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 H c2, unconventional order, high T c, upward curvature of H c2 z (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. CDW / SDW band structure Energy spectrum in the CDW /SDW state Perfect nesting condition: Empty states 22 kyky E  Electron Hamiltonian in the mean field approximation: The order parameter is a number for CDW, and a spin operator for SDW: 22 kxkx E  Energy band diagrams 7 The energy gap in DW state prevents from SC

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

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

5. Coexistence of CDW and superconductivity in NbSe 3 Fermi surfacePhase diagram of NbSe 3 Phys. Rev. B 64, 235119 (2001) S. Yasuzuka et al., J. Phys. Soc. Jpn. 74, 1782 (1982) 4b

6. Coexistence of CDW and superconductivity in sulfur Fermi surface Phase diagram of sulfur O. Degtyareva et al., PRL 99, 155505 (2007 ) 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. 4a

7. Experimental phase diagrams in organic metals 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 (TMTSF) 2 PF 6 : T.Vuletic et al., Eur. Phys. J. B 25, 319 (2002)  -(BEDT-TTF) 2 KHg(SCN) 4 : D. Andres et al., Phys. Rev. B 72, 174513 (2005) 4

8. Quasi-1D metals and Peierls instability Nesting vector Q N Fermi surface Electron dispersion in quasi-1D metals (tight-binding approximation) External pressure increases the antinesting term t’ y and damps the DW. Nesting condition: kxkx kyky 4 antinesting term (TMTSF) 2 PF 6 What is the structure of coexisting SC and DW?

9. Macroscopic coexistence of superconductivity or normal metal with DW 29b SC insulator This model explains the anomalous increase of H c2 and its upward curvature only if the domain size d S >  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. dSdS soliton band 22 kyky E  I. J. Lee et al, PRL 88, 207002 (2002)

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

11. 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 H c2 for SC on the CDW and SDW background. P1 This procedure allows to investigate the superconducting properties on the DW background and to explain many experimental observations !

12. DoS in the open-pocket scenario (DW-SC separation in the momentum space) The density of states (DoS) in the density wave (DW) state with open pockets remains large in DW: 0  00 00 ()()  D2 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 T c SC even higher than without DW Empty band 22 kyky E  ungapped pockets of size  [ P.D. Grigoriev, “Properties of superconductivity coexisting with a density wave with small ungapped FS parts”, Phys. Rev. B 77, 224508 (2008). ] 1

13. Suppression of spin-singlet SC by SDW background appears in both models in agreement with experiments [ L.P. Gor'kov, P.D. Grigoriev, Phys. Rev. B 75, 020507 (2007) ] I.J. Lee, P. M. Chaikin, M. J. Naughton, PRB 65, 18050(R ) (2002) ! Critical magnetic field and the Knight shift in (TMTSF) 2 PF 6 in the superconductivity-SDW coexistence phase confirm the triplet paring. The absence of gap nodes suggests p x symmetry of order parameter. Knight shift does not change as temperature decreases: absorption I.J. Lee et al., PRB 68, 092510 (2003) Critical magnetic field H c exceeds ~5 times the paramagnetic limit: 2

14. Equations for SC instability in SDW phase If we introduce the diagonal and non-diagonal Cooper bubbles: the self-consistency equations for superconductivity rewrite: 13 SDW spin structure R L f LR = R L f RL + L R f RL = L R f LR + R R L LR L f LR L RR R R L L L L R f RL R L L R 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. Triplet superconductivity in SDW or CDW. The triplet superconducting order parameter is Using the commutation identity for triplet pairing with we obtain the SC equation on SDW background: The self-consistency equations for superconductivity: For one obtains while forone has Infrared singularities cancel each other as for singlet SC on SDW. Infrared singularities do not cancel. 15

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

17. Electron dispersion in the ungapped FS pockets on the DW background is strongly changed 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 27a 3

18. Result for H c2 z on uniform DW background For some dispersion For tight-binding dispersion with only two harmonics where is the size of the new FS pockets. In all cases, since the size of new FS Hence, H c2 diverges as P  P c1 : which agrees well with experiment. 13 the constant C 1 depends on electron dispersion. [ P.D. Grigoriev, Phys. Rev. B 77, 224508 (2008). ] 3

19. Critical magnetic field in the coexistence phase (TMTSF) 2 PF 6 : J. Lee, P. M. Chaikin and M. J. Naughton, PRL 88, 207002 (2002) 26 ! The critical magnetic field H c2 has very unusual temperature and pressure dependence.  -(BEDT-TTF) 2 KHg(SCN) 4 : D. Andres et al., Phys. Rev. B 72, 174513 (2005) CDW + superconductivity:

