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480
the
superexchange interaction becomes destroyed
with
increasing
the
hole
concentration in
the
overdoped regime,
the
K-S model
is
considered
not
to
hold beyond a
certain
hole concentration
XO'
As a result
FS
will change
from small ones
to
a large one beyond
XO'
In
the
case of LSCO
we
think
that
Xo
is 0.2 from experimental results of Loram et
al
23
and
of Nakano et
al
24.
4.
High
temperature
superconductivity
4.1.
Spin-dependent
electron-phonon
interaction
In
this
subsection we describe
that
the
electron-phonon interaction in
the
K-S model depends on a spin direction of a hole carrier, in doing so,
we
describe only a
stream
of theoretical derivation. Therefore, for those who
have interests in
the
derivation of equations, please
read
chapters
13
and
14
in
our
Springer
text
book
entitled
"Theory of
Copper
Oxide Superconduc-
tors"
6.
On
the
basis of
the
K-S model we have shown
that
the
interplay
between electron-phonon interaction
and
the
AF
order leads
to
the
d-wave
phonon-involved mechanism in
LSCO
26,27,6.
As seen in figure
7,
the
atomic wave function for up-spin coincides
with
that
for down-spin, if it
is
displaced by vector a
(a
is
Cu-O-Cu distance).
Thus,
the
wavefunctions of a hole-carrier
with
up
and
down spin in K-S
model have
the
following phase relation:
(2)
From
this
relation (2),
the
electron-phonon interaction
matrix
elements
from
states
k
to
k'
with
up
spin
scattered
by phonon
with
wave vector
q,
V
l'
(k, k',
q),
has
the
following spin-dependent property:
V
l'
(k,
k',
q)
=
exp(iK
.
a)V
1 (k,
k',
q),
(3)
where K = k -
k'
- q is a reciprocal lattice vector in
the
AF
Brillouin zone
and
a is a
Cu-O-Cu
distance. Since a reciprocal lattice vector in
the
AF
Brillouin zone is expressed as K =
(mr/a,m1f/a,O)
with
n+m
= even,
the
electron-phonon interactions for
up
spin
and
down spin
may
have a
different sign depending on a value of
K.
4.2.
Effective
inter-hole
interaction
via
phonon
From
the
relation (3)
and
K = (n1f/a,m1f/a,O)
with
n+m
= even,
the
effective interactions of a
pair
of holes from
(k
1',
- k
1)
to
(k'
1',
-
k'
1),
-
a
(a)
(b)
481
Figure
7.
Sche
matic
picture
showing
th
e
ph
a
se
relation
betwe
en
the
wav
e
functions
of
a
hole-carrier
with
up
and
down
spin
on
the
extended
two-story
house
mod
el for
La2-xSrxCu04
scattered
by phonon
with
wave vector
q,
is expressed as
v
l'
(k,
k'
,
q)V
1
(-k,
-k',
q)
=
exp(iK
.
a)1V
l'
(k, k',
q)12.
(4)
Since
exp(iK
.
a)
=
+1
for n = even
and
exp(iK
. a) =
-1
for n = odd,
the
effective interaction for forming a Cooper
pair
becomes
attractive
for
n = even while repulsive for n = odd.
This
remarkable result in
the
K-S
model leads
to
the
superconducting
gap
of d
x
2_y2
symmetry.
In
figure 8 we show how
attractive
and
repulsive subprocesses compete
each
other
in a
scattering
process of a conduction hole
by
phonon. Suppose
a hole occupies
the
state
A in figure
8.
A hole is
scattered
by phonon
with
wavevector q from A
to
shaded
region because q is in
normal
BZ.
So
that
,
scattering
process from
the
state
A
to
the
state
B consists
of
two kinds of subprocesses. One subprocess
is
that
a hole is
scatt
ered
by
phonon
with
wavevector q from A
to
state
C'
on
FS. Since
state
C'
is
transferred
to
an
equivalent
state
B by
the
translation
of a reciprocal lattice
vector
(-Ql
- Q2),
the
effective interaction for
this
subprocess is
attractive
482
q: phonon wave vector
QIt Q2:
AF
reciprocal lattice vectors
-
umklapp
sub-process
(repulsive)
..........•
normal
sub-process
(attractive)
Figure
8.
