1

The investigation of charge ordering in colossal magnetoresistance

Shih-Jye SunDepartment of Applied Physics

National University of Kaohsiung

2005/9/30 in NCKU

2

Colossal Magnetoresistance

La1-x(Ca,Sr…)xMnO3

)0(

)0()(

HR

HRHRMR

3

31 MnOSrLa xx

Urushibara et al (1995) Cheong and Hwang (1999)

Phase diagram of CMR

4

2p

（ A）

(1)eg

t2g

Mn3+

(2)

O2-

2p(3)

eg

t2g

Mn4+

（ B）

O2-

2p(3)

(2)eg

t2g

Mn3+

(1)eg

t2g

Mn3+

（ C）

(2)eg

t2g

Mn3+

(1)

O2-

(3)eg

t2g

Mn4+

Double exchange mechanism

5

John Teller distortion

6

The motivation

para-insulator(PI)

CO

AFM

FI

CO

PI

x

Temp

x~0.2 0.5<x<0.85

La1-xCaxMnO3

TC TCO

TCOTN

TC

χ TC(TCO or TN)

T

Susceptibility instability

I

II

III

From region I to II and II to III

7

Theoretical formulas derivation

iiiU

ijjiijV

kkkkK

VUK

nUnH

nnVH

CCH

HHHH

,,

,,,

2

1

Hamiltonian:

(kinetic energy)

(inter-Coulomb repulsion)

(on-site Coulomb repulsion)

Local spin

Itinerant spin

8

Hamiltonian in momentum representation

ipk

kkkk

k p

ip

i

Rpkipk

ij k

ij

p

ipRpkipk

p

Ripp

ij k

RikkV

eN

nnN

V

eennN

V

RReennN

Ven

Nen

NVH

i

iji

1

)( 11

)(2

)(2

k

kkk

kkU nnN

Unn

N

UH ,,,,

p

pkpk CCnwhere

9

Greens function for susceptibilities

),,(1

)0,0(,,1

)(

)0,0(),(1

)(

);(),;()(;

,

,,,

tqkeN

enCCN

ti

enqnN

ti

tRntRntitRR

c

k q

Riq

k q

Riqkqk

q

Riq

jijic

i

i

i

Charge-charge susceptibility

)0,0(;)0,0(,)(),,( ,,,, nCCnCCtitqkwhere kqkkqkc

10

Spin-spin susceptibility

),,(1

)0,0(,1

)();(

)(

)0,0(),(1

)();(),;()();(

,,,

,,

tqkeN

eCCN

titRR

CCq

eqN

titRtRtitRR

s

qk

RiqRiq

k qkqkji

s

kkqk

q

Riqjiji

s

ii

i

)0,0(,)0,0(,)(),,( here,,,,

kqkkqks CCCCtitqkw

11

Equation of motion method

)0,0(;,)0,0(,)();,( ,,,, nHCCnCCttqkdt

di kqkkqk

c

VkqkUkqkKkqkkqk HCCHCCHCCHCC ,,,, ,,,,,,,,

,,

,,,,

',',,,',',,',,

', ,',,',',',,

',,',',',,

',,',',,',',,,

',',',,,,,

,,

,,

,,

,,

kqkqkk

kqkqkkqkk

ppqkkppkpqkp

p kpqkpppqk

pkpppk

p

pkppqkppkqkp

pppkqkpKkqk

CC

CCCC

CCCC

CCCCCCC

CCCCCC

CCCCCCCC

CCCCHCC

(1) (2) (3)

(1)

12

)4(

,,'

,,','

)3(

,,'

,,','

)2(

,,''''

,'',,

)1(

,,''''

,'',,

' '' ,'',,,'',',','',,'',,','

,',,'','',,',',,'','',',

' '' ,,'','',,'','',,,','

,'','',,',',,',',,

' '','','',,,',','','',',',,

' '','','',',',,

,,

,,,,

,,,

,,,,

,,

,

kp k

pqkkpkpkp k

qkkpk

kp k

pkkqpkkp k

pkpkqk

p k k kqkkpkkpkpkkkqkkpk

kqkkpkkpkpkkkpkkqk

p k k kkpkqkkpkkqkkpk

kpkkkpkqkkpkkqk

p k kkpkkqkkpkkpkkpkkqk

p k kkpkkpkkqk

Ukqk

CCCCN

UCCCC

N

U

CCCCN

UCCCC

N

U

CCCCCCCC

CCCCCCCC

N

U

CCCCCCCCCC

CCCCCCCCCC

N

U

CCCCCCCCCCCCN

U

CCCCCCN

U

HCC(2)

13

',,,,'

',,',',

'',,,,'',''

'',,'','',,)1(

kkqkk

kkqkqk

p kpkqkkpk

p kkpkpkqk

RPA

CCfN

UCCf

N

U

CCCCN

UCCCC

N

U

,,,,, kqkqk fCCwhere

,,'

,',',,''

,',

'',,,,'',''

'',,'','',,)2(

kk

qkkkk

qkk

p kkPqkkpk

p kkPkkpqk

CCfN

UCCf

N

U

CCCCN

UCCCC

N

U

,,''

,',,,'

,,'

',,',',,

',,,,',')3(

kk

qkqkkk

qkk

p kkPkpkqk

p kpkqkkpk

CCfN

UCCf

N

U

CCCCN

UCCCC

N

U

,,''

,',,,'

,,'