20. 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 22 kyky E  ungapped pockets The antinesting dispersion soliton band 22 kyky E  2. Soliton phase (non-uniform). The SDW order parameter depends on the coordinate along the 1D chains: or 29 [ P.D. Grigoriev, PRB 77, 224508 (2008) ]

21. Energy of soliton phase in Q1D case where n is the soliton wall linear density, is the soliton wall energy per chain, is the width of center allowed band (appearing due to periodic domain walls) and gives the soliton wall interaction energy. Soliton phase linear energy: 35 [S.A. Brazovskii, L.P. Gor'kov, A.G. Lebed', Sov. Phys. JETP 56, 683 (1982)] Boundaries E_ of the soliton level band 22 kyky E Schematic picture of energy bands  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 ! Then the soliton phase is the thermodynamically stable state.

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

23. Energy of soliton phase (intermediate general case dispersion) For the intermediate electron dispersion the interval of soliton phase can be about 10% of p C in agreement with experiment in (TMTSF) 2 PF 6. The SDW–SP transition at p C1 is of the second kind while the SP–Metal transition at p C is of the first kind also in agreement with experiment. The domain phase observed in (TMTSF) 2 PF 6 may be the soliton phase. 37 [ L.P. Gor'kov, P.D. Grigoriev, Europhysics Letters 71, 425 (2005) ]

24. Superconductivity in the soliton phase (suppression of spin-singlet SC by SDW background) The Green functions in the soliton phase are 4x4 matrices: 38 Self-consistency Gor’kov equations for superconductivity in soliton phase: R L f LR = R L f RL + L R f RL = L R f LR + R R L LR L f LR L RR R R L L L L R f RL R L 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. 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 : 40

26. Width of soliton band in Q1D metals where the soliton wall linear density From the soliton phase linear energy one obtains the width of the soliton band: and In the tight-binding model with only two harmonics near the transition at P = P c1 ( where 2t’ b =  0 ) and 41

27. Upper critical field in SC state on soliton-phase background. Result: close to Tc For tight binding dispersion where The width of the soliton band and H c2 diverges as P  P c1 : which agrees well with experiment. and the constant C 1s depends on the electron dispersion. 42

28. Upward curvature of H c2 z (T) Solitons create a layered structure, which is described by the Lawrence-Doniach model of 1D Josephson lattice. 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 H c2 in a superconducting slab: where d s = s is the interlayer distance. SC insulator s Upper critical field in this Josephson lattice is

29. Upper critical field H c2 z in  -(BEDT-TTF) 2 KHg(SCN) 4  -(BEDT-TTF) 2 KHg(SCN) 4 : D. Andres et al., Phys. Rev. B 72, 174513 (2005) CDW + superconductivity: T c SC <T c DW 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 H c2 z behavior

30. Origin of hysteresis. 44 The observed hysteresis in resistance at temperature change can be explained in both scenarios. For open-pocket scenario of DW 1 hysteresys is due the shift of the DW wave vector at P>P c1 Phase diagram In the soliton scenario of DW 1 the hysteresys is due the sliding of soliton walls.

31. Conclusions I.There are, at least, 2 possible structures of a DW 1 state, where superconductivity coexists microscopically with density wave. II.The SC properties of such state are investigated for both structures: 1). The DoS on the Fermi level in DW 1 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 P c1, where SC first appears, and shows unusual temperature (upward curvature) and pressure dependence. III. The results agree with experiment in organic metals (TMTSF) 2 PF 6 and  -(BEDT-TTF) 2 KHg(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) 2 PF 6 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. 1.We developed the theory, describing superconductivity on SDW or CDW background when T c DW >>T c SC in quasi-1D compounds with one conducting band. 2.There are two possible microscopic structures of DW 1 phase, where SC may coexist microscopically with DW: (1) uniform structure with ungapped states in momentum space (open pockets); (2) non-uniform soliton phase. 3.The DoS at the Fermi level in DW 1 state in both scenarios is rather high, which makes T C SC on DW background comparable with T C SC in pure SC state. The enhancement of the e-e interaction by critical fluctuations may increase T c SC even to the value higher than without DW. 4.The upper critical field is calculated in both scenarios and shown to considerably exceed the usual H c2. It diverges at critical pressure P c1, where SC first appear, and shows unusual temperature (upward curvature) and pressure dependence. 5.The SDW background strongly damps singlet SC. The SC, appearing on SDW background in metals with single conducting band, should be triplet. 6.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) 2 PF 6 and  -(BEDT-TTF) 2 KHg(SCN) 4. Conclusions 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) 2 PF 6 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 [ 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). ] Here SC insulator s