Competition
of
attractive
and
repulsive
subproc
esses
in
the
same
scattering
process
of a
carrier
hole
by
phonon
from
stat
e A
to
B
in
the
AF
Brillouin
zone. A hole
is first
scatt
ered
by
phonon
to
the
point
in
th
e
shadowed
area
,
and
then
to
the
point
equivalent
to
it.
according
to
equation
(4)
with
n =
2.
The
other
subprocess
is
that
the
hole
at
the
state
A
on
FS
is also
scattered
from A
to
state
C
by
phonon
with
wavevector
q.
Since
th
e
state
C is equivalent
to
state
B
by
the
translation
of
a reciprocal
lattice
ve
ctor
( -
Ql),
the
effective
interaction
is
repul
sive by
equation
(4)
with
n = m =
1.
State
C
I
is inside
an
AF
Brillouin zone while
state
C is
outside
th
e
AF
Brillouin zone, one
may
say
that
a
scattering
process from A
to
B
via
C
I
is a
Normal
scattering
while a
scattering
process from A
to
B
via
Cis
Umklapp
scattering.
Normal
and
Umklapp
scattering
have different sign, so
that
the
effective
int
e
raction
between
holes
at
state
A
and
B becomes
attractive
or
repulsive
depending
on
the
strengths
of
two
scatterings.
These
str
e
ngths
vary
by
depending
a wave
vector
q.
As
an
example
of
calculated
results, we
present
the
calculated
re-
sults
of
the
k
and
kl
de
pendence
of
th
e
electron-phonon
spectral
func
tion
0;2
Fr 1
([1,
k, k/)
with
spin-singlet for
on
e of
out-of-plan
e modes, A
1g
mode
(see
the
lower left
corner
ofthe
Fig.
9(a))
in
LSCO
with
tetragonal
symme-
try
in
figure 9, where
Fr
1
([1,
k, k/) is
th
e
momentum-dependent
density
of
phonon
st
a
tes
and
0;2
is
th
e
square
of
th
e electron-
phonon
coupling
constant
483
La
(b)
attractive
(a)
repulsive repulsive
(C)
Figure
9.
Calculated
result
of a
2
P,
1
(0,
k,
k')
and
a
gap
function
for
one
of
out-of-plane
phonon
mod
es,
Al
g
mode
in
LSCO
as
a
function
of
k'
26
,
6.
From
the
Fig. 9(
c)
we
can
see
that
the
momentum-dependent
spectral
function varies by
taking
values
with
+
and
- sign, when
the
wave vector
k'
changes from
the
section
CD
of
small
FS
to
the
section @)while k is fixed.
This
is clearly a d-wave behavior.
By
using
this
spectral
function,
we
can
obtain
the
Fig. 9(c),
the
obtained
d-component
gap
function 6.(k) vary as
a function of
(cos(kxa) - cos(kya))
and
the
wavefunction
of
a Cooper
pair
has a
spatial
extension
of
d
x
2_y2
symmetry.
Kamimura
et
al calculated
the
electron-phonon
spectral
function
a
2
F(n)
for all
phonon
modes
of
LSCO
with
tetragonal
symmetry,
and
their
calculated result
of
d-wave
component
of
th
e
spectral
function for
LSCO
with
tetragonal
symmetry, a
2
F i 1
(2)
(0,), is shown
in
figure 10 as a
function
of
phonon
frequency
n.
The
phonon
modes of
LSCO
with
tetrag-
onal
symm
e
try
are classified
into
in-plane modes
and
out-of-plane modes
with
regard
to
a CU02 plane.
Figure
10 shows
that
the
out-of-plane modes
484
(0
La
~
c:
Phonon
Energy n
(m
eV)
~
~
"'-
Example
of
~
phonon mode
""
~
a
...
s:l
0
.9
M
...,
u
that
contributes
&
C"I
to positive a
2F
Cil
I-<
(b)
•
...,
~
u
Q)
p.
0
r:n
ci
100
La
Example
of
phonon mode
...
0
that
contributes
(a)
to
negative a
2F
(C)
Figure
10.
The
d-wave
component
of
a
2
F(O)
calculated
for
tetragonal
LSCO
with
an
optimum
doping
such as
the
A
1g
mode
in figure 9 yield positive
spectral
function so
that
they
contribute
to
the
formation
of
Cooper
pairs by acting as
an
attractive
force while
the
in-plane modes yield negative
spectral
function so
that
these
do
not
contribute
to
the
formation
of
Cooper
pairs by
acting
as a repulsive
force.
