First

Principles Prediction of Tc (Takada)
ISSP Workshop/Symposium: MASP 2012
1
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
14:00

15:30,
Thursday 28 June 2012
First

Principles Prediction of Tc (Takada)
Outline
2
1. Introduction
2. Electron

phonon system in the Green’s

function Approach
o
Eliashberg theory
and the Eliashberg function
a
2
F
(
W
)
o
Problem about the smallness parameter
Q
D
/
E
F
Uemura Plot
o
Eliashberg theory with
vertex correction in GISC
3. G
0
W
0
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
K
ij
o
Introduction of
pairing kernel
g
ij
as
an analogue of e
xchange

correlation kernel
f
xc
in time

dependent density functional
theory
6. Summary
2
First

Principles Prediction of Tc (Takada)
Introduction
3
3
○
Discovery of novel superconductors
novel physical properties and/or phenomena
◎
High

T
c
superconductors
By far the most interesting property is
T
c
itself!
Why don’t we investigate this quantity directly?
○
An ultimate goal in theoretical high

T
c
business
Develop a reliable scheme for a
first

principles
prediction of
T
c
, with using only information
on constituent atoms.
○
For the time being, we shall be content with
an
accurate estimation of
T
c
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
4
4
Hamiltonian
Nambu Representation
Green’s Function
Off

diagonal part
Anomalous Green’s Function
:
F
(
p
,i
w
p
)
First

Principles Prediction of Tc (Takada)
Exact Self

Energy
5
5
Formally exact equation to determine the self

energy
Bare electron

electron interaction
Effective electron

electron interaction
Direct extension of the Hedin’s set of equations !
Polarization function
First

Principles Prediction of Tc (Takada)
Eliashberg
Theory
6
6
Basic assumption:
Q
D
/
E
F
≪
1
(2) Separation between phonon

exchange & Coulomb parts
(1) Migdal Thorem:
neglect for a while
↑
P
(
q
,
i
w
q
)
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
7
7
(2) Gap Equation at
T
=
T
c
(1) Equation to determine the Renormalization Function
Function
l
(
n
) with
n
: an integer
Cutoff function
h
p
(
w
c
) with
w
c
of the order of
Q
D
First

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

Principles Prediction of Tc (Takada)
Eliashberg
Function
9
9
ab initio
calculation of
a
2
F
(
W
)
First

Principles Prediction of Tc (Takada)
MgB
2
10
10
Two

gap typical BCS superconductor with
T
c
=40.2K
with aid of
E
2
g
phonon modes in the B

layer
A
lB
2
（
P6/mmm
)
a = 3.09
Å
、
c = 3.52
Å
B

B distance=1.78
Å
l
arger
than
1.67
Å
in boron
solids
First

Principles Prediction of Tc (Takada)
Uemura Plot
11
11
Will high

T
c
be obtained
under the condition of
Q
D
/
E
F
≪
1?
←
乯琠慴N慬a!
In the phonon mechanism,
T
c
/
Q
D
is
known to be less than about 0.05.
Because
T
c
/
E
F
=(
T
c
/
Q
D
)(
Q
D
/
E
F
), this
indicates that
Q
D
/
E
F
should be of
the order of unity. Thus
interesting
high

T
c
materials cannot be studied
by the conventional Eliashberg
theory!!
Need to develop a theory applicable to the case of
Q
D
/
E
F
~
1.
First

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

Principles Prediction of Tc (Takada)
Gap Equation in GISC
13
13
Gap Equation with the vertex correction without
m
*
Gauge

Invariant Self

Consistent (GISC) determination of
Z
(i
w
p
)
Main message obtained from this study:
For
Q
D
~
E
F
, G
0
W
0
is much better than
GW (= Eliashberg theory) in calculating
T
c
.
Let us go with
G
0
W
0
in the first place!
Model Eliashberg Function
First

Principles Prediction of Tc (Takada)
Gap Equation in G
0
W
0
Approximation
14
14
Derive a gap equation in G
0
W
0
in which
Z
p
(i
w
p
)=1,
c
p
(i
w
p
)=0.
cf
. YT, JPSJ
45
, 786 (1978); JPSJ
49
, 1267 (1980).
Analytic continuation:
First

Principles Prediction of Tc (Takada)
BCS

like Gap Equation
15
15
The pairing interaction
can be determined
from first principles.
BCS

like gap equation obtained
by integrating
w

variables
No assumption is made for pairing symmetry.
First

Principles Prediction of Tc (Takada)
SrTiO
3
16
16
◎
Ti 3d electrons (near the
G
point in the BZ) superconduct with
the exchange of the soft ferroelectric phonon mode
cf
. YT, JPSJ
49
, 1267 (1980)
First

Principles Prediction of Tc (Takada)
Graphite Intercalation Compounds
17
17
CaC
6
KC
8
:
T
c
= 0.14K
[
Hannay
et al
.,
PRL
14
, 225(1965)]
CaC
6
:
T
c
=
11.5K [Weller
et al
.,
Nature Phys.
1
, 39(2005);
Emery et al.,
PRL
95
, 087003(2005)]
up to 15.1K under pressures [
Gauzzi
et al
.,
PRL
98
, 067002(2007)]
We should know the reason why
T
c
is enhanced
by
a hundred times
by just changing K with Ca?
First

Principles Prediction of Tc (Takada)
Electronic Structure
18
18
Band

structure calculation
:
KC
8
:
[Ohno
et al
.,
JPSJ
47
, 1125(1979); Wang
et al
.,
PRB
44
, 8294(1991)]
LiC
2
:
[Csanyi
et al
.,
Nature Phys.
1
, 42 (2005)]
CaC
6
,YbC
6
: [Mazin,
PRL
95
,227001(2005);Calandra & Mauri,
PRL
95
,237002(2005)
]
I
mportant 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
19
19
This model was proposed in
1982
for explaining superconductivity
in KC
8
:
YT, JPSJ
51
, 63 (1982)
In
2009
, it was found that the same
model also worked very well for CaC
6
:
YT, JPSJ
78
, 013703 (2009).
First

