF6 under pressure", Phys. Rev. B 75, 020507 (2007). 3). P.D. Grigoriev, “Properties of superconductivity coexisting with a density wave with small ungapped FS parts”, Phys. Rev. B 77, 224508 (2008). 4). P.D. Grigoriev, “Superconductivity on the density wave background with soliton-wall structure”, in preparation.

  32. Thank you for the attention !

  33. Conclusions • We developed the theory, describing superconductivity on SDW or CDW background when TcDW>>TcSC in quasi-1D compounds with one conducting band. • There are two possible microscopic structures of DW1 phase, where SC may coexist microscopically with DW: (1) uniform structure with ungapped states in momentum space (open pockets); (2) non-uniform soliton phase. • The DoS at the Fermi level in DW1 state in both scenarios is rather high, which makes TCSC on DW background comparable with TCSC in pure SC state. The enhancement of the e-e interaction by critical fluctuations may increase TcSC even to the value higher than without DW. • The upper critical field is calculated in both scenarios and shown to considerably exceed the usual Hc2. It diverges at critical pressure Pc1, where SC first appear, andshowsunusual temperature (upward curvature) and pressure dependence. • The SDW background strongly damps singlet SC. The SC, appearing on SDW background in metals with single conducting band, should be triplet. • The hysteresis of R(T) may appear in both scenarios (for different reasons). 7.The results obtained are in good agreement with experimental observations in organic metals (TMTSF)2PF6 and -(BEDT-TTF)2KHg(SCN)4 . Publications: 1). L.P. Gor'kov, P.D. Grigoriev, "Soliton phase near antiferromagnetic quantum critical point in Q1D conductors", Europhys. Lett. 71, 425 (2005). 2). L.P. Gor'kov, P.D. Grigoriev, " Nature of superconducting state in the new phase in (TMTSF)2PF6 under pressure", Phys. Rev. B 75, 020507 (2007). 3). P.D. Grigoriev, “Properties of superconductivity coexisting with a density wave with small ungapped FS parts”, Phys. Rev. B 77, 224508 (2008). 4). P.D. Grigoriev, “Superconductivity on the density wave background with soliton-wall structure”, in preparation.

  34. Lawrence-Doniach model insulator SC [ Lawrence, W. E., and Doniach, S., in Proceedings of the 16th International Conference on Low Temperature Physics, ed. E. Kanda, Kyoto: Academic Press of Japan, p. 361 (1971). ] s Here

  35. Lawrence-Doniach model (2). Introducing The lowest eigenvalue of this equation gives upper critical field:

  36. 44 Which of the two proposed microscopic structures appears in the experiment? The observed hysteresis in resistance for increasing and decreasing magnetic field suggests the soliton phase (spatial inhomogeneity in the form of microscopic domains). The high upper critical field Hc2 suggests the domain size is much less than the SC coherence length, because for a SC slab This means, that superconducting domains must be microscopically narrow, supporting that the soliton scenario takes place.

  37. 45 NMR experiments in (TMTSF)2PF6 NMR absorption line Red= normal state; Blue= zero width; Black=wide soliton. Lineshapesfor incommensurate SDWs, with different solitonwidths, using hyperbolic tangent function for describing solitons. Stuart Brown et al., UCLA, Dresden, 2005.

  38. Upward curvature ofHc2(T) CDW + superconductivity: -(BEDT-TTF)2KHg(SCN)4:D. Andres et al., Phys. Rev. B 72, 174513 (2005) (TMTSF)2PF6:J. Lee, P. M. Chaikin and M. J. Naughton, PRL 88, 207002 (2002) The upward curvature of Hc2(T) also suggests the soliton structure

  39. 21 Model with two coupling constants in e-e interactions for forward and backward scattering Electron Hamiltonian is , where the free-electron part And the e-e interaction has two coupling constants for forward and backward scattering: (keeps electrons on the same FS sheet) (scatters electrons to the opposite FS sheet) where The CDW or SDW onset is due to the interaction with Q=QN only, while the SC onset is due to the interaction with all other Q. Therefore, the same interaction constants lead to both DW and SC.