4.3.
Calculated
results
of
the
hole-concentration
dependence
of
Tc
and
of
isotope
effect
Before calculating T
e
,
We would like
to
point
out
that
a
mean
free
path
Co
in
a metallic region, where K-S model
stands
up
all right, is much
expanded
from
th
e correlation
length
AS
by
the
spin fluctuation effect. As is shown
485
in
figure 11, localized spins
in
the
boundary
of
spin
correlation
region
are
frustrated
and
spin-flip
time,
Ts
cv
1O-14
sec
, is
shorter
than
traveling
time
of
a hole over
the
spin-correlated region,
TF
cv
1O-13
sec
.
So
that,
the
spin
correlated
region moves
with
the
dopant
hole
and
a
mean
free
path
is
much
expanded
from
the
correlation length.
bIg
spins
do
nt
hole
::
:
::
::::
::::.
:
-::
~
::::::::::::
:
::::::::
:
:::
- :
-
:::::::
:
::
:
:::::::
:
::::::
:
8
(a)
RVB
stat
e
::
: :
::::::::::
:
:
:::
: - :
_::
:~
:,
~
.o:;:::::::
:::: ::::
:
:
__
--:::
::
:
:::
:
:::::::
~
...,
t-HI--++--
-
--
spins
(b)
RVB
state
Figure
11.
Model
of
expansion
of
metallic
region.
When
a
dopant
hole
moves, localized
spin
flips
on
a
boundary
and
the
metallic
region moves
with
a
dopant
hole.
As
mentioned
previous section, we have shown
that
even
though
the
electron-phonon
interaction
is involved in
the
mechanism
of superconduc-
tivity,
the
interplay
of
the
AF
order
and
phonon
mechanism
in
the
K-S
model
creates
superconductivity
of "d-wave
symmetry",
when
a
system
is
infinite
and
homogeneous. Now we will
present
the
calculated
results
of
Tc
and
the
isotope effect for
LSCO
with
tetragonal
symmetry,
considering
that
the
metallic region is finite.
Kamimura
et al
26,27,6
calculated
the
hole-
concentration
dependence
of
Tc
and
the
isotope effects for
LSCO
using a
slightly modified McMillan
Tc-equation
28,
and
their
calculated
results
of
Tc
486
fa
(x=O
.15)
~[K]
= 300A
0
•
Exp
.
by
~
Ta
kag
i
et
al.
0
(Y')
~
TJK]
0
0
0
~
(Y')
0
(Y')
0
0
N
0
N
0
0
0
,....,
,....,
isotope effect a
o
N
O
L-
-L
__
~~
__
-L
~
L-~
__
~
O
0
L-~
__
~
~~~~~
.
~~~
~
0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3
Hole
Concentration
x
Hole
Concentration
x
(a)
(b)
Figure
12. (1)
The
calculated
result
on
the
hole-concentration
dependence
of
Tc(x)
for
LSCO
and
the
mean
free
path
£0
. (2)
The
calculated
isotope
effect a
and
Tc
curve
(thick
line) for te
tragonal
LSCO
as
a
function
of
x
and
the
isotope effect
ex
are shown in figure 12.
The
result of isotope effect
is compared
with
experimental results by Crawford
et
al 29
and
Ronay
et
al
30.
Here
the
isotope effect
ex
is
defined by
d log
Te
ex
= -
dlogM'
where M denotes
the
mass of
constituent
atoms.
In
their
calculation a well-
known relation between
the
Debye frequency
and
a mass,
DD
'"
M -
O
.
5
,
is
used.
Further,
in
their
calculation,
the
masses of all
constituent
atoms
have
been changed by
the
mass
ratio
of
18
0
to
16
0.
Thus
their
calculated results
may be overestimated.
Here
we
should make a
remark
on
the
calculated results of Te.
On
the
K-S
mod
el
the
size of a metallic
state
is finite.
In
order
to
obtain
the
observed value of 40 K for
the
highest
Tc
of LSCO
at
the
optimum
doping
(x
= 0.15),
Kamimura
et
al
26,27,6
hav
e chosen
the
size of a metallic region
£0
as shown in figure
12
.
487
From
the
calculated
results
of
the
isotope
effect shown
in
figure 12,
the
following conclusions emerge:
The
isotope
effect
on
Tc
in
LSCO
depends
on
the
hole
concentration
critically,
and
the
isotope
constant
0:
is
remarkably
large
near
the
onset
concentration
of
superconductivity
while
it
is
small
around
the
optimum
concentration.
