Seminar @ Academia Sinica

On the First-Principles Determinationof the Superconducting Transition Temperature
Yasutami Takada
The Institute for Solid State Physics, The University of Tokyo
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581
takada@issp.u-tokyo.ac.jp
505B Meeting Room, 5F, IRBST
Research Center for Applied Sciences, Academia Sinica, Taipei
14:00-16:00, Wednesday, 21 December 2016

cf. YT, Reference Module in Materials Science and Materials Engineering(Elsevier, 2016) doi: 10.1016/B978-0-12-803581-8.00774-8.
YT, Molecular Physics 114, 1041 (2016).
On Tc from first principles (Takada)

1Outline
1. Introduction
Fröhlich polaron model
2. Electron-phonon system in the Green’s-function approach2
 Eliashberg theory and the Eliashberg function  F()Problem about the smallness parameter D/EF
Eliashberg theory with vertex correction in GISC
3. G0W0 approximation to the Eliashberg theory
 BCS-type gap equation derived from first principles
 STO (plasmon-phonon mechanism Uemura Plot) and GIC
4. Connection with DFT for superconductors (SCDFT)
Functional form for pairing interaction Kij
Introduction of pairing kernel gij as an analogue of exchange
correlation kernel fxc in time-dependent DFT (TDDFT)
5. Superconductors with short coherence length Hubbard-Holstein model and alkali-doped fullerenes
6. SummaryOn Tc from first principles (Takada)

2

Polar Crystals◎ Late 70’s: High-Tc superconductivity in CuCl and CdS?
 Investigate the possibility of high-Tc superconductivity
in doped polar crystals with low carrier density in termsof bipolaron Bose-Einstein condensation (BEC) A research subject given by Prof. Al Overhauser (1925-2011).

Many-electron Fröhlich polaron model

On Tc from first principles (Takada)

3

Fröhlich Polaron Model◎ One-electron problemFröhlich polaron Diagrammatic QMC: Mishchenko et al. , PRB62, 6317 (2000).

◎ Two-electron problem  Interaction between electrons

 Bipolaron (i.e., preformed pair) is not constructed unless ≫1YT, PRB26, 1223(1983); Hiramoto-Toyozawa, JPSJ, 1985

“Bipolaron high-Tc superconductivity” is not expected to occur.
◎ Many-electron problem

 Cooper pairs can be formed even for small , but how about Tc?
On Tc from first principles (Takada)4

General Model for Electron-Phonon System
Hamiltonian

Nambu Representation

Singlet pairing

Green’s Function

Off-diagonal part Anomalous Green’s Function: F(p,ip)
On Tc from first principles (Takada)

5Exact 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 exact set of equations !
On Tc from first principles (Takada)

6

Eliashberg Theory
Basic assumption: D/EF ≪ 1(1) Migdal Thorem:(2) Separation between phonon-exchange & Coulomb parts

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

On Tc from first principles (Takada)

7

Renormalization Function and Gap Function(1) Equation to determine the Renormalization Function(2) Gap Equation at T=Tc

Function (n) with n: an integer

Cutoff function p(c) with c of the order of D

On Tc from first principles (Takada)8

Inclusion of Coulomb Repulsion(1) Equation to determine the Renormalization Function← Invariant!
(2) Gap Equation← Revised
Coulomb pseudopotential

On Tc from first principles (Takada)

9Eliashberg Function
Method for ab-initio calculation of 2F()Package: Quantum Espresso
k-points：48x48x48
q(=k’-k)-points：16x16x16

On Tc from first principles (Takada)

10Consideration on the Ward Identity

Include  in terms of GW theory for the normal state↖ YT, PRL87, 226402 (2001).
Ward identity
In accordance with the Eliashberg theory, after averaging over p
dependence, we obtain GISC scheme: YT, J. Phys.
Chem. Solids54, 1779 (1993).
Comments will be given later on (1) case of p’=p and (2) consideration on theeffects of Anderson-Bogoliubov-Nambu-Goldstone & Higgs-Anderson modes.
On Tc from first principles (Takada)11

Gap equations in GISC & G0W0
YT, J. Phys. Chem. Solids 54, 1779 (1993)
Gauge-Invariant Self-Consistent (GISC) determination of Z(ip)

Gap equation in GISC

Gap equation in G0W0 in which we take Z(ip)=1.