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

36. Which of the two proposed microscopic structures appears in the experiment? 44 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 H c2 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. NMR experiments in (TMTSF) 2 PF 6 Lineshapes for incommensurate SDWs, with different soliton widths, using hyperbolic tangent function for describing solitons. Stuart Brown et al., UCLA, Dresden, 2005. Red= normal state; Blue= zero width; Black=wide soliton. NMR absorption line 45

38. Upward curvature of H c2 (T) (TMTSF) 2 PF 6 : J. Lee, P. M. Chaikin and M. J. Naughton, PRL 88, 207002 (2002)  -(BEDT-TTF) 2 KHg(SCN) 4 : D. Andres et al., Phys. Rev. B 72, 174513 (2005) CDW + superconductivity: The upward curvature of H c2 (T) also suggests the soliton structure

39. 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: The CDW or SDW onset is due to the interaction with Q=Q N 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. where (keeps electrons on the same FS sheet) (scatters electrons to the opposite FS sheet) 21

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

41. Previous theoretical results on SC+DW. 4t 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); H c2 : A. M. Gabovich and A. S. Shpigel, Phys. Rev. B 38, 297 (1988). ] 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). ] 3). Proximity to the Peierls (DW) instability increases the effective e-e interaction g(Q) with the wave vector Q  Q N : The RPA result gives

42. 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 T c is higher than expected (not exponentially smaller than T c 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 H c2 may be strongly enhanced as compared to SC without DW. P1

43. 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 H c2 for SC on the CDW and SDW background. P1

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

45. Electron dispersion in the ungapped FS pockets on the DW background in tight-binding approximation Small ungapped pockets on a FS sheet get formed when the antinesting term in the electron dispersion exceeds DW energy gap. The quasi-particle dispersion in these small pockets where The important contribution to Cooper logarithm and to SC properties comes from the ungapped electron states on the Fermi level. 27a Empty band 22 kyky E  ungapped pockets 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: This gives where g 0 (Q)<<1 is the bare interaction, and the susceptibility may diverge at some (nesting) wave vector, so that Then the new coupling also diverges at some Q. The original coupling g 0 (Q) may be more complicated (include spin). Then the renormalized coupling includes all components of g 0 (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) Y. Tanaka and K. Kuroki, PRB 70, 060502(R) (2004) 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)

48. 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 D1

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

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

51. Nature of superconductivity in (TMTSF) 2 PF 6 I.J. Lee, P. M. Chaikin, M. J. Naughton, PRB 65, 18050(R ) (2002) ! Critical magnetic field and the Knight shift in (TMTSF) 2 PF 6 in the superconductivity-SDW coexistence phase confirm the triplet paring: Knight shift does not change as temperature decreases: absorption I.J. Lee et al., PRB 68, 092510 (2003) Critical magnetic field H c exceeds ~5 times the paramagnetic limit: 19

52. Result2: Comparison of triplet Tc in normal metal, CDW and SDW background for 1. Normal metal background: and 2. CDW background: and 3. SDW background at : which gives: Not too small. 18

53. 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) Y. Tanaka and K. Kuroki, PRB 70, 060502(R) (2004) 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)

54. Enhancement of the e-e coupling helps to SC (TMTSF) 2 PF 6 : T.Vuletic et al., Eur. Phys. J. B 25, 319 (2002)  -(BEDT-TTF) 2 KHg(SCN) 4 : D. Andres et al., Phys. Rev. B 72, 174513 (2005) 3 SC transition temperature considerably increase as the DW instability is approached. This increase is attributed to the critical fluctuation.

55. Two mechanisms of microscopic coexistence of superconductivity or normal metal with SDW 1. Ungapped pockets of FS. Empty band 22 kyky E  ungapped pockets The antinesting dispersion soliton band 22 kyky E  2. Soliton phase (non-uniform). The SDW order parameter depends on the coordinate along the 1D chains: or 29

56. Solitons in CDW or SDW. 31 Soliton phase. The SDW order parameter depends on the coordinate along the 1D chains: Boundaries E_ of the soliton level band 22 kyky E Schematic picture of energy bands in the soliton phase in Q1D case.  The soliton level band is only half-filled and the system gains the energy due to the dispersion along k y, which can be greater than the soliton wall energy cost Each soliton costs energy

57. Upward curvature of H c2 (T) Solitons create a layered structure, which is described by the Lawrence-Doniach model of 1D Josephson lattice. 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 H c2 in a superconducting slab: where s is the interlayer distance. SC insulator s Upper critical field in this Josephson lattice is

58. Lawrence-Doniach model [ 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). ] Here SC insulator s

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

60. Gor’kov equations with forward and backward scattering R L f LR = R L f RL + L R f RL = L R f LR + R R L LR L L RR R R L L L L R f RL + R L L R R L L RR L f LR ; R LR RL L L R f LR + L R R L L R f RL L R L R backward scatteringforward scattering Self-consistency equations for superconductivity order parameter: In analytical form this rewrites: 22

