Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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470
Cu
dz
2
~
II
P "
la\g>
antibonding orbital
(a)
o
Cu
0°
11
0
.1
•
I
big>
bonding orbitals
(b)
Figure
1.
Spatial
extension
of
lai
g
)
antibonding
orbital
and
I
bl
g
)
bonding
orbital
Undoped
copper
oxide
La2Cu04
is
an
anti
ferromagnetic
Mott
insulator,
in which
an
electron correlation plays
an
important
role.
Thus
we
may
say
that
undoped
cuprates
are
governed
by
Mott
physics. Most
of
models of
high
temperature
superconductivity,
assumed
that
the
doped
holes
itinerate
through
an
orbital
of b
1g
symmetry
extended
over a CU02
plane
in
the
systems
consisting
of
the
CU06
octahedrons
elongated
by
the
JT
effect.
Those
models
are
called "single-component
theory",
because
they
consider
only
the
orbitals
of hole carriers
extend
only over a CU02 plane.
This
single-
component
theory, however,
met
a serious difficulty
that,
in
the
presence
of
anti-ferromagnetic (AF)
order
constructed
from
the
localized spins,
the
hole carriers
cannot
move smoothly.
In
order
to
remove
this
difficulty, RYB
model
2,
t-J
model
3,
etc. were proposed.
When
Sr2+ ions
are
substituted
for La3+ ions
in
LSCO, first-principles
calculations showed
that
the
apical oxygen in
the
CU06
octahedrons
tend
to
approach
towards
Cu2+ ions
4,5.
As a
result
the
CU06
octahedron
elongated
by
the
JT
effect
shrink
by
doping
holes.
This
deformation
against
the
JT
distortion
is called
"anti-J
ahn-
Teller effect"
6.
471
By
this
effect,
the
energy
separation
between
the
two kinds
of
orbital
states,
the
ai
g
anti-bonding
orbital
state
and
the
bIg
bonding
orbital
state,
becomes smaller
7,8.
The
spatial
extension
of
the
ai
g
anti-bonding
orbital
and
the
bIg
bonding
orbital
is shown in figure
l.
By
taking
account
of
the
anti-Jahn-
Teller effect
Kamimura and
Suwa
proposed
that
one
must
consider
the
ai
g
and
bIg
states
equally in forming
the
metallic
state
of
cuprates,
and
they
constructed
a metallic
state
coex-
isting
with
the
local
AF
order.
This
model is now called
"Kamimura-Suwa
model" (K-S model) following
the
two
authors'
names
of
the
paper
by
H.
Kamimura
and
Y. Suwa
9.
In
this
paper
we
will discuss two
important
results
obtained
from
the
interplay
of
Jahn-
Teller Physics
and
Mott
Physics.
One
is concerned
with
the
Fermi
surface.
On
the
basis
of
the
K-S
model
we will show
that
the
"Fermi arcs" observed
in
ARPES
should
be
one
of
the
edges
of
real small
Fermi pockets
in
the
nodal
region.
This
prediction
is shown
to
be
consistent
with
recent
ARPES
experimental
results
reported
by
Tanaka
et al 11
and
by
Meng
et al
22
.
As
the
second
important
result
obtained
from
the
interplay
of
Jahn-
Teller Physics
and
Mott
Physics, we will show
that
the
mechanism
of
su-
perconductivity
is
caused
by
the
interplay
of
strong
electron-phonon
in-
teractions
and
an
AF
order.
It
is shown
that
(1)
the
characteristic
phase
difference
of
wave functions between up-
and
down-spin
carriers
in
the
pres-
ence
of
the
AF
order
leads
to
the
superconducting
gap
of
d
x
2_y2
symmetry
even
in
the
phonon-involved mechanism, (2)
the
out-of
plane
phonon
modes
in
LSCO
with
tetragonal
symmetry
contribute
to
the
formation
of
Cooper
pairs
while
the
in-plane modes
do
not,
(3)
Tc
is higher
in
a
cuprate
with
CU05
pyramid
than
that
with
CU06
octahedron
because
the
number
of
out-of plane
modes
which
contribute
to
d-wave
superconductivity
is larger
in
the
former
than
in
the
latter,
and
(4)
the
calculated
hole-concentration
dependences
of
Tc
and
of
the
isotope effects for
LSCO
are
consistent
with
recent
experimental
results.
