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
Thursday 28 June 2012
First-Principles Prediction of Tc (Takada)
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
6. Summary
First-Principles Prediction of Tc (Takada)
○ 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)
Model Electron-Phonon System
Nambu Representation
Green’s Function
Off-diagonal part Anomalous Green’s Function: F(p,iwp)
First-Principles Prediction of Tc (Takada)
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)
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)
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)
Inclusion of Coulomb Repulsion
(1) Equation to determine the Renormalization Function
← Invariant!
(2) Gap Equation
← Revised
Coulomb pseudopotential
First-Principles Prediction of Tc (Takada)
Eliashberg Function
ab initio calculation of a2F(W)
First-Principles Prediction of Tc (Takada)
Two-gap typical BCS superconductor with Tc=40.2K
with aid of E2g phonon modes in the B-layer
a = 3.09Å、c = 3.52Å
B-B distance=1.78Å larger
than 1.67Åin boron
First-Principles Prediction of Tc (Takada)
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
Need to develop a theory applicable to the case of QD/EF ~ 1.
First-Principles Prediction of Tc (Takada)
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)
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)
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)
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)
◎ 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)
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)]
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)
Electronic Structure
Band-structure calculation:
[Ohno et al., JPSJ47, 1125(1979); Wang et al., PRB44, 8294(1991)]
[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)
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)
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)
Effective Electron-Electron Interaction in RPA
First-Principles Prediction of Tc (Takada)
Calculated Results for Tc
Layer separation d
~ 5.5A
~ 4.5A
Branching ratio
~ 0.6
~ 0.15
Band mass
~ me (s-like) ~ 3me (d-like)
cf. Atomic mass mM is about the same.
First-Principles Prediction of Tc (Takada)
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)
Dynamical Pairing Correlation Function
Conventional approach
First-Principles Prediction of Tc (Takada)
Reformulation of Qsc(q,w)
In ~g, both self-energy renormalization
and vertex corrections are included.
First-Principles Prediction of Tc (Takada)
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)
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)
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)
Hubbard-Holstein Model
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)
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)
Hypothetical Hole-Doped C60
Hole-doped C60 : Carriers will be in the
fivefold hu valence band.  a =3
First-Principles Prediction of Tc (Takada)
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)
Hypothetical Doped C36
If solid C36 is successfully doped  a =4
First-Principles Prediction of Tc (Takada)
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)
Pairing Interaction in Weak-Coupling Region
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)
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)
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!
First-Principles Prediction of Tc (Takada)
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)

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