  40. 27 Calculation of upper critical field when superconductivity coexists with CDW or SDW We use the Ginzburg-Landau approximation: then where [ L.P. Gor'kov and T.K. Melik-Barkhudarov, JETP 18, 1031(1963) ]

  41. 4t Previous theoretical results on SC+DW. DW reduces the SC transition temperature since it creates an energy gap on the part or on the whole Fermi surface. [ K. Levin, D. L. Mills, and S. L. Cunningham, Phys. Rev. B 10, 3821 (1974); C. A. Balseiro and L. M. Falicov, Phys. Rev. B 20, 4457 (1979). ] Model with initially imperfect nesting or with several conducting bands. ( CDW leaves some electron states on the Fermi level and does not affect the dispersion of the unnested parts of Fermi surface. ) [ General properties:K. Machida, J. Phys. Soc. Jpn. 50, 2195 (1981); Hc2 : A. M. Gabovich and A. S. Shpigel, Phys. Rev. B 38, 297 (1988). ] 3). Proximity to the Peierls (DW) instability increases the effective e-e interaction g(Q) with the wave vector Q QN: The RPA result gives

  42. P1 Why the proposed approach is different? In fact, the DW may considerably change the quasi-particle dispersion even on the ungapped parts of Fermi surface ! New properties in DW superconductors appear: 1). SC transition temperature Tc is higher than expected (not exponentially smaller than Tc without DW). With renormalization of the coupling constant g(Q) by critical fluctuations it may be even higher than without DW. 2). The upper critical field Hc2 may be strongly enhanced as compared to SC without DW.

  43. P1 Procedure of the theoretical analysis Step 1: Describe the DW in the mean field approximation. a). Calculation of the quasi-particle energy spectrum and Green functions as function of pressure (imperfect nesting). b). Renormalization of the e-e coupling by the DW critical fluctuations. Step 2: Describe SC with the new quasi-particle spectrum and new e-e interaction potential. a). Estimate the SC transition temperature with new quasi-particle energy spectrum and new e-e interaction potential. b). Consider the influence of the spin-structure of SDW on SC. c). Calculate the upper critical field Hc2 for SC on the CDW and SDW background.

  44. H Model for a quasi-1D metal Dispersion relation of electrons in quasi-1D metals in magnetic field imperfect nesting term Hamiltonian where the free-electron term and the electron-electron interaction is given by For CDW or SDW UCand US are just the charge and spin coupling constants (being taken at the wave vector transfer Q=QN). For SC the functional dependence of UC (Q) and US (Q) is important (it determines the type of pairing). The couplings have maximum at the wave vector transfer Q=QN (the backward scattering is enhanced).

  45. 27a Electron dispersion in the ungapped FS pockets on the DW background in tight-binding approximation The important contribution to Cooper logarithm and to SC properties comes from the ungapped electron states on the Fermi level. E Empty band Small ungapped pockets on a FS sheet get formed when the antinesting term in the electron dispersion exceeds DW energy gap. 2  The quasi-particle dispersion in these small pockets ungapped pockets ky where Effective mass

  46. Enhancement of the e-e coupling by the proximity to DW transition (critical fluctuations) In RPA the renormalized e-e interaction is given by the sum of diagrams: = + + .. where g0(Q)<<1 is the bare interaction, This gives and the susceptibility may diverge at some (nesting) wave vector, so that Then the new coupling also diverges at some Q. The original coupling g0(Q) may be more complicated (include spin). Then the renormalized coupling includes all components of g0(Q). The new coupling g(Q) is strongly Q-dependent, being considerably changed only in the vicinity of the DW wave-vector. Therefore, the SC coupling doesn’t change for almost the whole FS except “hot spots”.

  47. The enhancement of e-e coupling depends very strongly on the bare e-e interaction (example) Consider the Hubbard model with two coupling functions U and V(Q) Then the RPA gives the following renormalization of the couplings in the superconducting singlet and triplet channels: where the spin and charge susceptibilities and The renormalized SC couplings depend very strongly on the bare interaction U and V(Q) Y. Tanaka and K. Kuroki, PRB 70, 060502(R) (2004)

  48. D1 The density of states at the Fermi level (1) Without DW the DoS in Q1D metal is In the presence of DW or where and for small FS pockets

  49. 17 Result1: Comparison of singlet Tc on metallic, CDW and SDW states without change of e-e interaction 1. Normal metal background: and  is the size of the ungapped parts of FS 2. CDW background: and Not too small. 3. SDW background: which gives very low Tc:

  50. Why the spin structure of SDW background suppresses the spin-singlet superconductivity (illustration) Spin-dependent scattering:the sign of the scattered electron wave function depends on its spin orientation. spin-triplet SC pair -QN singlet SC pair after scattering by SDW Direct SC singlet pairing The two-electron wave function acquires “” sign after scattering by SDW if the electron spins in this pair look in opposite directions. QN Fermi surface Nesting vector QN This affects only the infrared divergence in the Cooper logarithm. The ultraviolet divergence remains unchanged. 16

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