Principles Prediction of Tc (Takada)
Model Hamiltonian
20
First

principles Hamiltonian for polar

coupling layered crystals
cf
. YT,
J. Phys. Soc. Jpn.
51
, 63 (1982)
20
First

Principles Prediction of Tc (Takada)
Effective Electron

Electron Interaction in RPA
21
First

Principles Prediction of Tc (Takada)
Calculated Results for
T
c
22
K
Ca
Valence
Z
1
2
Layer separation
d
~ 5.5A ~ 4.5A
Branching ratio
f
~ 0.6 ~ 0.15
Band mass
m
*
~
m
e
(s

like)
~
3
m
e
(d

like)
cf.
Atomic mass
m
M
is about the same.
First

Principles Prediction of Tc (Takada)
Perspectives for Higher
T
c
23
◎
Two key controlling parameters:
Z
and
m
*
.
◎
T
c
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
Conventional approach
Q
sc
(
q
,
w
)
First

Principles Prediction of Tc (Takada)
25
Reformulation of
Q
sc
(
q
,
w
)
In g, both self

energy renormalization
and vertex corrections are included.
~
First

Principles Prediction of Tc (Takada)
26
x
0
in the BCS Theory
High

T
c
Inevitably associated with short
x
0
Formulate a scheme to calculate the pairing interaction
from the zero

x
0
limit in real

space approach.
a
0
: lattice constant
First

Principles Prediction of Tc (Takada)
27
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
x
0
.
If
x
0
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 C
60
molecules
2) Superconductivity appears with
T
c
=18

38K in the half

filled
threefold narrow conduction bands (bandwidth
W
0.5eV
)
derived from the
t
1u

levels in each C
60
molecule.
3) The phonon mechanism with high

energy (
w
0
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

2
aw
0
with
a
the electron

phonon coupling strength (
a
2
).
U 2
aw
0
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, PRL
84
, 1276 (2000)], but it is
not the case for discussing superconductivity [Cappelluti, Paci,
Grimaldi, & Pietronero, PRB
72
, 054521 (2005)].
The simplest possible model to describe this situation is:
, because
x
0
is very
short (less than 2
a
0
)
.
cf
. YT, JPSJ
65
, 1544, 3134 (1996).
First

Principles Prediction of Tc (Takada)
30
Electron

Doped
C
60
According to the band

structure calculation:
The difference in
T
c
induced by that of the crystal structure
including Cs
3
C
60
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)
31
Hypothetical Hole

Doped
C
60
Hole

doped C
60
: Carriers will be in the
fivefold
h
u
valence band.
a
=3
First

Principles Prediction of Tc (Takada)
32
Case of Even Larger
a
What happens for
T
c
, if
a
becomes even larger than 3?
A larger
a
is expected in a system
with a smaller number of
p

electrons
N
p
:
A. Devos & M
Lannoo, PRB
58
, 8236 (1998).
Case of C
36
is interesting:
a
=4
The C
36
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, PRL
81
, 697 (1998).
First

Principles Prediction of Tc (Takada)
33
Hypothetical Doped C
36
If solid C
36
is successfully doped
a
=4
First

Principles Prediction of Tc (Takada)
SCDFT
34
34
Extension of DFT to treat superconductivity (
SCDFT
)
Basic
variables:
n
(
r
)
and
c
(
r
,
r’
)
cf
. Oliveira, Gross & Kohn, PRL
60
, 2430 (1988).
First

Principles Prediction of Tc (Takada)
Pairing Interaction in Weak

Coupling Region
35
35
Remember:
The homogeneous electron gas
is useful in constructing
a practical and useful form for
V
xc
(
r
;[
n
(
r
)]):
LDA, GGA etc.
Let us consider the same system for constructing
K
ij
in the weak

coupling region
.
G
0
W
0
calculation will be enough!
First

Principles Prediction of Tc (Takada)
K
ij
in the Weak

Coupling Region
36
36
i
*
: time

reversed orbital of the KS orbital
i
Good correspondence!
For the problem of determining
T
c
, the KS
orbitals can be determined uniquely as a
functional of the exact normal

state
n
(
r
).
Scheme for determining
T
c
in
inhomogeneous
electron systems in the weak

coupling region
First

Principles Prediction of Tc (Takada)
K
ij
in the Strong

Coupling Region
37
37
Q
sc
in terms of KS orbitals
In the strong

coupling region, the
W

dependence of g will be weak.
~
Weak

coupling case
Use
g
ij
instead of
V
ij
in the general case!
~
Note: g corresponds to
f
xc
in TDDFT!
~
First

Principles Prediction of Tc (Takada)
38
Summary
1
0
Review the Green’s

function approach to the calculation
of the superconducting
T
c
.
2
0
The Eliashberg theory is good for phonon mechanism of
superconductivity, but not good for high

T
c
materials.
3
0
For weak

coupling superconductors, G
0
W
0
is applicable
to both phonon and/or electronic mechanisms.
4
0
Clarified the mechanism of superconductivity in GIC,
especially the difference between KC
8
and CaC
6
.
5
0
Proposed a calculation scheme to treat strong

coupling
superconductors, if the coherence length is short.
6
0
Addressed fullerites in this respect and find that
T
c
might exceed 100K.
7
0
Connection is made to the density functional theory for
superconductivity; especially
a new functional form for
the pairing interaction is proposed.
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