Theory for Reliable First-Principles Prediction of the Superconducting

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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


c = 3.52






B
-
B distance=1.78


l
arger
than
1.67

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.