1. Interplay between magnetism and superconductivity in Fe-pnictides Institute for Physics Problems, Moscow, June 30, 2010 Andrey Chubukov University of Wisconsin

2. IRON AGE (1200 - 550 B.C.E.) RFeAsO (1111) R = La, Nd, Sm, Pr, Gd AFe 2 As 2 (122) A = Ba, Sr, Ca LiFeAs (111) Fe-Pnictides: Binary componds of pnictogens A pnictogen – an element from the nitrogen group N,P, As,Sb,Bi LaOFeP Fe(SeTe) T c,max =57K (SmFeAsO)

3. Hideo Hosono, TITechFe-pnictides: May 2006 2006 2008

4. LaFeAsO 1-x F x La 1-x Sr x FeAsO SmFeAsO 1-x F x CeFeAsO 1-x F x PrFeAsO 1-x F x NdFeAsO 1-x F x GdFeAsO 1-x F x SmFeAsO 1-x F x SmFeAsO 1-x GdFeAsO 1-x Gd 1-x Th x FeAsO DyFeAsO 1-x F x TbFeAsO 1-x F x Tb 1-x Th x FeAsO Ba 1-x K x Fe 2 As 2 Sr 1-x K x Fe 2 As 2 Eu 1-x La x Fe 2 As 2 Ca 1-x Na x Fe 2 As 2 Eu 1-x K x Fe 2 As 2 Li 1-x FeAs  -FeSe 1-x BaNi 2 P 2 LaO 1-x NiBi LaOFeP LaO 1-x F x FeP LaONiP  -FeSe SrNi 2 As 2 BaCo x Fe 2-x As 2 SrCo x Fe 2-x As 2 BaNi x Fe 2-x As 2 FeSe 0.5 Te 0.5 2008 T c (K) courtesy of J. Hoffman

5. Phase diagram: magnetism and superconductivity BaFe 2 (As 1-x P x ) 2 Fernandes et al Matsuda et al Luetkens et al

6. Crystal structure 2D Fe-As layers with As above and below a square lattice formed by Fe LaFeAsO Folded BZ Unfolded BZ

7. Pnictides Cuprates Are pnictides similar to cuprates? Parent compounds are antiferromagnets Superconductivity emerges upon doping

8. Pnictides Cuprates Are pnictides similar to cuprates? metal Mott insulator/ Heisenberg AFM

9.  Metallic behavior of parent compounds Metallic behavior of parent compounds TNTN

10. Resistivity in the cuprates parent compounds (magnetic)

11. Band theory calculations agree with experiments Band theory calculations agree with experiments Lebegue, Mazin et al, Singh & Du, Cvetkovic & Tesanovic… 2 circular hole pockets around (0,0) 2 elliptical electron pockets around  ) (folded BZ),  or  and  (unfolded BZ) Electron Fermi surface Hole Fermi surface

12. NdFeAs(O 1-x F x ) (x=0.1) A. Kaminski et al. Hole pockets near (0,0) Electron pockets near (  ) dHVa ARPES LaFeOP A. Coldea et al, Ba 06 K 04 Fe 2 As 2 H. Ding et al. A. Kordyuk et al LiFeAs

13. Fe-pnictides are doped metals High Tc cuprates are doped Mott insulators

14. A simple way to understand the difference between cuprates and pnictides Cuprates: Tesanovic, Physics 2, 60 (2009) one hole in a filled d-shell (1 “free” fermion per cite: half-filling) U Only when doped with holes (or electrons) do cuprates turn into superconductors 9

15. Pnictides: 4 holes per cite – multiband structure. chemical potential lies in the gap LaOFeAs

16. Itinerant approach Interacting fermions with hole and electron Fermi surfaces, no localization of electronic states What are generic, model-independent features of Fe-pnictides?

