1. First-Principles Prediction of Tc (Takada) ISSP Workshop/Symposium: MASP 2012 1 Theory for Reliable First-Principles Prediction of the Superconducting T c Yasutami Takada Institute for Solid State Physics, University of Tokyo 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan Seminar Room A615, ISSP, University of Tokyo 14:00-15:30, Thursday 28 June 2012

2. First-Principles Prediction of Tc (Takada) Outline 2 1. Introduction 2. Electron-phonon system in the Green’s-function Approach Eliashberg theory  2 F(  )  Eliashberg theory and the Eliashberg function  2 F(  )  D /E F Uemura Plot  Problem about the smallness parameter  D /E F  Uemura Plot vertex correction in GISC  Eliashberg theory with vertex correction in GISC 3. G 0 W 0 approximation to the Eliashberg theory STOGIC  STO and GIC 4. Superconductors with short coherence length alkali-doped fullerenes  Hubbard-Holstein model and alkali-doped fullerenes 5. Connection with density functional theory for supperconductors pairing interaction K ij  Functional form for pairing interaction K ij pairing kernel g ij  Introduction of pairing kernel g ij as an analogue of exchange-  correlation kernel f xc in time-dependent density functional  theory 6. Summary 2

3. First-Principles Prediction of Tc (Takada) Introduction 33 Discovery of novel superconductors ○ Discovery of novel superconductors  novel physical properties and/or phenomena ◎ High-T c superconductors  By far the most interesting property is T c itself!  Why don’t we investigate this quantity directly? ○ An ultimate goal in theoretical high-T c business  Develop a reliable scheme for a first-principles  Develop a reliable scheme for a first-principles prediction of T c, with using only information prediction of T c, with using only information on constituent atoms. on constituent atoms. ○ an accurate estimation of T c on a suitable microscopic model Hamiltonian without  *. ○ For the time being, we shall be content with an accurate estimation of T c on a suitable microscopic model Hamiltonian for the electron-phonon system without employing such phenomenological adjustable parameters as  *.

4. First-Principles Prediction of Tc (Takada) Model Electron-Phonon System 44 HamiltonianHamiltonian Nambu Representation Green’s Function Anomalous Green’s Function F(p,i  p ) Off-diagonal part  Anomalous Green’s Function: F(p,i  p )

5. First-Principles Prediction of Tc (Takada) Exact Self-Energy 55 Formally exact equation to determine the self-energy Bare electron-electron interaction Effective electron-electron interaction Direct extension of the Hedin’s set of equations ! Polarization function

6. First-Principles Prediction of Tc (Takada) Eliashberg Theory 66 Basic assumption:  D /E F 1 Basic assumption:  D /E F ≪ 1 (2) Separation between phonon-exchange & Coulomb parts (1) Migdal Thorem: neglect for a while↑  (q,i  q )   (q,0): perfect screening ↑ (3) Introduction of the Eliashberg function (4) Restriction to the Fermi surface & electron-hole symmetry

7. First-Principles Prediction of Tc (Takada) Renormalization Function and Gap Function 77 (2) Gap Equation at T=T c (1) Equation to determine the Renormalization Function Function (n) with n: an integer Cutoff function  p (  c ) with  c of the order of  D

8. First-Principles Prediction of Tc (Takada) Inclusion of Coulomb Repulsion 88 (2) Gap Equation (1) Equation to determine the Renormalization Function Coulomb pseudopotential ← Invariant! ← Revised

9. First-Principles Prediction of Tc (Takada) Eliashberg Function 99 ab initio calculation of  2 F(  )

10. First-Principles Prediction of Tc (Takada) MgB 2 10 Two-gap typical BCS superconductor with T c =40.2K with aid of E 2g phonon modes in the B-layer Two-gap typical BCS superconductor with T c =40.2K with aid of E 2g phonon modes in the B-layer AlB 2 （ P6/mmm) a = 3.09 Å、 c = 3.52 Å B-B distance=1.78 Å larger than 1.67 Å in boron solids