61. Equations on Tc with forward and backward scattering 1. Normal metal or CDW background. Singlet SC equation Triplet SC equation 2. Superconductivity on SDW background. Singlet SC equation Triplet SC equation 23

62. Discussion 1). Usually, the coupling constants, g f, g b, have the same sign, and Hence, in the normal-metal state SC is usually singlet. On CDW background the triplet order is even less favorable. 2). On SDW background the spin structures of SC and SDW order parameters interfere, which leads to different self-consistency equations: The non-diagonal block of the Cooper bubble enters with the opposite sign and cancels the infrared singularity from the diagonal block. This leads to the strong reduction of Tc for singlet SC in SDW. This cancellation happens for singlet SC but may not happen for triplet. 24

63. Outlook The proposed study opens a new field in the investigation of density-wave superconductors rather than closes this problem. 1.There are many other DW superconductors. 2.Most results obtained qualitatively and require further elaboration. 3.The results depend on a particular electron dispersion. 4.Many other properties are left for investigation. 5.More complicated models can be studied (with more complex e-e interaction and impurity scattering, etc.)

64. The Green functions in the uniform SDW state. The equations for the Green functions in the SDW state In the matrix form these equations rewrite: where the matrix Green function Diagonalization of the 2x2 matrix Hamiltonian gives the new energy spectrum: where 10

65. Expressions for the electron Green functions in the SDW state The diagonal elements of the Green function matrix: The non-diagonal elements of the Green function matrix: 11

66. Equations for superconducting instability R L f LR = R L f RL + L R f RL = L R f LR + R R L LR L f LR L RR R R L L L L R f RL R L L R The Gor’kov functions at t 1 =t 2 +0 : In the presence of SDW or CDW the SC equations contain two additional terms, coming from non-diagonal elements in the Green functions: In the normal metal state (without SDW or CDW) the SC self-consistency equation in diagram form R L f LR = R L f RL L R f RL = L R f LR R R L L R R L L In the uniform phase 12

67. Equations for SC instability in SDW phase With backward scattering only the SC equation are due to SDW spin structure If we introduce the diagonal and non-diagonal Cooper bubbles: the self-consistency equations for superconductivity rewrite: 13

68. Singlet superconductivity in SDW or CDW. The spin-singlet superconducting order parameter Using the commutation identity for spin-singlet pairing with we obtain the SC equation on SDW background: The self-consistency equations for superconductivity: The SC equations on the CDW background would be 14

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

70. Illustration of the cancellation of different contributions to the SC order parameter on the SDW background Nesting vector Q N Fermi surface Direct SC pairing SC pairing after scattering by SDW wave vector -Q N QNQN The two-electron wave function acquires “  ” sign after scattering by SDW if the electron spins in this pair look in opposite directions. This affects only the infrared divergence in the Cooper logarithm. The ultraviolet divergence remains unchanged. Spin-dependent scattering: the sign of the scattered electron wave function depends on its spin orientation. 16

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

72. Result2: Comparison of triplet Tc in normal metal, CDW and SDW background for 1. Normal metal background: and 2. CDW background: and 3. SDW background at : which gives: Not too small. 18

73. Publications Publications. 1). L.P. Gor'kov, P.D. Grigoriev, "Soliton phase near antiferromagnetic quantum critical point in Q1D conductors", Europhysics Letters 71, 425 (2005). 2). L.P. Gor'kov, P.D. Grigoriev, " Nature of superconducting state in the new phase in (TMTSF) 2 PF 6 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).

74. Summary 1.We developed the theory, describing superconductivity on SDW or CDW background when T c SDW >>T c SC in quasi-1D compounds with one conducting band. 2.There are two possible microscopic structures of superconducti-vity, coexisting with CDW or SDW in quasi-1D metals with one conducting band: (1) uniform structure with ungapped states in momentum space; (2) non-uniform soliton phase. 3.The DoS at the Fermi level in the DW phase with open pockets is the same as in the metallic state, which makes the SC transition temperature to be rather high. The enhancement of the e-e interaction by the Peirls instability may increase T c SC even to the value higher than without DW. 4.The upper critical field is calculated in both scenarios and shown to considerably exceed the usual H c2, diverging at critical pressure and showing unusual temperature and pressure dependence. 5.The SDW background strongly damps singlet SC. The SC, appearing on SDW background should be triplet. 6.The proposed models and approach to study these models open new scope to investigate the coexistence of SC with DW also in many existing DW superconductors. 7. The results obtained are in good agreement with experimental observations in organic metals (TMTSF) 2 PF 6 and  -(BEDT-TTF) 2 KHg(SCN) 4. 46