17. I will skip magnetism and focus only on superconductivity Magnetic and superconducting properties are both interesting

18. Superconductivity

19. Electron-phonon interaction is probably too weak Boeri, Dolgov, & Golubov, Singh & Du =0.44 for Al Too small to account for Tc ~ 50K

20. I. Mazin et al., PRL 101, 057003 (2008) How about using the “analogy” with the cuprates and assume that the pairing is mediated by (spin) fluctuations Pairing due to el-el interaction Cuprates 0 Mazin et al, Kuroki et al…. peaked at (  ) Pnictides sign-changing s-wave gap (s +- )

21. Some experiments are consistent with the sign-changing, no-nodal s +- gap

22. Almost angle-independent gap (consistent with no-nodal s-wave) NdFeAsO 1-x F x T. Kondo et al., arXiv:0807.0815 1a. Photoemission in 1111 and 122 FeAs D. Evtushinsky et al Ba 1-x K x Fe 2 As 2

23. 1b. Penetration depth behavior in 1111 and 122 FeAs Carrington et al “Exponential” behavior at low T (or, at least, a very flat behavior) Y. Matsuda SmOFeAs Ba1-xKxFe2As2 Ba 0.46 K 0.56 Fe 2 As 2

24. 1c. Neutron scattering – resonance peak below 2D D. Inosov et al. Eremin & Korshunov Scalapino & Maier The “plus-minus” gap is the best candidate s +- gap

25. Other experiments, however, indicate that the gap may have nodes

26. 2a. The behavior of LaOFeP, Tc =5K Carrington et al Linear in T dependence at small T Penetration depth Thermal conductivity Matsuda et al Residual term in  /T, H 1/2 field dependence Consistent with line nodes

27. 2b. The behavior of BaFe 2 (As 1-x P x ) 2, Tc =30K Y. Matsuda et al (BaK)FeAs BaFe(AsP) Consistent with line nodes

28. 2c. Residual term in c-axis thermal conductivity of overdoped Co-Ba122 c-axis conductivity a-axis conductivity J-Ph. Reid et al

29. Back to a simple reasoning Problem: how to get rid of an intra-band Coulomb repulsion? Cuprates Pnictides Coulomb repulsion cancels out, only d-wave,  interaction matters Intra-band repulsion does not cancel and has to be overtaken by a  interaction

30. hole FS u4u4 u3u3 electron FS Intra-band repulsion u 4 Pair hopping  interaction u4u4 If the intra-band repulsion (u 4 ) is stronger than the pair hopping (u 3 ), the pairing interaction is repulsive s +- superconductivity, a simple RPA Pairing interactions: A toy model: one hole and one electron FSs.

31. How to overcome intra-pocket Coulomb repulsion is the most essential part of the theory of itinerant superconductivity in Fe-pnictides

32. Electronic Structure: a zoo of p- and d-states bands L. Boeri, O.V. Dolgov, and A.A. Golubov, PRL 101, 026403 (2008) Strong hybridization of all Fe-orbitals 10 Fe d-bands in the window of +- 2 eV 12 O and As p-bands from -6 eV to -2 eV La f-bands at around -3 eV I. Pnictides are multi-orbital systems

33. What multi-orbital/multi-band structure means for our story? Interactions between low-energy fermions are dressed by orbital coherence factors and in general have strong angular dependence. Effective intra-band interaction: Effective inter-band interaction: (previously we only had u 3 ) Exercise in trigonometry with s-wave interactions: This is still s-wave!

34. Solve three coupled BCS equations for the gaps: When is non-zero, the solution exists for arbitrary u 3 /u 4 and for arbitrary sign and magnitude of, When is zero, the solution for Tc exists only for u 3 >u 4 but the gap along electron FS has nodes for u 3 >u 4.

35. 1 s +- gap with the nodes on electron FS s +- gap with no nodes d-wave gap with the nodes on both FS Coulomb repulsion is the strongest – the s +- gap with the nodes along electron FS is most likely scenario Superconducting phase diagram

36. How can the system develop a no-nodal gap (still of s +- symmetry) ? SmOFeAs

37. RG helps: u 4 and u 3 are bare interactions,at energies of a bandwidth For SC we need interactions at energies smaller than the Fermi energy E E F ~ 0.1 eV W ~3-4 eV | | 0 Couplings flow due to renormalizations by particle-particle and particle-hole bubbles

38. We know this story for conventional (phonon) superconductors E F ~ W DD 0 E Coulomb repulsion is renormalized down by the renormalization in the particle-particle channel (McMillan-Tolmachev renormalization) If only Cooper channel is involved, this cannot change a repulsion into an attraction In our case, there are renormalizations in both particle-particle AND particle hole channel. This implies that we need to construct parquet RG to analyze the system flow between W and E F (H. Shultz, Dzyaloshinskii & Yakovenko)

39. One-loop parquet RG u 3 is pushed up by u 1, which gives rise to SDW Intra-band repulsion u 4 Pair hopping  interaction Inter-band density-density and exchange interaction SDW u0Lu0L If the tendency towards SDW is strong, u 3 becomes larger than u 4, leading to an s +- superconductivity with no-nodal gap