The
organization
of
the
present
paper
is
the
following:
It
consists
of
three
parts.
In
the
first
part
(Section
2)
we will review briefly
the
K-S
model.
In
the
second
part
(Section 3) we will give
the
key features
of
the
many-body-effects
included
energy
bands
and
Fermi surfaces.
And
in
the
final
part
(Section 4) we will show how
the
d-symmetry
superconducting
gap
appears
from
the
K-S model. Section 5 is devoted
to
conclusion
and
concluding remarks.
472
2.
Brief
review
of
the
Kamimura-Suwa
(K-S)
model
2.1.
Anti-Jahn-Teller
Effect
When
Sr2+ ions
are
substituted
for La3+ions in LSCO, one
may
think
intu-
itively
that
apical oxygen
in
the
CU06
octahedrons
tend
to
approach
toward
central
Cu
2
+ ions
in
order
to
gain
the
attractive
electrostatic
energy.
The-
oretically
it
was shown
by
the
first-principles
variational
calculations
of
the
spin-density-functional
approach
4,5
that
the
optimized
distance
between
apical
0
and
Cu
in
LSCO
which minimizes
the
total
energy
of
LSCO
de-
creases
with
increasing Sr
concentration.
As a
result
the
elongated
CU06
octahedrons
by
the
Jahn-
Teller
(JT)
interactions
shrink
by
doping
holes.
We have called
this
shrinking
effect
against
the
Jahn-
Teller
distortion
"anti-
J
ahn-
Teller effect"
6.
2.2.
The
energy
landscape
of
CU06
octahedron
In
order
to
clealify
the the
character
of
carriers hole,the first discribe
many
electron
state
CU06
octahedrons.
Figure
2 shows
the
energy-level
landscape
starting
from
the
e
g
and
t2g
orbitals
of
a Cu2+ ion in a regular CU06
octahedron
with
octahedral
sym-
metry
shown
at
the
left column in
the
case
of
LSCO.
By
the
JT
effect
the
Cu
e
g
orbital
state
splits
into
aIg
and
bIg
orbital
states,
which form
antibonding
and
bonding
molecular
orbitals
with
oxygen p orbitals.
The
molecular
orbitals
are
denoted
by
ai
g
,
aIg,
big
and
bIg,
as
shown
at
the
middle
column.
In
an
undoped
case, 7 electrons occupy
these cluster
orbitals, so
that
the
highest occupied
big
state
is
half
filled,
resulting
in
an
S =
1/2
state,
where
a
big
orbital
has
Cu
d
x
2_y2
character.
Following
Mott
physics, we
introduce
the
Hubbard
U
interaction
(U
= 10e V)
as
a
strong
electron-correlation
effect.
Then
the
big
state
splits
into
the
lower
and
upper
Hubbard
bands
denoted
by
L.H.
and
V.H. in
the
figure,
and
this
gives rise
to
a localized
spin
around
a
Cu
site.
These
localized spins form
the
antiferromagnetic
(AF)
order
by
the
superexchange
interaction
via
an
intervening 0
2
- ion
in
undoped
La2Cu04.
Now let
us
remove
one
electron
from
the
present system.
In
other
words,
let us consider
the
case
that
one
hole is
introduced
to
this
CU06
cluster
which
has
an
up-spin
electron
as
the
localized spin.
Due
to
the
strong
electron-electron
interaction,
it
is
not
the
big
electron
which is removed
from
the
original electron configuration
as
expected
from one-electron en-
ergy
diagram.