On Tc from first principles (Takada)

12

Comparison of Tc for Al among Three Schemes
Plot of Tc as afunction of *with =0.409

fcc AlYT, Mol. Phys.
114, 1041 (2016).

Conclusion(1) Tc & :
GISC G0W0 
GW (Eliashberg)(2) Z :GISC GW (Eliashberg)
G0W0On Tc from first principles (Takada)

13Comments(1) What is the actual meaning of Migdal’s theorem?
In the calculation of Tc, if all the vertex corrections are neglected, we should
neglect all the self-energy correction as well, in order to satisfy the Ward identity.
(2) Polaron effect
The Ward identity determines the vertex function for p’ ≠p as

but it is useless for p’ =p； instead, it is known as  =[1/]3.
If the polaron effect is large, it is much different from  in the limit of p’p.
cf. YT & T. Higuchi, PRB52, 12720 (1995).
(3) Effect of Anderson-Bogoliubov-Nambu-Goldstone mode on Tc◎ Amplitude mode（Higgs mode） does not change Tc；
cf. T. Gherghetta & Y. Nambu, PRB49, 740 (1994).
◎ Phase mode, the energy scale of which is that of the plasmon, decreases Tc；
cf. YT, J. Phys. Chem. Solids 54, 1779 (1993).
On Tc from first principles (Takada)

14

Gap Equation in G0W0 Approximation
Derive a gap equation in G0W0 in which Zp(ip)=1, p(ip)=0.

cf. YT, JPSJ45, 786 (1978); JPSJ49, 1267 (1980).

Analytic continuation:

On Tc from first principles (Takada)

15

BCS-like Gap Equation

BCS-like gap equation obtainedby integrating out -variables
The pairing interaction
can be determinedfrom first principles.
No assumption is made for pairing symmetry.
On Tc from first principles (Takada)16Summary for Tc from First Principles in G0W0First-principles calculation of Tc without employing
the concept of the Coulomb pseudopotential *.
 Exactgap equation

Here the smallness parameter is not D/EF, but the coupling constant.
◎ Derivation of this gap equation is given in: YT, JPSJ 45, 786 (1978) , but I did
not understand the real meaning of this pairing interaction until late 2000’s.
On Tc from first principles (Takada)

17SCDFT
Extension of DFT to treat superconductivity (SCDFT)

 Basic variables: n(r) and (r,r’)cf. Oliveira, Gross & Kohn, PRL60, 2430 (1988).

On Tc from first principles (Takada)

18

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 afunctional of the exact normal-state n(r).

Scheme for determining Tc in inhomogeneouselectron systems in the weak-coupling region
On Tc from first principles (Takada)

19

Merits of This Functional Form
From the side of the Green’s function approach: For normal properties, we recently come to know that G0W0 ismuch better than GW ( corresponding to the Eliashberg
theory in superconductivity). As long as we do not know abouta proper form of the vertex function and D/EF is not smallenough, G0W0 should be adopted instead of GW,
recommending to use this functional form for the pairing
interaction for a wide range of applications.

From the side of SCDFT:
 This functional is an analogue to LDA for calculating normalstate properties. Although it do not always provide accurateenough information, LDA still always gives useful information
at least for the initial investigation of actual materials. Thesame would be true for first-principles calculation ofTc with this functional.
On Tc from first principles (Takada)

20

SrTiO3: Old Results◎ Ti 3d electrons (near the  point in the BZ) superconduct with the exchangeof the soft ferroelectric phonon mode

cf. YT, JPSJ49, 1267 (1980)

m* (band-edge mass)is the only parameterin determining Tc.

m* =1.8me ismost probable.
Pressure effects
are investigatedwith this m*.

