Transcript PowerPoint プレゼンテーション

S. Maekawa
(IMR, Tohoku University)
Spin, Charge and Orbital and their Excitations
in Transition Metal Oxides
(Hong Kong, Dec. 18, 2006)
Contents:
i) Spin-charge separation in one-dimensional cuprates,
ii) Non-linear optical response due to spin-charge separation,
iii) Orbital in High Tc cuprates,
iv) Anomalous transport properties due to orbital,
v) Thermo-electric response due to spin and orbital,
Internal degrees of freedom of electron
Spin
Magnet
Charge
Electric Current
Orbital (Shape of wave function: Shape of electron)
z
Oxygen
y
x
d(xy)
d(x2y2)
d(3z2r2)
d(yz)
d(zx)
Hong Kong Conference
December 18, 2006
Anomalous Electronic Lattices
in Cobaltates
S. Maekawa, W. Koshibae and N. Bulut
(IMR, Tohoku University, Sendai)
Co - Oxides in triangular lattice
(NaxCoO2 and NaxCoO2・yH2O)
i) Review of Unconventional properties
ii) Orbital degeneracy in the frustrated lattice
crystal lattice vs. electron lattice
unconventional properties
•Crystal Structure
In NaxCoO2,
x Co3+ (3d6) and (1  x) Co4+ (3d5) in CoO6 units
CoO2 layer
CoO6
octahedron
Na layer
CoO2 layer
Na layer
CoO2 layer
edge-shared CoO6 units
Superconductivity in water-intercalated NaxCoO2·yH2O
H2O
Na layer
CoO2 layer
K. Takada, H. Sakurai,
E. TakayamaMuromachi, F. Izumi,
R.A. Dilanian, T.
Sasaki, Nature 422,
53 (2003).
In cubic CoO6 units,
Co3+ e
Co4+
g
NaxCoO2:
t2g
Co3+ (3d6)
S=0
Co4+ (3d5)
S = 1/2
5 - 3d orbitals
z
eg
y
x
d(x2y2)
d(3z2r2)
t2g
d(xy)
d(yz)
d(zx)
Anomalous physical properties in CoO2 layer:
i.
ii.
iii.
iv.
v.
vi.
Giant Hall effect at T  R.T.
NaxCoO2
(Y. Wang, et al., cond-mat/0305455)
Ferromagnetism
[Bi2xPbxSr2O4]yCoO2, Tc ~ 3.2 K
(I. Tsukada et al., J. Phys. Soc. Jpn. 70, 834 (’01).)
Giant thermopower at T  R.T.
NaxCoO2
(I. Terasaki, Y. Sasago, and K. Uchinokura, PRB56, 12685(’97).)
[Bi2xPbxSr2O4]yCoO2
(T. Yamamoto et al., Jpn. J. Appl. Phys. 39, L747 (’00).)
Ca3Co4O9
(A. C. Masset et al., PRB62, 166 (’00).)
Superconductivity
NaxCoO2·yH2O
(K. Takada et al., Nature 422, 53 (’03).)
Charge ordering
NaxCoO2
(Foo et al., cond-mat/0312174)
Antiferromagnetism
Na0.5CoO2
(T. Uemura et al.)
I. Terasaki, Y. Sasago, and K. Uchinokura, PRB56, 12685(’97).
Y. Wang et al., cond-mat/0305455
Novel physics in CoO2 layer with triangular structure
1. Kagomé lattice hidden in CoO2 layer
(WK and SM: PRL 91, 257003 (’03), NB, WK and SM: PRL 95, 037001 (05))
2. Anomalous physical properties:
- Superconductivity (G. Khaliullin, WK and SM: PRL93, 176401(’04))
- Hall effect (WK, A. Oguri and SM: unpublished)
- Thermopower and Nernst effect
(WK and SM: PRL 87, 236603 (’01). )
t2g orbital degeneracy in edge-shared CoO6 units
•Kagomé in triangular lattice
CoO2 layer
Co
z
90
degrees
y
x
Edge shared
octahedra
+
+

d(xy)
+

+

d(xy)
Co
OK to GO ! 2p
 x

+
+
+

OK to GO !
d(xy)
NO GO !