Instead,
there
are
two possibilities
of
removing
an
electron;
3d
CU
2 1
/
/
/
/
/
U.H.
fl······
/
/
/
Electron
/"
energy
/ I A-site II B-site I
\
,
\
\
\
\
Exchange
interaction
3eV
--
---------------
~
.
....
+
\
\
\
\
L.H.
\.
~
••.
••
'1
-"
Localized spin
473
Figure
2.
Energy
diagram
showing
the
interplay
of
the
Mott
physics
(electron
correla-
tion
and
exchange
interaction
between
the
spins
of
a
doped
hole
and
a
localized
hole)
and
of
the
J
ahn-
Teller
physics
(anti-JT
effects)
(Results
of
the
first-principles
calculations
by
Kamimura
and
Eto
7,8).
the
bIg
and
ai
g
states.
Since
the
multiplet
energy
of
the
six
electron
state
with
an
absent
big
electron becomes
very
large
due
to
the
Hubbard
repulsion
of
bIg,
other
multiplet
states
become
more
favorable.
That
is,
the
many-electron
states
with
one
electron
occupying
the
big
state
and
the
other
one in
the
bIg
or
ai
g
orbital
with
other
doubly
occupied orbitals.
The
former
state
must
have
total
spin
S = 0
due
to
the
exchange
interaction
between bIg
and
big
states
while
the
latter
has
spin
triplet
structure
because
of
Hund's
interaction
between
big
and
ai
g
states.
In
the
right
column
of figure 2,
the
present
situation
is described
by
assigning effective "one-electron energy"
to
each eg-electrons in a CU06
octahedron.
For
example, removing
an
up-spin
electron
form
an
ai
g
orbital
costs O.5eV
more
than
removing a down-spin
electron
from
the
same
orbital
as
shown
in
the
figure.
Next,
we show
the
energy
landscape
in "hole
picture"
in figure
3.
As is
474
~
/
Localized
sp
in /
\
\
\
\
\
\
--
-----
-
---
\ -
..
.
..
+
\
\
L.H.
\.
.••. .
.....
Local
i
zed
spin
Localized
spin
Hole
energy
Figure
3_
Energy
diagram
in
hol
e
picture,
when
one
down-spin
hole is
doped
into
CU06
cluster
embedded
in
LSCO
material.
shown in figure 3,
when
one
down-spin hole is
doped
,
it
occupies bIg
orbital
in
A-site.
Then
this
hole
can
transfer
from bIg
orbital
at
A-site
to
ai
g
orbital
at
B-site.
While
the
localized
up-spin
at
A-
and
the
localized down-spin
at
B-sites
occupy
U.H.
band,
in
the
hole
picture.
So
that,
the
dopant
down-
spin
hole
in
ai
g
orbital
at
B-site becomes
parallel
to
the
localized
spin
in
the
localized big
orbital
by
Hund's
coupling.
Thus
this
spin-triplet
multiplet
is called
Hund's
coupling 3B
Ig
.
On
the
other
hand,
the
dopant
down-spin
hole
in
bIg
orbital
at
A-site becomes
anti-parallel
to
the
localized
spin
in
the
localized big
orbital.
This
spin-singlet
multiplet
is called Zhang-Rice
1 A
lg
.
The
spatial
distribution
of
these
two
kinds
of
multiplets
are
shown
in
the
upper
part
of
figure 3.
By
the
first-principles
cluster
calculations
which
takes
into
account
the
Madelung
potential
due
to
all
the
ions
surrounding
a CU06
cluster
in
LSCO,
Kamimura
and
Eto
showed
that
the
lowest-state
energies of
these
two mul-
tiplets
are
nearly
equal
wh
en
both
the
Hund's
coupling
and
the
spin-singlet
475
exchange
interaction
are
taken
into account
7,8.
