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ISSP Workshop/Symposium: MASP 2012

ISSP Workshop/Symposium: MASP 2012. 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

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ISSP Workshop/Symposium: MASP 2012

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  1. ISSP Workshop/Symposium: MASP 2012 Theory for Reliable First-Principles Prediction of the Superconducting Tc 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 First-Principles Prediction of Tc (Takada)

  2. Outline • 1. Introduction • 2. Electron-phonon system in the Green’s-function Approach • o Eliashberg theory and the Eliashberg function a2F(W) • o Problem about the smallness parameter QD/EF Uemura Plot • o Eliashberg theory with vertex correction in GISC • 3. G0W0 approximation to the Eliashberg theory • oSTOandGIC • 4. Superconductors with short coherence length • oHubbard-Holstein model and alkali-doped fullerenes • 5. Connection with density functional theory for supperconductors • o Functional form for pairing interaction Kij • o Introduction of pairing kernel gijasan analogue of exchange- • correlation kernel fxc in time-dependent density functional • theory • 6. Summary 2 First-Principles Prediction of Tc (Takada)

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

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

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

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

  7. Renormalization Function and Gap Function (1) Equation to determine the Renormalization Function (2) Gap Equation at T=Tc Function l(n) with n: an integer Cutoff function hp(wc) with wc of the order of QD 7 First-Principles Prediction of Tc (Takada)

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

  9. Eliashberg Function ab initio calculation of a2F(W) 9 First-Principles Prediction of Tc (Takada)

  10. MgB2 Two-gap typical BCS superconductor with Tc=40.2K with aid of E2g phonon modes in the B-layer AlB2（P6/mmm) a = 3.09Å、c = 3.52Å B-B distance=1.78Å larger than 1.67Åin boron solids 10 First-Principles Prediction of Tc (Takada)

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

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

  13. Gap Equation in GISC Gauge-Invariant Self-Consistent (GISC) determination of Z(iwp) Gap Equation with the vertex correction without m* Model Eliashberg Function Main message obtained from this study: For QD ~ EF , G0W0 is much better than GW (= Eliashberg theory) in calculating Tc.  Let us go with G0W0 in the first place! 13 First-Principles Prediction of Tc (Takada)

  14. Gap Equation in G0W0 Approximation Derive a gap equation in G0W0in which Zp(iwp)=1, cp(iwp)=0. cf. YT, JPSJ45, 786 (1978); JPSJ49, 1267 (1980). Analytic continuation: 14 First-Principles Prediction of Tc (Takada)

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

  16. SrTiO3 ◎ Ti 3d electrons (near the G point in the BZ) superconduct with the exchange of the soft ferroelectric phonon mode cf. YT, JPSJ49, 1267 (1980) 16 First-Principles Prediction of Tc (Takada)

  17. Graphite Intercalation Compounds KC8: Tc= 0.14K [Hannayet al., PRL14, 225(1965)] CaC6: Tc= 11.5K [Weller et al., Nature Phys. 1, 39(2005); Emery et al., PRL95, 087003(2005)] up to 15.1K under pressures [Gauzziet al., PRL98, 067002(2007)] CaC6 We should know the reason why Tc is enhanced by a hundred times by just changing K with Ca? 17 First-Principles Prediction of Tc (Takada)

  18. Electronic Structure Band-structure calculation: KC8: [Ohno et al., JPSJ47, 1125(1979); Wang et al., PRB44, 8294(1991)] LiC2: [Csanyi et al., Nature Phys.1, 42 (2005)] CaC6,YbC6: [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. 18 First-Principles Prediction of Tc (Takada)

  19. Microscopic Model for GICs This model was proposed in 1982 for explaining superconductivity in KC8: YT, JPSJ 51, 63 (1982) In2009, it was found that the same model also worked very well for CaC6: YT, JPSJ 78, 013703 (2009). 19 First-Principles Prediction of Tc (Takada)