OK to GO !
2px


+
+
O
+
 d(zx)
Martin Indergand, Yasufumi Yamashita, Hiroaki Kusunose, Manfred Sigrist，
( cond-mat/0502116)
•Hopping of a 3d electron via O2p orbital
xy
xy
yz
zx





yz
zx
t

t



z
x
CoO2 layer
yz
xy
yz
zx
xy
yz
zx

t



t





xy
xy
yz
zx



t

yz
zx
t




y
xy
zx
yz
zx
The triangular lattice of Co ions
is resolved into
four Kagomé lattices
(green, yellow, red and white)
for the electronic states.
WK & SM, PRL91, 257003 (’03).
xy
zx
yz
C is the zero-frequency magnetic correlation function between two nearest-neighbor sites i and j
on the triangular or the kagome lattices,

C   d m z  ri ,  m z  rj  .
0
Here, C is shown for the kagome and the triangular lattices at n =1.15 for U  8 | t | and 4 | t | .
The results for the kagome lattice were obtained for 6  6 (filled points) and 4  4 (empty points) unit cells.
The results for the triangular lattice were obtained for 12 12 (filled points) and 8  8 (empty points) lattices.
I. Terasaki, Y. Sasago, and K. Uchinokura, PRB56, 12685(’97).
Y. Wang et al., cond-mat/0305455
•Hall coefficient
RH  lim RH  
+0

 xy  
 
RH   


B   xx    yy     xy    yx   
B 0
ie 2 
1
  J , H  , H  , J y  +


 xy   
J
,
J
+

2   x y
2  x
V 

ie 2 
1
 J x , H  , J x  +
 xx   

+
xx

2
V 







*
RH
RH   
1   H  
a high frequency “residue” RH*

Jx , J y 
iV


*
RH
 lim RH    lim  
2

B0  Be2
 xx


.


Shastry, Shraiman & Singh, PRL70, 2004 (’93);
Kumar & Shastry, PRB68, 104508 (’03).

Jx , J y 
iV


*
RH
 lim RH    lim  
2

B0  Be2
 xx

Jy
Jx
H  t
Jx , J y 


1 
1
2 1
3
Tr 1  H +   H     H  +
Z 
2!
3!



 J x , J y 


These contributions are absent !!
Jx , J y 


t 
t
k BT
,
 xx

.


t
k BT
*
RH
k BT
t
Difference of R*H between square and triangular lattices

Jx , J y 
iV


*
RH
 lim RH    lim  
2

B0  Be2
 xx

High temperature expansion
Jx , J y 


charge carrier
1 
1
2 1
3
Tr 1  H +  H   H  +
Z 
2!
3!
H  t
H  t
H t
t 
c
†
i c j



J
,
J
  x y 


+ h.c.

Doubly occupied states
are excluded.
Jy
t
k BT

ij 
Jx
 xx

.


*
RH
const.

Jx , J y 
iV


*
RH
 lim RH    lim  
2

B0  Be2
 xx

High temperature expansion
Jx , J y 


1 
1
2 1
3
Tr 1  H +  H   H  +
Z 
2!
3!
Jy
Jx
H  t
 xx
t 
t
k BT
*
RH
k BT
t

.





J
,
J
  x y 


a high frequency “residue” RH*

3
Jx, J y 
ia
N


*
0
RH  lim RH    lim  
2

B0 
Be2
 xx

Jx
Jy

.


H  t
H  t
Jx, J y 


Jx
Jy
H  t
1 
1
2 1
3
Tr 1  H +  H    H  +
Z 
2!
3!
H  t
H  t



 J x , J y 


…..
…..
high frequency “residue” RH*
k T
t
*
RH
 A B + B +C
+
t
k BT
RH* (in units of v/de)
0.4
0.3
Kagomé lattice
0.2
0.1
triangular lattice
0.0
0.0
0.5
1.0
1.5
kBT / t
t ~ 25K
RH*
k BT
t
WK, Oguri & SM, unpublished.
Large Thermopower in NaCo2O4
I. Terasaki, Y. Sasago, and K. Uchinokura, PRB56, 12685(’97).
r (mWcm)
200
Q (mV/K)
Small r
100
0
•Key of Large Thermopower
Spin and Orbital
Degrees of Freedom
in Co3+(3d6 ) and Co4+(3d5 )
in-plane resistivity
Basic unit
Thermopower
80
Co
Large Q
40
3d orbitals
0
0
100
200
300
CoO6
octahedron
O
eg
Temperature(K)
t2g
Orbital degree of freedom
W. Koshibae and S. Maekawa，PRL87, 236603 (’01
•Thermoelectric material
Thermopower
electricity
heat
Large Thermopower (Q) &
Small Resistivity (r are
required.
•Figure of Merit Z = Q2/rk
k: thermal conductivity)
Figure of Merit Z [K1]
102
n-Bi2Te3 (n)
GeTe3-AgSbTe2 alloy (p)
PbTe (n)
103
ZT = 1
NaCo2O4n-SiGe
(p) [n]
n-FeSi2 (n)
B9C+Mg (p)
104
500
1000
T [K]
1500
•Galileo: NASA's Spacecraft
Radioisotope
Themroelectric Generator
•SEIKO THERMIC
•CITIZEN
ECO-DRIVE THERMO
Thermo-electric materials:
Heat→Electricity
Garbage burning plant
Heat of car
Electricity→Heat
Refrigerator
Thermo-electric materials：
No vibration (no moving part),
Easy to miniaturize,
Gentle to environment.
Thermopower at high temperatures:
M12 / M11 m
Q
+
eT
eT
M 11  T
zz