Consequently
the
energy
difference between
the
highest occupied
orbital
states
ai
g
in
3B
1g
multi-
plet
and
b
1g
in
1 A
1g
multiplet
becomes only O.leV for
the
optimum
doping
(x = 0.15)
in
La2-xSrxCu04'
Thus,
when
the
two CU06 clusters
with
lo-
calized
up
and
down spins
are
placed
at
the
nearest neighbors,
these
states
are easily mixed by
the
transfer
interaction
between
ai
g
and
b
1g
orbitals,
which is
about
0.3eV for LSCO.
2.3.
Kamimura-Suwa
model
(K-S
model)
By
using
the
calculated
results
of
Kamimura
and
Eto
7,8
and
assuming
that
the
localized spins form
an
AF
order
in a spin-correlated region,
Kamimura
and
Suwa 9
constructed
a metallic
state
of
LSCO
for
its
underdoped
regime.
The
feature
of
the
K-S model is
that
the
hole-carriers in
the
under
doped
regime
of
LSCO form a metallic
state,
by
taking
the
Hund's
coupling
triplet
and
the
Zhang-Rice singlet
alternately
in
the
presence of
the
local
AF
order
without
destroying
the
AF
order.
In
order
to
understand
the
coexistence of
AF
order
and
metallic
state,
we will show
the
cartoon
(figure 4), where
the
localized hole
state
and
carrier
hole
state
has
been
compared
with
1st
and
2nd
story.
In
figure 4
the
first
story
of a
Cu
house
with
(yellow) roof is occupied by
the
Cu
localized spins,
which form
the
AF
order
in
the
spin-correlated region by
the
superexchange
interaction.
The
second
story
in
a
Cu
house consists
of
two floors, lower
ai
g
floor
and
upper
b
1g
floor.
These
second stories between neighboring
eu
houses
are
connected by Oxygen rooms
with
(blue-color) roof.
In
the
second
story
a hole-carrier
with
up
spin
enters
into
the
ai
g
floor
at
the
left-
hand
eu
house
due
to
Hund's
coupling
with
eu
localized up-spin
in
the
first
story
(Hund's
coupling
triplet),
as shown
in
the
extreme
left column of
the
figure.
By
the
transfer
interaction,
the
hole is
transferred
into
the
b
1g
floor
at
the
neighboring
eu
house (the second from
the
left)
through
the
Oxygen room, where
the
hole
with
up
spin forms a spin-singlet
state
with
a localized down
spin
at
the
second
Cu
house from
the
left (Zhang-Rice
singlet).
Thus
the
AF
order
is preserved when a hole-carrier itinerates.
The
important
feature of K-S model is
the
coexistence
of
the
AF
order
and
a
metallic (superconducting)
state
in
the
underdoped
regime.
This
feature is
different from
that
of
the
single-component theory.
476
dopant hole 5 = 1 5 = 0 5 = 1 5 = 0 5 = 1 5 = 0
with up
spin
Up-spin Down
o 0
Up- Down- Up-
o 0
Down-
o
r-
spin-correlated region
--I
(antiferromagnetic order)
big
floor
2
nd
story
1
st
story
Figure
40
An
extended
two-story
house
model
(K-S
model)
2.4.
