The High T c Superconductivity Puzzle.

1. The High Tc Superconductivity Puzzle. Adriana Moreo Dept. of Physics and ORNL University of Tennessee, Knoxville, TN, USA. Supportedby NSF grant DMR-0706020.

2. Heike Kammerlingh Onnes discovers superconductivity In Hg. Tc=4.2K 1908 Superconductivity Timeline

3. H H What is Superconductivity? • Resisitivity vanishes below Tc. • Normal conductor: induced current dissipates as heat in seconds. • Superconductor: induced current last for years (decay constant >109 years). • No magnetic field in its interior: Meissner effect. • Normal conductor: perfect conductor with R=0 is penetrated by an external H-field. • Superconductor: spontaneously generates surface currents that opposes the external H-field. Hg T>Tc J T<Tc SC PC

4. Heike Kammerlingh Onnes discovers superconductivity In Hg. Tc=4.2K 1908 1958 Bardeen, Cooper, and Schrieffer develop BCS theory. Superconductivity Timeline

5. k -k What causes SC in Hg?BCS Theory • Electrons form pairs. • Electron-phonon interaction is the “glue”. • Only electrons within a shell around the FS form pairs. • Pairs are rotationally invariant. U Coulomb repulsion Normal State Cooper Pair

6. BCS Superconductors • Metals. • Quest towards higher Tc not very successful. • Highest Tc = 23.2K in Nb3Ge (1973).

7. Heike Kammerlingh Onnes discovers superconductivity In Hg. Tc=4.2K Bednorz and Muller discover high Tc Cuprates. 1986 Bardeen, Cooper, and Schrieffer develop BCS theory. Superconductivity Timeline 1908 1958

8. High Tc Cuprates • Discovered in 1986 by Bednordz and Muller. • Tc~30K in La2-xBaxCuO4. • Ceramics with CuO planes. • AF insulators for x=0. • Tc= 90K in YBaCu3O7. • Highest Tc~130K for HgBa2Ca2Cu3O6+d.

9. Cuprates: Unconventional SC • The SC gap has nodes. • D-wave symmetry.

10. Models • t-J model or Hubbard model with large U (strong Coulomb repulsion). • One orbital:dx2-y2 • AF for undoped. • D-wave pairing trend. • Correct FS shape. t J

11. Mechanism: Magnetism friend or foe? • Electron-Phonon? • Tc is too high. • E-ph too weak to overcome strong Coulomb repulsion. • Magnetism? • Does it provide the “glue”? • Or does it need to go away to allow pairing? We still do not know the answer!

12. Heike Kammerlingh Onnes discovers superconductivity In Hg. Tc=4.2K Bednorz and Muller discover high Tc Cuprates. 2001 1986 MgB2 is discovered at BNL, NIST, and University of Oslo. Tc=39K Bardeen, Cooper, and Schrieffer develop BCS theory. Superconductivity Timeline 1908 1958

13. Fe based superconductors are discovered in Japan. Tc=56K. Heike Kammerlingh Onnes discovers superconductivity In Hg. Tc=4.2K Bednorz and Muller discover high Tc Cuprates. 2001 2007 1986 MgB2 is discovered At BNL, NIST, and University of Oslo. Tc=39K Bardeen, Cooper, and Schrieffer develop BCS theory. Superconductivity Timeline 1908 1958

14. Quaternary oxypnictides: LnOMPn (Ln: La, Pr; M:Mn, Fe, Co, Ni; Pn: P, As). Fe –As planes. La-O planes. Fe form a square lattice. F replaces O and introduces e- in Fe. F doped LaOFeAs

15. Parent compound • Long range magnetic order. • Metal. • Order parameter: suggests small to intermediate U and J. De la Cruz et al., Nature 453, 899 (2008). See also McGuire et al., cond-mat:0804.0796; Dong et al., cond-mat:0803.3426 and others.

16. Theory • Band Structure: 3d Fe orbitals are important. (LDA) • dxz and dyz most important close to eF. (Korshunov et al., cond-mat:0804.1793). • Metallic state. • Possible itinerant magnetic order. Singh et al., cond-mat: 0803.0429; Xu et al., cond-mat:0803.1282; Giovannetti et al., cond-mat: 0804.0866; and several others.

17. Fermi Surface LDA • Two hole pockets at G point. • Two electron pockets at M. • dxz and dyz orbitals (with some dxy hybridization). ARPES Liu et al., cond-mat: 0806.2147 NdFeAsO1-xFx Singh et al., PRL100, 237003 (2008).

18. Numerical Simulations (Daghofer et al., PRL101, 237004 (2008)). • Relevant degrees of freedom need to be identified. • Construct the minimal model. • Exact diagonalization and Variational Cluster Approximation (Daghofer) using a small cluster. • Successful with the cuprates: found magnetic order and correct pairing symmetry.

19. Minimal Model • Consider the Fe-As planes. • Two d orbitals dxz and dyz based on LDA and experimental results. • Consider electrons hopping between Fe ions via As bridge. • Square Fe lattice. • Interactions: Coulomb and Hund.

20. Hoppings Obtain from Slater-Koster overlap integrals between Fe-d and As-p orbitals and Fe-As-Fe hopping.

21. Fitted Hoppings (from Raghu et al.) • t1=-1 • t2=1.3 • t3=-.85 • t4=-.85

22. Coulomb interactions Largest cluster that can be studied with Lanczos techniques.

23. Numerical results: undoped limit • U<1 for metal in undoped case. • If J=U/4, U<1 to reproduce experimental order parameter in parent compound. • S(k) peaks at (0,p) and (p,0). • Similar results for fitted hoppings.

24. Electron Doping (Moreo et al., PRB79, 134502 (2009)) Spin singlet B2g orbital A1g spacial Intra-orbital with A1g symmetry Inter-orbital pairing with B2g symmetry Spin triplet

25. Latest Developments • Mean Field Calculations Performed in multiorbital Models. • Three Orbital Model Presented. • Symmetry Based Study of the possible pairing operators in a full 5 orbital model. • References: • Daghofer et al., PRL101, 237004 (2008). • Yu et al., PRB79, 104510(2009). • Moreo et al., PRB79, 134502 (2009). • Moreo et al., PRB80, 104507 (2009). • Daghofer et al, submitted to PRB (2009).

26. Conclussions • Numerical Techniques provide Crucial guidance in systems where electrons interact strongly. • Magnetism and orbital order need to be studied in order to understand the mechanism of SC. • The knowledge and the techniques developed from the study of SC can be applied to the study of many other problems in materials.