0

k
0
zz

0
p
dt d tr r0 j1 ( t  i) j1
particle current
M 12  T
density matrix

energy flux operator
k
p
dt d tr r0 j2 ( t  i) j1
0
High temperature
independent of T
m

eT
chemical potential
entropy
FI
HK
m
S

T
N
FI
HK
1 S

e N
E ,V
E ,V
Entropy per carrier
S=kBlng
g: total number
of the states
number of electrons

F I
H K
k B  ln g
e N
E ,V
At high temperatures:
Spin and Orbital Degrees of Freedom
based on the Strong Coulomb Interaction
Key of Large Thermopower
F I
H K
k B  ln g
Q
e N
E ,V
kB
kB
kB
x
  ln ge + ln gh  ln
e
e
e (1  x )
Spin and Orbital
•Thermopower in NaCo2O4
Co3+
ge
Co4+
gh
Charge
x = 0.5
Co3+eg
Co4+
t2g
ge
=1
gh
Q = 154 mV/K
=6
•Summary
The degeneracy induced by
Spin and Orbital degrees of freedom
kB ge
Q   ln
e gh
kB
x
 ln
e (1  x )
degeneracy of
Co3+ and Co4+
Charge
Heikes Formula
•Other Transition Metal Oxides
ge / g h
kB/eln(ge/gh)
Ti3+(3d1), Ti4+(3d0)
6/1
154 mV/K
V3+(3d2), V4+(3d1)
9/6
35 mV/K
Cr3+(3d3), Cr4+(3d2)
4/9
70 mV/K
Mn3+(3d4), Mn4+(3d3)
Rh3+(4d6), Rh4+(4d5)
10 / 4
1/6
79 mV/K
154 mV/K
Large thermopower is also expected!
Experimental Group … 1
Y. Ono
New thermoelectric material
- delafossite-type Mg-doped chromium oxides -
Cu
• We have studied high-temperature
thermoelectric properties of CuCr1-xMgxO2
(x=0-0.05) between 300 K and 1100 K.
CrO2
• CuCr1-xMgxO2 thin film prepared by pulsed
laser deposition technique was oriented to caxis, perpendicular to the sapphire substrate.
Cu
CrO2
Cu
(1-x)Cr3+ + x Cr4+
CrO2
3d3
Cu
eg
Crystal structure of CuCrO2
t2g
3d2
Sr1xRh2O4
Rh3+ (4d6) and Rh4+ (4d5)
Large Thermopower
Y. Okamoto, M. Nohara, F. Sakai and H. Takagi
J. Phys. Soc. Jpn. 75, 023704 (’06).
Thermopower (Q) at    (cf. B. Sriram Shastry, PRB73, 085117(’06).)
Electron dope U =  Hubbard model on the kagomé lattice
kB   x 
1+ x
2 2 
Q* 
ln  6
+O  t 
  t

e   1 x 
4

 
NaCo2O4, x ~ 0.5, t ~ +100K
kB t 1 + x
Q* 154[mV/K] 
e k BT 4
154 mV/K
150
100
Q*
50
0
0
100
T [K]
200
300
•Thermo-electric response tensor at   0, (t)  0
at high-temperatures,
the tensor M 12  diagonal, M 12  1
 RH B  e
e
m    r
m 
1 12

lim      M   2 +
1   
1
 2 +
R
B
r
eT
eT
0 
T
   H
 T




Q
 eRH / T 2 B 




Q
 eRH / T 2 B



 Q  NB 

Q 
 NB


Nernst coefficient N  RH / T2  1 / T
RH is positive and
linear in T at high temperature.
In conclusion;
It is of crucial importance to see the electronic lattice
hidden in the frustrated crystal lattice.