40. 1 s +- gap with the nodes on electron FS s +- gap with no nodes d-wave gap with the nodes on both FS If RG renormalization is strong, Coulomb interaction is overshadowed, and the s +- gap may have no nodes along the electron FS Superconducting phase diagram Less SDW fluctuations Underdoped 1111, 122 FeAs LaFeOP, Overdoped FeAs

41. The role of spin fluctuations SC sf Spin-fluctuation exchange is an additional dynamical interaction, which acts in parallel with the pair hopping. The real issue is what sets the upper cutoff for the pairing: the scale at which u 3 =u 4, or the dynamics of the spin-fluctuatioon exchange

42. Interplay between magnetism and superconductivity We can re-write these as equations for SDW, CDW, and SC vertices Interactions in all channels grow - -

43. One-loop RG Flow – all channels SDW with real order parameter Extended s-wave CDW with imaginary order parameter Above E F Lower boundary for parquet RG is the Fermi energy, E F O(6) fixed point: 3 for SDW, 2 for SC, 1 for CDW

44. Below E F – decoupling between SDW and SC channels For a perfect nesting, whichever vertex is larger at E F, wins

45. Perfect nesting – SDW wins Non-perfect nesting –SDW vertex remains the strongest, but the SDW instability is cut, and s +- SC wins

46. Either first-order transition between SDW and SC states, or co-existence, depending on parameters doping depending on the degree of ellipticity of electron FSs Fernandes, Schmalian, et al, Vorontsov, Vavilov, AC

47. How important is the fact that there are 5 bands and the orbital structure changes along the FS? Coldea et al, dHvA Mazin & Schmalian Graser, Maier, Hirschfeld, Scalapino Thomale, Platt, Hanke, Bernevig. D-H Lee, Vishvanath, Honerkamp, Benfatto, Grilli, Kuroki, Imada… At least 2 hole and two electron FSs additional 3D FSs Detailed studies:

48. These extra features give rise to some qualitative changes, but do not change the overall picture of s +- pairing IIa. RG for 2 hole and 2 electron FSs more parameters, more equations 1hole 1 electron FSs 2 hole 2 electron FSs Still, SC vertex changes sign under RG Thomale et al Maiti, AC

49. Summary: the phase diagram: 1 s +- gap with the nodes on electron FS s +- gap with no nodes Less SDW fluctuations Underdoped 1111, 122 FeAs LaFeOP, Overdoped FeAs Coulomb repulsion Angle dependence of the interaction Position depends on interactions, FS geometry, etc…

50. Conclusions: Superconductivity is the result of the interplay between intra-pocket repulsion and the pair hopping. If the tendency towards SDW is strong, pair hopping increases in the RG flow, and the system develops an s +- gap without nodes, once SDW order is eliminated by doping. If the tendency towards SDW is weaker, intra-pocket repulsion remains the strongest. The system still becomes an s +- supercnductor, but the gap has nodes along the two electron Fermi surfaces. Fe-pnictides are itinerant systems, no evidence for Mott physics

51. THANK YOU

52. IIb. Does the solution for s+- gap with nodes exists for 2 hole and 2 electron FSs when the Coulomb repulsion is the strongest (i.e., when RG does not work)? more parameters, more equations 2 hole 2 electron FSs, hence two hole two electron Still, the solution for s +- gap with nodes along electron FS does exist! The equation on L= log Tc becomes 4 th order 1,2

53. Matano et al Knight shift NMR relaxation rate Non-exponential behavior! 1. NMR and Knight shift in 1111 FeAs

54. 3. Penetration depth behavior in 1111 and 122 FeAs Carrington et al “Exponential” behavior at low T (or, at least, a very flat behavior) Y. Matsuda SmOFeAs Ba1-xKxFe2As2 Ba 0.46 K 0.56 Fe 2 As 2

55. A “derivation”: take a toy model of two electron orbits + hybridization + Hubbard-type repulsive interactions between fermions from the same orbital (U 11 ) and from different orbitals (U 12 ) +

56. The couplings depend on the energies of the two orbitals, and on hybridization All coupling are positive (repulsive), and intra-band repulsion is stronger than the inter-band pair hopping

58. How we proceed in the cuprates: nodal antinodal Abanov et al 09 actual gap