On Tc from first principles (Takada)

21

SrTiO3:Compare with New Experiment
Recent experimental result:
Lin, Zhu, Fauque, & Behnia, PRX3, 021002 (2013)

YT, JPSJ49, 1267 (1980)

On Tc from first principles (Takada)

22

Plasmon-Polar Phonon Combined Mechanism
Calculated Tc for the many-electron Fröhlich model
YT, JPSJ 49, 1267 (1980)

・Optimum density exists with Tcmax ~ (5-10)m*/∞2 K.
・At very low densities, the polar phonons do not contribute;i.e., only the plasmons contribute to give Tc/EF ~ 0.04.
On Tc from first principles (Takada)

23

Uemura Plot
Uemura Plot:
Plot of Tc as a function of EF
obtained by the penetration depth
Y. J. Uemura et l., PRL66, 2665 (1991);
Nature 352, 605 (1991).

Will highhigh-Tc be obtained under
the condition of D/EF ≪ 1?
← Not at all!
In the phonon mechanism, Tc/D isknown to be less than about 0.05.
Because Tc/EF =(Tc/D )(D/EF), thisindicates that D/EF should be of theorder of unity. Thus interestinghigh-Tc materials cannot be
studied by the conventional
Eliashberg theory!!
On Tc from first principles (Takada)

24

Plasmon as Universal Mechanism
The universal value Tc/EF ~ 0.04
is reproduced in the plasmon
mechanism!
YT, JPSJ61, 3849 (1992).

Why is this the case?
 All the metallic
low-carrier density
systems are describedby the Hamiltonian ofthe electron gas, aslong as pF-1 ≪a0.
On Tc from first principles (Takada)

25

More on Superconductivity in the Electron Gas
1)
2)
3)
4)
5)
6)

Kohn-Luttinger, PRL 15, 524 (1965): Stress on Friedel oscillation; l > 0; Tc<<1K
YT, JPSJ 45, 786 (1978): G0W0 approximation; Tc ~ 1-10K at low densities
Rietschel-Sham, PRB 28, 5100 (1983): GW0approximation; Tc ~100K
Grabowski-Sham, PRB 29, 6132 (1984): Inclusion of 1st-order vertex; no SC
Shirron-Ruvalds, PRB 34, 7596 (1986): With static local-field correction
YT, PRB 37, 155 (1988): Variational approach with correlated BCS function;
Inclusion of more than 40 vertex corrections; Similar results with that in G0W07) Kuechenhoff-Woelfle, PRB 38, 935 (1988): Obtain the effective interaction bysolving the Bethe-Salpeter equation; Extension of the approach in 1).
8) Canright-Vignale, PRB 39, 2740 (1989): 2D version of 5).
9) YT, PRB 47, 5202 (1993): Same approach as in 5), but extended to much lower
densities to obtain the similar result as that in G0W0.
10) Richardson-Ashcroft, PRL 78, 118 (1997); PRB 55, 15130 (1997): Same approachas that in 5); Improvement on the local-field correction and simultaneouslyincluding the phonon effects to apply to the metallic hydrogen.
11) Gross group; PRL 100, 257001 (2008); PRB 81, 134505 & 134506 (2010):

SCDFT applied to metallic hydrogen with Tc over 200K.
12) Akashi & Arita, PRL111, 057006 (2013); Verification of plasmon effect in
actual systems: Li & Ca, and also recently in Li doped ZrNCl & HfNCl.
On Tc from first principles (Takada)26

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?
On Tc from first principles (Takada)

27

Electronic Structure
Band-structure calculation:
Band[Ohno et al., JPSJ47, 1125(1979); Wang et al., PRB44, 8294(1991)]
KC8:
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..

On Tc from first principles (Takada)28

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 samemodel also worked very well for CaC6: YT, JPSJ 78, 013703 (2009).
On Tc from first principles (Takada)

29

Model HamiltonianFirst-principles Hamiltonian for polar-coupling layered crystals
cf. YT, J. Phys. Soc. Jpn. 51, 63 (1982)

On Tc from first principles (Takada)

30

Effective Electron-Electron Interaction in RPA

On Tc from first principles (Takada)31

Calculated Results for Tc

YT, JPSJ 78,013703 (2009).
KCa
ValenceZ
12Layer separation d~ 5.5A~ 4.5A
Branching ratio
f~ 0.6~ 0.15
~ me (s-like) ~ 3me (d-like)Band massm*cf. Atomic mass mM is about the same.
On Tc from first principles (Takada)32

Perspectives for Higher Tc

YT, JPSJ 78,013703 (2009).