11. First-Principles Prediction of Tc (Takada) Uemura Plot 11 Will high-T c be obtained under the condition of  D /E F 1? Will high-T c be obtained under the condition of  D /E F ≪ 1? ← Not at all! In the phonon mechanism, T c /  D is known to be less than about 0.05. Because T c /E F =(T c /  D )(  D /E F ), this indicates that  D /E F should be of the order of unity. Thus interesting high-T c materials cannot be studied by the conventional Eliashberg theory!! In the phonon mechanism, T c /  D is known to be less than about 0.05. Because T c /E F =(T c /  D )(  D /E F ), this indicates that  D /E F should be of the order of unity. Thus interesting high-T c materials cannot be studied by the conventional Eliashberg theory!! Need to develop a theory applicable to the case of  D /E F 1. Need to develop a theory applicable to the case of  D /E F ~ 1.

12. First-Principles Prediction of Tc (Takada) Return to the Exact Theory 12 How should we treat the vertex function?  “GW  ” Reformulate the Eliashberg theory with including this vertex function. cf. YT, in “Condesed Matter Theories”, Vol. 10 (Nova, 1995), p. 255 Ward Identity If we take an average over momenta in accordance with the Eliashberg theory, we obtain:

13. First-Principles Prediction of Tc (Takada) Gap Equation in GISC 13 Gap Equation with the vertex correction without  * Gauge-Invariant Self-Consistent (GISC) determination of Z(i  p ) Main message obtained from this study: For  D ~ E F, G 0 W 0 is much better than GW (= Eliashberg theory) in calculating T c. For  D ~ E F, G 0 W 0 is much better than GW (= Eliashberg theory) in calculating T c.  Let us go with G 0 W 0 in the first place! Main message obtained from this study: For  D ~ E F, G 0 W 0 is much better than GW (= Eliashberg theory) in calculating T c. For  D ~ E F, G 0 W 0 is much better than GW (= Eliashberg theory) in calculating T c.  Let us go with G 0 W 0 in the first place! Model Eliashberg Function

14. First-Principles Prediction of Tc (Takada) Gap Equation in G 0 W 0 Approximation 14 Derive a gap equation in G 0 W 0 Derive a gap equation in G 0 W 0 in which Z p (i  p )=1,  p (i  p )=0. cf. YT, JPSJ45, 786 (1978); JPSJ49, 1267 (1980). Analytic continuation:

15. First-Principles Prediction of Tc (Takada) BCS-like Gap Equation 15 The pairing interaction can be determined from first principles. BCS-like gap equation obtained by integrating  -variables No assumption is made for pairing symmetry.

16. First-Principles Prediction of Tc (Takada) SrTiO 3 16 ◎ Ti 3d electrons (near the  point in the BZ) superconduct with the exchange of the soft ferroelectric phonon mode cf. YT, JPSJ49, 1267 (1980)

17. First-Principles Prediction of Tc (Takada) Graphite Intercalation Compounds 17 CaC 6 KC 8 : T c = 0.14K [Hannay et al., PRL14, 225(1965)] CaC 6 : T c = 11.5K [Weller et al., Nature Phys. 1, 39(2005); Emery et al., PRL95, 087003(2005)] up to 15.1K under pressures [Gauzzi et al., PRL98, 067002(2007)] KC 8 : T c = 0.14K [Hannay et al., PRL14, 225(1965)] CaC 6 : T c = 11.5K [Weller et al., Nature Phys. 1, 39(2005); Emery et al., PRL95, 087003(2005)] up to 15.1K under pressures [Gauzzi et al., PRL98, 067002(2007)] We should know the reason why T c is enhanced by a hundred times by just changing K with Ca? We should know the reason why T c is enhanced by a hundred times by just changing K with Ca?

