S. Maekawa(IMR, Tohoku University)

Spin, Charge and Orbital and their Excitations in Transition Metal Oxides

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, 　　

(Hong Kong, Dec. 18, 2006)

Internal degrees of freedom of electron

Spin Magnet

Charge Electric Current

z

xy

Oxygen

d(3z2r2)d(x2y2)

d(xy) d(yz) d(zx)

Orbital (Shape of wave function: Shape of electron)

Hong Kong Conference

December 18, 2006

Anomalous Electronic Latticesin 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

x Co3+ (3d6) and (1 x) Co4+ (3d5) in CoO6 units

CoO6

octahedron

•Crystal Structure

CoO2 layer

edge-shared CoO6 units

Na layer

CoO2 layer

Na layer

CoO2 layer

In NaxCoO2,

K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R.A. Dilanian, T. Sasaki, Nature 422, 53 (2003).

Superconductivity in water-intercalated NaxCoO2·yH2O

Na layer

CoO2 layer

H2O

In cubic CoO6 units,

Co3+eg

t2g

Co4+

Co3+ (3d6)S = 0

Co4+ (3d5)S = 1/2

z

x

y

d(3z2r2)d(x2y2)

d(xy) d(yz) d(zx)

5 - 3d orbitals

eg

t2g

NaxCoO2:

Anomalous physical properties in CoO2 layer:

i. Giant Hall effect at T R.T. NaxCoO2

(Y. Wang, et al., cond-mat/0305455)ii. Ferromagnetism

[Bi2xPbxSr2O4]yCoO2, Tc 3.2 K(I. Tsukada et al., J. Phys. Soc. Jpn. 70, 834 (’01).)

iii. Giant thermopower at T R.T.NaxCoO2

(I. Terasaki, Y. Sasago, and K. Uchinokura, PRB56, 12685(’97).)[Bi2xPbxSr2O4]yCoO2

(T. Yamamoto et al., Jpn. J. Appl. Phys. 39, L747 (’00).) Ca3Co4O9

(A. C. Masset et al., PRB62, 166 (’00).)iv. Superconductivity

NaxCoO2·yH2O(K. Takada et al., Nature 422, 53 (’03).)

v. Charge ordering NaxCoO2

(Foo et al., cond-mat/0312174)vi. 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

CoO2 layer

Edge shared octahedra

90 degrees

O

Co

Co

x y

z

2px

d(xy)

d(zx)

+

+

+

2px

d(xy)+

d(xy)

OK to GO ! OK to GO !

NO

GO

!

OK

to GO

!

•Kagomé in triangular lattice

Martin Indergand, Yasufumi Yamashita, Hiroaki Kusunose, Manfred Sigrist ， ( cond-mat/0502116)

xyyz zx

•Hopping of a 3d electron via O2p orbital

x y

z

xy yz zx

xy t

yz

zx t

xy yz zx

xy t

yz t

zx

xy yz zx

xy

yz t

zx t

CoO2 layer

xyyz zxxyzx yz

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).

0

is the zero-frequency magnetic correlation function between two nearest-neighbor sites and

on the triangular or the kagome lattices,

Here, is shown for the kagome and

, .z zi jC d m r

i

r

j

C

m

C

the triangular lattices at =1.15 for 8 | | and 4 | | .

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

lattic

n U t t

e 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

0

limH HR R

•Hall coefficient

a high frequency “residue” RH*

*2 20

,lim lim .

x yH H

Bxx

J JiVR R

Be

0

xyH

xx yy xy yx B

RB

2

2 2

2

2

*

1, , , ,

1, ,

1

xy x y x y

xx xx x x

HH

H

ieJ J J H H J

V

ieJ H J

V

RR

Shastry, Shraiman & Singh, PRL70, 2004 (’93); Kumar & Shastry, PRB68, 104508 (’03).

2 3

,

1 1 1Tr 1 ,

2! 3!

x y

x y

J J

H H H J JZ

H t

JxJy

These contributions are absent !!

, ,x y xxB B

t tJ J t

k T k T * B

Hk T

Rt

*2 20

,lim lim .

x yH H

Bxx

J JiVR R

Be

H t

Jx

Jy

H

t

Difference of R*H between square and triangular lattices

charge carrier

* .HR const

*2 20

,lim lim .

x yH H

Bxx

J JiVR R

Be

xxB

tt

k T

High temperature expansion

2 31 1 1, Tr 1 ,

2! 3!x y x yJ J H H H J JZ

† . .jiij

H t c c h c

Doubly occupied states are excluded.

* BH

k TR

t

H t

JxJy

*2 20

,lim lim .

x yH H

Bxx

J JiVR R

Be

High temperature expansion

2 31 1 1, Tr 1 ,

2! 3!x y x yJ J H H H J JZ

xxB

tt

k T

a high frequency “residue” RH*

3

* 02 20

,lim lim .

x yH H

Bxx

J Jia NR R

Be

2 3

,

1 1 1Tr 1 ,

2! 3!

x y

x y

J J

H H H J JZ

Jy

Jx

Jy

H t

H t

Jx

H t

H tH t

…..

…..