◎ Two key controlling parameters: Z and m*.
◎ Tc will be raised by a few times from thecurrent value of 15K, but never go beyond100K.
On Tc from first principles (Takada)33

強結合電子フォノン系の場合
=1.023

=1.504

On Tc from first principles (Takada)34

Dynamical Pair Correlation Function
Qsc(q,); Qsc(q = 0,) at T=Tc
Green’s-Function approach:

DFT approach:In this definition of g, both vertex and self-energy corrections are
simultaneously included, in good corresponding to the Ward identity.
On Tc from first principles (Takada)35

Kij in the Strong-Coupling Region
Qsc in terms of KS orbitals

Weak-coupling caseUse ~gij instead of Vij in the general case!
In the strong-couplingstrongregion, the~-dependence of g will be weak.
Note: ~g corresponds to f in TDDFT!
xc

On Tc from first principles (Takada)360 in the BCS Theory

a0: lattice constantHigh-Tc  Inevitably associated with short 0

Formulate a scheme to calculate the pairing interaction
from the zero-0 limit in real-space approach.
On Tc from first principles (Takada)37

Evaluation of the Pairing Interaction

Basic observation: The essential physics of electron
pairing can be captured in an N-site system, if thesystem size is large enough in comparison with 0.
 If 0 is short, N may be taken to be very small.
On Tc from first principles (Takada)38

Fullerene Superconductors◎ Alkali-doped fullerene superconductors
1) Molecular crystal composed of C60 molecules2) 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 (0 0.2eV)
intramolecular phonons is believed to be the case, although
the intramolecular Coulomb repulsion U is also strong and isabout the same strength as the phonon-mediated attraction-20 with  the electron-phonon coupling strength ( 2).
 U 20cf. O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997).

~~~~~On Tc from first principles (Takada)39

Hubbard-Holstein Model
Band-multiplicity:
It may be important in discussing the absence of Mott insulatingphase [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 2a0) .

cf. YT, JPSJ65, 1544, 3134 (1996).
On Tc from first principles (Takada)40

Electron-Doped C60
According to the bandstructure calculation:

The conventional electronphonon parameter isabout 0.6 for =2.

The difference in Tc induced by that of the crystal structure
including Cs3C60 under pressure [Takabayashi et al., Science323, 1589 (2009)] is successfully incorporated by that in .
On Tc from first principles (Takada)

41

Hypothetical Hole-Doped C60
Hole-doped C60 : Carriers will be in thefivefold hu valence band.  =3On Tc from first principles (Takada)42

Case of Even Larger What happens for Tc, if becomes even larger than 3?
A larger  is expected in a systemwith a smaller number of electrons N: A. Devos & M
Lannoo, PRB58, 8236 (1998).

Case of C36 is interesting: =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).
On Tc from first principles (Takada)

43

Hypothetical Doped C36
If solid C36 is successfully
doped  system with =4
 Room-temperature
superconductor!

cf. For aromatic materials, the bond-phonon model like the SuSchrieffer-Heeger one seems to be more realistic;
the case of K3 picene was studied in
M. Casula et al., PRL107, 117006 (2011);
PRB86, 075445 (2012).
On Tc from first principles (Takada)44

Summary
10 In the weak-coupling region, the scheme in G0W0 provides
more accurate results than the conventional Eliashberg
scheme. In particular, the former determines the Coulomb
pseudopotential * from first principles.
20 A scheme based on SCDFT also provides Tc with similar
accuracy for the phonon mechanism without the empiricalparameter *, but its current functional form for the pairing
interaction is not good enough to treat the phononmechanism and the electronic mechanisms on the samefooting, contrary to the G0W0 scheme.
30 In the strong coupling and/or correlation region, much morework is needed in constructing the scheme for Tc from firstprinciples.
On Tc from first principles (Takada)45