18. First-Principles Prediction of Tc (Takada) Electronic Structure 18 Band-structure calculation: KC 8 : [Ohno et al., JPSJ47, 1125(1979); Wang et al., PRB44, 8294(1991)] LiC 2 : [Csanyi et al., Nature Phys.1, 42 (2005)] CaC 6,YbC 6 : [Mazin,PRL95,227001(2005);Calandra & Mauri,PRL95,237002(2005)] Important common features (1) 2D- and 3D-electron systems coexist. (2) Only 3D electrons (considered as a 3D homogeneous electron gas with the band mass m * ) in the interlayer state superconduct. Band-structure calculation: KC 8 : [Ohno et al., JPSJ47, 1125(1979); Wang et al., PRB44, 8294(1991)] LiC 2 : [Csanyi et al., Nature Phys.1, 42 (2005)] CaC 6,YbC 6 : [Mazin,PRL95,227001(2005);Calandra & Mauri,PRL95,237002(2005)] Important common features (1) 2D- and 3D-electron systems coexist. (2) Only 3D electrons (considered as a 3D homogeneous electron gas with the band mass m * ) in the interlayer state superconduct.

19. First-Principles Prediction of Tc (Takada) Microscopic Model for GICs 19 This model was proposed in 1982 for explaining superconductivity in KC 8 : YT, JPSJ 51, 63 (1982) In 2009, it was found that the same model also worked very well for CaC 6 : YT, JPSJ 78, 013703 (2009). This model was proposed in 1982 for explaining superconductivity in KC 8 : YT, JPSJ 51, 63 (1982) In 2009, it was found that the same model also worked very well for CaC 6 : YT, JPSJ 78, 013703 (2009).

20. First-Principles Prediction of Tc (Takada) Model Hamiltonian 20 First-principles Hamiltonian for polar-coupling layered crystals cf. YT, J. Phys. Soc. Jpn. 51, 63 (1982) 20

21. First-Principles Prediction of Tc (Takada) Effective Electron-Electron Interaction in RPA 21

22. First-Principles Prediction of Tc (Takada) Calculated Results for T c 22 K Ca Valence Z 1 2 Layer separation d ~ 5.5A ~ 4.5A Branching ratio f ~ 0.6 ~ 0.15 Band mass m * m e 3m e Band mass m * ~ m e (s-like) ~ 3m e (d-like) cf. Atomic mass m M is about the same.

23. First-Principles Prediction of Tc (Takada) Perspectives for Higher T c 23 Zm * ◎ Two key controlling parameters: Z and m *. ◎ T c will be raised by a few times from the current value of 15K, but never go beyond100K. Zm * ◎ Two key controlling parameters: Z and m *. ◎ T c will be raised by a few times from the current value of 15K, but never go beyond100K.

24. First-Principles Prediction of Tc (Takada) 24 Dynamical Pairing Correlation Function Conventional approach Q sc (q,  )

25. First-Principles Prediction of Tc (Takada) 25 Reformulation of Q sc (q,  ) In g, both self-energy renormalization and vertex corrections are included. ~

26. First-Principles Prediction of Tc (Takada) 26  0 in the BCS Theory High-T c  Inevitably associated with short  0 Formulate a scheme to calculate the pairing interaction from the zero-  0 limit in real-space approach. a 0 : lattice constant

27. First-Principles Prediction of Tc (Takada) 27 Evaluation of  the Pairing Interaction Basic observation Basic observation: The essential physics of electron pairing can be captured in an N-site system, if the system size is large enough in comparison with  0. If  0 is short, N may be taken to be very small.  If  0 is short, N may be taken to be very small.

28. First-Principles Prediction of Tc (Takada) 28 Fullerene Superconductors ◎ Alkali-doped fullerene superconductors 1) Molecular crystal composed of C 60 molecules 2) Superconductivity appears with T c =18-38K in the half-filled W 0.5eV threefold narrow conduction bands (bandwidth W 0.5eV) derived from the t 1u -levels in each C 60 molecule.  0 0.2eV 3) The phonon mechanism with high-energy (  0 0.2eV) intramolecular phonons is believed to be the case, although the intramolecular Coulomb repulsion U is also strong and is about the same strength as the phonon-mediated attraction  2 -2  0 with  the electron-phonon coupling strength (  2).  U 2  0 cf. O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997). ~ ~ ~ ~ ~

29. First-Principles Prediction of Tc (Takada) 29 Hubbard-Holstein Model Band-multiplicity: It may be important in discussing the absence of Mott insulating phase [Han, Koch, & Gunnarsson, PRL84, 1276 (2000)], but it is not the case for discussing superconductivity [Cappelluti, Paci, Grimaldi, & Pietronero, PRB72, 054521 (2005)]. The simplest possible model to describe this situation is:, because  0 is very short (less than 2a 0 ). cf. YT, JPSJ65, 1544, 3134 (1996).