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

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

  22. Calculated Results for Tc K Ca Valence Z1 2 Layer separation d ~ 5.5A ~ 4.5A Branching ratio f ~ 0.6 ~ 0.15 Band mass m*~ me(s-like)~3me(d-like) cf. Atomic mass mM is about the same. First-Principles Prediction of Tc (Takada)

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

  24. Dynamical Pairing Correlation Function Qsc(q,w) Conventional approach First-Principles Prediction of Tc (Takada)

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

  26. x0 in the BCS Theory a0: lattice constant High-Tc Inevitably associated with short x0 Formulate a scheme to calculate the pairing interaction from the zero-x0 limit in real-space approach. First-Principles Prediction of Tc (Takada)

  27. Evaluation ofthe Pairing Interaction 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 x0.  If x0 is short, N may be taken to be very small. First-Principles Prediction of Tc (Takada)

  28. Fullerene Superconductors • Alkali-doped fullerene superconductors 1) Molecular crystal composed of C60 molecules 2) Superconductivity appears with Tc =18-38K in the half-filled threefold narrow conduction bands (bandwidth W 0.5eV) derived from the t1u-levels in each C60 molecule. 3) The phonon mechanism with high-energy (w0 0.2eV) intramolecular phonons is believed to be the case, although the intramolecular Coulomb repulsionUis also strong and is about the same strength as the phonon-mediated attraction -2aw0 with athe electron-phonon coupling strength (a 2).  U 2aw0 cf. O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997). ~ ~ ~ ~ ~ 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 x0 is very short (less than 2a0) . cf. YT, JPSJ65, 1544, 3134 (1996). First-Principles Prediction of Tc (Takada)

  30. Electron-DopedC60 According to the band-structure calculation: The conventional electron- phonon parameter l is about 0.6 for a=2. The difference in Tc induced by that of the crystal structure including Cs3C60 under pressure [Takabayashi et al., Science 323, 1589 (2009)] is successfully incorporated by that in. First-Principles Prediction of Tc (Takada)

  31. Hypothetical Hole-DopedC60 Hole-doped C60 : Carriers will be in the fivefold hu valence band.  a =3 First-Principles Prediction of Tc (Takada)

  32. Case of Even Larger a What happens for Tc, if a becomes even larger than 3? A larger a is expected in a system with a smaller number of p-electrons Np: A. Devos & M Lannoo, PRB58, 8236 (1998). Case of C36 is interesting: a=4 The C36 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). First-Principles Prediction of Tc (Takada)

  33. Hypothetical Doped C36 If solid C36 is successfully doped  a =4 First-Principles Prediction of Tc (Takada)

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

  35. Pairing Interaction in Weak-Coupling Region Remember: The homogeneous electron gas is useful in constructing a practical and useful form for Vxc(r;[n(r)]):  LDA, GGA etc. Let us consider the same system for constructing Kij in the weak-coupling region.  G0W0 calculation will be enough! 35 First-Principles Prediction of Tc (Takada)

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

  37. Kij in the Strong-Coupling Region Qsc in terms of KS orbitals Weak-coupling case ~ Use gij instead of Vij in the general case! In the strong-coupling region, the W-dependence of g will be weak. ~ ~ Note: g corresponds to fxc in TDDFT! 37 First-Principles Prediction of Tc (Takada)

  38. Summary 10 Review the Green’s-function approach to the calculation of the superconducting Tc. 20The Eliashberg theory is good for phonon mechanism of superconductivity, but not good for high-Tc materials. 30For weak-coupling superconductors, G0W0 is applicable to both phonon and/or electronic mechanisms. 40Clarified the mechanism of superconductivity in GIC, especially the difference between KC8 and CaC6. 50Proposed a calculation scheme to treat strong-coupling superconductors, if the coherence length is short. 60Addressed fullerites in this respect and find that Tc might exceed 100K. 70Connection is made to the density functional theory for superconductivity; especially a new functional form for the pairing interaction is proposed. First-Principles Prediction of Tc (Takada)

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