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) 1 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 o STO and GIC 4. Superconductors with short coherence length o Hubbard-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 gij as an analogue of exchangecorrelation kernel fxc in time-dependent density functional theory 6. Summary First-Principles Prediction of Tc (Takada) 2 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 Hamiltonian for the electron-phonon system without employing such phenomenological adjustable parameters as m*. First-Principles Prediction of Tc (Takada) 3 Model Electron-Phonon System Hamiltonian Nambu Representation Green’s Function Off-diagonal part Anomalous Green’s Function: F(p,iwp) First-Principles Prediction of Tc (Takada) 4 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 ! First-Principles Prediction of Tc (Takada) 5 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 First-Principles Prediction of Tc (Takada) 6 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 First-Principles Prediction of Tc (Takada) 7 Inclusion of Coulomb Repulsion (1) Equation to determine the Renormalization Function ← Invariant! (2) Gap Equation ← Revised Coulomb pseudopotential First-Principles Prediction of Tc (Takada) 8 Eliashberg Function ab initio calculation of a2F(W) First-Principles Prediction of Tc (Takada) 9 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 First-Principles Prediction of Tc (Takada) 10 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. First-Principles Prediction of Tc (Takada) 11 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 First-Principles Prediction of Tc (Takada) 12 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! First-Principles Prediction of Tc (Takada) 13 Gap Equation in G0W0 Approximation Derive a gap equation in G0W0 in which Zp(iwp)=1, cp(iwp)=0. cf. YT, JPSJ45, 786 (1978); JPSJ49, 1267 (1980). Analytic continuation: First-Principles Prediction of Tc (Takada) 14 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. First-Principles Prediction of Tc (Takada) 15 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) First-Principles Prediction of Tc (Takada) 16 Graphite Intercalation Compounds KC8: Tc = 0.14K [Hannay et 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 [Gauzzi et al., PRL98, 067002(2007)] CaC6 We should know the reason why Tc is enhanced by a hundred times by just changing K with Ca? First-Principles Prediction of Tc (Takada) 17 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. First-Principles Prediction of Tc (Takada) 18 Microscopic Model for GICs This model was proposed in 1982 for explaining superconductivity in KC8: YT, JPSJ 51, 63 (1982) In 2009, it was found that the same model also worked very well for CaC6: YT, JPSJ 78, 013703 (2009). First-Principles Prediction of Tc (Takada) 19 Model Hamiltonian First-principles Hamiltonian for polar-coupling layered crystals cf. YT, J. Phys. Soc. Jpn. 51, 63 (1982) First-Principles Prediction of Tc (Takada) 20 Effective Electron-Electron Interaction in RPA First-Principles Prediction of Tc (Takada) 21 Calculated Results for Tc K Ca Valence Z 1 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) 22 Perspectives for Higher Tc ◎ Two key controlling parameters: Z and m*. ◎ Tc will be raised by a few times from the current value of 15K, but never go beyond100K. First-Principles Prediction of Tc (Takada) 23 Dynamical Pairing Correlation Function Qsc(q,w) Conventional approach First-Principles Prediction of Tc (Takada) 24 Reformulation of Qsc(q,w) In ~g, both self-energy renormalization and vertex corrections are included. First-Principles Prediction of Tc (Takada) 25 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) 26 Evaluation of the 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) 27 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 repulsion U is also strong and is about the same strength as the phonon-mediated attraction -2aw0 with a the electron-phonon coupling strength (a 2). U 2aw0 cf. O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997). ~ ~ ~ ~ ~ First-Principles Prediction of Tc (Takada) 28 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) 29 Electron-Doped C60 According to the bandstructure calculation: The conventional electronphonon 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) 30 Hypothetical Hole-Doped C60 Hole-doped C60 : Carriers will be in the fivefold hu valence band. a =3 First-Principles Prediction of Tc (Takada) 31 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 pelectrons 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) 32 Hypothetical Doped C36 If solid C36 is successfully doped a =4 First-Principles Prediction of Tc (Takada) 33 SCDFT Extension of DFT to treat superconductivity (SCDFT) Basic variables: n(r) and c(r,r’) cf. Oliveira, Gross & Kohn, PRL60, 2430 (1988). First-Principles Prediction of Tc (Takada) 34 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! First-Principles Prediction of Tc (Takada) 35 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 First-Principles Prediction of Tc (Takada) 36 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 f in TDDFT! xc First-Principles Prediction of Tc (Takada) 37 Summary 10 Review the Green’s-function approach to the calculation of the superconducting Tc. 20 The Eliashberg theory is good for phonon mechanism of superconductivity, but not good for high-Tc materials. 30 For weak-coupling superconductors, G0W0 is applicable to both phonon and/or electronic mechanisms. 40 Clarified the mechanism of superconductivity in GIC, especially the difference between KC8 and CaC6. 50 Proposed a calculation scheme to treat strong-coupling superconductors, if the coherence length is short. 60 Addressed fullerites in this respect and find that Tc might exceed 100K. 70 Connection 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) 38

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