* BH

B

k T tR A B C

t k T high frequency “residue” RH*

0.0 0.5 1.0 1.50.0

0.1

0.2

0.3

0.4R

H*

(in

unit

s of

v/

e)

kBT / t

triangular lattice

WK, Oguri & SM, unpublished.

Kagomé lattice

t ~ 25K* BH

k TR

t

200

100

0

80

40

0

in-plane resistivity

Thermopower

(

cm

)Q

(V

/K)

0 100 200 300Temperature(K)

I. Terasaki, Y. Sasago, and K. Uchinokura, PRB56, 12685(’97).

Small

Large Thermopower in NaCo2O4

Large Q

Spin and Orbital Degrees of Freedom

in Co3+(3d6 ) and Co4+(3d5 )

CoO6

octahedron

OCo

Basic unit

•Key of Large Thermopower

eg

t2g

Orbital degree of freedom

3d orbitals

W. Koshibae and S. Maekawa ， PRL87, 236603 (’01).

•Thermoelectric material

heat

electricity

Th

erm

opow

er

Large Thermopower (Q) & Small Resistivity (are required.

n-SiGe [n]

500 1000 1500

T [K]

Fig

ure

of M

erit

Z [

K

] n-Bi2Te3 (n)

GeTe3-AgSbTe2 alloy (p)

PbTe (n)

n-FeSi2 (n)B9C+Mg (p)

ZT = 1

NaCo2O4 (p)

•Figure of Merit Z = Q2/ thermal conductivity)

•Galileo: NASA's Spacecraft

Radioisotope Themroelectric Generator

•SEIKO THERMIC •CITIZEN ECO-DRIVE THERMO

Thermo-electric materials:

Heat→Electricity

Electricity→Heat

Thermo-electric materials： No vibration (no moving part),Easy to miniaturize,Gentle to environment.

Garbage burning plant Heat of car

Refrigerator

Thermopower at high temperatures:

QM M

eT eT

12 11/

M T dt d j t i j110 1 100

zz

tr ( )k p

independent of T

High temperature

particle currentenergy flux operator

eT

T

SN E V

FH IK,

chemical potentialentropy

FH IK1e

SN E V

,

number of electrons

FH IKke

gN E V

B ln

,

S=kBlngg: total number

of the states

M T dt d j t i j120 2 100

zz

tr ( )k p

density matrix

Entropy per carrier

Spin and Orbital Degrees of Freedom based on the Strong Coulomb Interaction

Key of Large Thermopower

Qke

gN

ke

gke

gke

xxE V

e h FH IK

B B B Bln

ln ln ln( ), 1

ge gh

•Thermopower in NaCo2O4

=1 =6ge gh

Co3+ Co4+

Co3+eg

t2g

Co4+

Q = 154 V/K

x = 0.5

ChargeSpin and Orbital

At high temperatures:

The degeneracy induced by Spin and Orbital degrees of freedom

degeneracy of

Co3+ and Co4+

Charge

ke

g

ge

h

B lnQ

ke

xx

B ln( )1

Heikes Formula

•Summary

•Other Transition Metal Oxides

Ti3+(3d1), Ti4+(3d0)

ge / gh

6 / 1 154 V/K

kB/eln(ge/gh)

V3+(3d2), V4+(3d1) 9 / 6 35 V/K

Mn3+(3d4), Mn4+(3d3) 10 / 4 79 V/KCr3+(3d3), Cr4+(3d2) 4 / 9 70 V/K

Large thermopower is also expected!Rh3+(4d6), Rh4+(4d5) 1 / 6 V/K

New thermoelectric material - delafossite-type Mg-doped chromium oxides -

• We have studied high-temperature thermoelectric properties of CuCr1-xMgxO2

(x=0-0.05) between 300 K and 1100 K.

• CuCr1-xMgxO2 thin film prepared by pulsed laser deposition technique was oriented to c-axis, perpendicular to the sapphire substrate.

Experimental Group … 1

(1-x)Cr3+ + x Cr4+ 3d23d3

t2g

eg

CrO2

Cu

Crystal structure of CuCrO2

CrO2

CrO2

Cu

Cu

Cu

Y. Ono

Y. Okamoto, M. Nohara, F. Sakai and H. TakagiJ. Phys. Soc. Jpn. 75, 023704 (’06).

Sr1xRh2O4

Rh3+ (4d6) and Rh4+ (4d5)

Large Thermopower

2 21* ln 6

1 4Bk x x

Q t O te x

NaCo2O4, x ~ 0.5, t ~ 100K

Electron dope U = Hubbard model on the kagomé lattice

1* 154[ V/K]

4B

B

k t xQ

e k T

0 100 200 3000

50

100

150 154 V/K

T [K]

Q*

Thermopower (Q) at (cf. B. Sriram Shastry, PRB73, 085117(’06).)

•Thermo-electric response tensor at 0, (t) 0

1 122 20

2

2

lim 1 1

/

/

H

H

H

H

R Be eM

R BeT eTT T

Q e R T B

e R T B Q

Q NB

NB Q

((((

Nernst coefficient RH / T2 1 / T

RH is positive and linear in T at high temperature.

at high-temperatures, 12 12the tensor diagonal, M M 1

(((

In conclusion;

It is of crucial importance to see the electronic lattice hidden in the frustrated crystal lattice.