30. First-Principles Prediction of Tc (Takada) 30 Electron-Doped  C 60 According to the band- structure calculation: The difference in T c induced by that of the crystal structure including Cs 3 C 60 under pressure [Takabayashi et al., Science 323, 1589 (2009)] is successfully incorporated by that in. The conventional electron- phonon parameter  is about 0.6 for  =2.

31. First-Principles Prediction of Tc (Takada) 31 Hypothetical Hole-Doped  C 60 Hole-doped C 60 : Carriers will be in the fivefold h u valence band.   =3

32. First-Principles Prediction of Tc (Takada) 32 Case of Even Larger  What happens for T c, if  becomes even larger than 3? A larger  is expected in a system with a smaller number of  - electrons N  : A. Devos & M Lannoo, PRB58, 8236 (1998). Case of C 36 is interesting:  =4 The C 36 solid has already been synthesized: C. Piskoti, J. Yarger & A. Zettl, Nature 393, 771 (1998); M. Cote, J.C. Grossman, M. L. Cohen, & S. G. Louie, PRL81, 697 (1998).

33. First-Principles Prediction of Tc (Takada) 33 Hypothetical Doped C 36 If solid C 36 is successfully doped   =4

34. First-Principles Prediction of Tc (Takada) SCDFT 34 Extension of DFT to treat superconductivity (SCDFT)  Basic variables: n(r) and  (r,r’) cf. Oliveira, Gross & Kohn, PRL60, 2430 (1988). Extension of DFT to treat superconductivity (SCDFT)  Basic variables: n(r) and  (r,r’) cf. Oliveira, Gross & Kohn, PRL60, 2430 (1988).

35. First-Principles Prediction of Tc (Takada) Pairing Interaction in Weak-Coupling Region 35 Remember: The homogeneous electron gas The homogeneous electron gas is useful in constructing a practical and useful form for V xc (r;[n(r)]):  LDA, GGA etc. K ij Let us consider the same system for constructing K ij in the weak-coupling region in the weak-coupling region.  G 0 W 0 calculation will be enough!

36. First-Principles Prediction of Tc (Takada) K ij in the Weak-Coupling Region 36   i * : time-reversed orbital of the KS orbital i Good correspondence! For the problem of determining T c, the KS orbitals can be determined uniquely as a functional of the exact normal-state n(r). Scheme for determining T c in inhomogeneous electron systems in the weak-coupling region Scheme for determining T c in inhomogeneous electron systems in the weak-coupling region

37. First-Principles Prediction of Tc (Takada) K ij in the Strong-Coupling Region 37 Q sc in terms of KS orbitals In the strong-coupling region, the  -dependence of g will be weak. ~ Weak-coupling case Use g ij instead of V ij in the general case! ~ Note: g corresponds to f xc in TDDFT! ~

38. First-Principles Prediction of Tc (Takada) 38 Summary 1 0 Review the Green’s-function approach to the calculation of the superconducting T c. 2 0 The Eliashberg theory is good for phonon mechanism of superconductivity, but not good for high-T c materials. 3 0 For weak-coupling superconductors, G 0 W 0 is applicable to both phonon and/or electronic mechanisms. 4 0 Clarified the mechanism of superconductivity in GIC, especially the difference between KC 8 and CaC 6. 5 0 Proposed a calculation scheme to treat strong-coupling superconductors, if the coherence length is short. 6 0 Addressed fullerites in this respect and find that T c might exceed 100K. 7 0 Connection is made to the density functional theory for superconductivity; especially a new functional form for the pairing interaction is proposed.