Effective
Hamiltonian
for
the
Kamimura-Suwa
model
(K-S)
model
The
following effective
Hamiltonian
is
introduced
in
order
to
describe
the
extended
two-story house (K-S)
model
by
Kamimura
and
Suwa 9
0
It
con-
sists
of
four
parts:
The
effective one-electron
Hamiltonian
Heff for
a~g(ai)
and
b1g(b
1
)
orbital
states,
the
transfer
interaction
between
neighboring
CU06
octahedrons
(CU05
pyramids)
H
tn
the
superexchange
interaction
between
the
Cu
d
x
2_y2
localized spins H
AF
,
and
the
exchange
interactions
between
the
spins
of
dopant
holes
and
of
d
x
2_y2
localized holes
within
the
same
CU06
octahedron
(CU05
pyramid)
Hex
0
Thus
we have
H
= Heff + H
tr
+ HAF + Hex
= L EmCJmuCimu + L tmn (CJmuCjnu +
hoc
o
)
i,m,a
(i,j),m,n,a
+JLSiOSj+LKmSi,moSi,
(1)
(i,j)
i,m
477
where
Em
(m
=
aig(ai)
or
b
1g
(bd)
represents
the
effective one-electron
energy
of
the aig(ai)
and
b
1g
(bd
orbital
states,
clma-
and
Cima-
are
the
creation
and
annihilation
operators
of
a
dopant
hole
with
spin
(J
in
the
i-th
CU06
octahedron
(i-th
CU05
pyramid),
respectively, tmn
the
effective
transfer
integrals
of
a
dopant
hole between
m-type
and
n-type
orbitals
of neighboring
CU06
octahedrons
(CU05 pyramids), J
the
superexchange
interaction
between
the
spins S i
and
S j
of
d
X
2
_y2
localized holes
in
the
big
(bi)
orbital
at
the
nearest neighbor
Cu
sites i
and
j
(J
> 0 for
AF
interaction),
and
K m
the
exchange integral for
the
exchange
interaction
between
the
spin
of
a
dopant
hole
Sim
and
the
d
X
2
_y2
localized spin S i in
the
i-th
CU06
octahedron
(i-th
CU05
pyramid).
The
values of
the
parameters
in
Eq. (1) for
the
case
of
LSCO are:
J = 0.1,
Ka*
=
-2.0,
Kb
l
= 4.0, ta* a* = 0.2, tb
l
b
,
= 0.4, ta* b
,
=
~_--;-_--,Ig~
g
19 19
g g
19
g
Jta~ga~gtbIgblg
'"
0.28,
Ea~g
=
0,
Eb
'g
= 2.6
in
units
of eV, where
the
values
of
Hund's
coupling exchange
constant
K
a
*
and
Zhang-Rice exchange
Ig
constant
K
b'g
are
taken
from
the
first principles
cluster
calculations for a
CU06
octahedron
in
LSCO
7,8,
and
the
energy difference
of
the
effective
one-electron energies between
ai
g
and
bIg
orbital
states,
Ea~g
-
Eb
,g
=
-2.6,
is
determined
so as
to
reproduce
the
energy difference between
the
3B
1g
and
1 A
1g
multiplets in LSCO in
the
MCSCF
cluster
calculations
8,
while those
of
tmn's
are
due
to
band
structure
calculations
4,5.
3.
Effective
energy
band
and
the
shape
of
Fermi
surface
Ushio
and
Kamimura
13,12
have
started
from
tight
binding Hamiltonian.
Then
they
have
obtained
the
effective many-body-effects included Hamil-
tonian
(1) by
taking
into
account
the
exchange
term
as a molecular field,
which is
determined
so
that
the
energies
of
1 A
1g
and
3B
1g
coincide
with
that
calculated by
Eto
and
Kamimura
8.
3.1.
Effective
energy
band
In
figure 5 (b),
the
calculated
many-body
effect included energy
band
struc-
ture
for up-spin (or down-spin)
dopant
holes for LSCO is shown for various
values
of
wave-vector k
and
symmetry
points
in
the
AF
Brillouin zone.
Here one should
note
that
the
energy
in
this
figure is
taken
for electron-
energy
but
not
hole-energy,
and
the
Hubbard
bands
for localized big holes
do
not
appear
in
this
figure.
In
the
undoped
La2Cu04,
all
the
energy
bands
in
figure 5
(2)
are
fully
occupied by electrons so
that
La2Cu04
is
an
insulator, consistent
with
ex-
478
AF
Brillouin zone
(a)
In
doping,
a
hole carrier begins
to
occupy from
11
point.
~
>~.----.-
~
0
~
~~-+
~
~
00
~ ~
o
....
~~~2j:s~~~==~
6L----L--L---L-~
('1C,
O
,O)
.t::..