Recently
Bishop
et al
32
published
a
review
article
on
recent
and
earlier
results
on
isotope
effects
in
cuprates,
pointing
out
that
the
isotope
effect
on
Tc
vanishes
at
optimum
doping
but
increases
with
decreasing
doping
level
to
be
substantially
larger
than
the
BCS
value
(0:
= 0.5)
at
the
border
to
the
AF
state.
This
experimental
result
is
consistent
with
the
theoretical
result
obtained
from
the
K-S
model
shown in figure 12.
In
this
section
we have
calculated
the
appearance
of d-wave suoercon-
ductivity, however,
because
of finite size effect, we consider
that
the
super-
conductivity
is
not
pure
d-wave
superconductivity
but
is
mixed
with
s-wave
superconductivity,
as
is
consistent
with
argument
by
professor Muller
31.
4.4.
Comment
on
why
Tc
is
higher
in
CU05
pyramid
than
in
CU06
octahedron
It
is well
known
that
the
transition
temperature
is
higher
in
CU05 pyra-
mid
than
in CU06
octahedron.
We discuss as for
this character
shortly.
As for
phonon
mode,
we will consider
only
out-of-plane mode.
For
ex-
ample,
Cu
out-of-plane
mode
shown in figure
13
contributes
to
d-wave
superconductivity
only
in
CU05
pyramid,
if we consider
the
q=O case.
It
does
not
contribute
to
d-wave
superconductivity
in CU06
octahedron
be-
cause
CU06
octahedron
has
a
symmetry
with
respect
to
CU02
plane
and
electron-phonon
interaction
cancels.
Thus
the
reason
why
Tc
is
higher
in
CU05
pyramid
than
in
CU06
octahedron,
is
because
the
number
of
out-of
plane
modes
which
contribute
to
d-wave
superconductivity
is larger.
5.
Conclusion
and
concluding
remarks
The
extended
two-story
house
model
(the
Kamimura-Suwa
(K-S) model)
has
clarified how
the
interplay
of
Mott
physics
and
Jahn-
Teller physics plays
an
important
role
in
determining
the
superconducting
as well as metallic
state
of
underdoped
cuprates.
In
this
paper
it
was first
pointed
out
for
under
doped
cuprates
that
Mott
physics gives rises
to
the
existence
of
local
antiferromagnetic
order
due
to
the
localized
spins
while
the
anti-J
ahn-
Teller
effect as a
central
issue
of
Jahn-
Teller physics
produces
the
existence of two
kinds
of
orbitals
parallel
and
perpendicular
to
a CU02
plane
which
are
ener-
488
CuD
6
octahedron
CuDs
pyramid
(a)
(b)
Figure
13.
Ex
a
mple
of
Cu
out
-
of-plane
mode,
which
contributes
to
d-wav
e
supercon-
ductivity
only
in
CU05
pyramid
.
getically nearby. As a
result
of
the
interplay
of
both
physics
the
K-S
model
has
shown
the
appearance
of
exchange
interactions
between
the
spins of a
localized hole
and
of
a
carrier
hole
in
underdoped
cuprat
es
and
that
these
exchange
interactions
play
an
important
role
in
producing
the
coexistence
of
superconductivity
and
antiferromagnetism
and
the
appearance
of d-wave
superconductivity
even in
the
phonon-involved mechanism.
Brief
review
of
these
facts
as
well as
the
K-S
model
has
been given
in
this
paper.
Acknowledgments
We would like also
to
thank
Dr. Wei-Sheng Lee
of
Stanford
University for
his valuable discussion
and
suggestions
with
regard
to
the
recent
ARPES
experim
e
ntal
re
sults
on
the
presence
of
two
distinct
energy
gaps
exhibiting
different doping
dependence
which
has
been
reported
in ref. 11 .
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J. G. Bednarz
and
K.
A.
Muller,
Z.
Phys. B
64
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P. W.
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erson, Science
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F.
C. Zhang
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T. M. Rice, Phys. Rev. B
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(1988).
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T.
Nakayama, A.
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Kamimura
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H.
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Matsuno
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T .
Hamada
, Theory
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H.
Kamimura
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Eto,
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Phys. Soc.
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59
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M.
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H.
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Phys. Soc.
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,
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Hamada
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K.
Tanaka,
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