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First-Principles Prediction of Tc (Takada)ISSP
Workshop/Symposium: MASP 2012*Theory for Reliable First-Principles
Prediction of the Superconducting TcYasutami Takada
Institute for Solid State Physics, University of Tokyo5-1-5
Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
Seminar Room A615,ISSP, University of Tokyo14:00-15:30, Thursday
28 June 2012
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Outline*1.
Introduction2. Electron-phonon system in the Greens-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 GISC3. G0W0
approximation to the Eliashberg theory o STO and GIC4.
Superconductors with short coherence length o Hubbard-Holstein
model and alkali-doped fullerenes5. Connection with density
functional theory for supperconductors o Functional form for
pairing interaction Kij o Introduction of pairing kernel gij as an
analogue of exchange- correlation kernel fxc in time-dependent
density functional theory6. Summary*
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Introduction**
Discovery of novel superconductors novel physical properties and/or
phenomena High-Tc superconductors By far the most interesting
property is Tc itself! Why dont 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)
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First-Principles Prediction of Tc (Takada)Model Electron-Phonon
System**HamiltonianNambu RepresentationGreens FunctionOff-diagonal
part Anomalous Greens Function: F(p,iwp)
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Exact
Self-Energy**Formally exact equation to determine the
self-energyBare electron-electron interactionEffective
electron-electron interactionDirect extension of the Hedins set of
equations !Polarization function
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Eliashberg
Theory**Basic assumption: QD/EF 1(2) Separation between
phonon-exchange & Coulomb parts (1) Migdal Thorem:neglect for a
whileP(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)
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First-Principles Prediction of Tc (Takada)Renormalization
Function and Gap Function**(2) Gap Equation at T=Tc (1) Equation to
determine the Renormalization FunctionFunction l(n) with n: an
integerCutoff function hp(wc) with wc of the order of QD
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Inclusion of Coulomb
Repulsion**(2) Gap Equation (1) Equation to determine the
Renormalization FunctionCoulomb pseudopotential Invariant!
Revised
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Eliashberg
Function**ab initio calculation of a2F(W)
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)MgB2**Two-gap typical
BCS superconductor with Tc=40.2Kwith aid of E2g phonon modes in the
B-layer
AlB2P6/mmm) a = 3.09c = 3.52
B-B distance=1.78 larger than 1.67in boron solids
First-Principles Prediction of Tc (Takada)
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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 theory!!Need to
develop a theory applicable to the case of QD/EF ~ 1.
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Return to the Exact
Theory**How should we treat the vertex function? GWGReformulate the
Eliashberg theory with including this vertex function. cf. YT, in
Condesed Matter Theories, Vol. 10 (Nova, 1995), p. 255Ward
IdentityIf we take an average over momenta in accordance with the
Eliashberg theory, we obtain:
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Gap Equation in
GISC**Gap Equation with the vertex correction without
m*Gauge-Invariant Self-Consistent (GISC) determination of
Z(iwp)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!Model Eliashberg Function
First-Principles Prediction of Tc (Takada)
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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)
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First-Principles Prediction of Tc (Takada)BCS-like Gap
Equation**The pairing interactioncan be determinedfrom first
principles.BCS-like gap equation obtained by integrating
w-variables No assumption is made for pairing symmetry.
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)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)
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First-Principles Prediction of Tc (Takada)Graphite Intercalation
Compounds**CaC6KC8: 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)
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First-Principles Prediction of Tc (Takada)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)
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First-Principles Prediction of Tc (Takada)Microscopic Model for
GICs**This model was proposed in 1982 for explaining
superconductivityin KC8: YT, JPSJ 51, 63 (1982) In 2009, it was
found that the samemodel also worked very well for CaC6: YT, JPSJ
78, 013703 (2009).
First-Principles Prediction of Tc (Takada)
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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)
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First-Principles Prediction of Tc (Takada)Effective
Electron-Electron Interaction in RPA*
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)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)
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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)
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First-Principles Prediction of Tc (Takada)*Dynamical Pairing
Correlation FunctionConventional approachQsc(q,w)
First-Principles Prediction of Tc (Takada)
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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)
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First-Principles Prediction of Tc (Takada)*x0 in the BCS
TheoryHigh-Tc Inevitably associated with short x0Formulate a scheme
to calculate the pairing interaction from the zero-x0 limit in
real-space approach. a0: lattice constant
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)*Evaluation of the
Pairing InteractionBasic 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)
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First-Principles Prediction of Tc (Takada)*Fullerene
SuperconductorsAlkali-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)
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First-Principles Prediction of Tc (Takada)*Hubbard-Holstein
ModelBand-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)
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First-Principles Prediction of Tc (Takada)*Electron-Doped
C60According to the band-structure calculation: 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 .The conventional
electron-phonon parameter l is about 0.6 for a=2.
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)*Hypothetical
Hole-Doped C60Hole-doped C60 : Carriers will be in the fivefold hu
valence band. a =3
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)*Case of Even Larger
aWhat 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=4The 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)
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First-Principles Prediction of Tc (Takada)*Hypothetical Doped
C36If solid C36 is successfully doped a =4
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)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)
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First-Principles Prediction of Tc (Takada)Pairing Interaction in
Weak-Coupling Region**Remember:The homogeneous electron gas is
useful in constructinga 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)
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First-Principles Prediction of Tc (Takada)Kij in the
Weak-Coupling Region**i*: time-reversed orbital of the KS orbital i
Good correspondence! 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
inhomogeneouselectron systems in the weak-coupling region
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)Kij in the
Strong-Coupling Region**Qsc in terms of KS orbitalsIn the
strong-coupling region, the W-dependence of g will be
weak.~Weak-coupling case Use gij instead of Vij in the general
case!~Note: g corresponds to fxc in TDDFT!~
First-Principles Prediction of Tc (Takada)
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First-Principles Prediction of Tc (Takada)*Summary10 Review the
Greens-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)
*********************************If we apply the same philosophy
to superconductivity, we can formulate a framework for determining
Tc exactly without accurately knowing any other physical
quantities. The extension of DFT can be done in something like
this. *****