(0,0,0)
(0,0,0)
G
1
('1C
/ 2,
'1C
/2,0) (0,0,
'1C)
(b)
Figure
5. (a)
The
Fermi
surface
of
hole-carriers
for x = 0.15
calculated
for
the
#1
band.
Here
the
kx
axis
is
taken
along
rG1,
corresponding
to
the
x-axis
(the
Gu-O-Gu
direction)
in
a
real
space.
(b)
The
many-body-effect
included
band-structure
for
up-
spin
dopant
holes,
obtained
by
Ushio
and
Kamimura
13,12.
In
this
figure
the
Gu-O-Gu
distance
a is
taken
to
be
unity.
perimental
results.
In
this
respect
the
present
effective energy
band
struc-
ture
is completely different from
the
ordinary
LDA energy
bands
16,17.
When
Sr
are
doped, holes begin
to
occupy
the
top
of
the
highest
band
in
figure 5 (b)
marked
by
#1
at
~
point
which corresponds
to
Clf
j2a, 7r j2a,
0)
in
the
AF
Brillouin zone.
At
the
onset
concentration
of
superconductivity,
the
Fermi level is
located
at
the
energy
of
E = 9.04
eV
just
below
the
top
of
the
#1
band
at
~,
which is a
little
higher
than
that
of
the
G
1
point.
Here
the
G
1
point
in
the
AF
Brillouin zone lies
at
(7rja,
0, 0),
and
corresponds
to
a saddle
point
of
the
van
Hove singularity.
3.2.
The
shape
of
Fermi
surface
Based
on
the
calculated
band
structure
shown
in
figure 5 (b), U shio
and
Kamimura
13,6
calculated
the
Fermi surfaces for
the
underdoped
regime
of
LSCO.
This
Fermi surface
structure
is also completely different from
that
479
of
an
ordinary
Fermi liquid picture,
in
which a Fermi surface is large.
In
fig-
ure 5 (a)
the
Fermi surface
structure
of
hole-carriers
calculated
for x = 0.15
is shown as
an
example, where
the
Fermi
surface (FS) consists of two
pairs
of
extremely
flat
tubes.
Thus
the
feature
of
FS
in
the
underdoped
regime
is a small
FS,
and
its
volume is
proportional
to
the
doping
concentration
of
dopant
holes.
1.
0
0.0
0.0
,
03
:,
05
. '
0.5
0.0
0.5
0.0
0.5
0.5
1.0
Figure
6.
Observed
doping
dependence
of
Fermi
arc
(the
inner
section
of
FS)
of
La2-xSrxCu04
from
x = 0.03
to
0.15,
observed
in
ARPES
by
T.
Yoshida
et al
20,21
The
calculated
results
based
on
the
K-S
model
for x = 0.05
and
x = 0.15
shown
by
thin
curves
are
superimposed
on
the
experimental
results
obtained
by
Yoshida
et
al.
Since
the
Bloch
function
of
top
band
has
Fourier
component
mainly
in
AF
BZ, only
the
inner
section
of
FS
might
be
observed.
ARPES
researchers
have called
it
"Fermi
arc".
Recently
the
Fermi
arcs
of
hole carriers have
been
observed for
the
underdoped
regime
of
LSCO
by
Yoshida et al
20,21.
In
figure 6
the
calculated
FS
for x = 0.05
and
x = 0.15
are
compared
with
ex-
perimental
results of
La2-xSrxCu04
with
x = 0.03, x = 0.05, x = 0.07
and
x = 0.15. As seen
in
the
figure,
the
agreement
on
the
doping
dependence
of
Fermi arcs
between
theory
and
experiment
is fairly good. Recently,
Meng
et.al.
22
report
ARPES
measurements
of
Bi2Sr2-xLaxCu06+8
(La-Bi2201)
that
reveal Fermi pockets,
supporting
our
calculated
Fermi surface.
We
extend
our
calculations
to
the
overdoped regime
of
LSCO. Since