',,',',,

',,,,',')4(

kk

qkkkk

qkk

p kkPkkpqk

p kkpqkkpk

CCfN

UCCf

N

U

CCCCN

UCCCC

N

U

Fermi-Dirac distribution

Wick’s theorem

Random Phase Approximation

14

,,',''

,,,,',''

,,

',,',',

',,,,'

',,',',

',,,,'

',,,','

',,',',

',,,,'

',,',',,,

-

,

kqkk

kqkkqkk

kqk

kkqkk

kkqkk

kkqkqk

kkqkk

kkqkk

kkqkk

kkqkk

kkqkqkUkqk

CCffN

UCCff

N

U

CCfN

UCCf

N

U

CCfN

UCCf

N

U

CCfN

UCCf

N

U

CCfN

UCCf

N

UHCC

p qkkkpkkpkpkkkqkkpk

kqkkpkkpkpkkkpkkqk

k kp

kkpkqkkpkkqkkpk

kpkp

kkpkqkkpkkqkk k

p

pkpkkqkkpkkpkkpkkqk

k kp

pkpkkpkkqk

k kp

Vkqk

CCCCCCCC

CCCCCCCC

N

V

CCCCCCCCCC

CCCCCCCCCCN

V

CCCCCCCCCCCCN

V

CCCCCCN

V

HCC

,'',,'',',''',,'',,','

',,'','',,'',,'','',',

' ''

,,'','',,'','',,,','

,'','',,',',,',',,' ''

,'','',,,',','','',',',,' ''

,'','',',',,' ''

,,

,,

,,

,,

,,

,

15

',,,'0

',',',

',,,',',',',,

..

)4(

',,,','

)3(

',,,','

)2(

'','','',,

)1(

'','','',,

)1(

kkqkk

kkqkqkq

p kpkqkkpkkpkpkqkp

APR

p kkpqkkpkp

p kpkqkkqkp

p kkpkkqpkp

p kkpkpkqkp

CCfN

VCCf

N

V

CCCCCCCCN

V

CCCCN

VCCCC

N

V

CCCCN

VCCCC

N

V

',,,'0

',',',

',,,',',',',,

)2(

kkqkk

kkqkkq

p kkpqkkpkkpkkpqkp

CCfN

VCCf

N

V

CCCCCCCCN

V

',',',

',,,'0

',',',,

',,,','

',,,','

)3(

kkqkqkq

kkqkk

p kkpkpkqkp

p kpkqkkpkp

p kpkqkkpkp

CCfN

VCCf

N

V

CCCCN

VCCCC

N

V

CCCCN

V

16

',',',

',,,'0

',',',,

',,,','

',,,','

)4(

kkqkkq

kkqkk

p kkpkkpqkp

p kkpqkkpkp

p kkpqkkpkp

CCfN

VCCf

N

V

CCCCN

VCCCC

N

V

CCCCN

V

',',',

',,,'0

',',',

',,,'0

',',',

',',',

',,,'0

',',',

,, ,

kkqkkq

kkqkk

kkqkqkq

kkqkk

kkqkqkq

kkqkkq

kkqkk

kkqkqkq

Vkqk

CCfN

VCCf

N

V

CCfN

VCCf

N

V

CCfN

VCCf

N

V

CCfN

VCCf

N

V

HCC

,',''

,,,,

2, kqk

kkqkqVkqk CCff

N

VHCC

17

,',''

,,

,,',''

,,,,',''

,,

,,,,

2

,

kqkk

kqk

kqkk

kqkkqkk

kqk

kqkqkkkqk

CCffN

V

CCffN

UCCff

N

U

CCHCC

',,',',,

',,',',,

,'

,',',,,'

,',',,

,,,,

)0,0(;2

)0,0(;2

)0,0(;)0,0(;

)0,0(;)0,0(,

);,(

kkqkkqkq

kkqkkqkq

kkqkkqk

kkqkkqk

kqkqkkkqk

nCCffN

VnCCff

N

V

nCCffN

UnCCff

N

U

nCCnCC

qkc

);(2),(

);,()0,0(;);,(

qffqVqffU

qknCCqk

ckqkckqk

cqkkkqk

c

Spin dependent in PI state

18

',,'','

',''

','

,,

,)0,0(,

pppqkkppppkpqk

ppkpppqkpppkqk

pppppkqkkqk

CCCC

CCCCCCCC

CCCCnCC

),()(211

),();,(1

),()(21);,(

),()(21

),()(2);,(

),()(2),();,();,(

)0,0(,',

,'',

',,

qqVUff

Nqqk

N

qqVUff

qk

qqVUff

qffqVUffqk

qffqVqffUqkffqk

ffCCCCnCC

c

k qkk

kqkc

k

c

c

qkk

kqkc

ckqk

ckqkkqk

cqkk

ckqk

ckqk

cqkkkqk

c

kqkpp

pqkkpppp

ppkpqkkqk

);(1

qff

N k qkk

kqk

令

19

);()(21);();( qqVUqq cc

);()(21

);();(

qqVU

qqc

PI to CO transition

Similarly, for spin-spin susceptibility

)0,0(,)0,0(,)(),,(

),,(1

)0,0(,1

)();(

)(

)0,0(),(1

)();(),;()();(

,,,,

,,,

,,

kqkkqks

s

qk

RiqRiq

k qkqkji

s

kkqk

q

Riqjiji

s

CCCCtitqk

tqkeN

eCCN

titRR

CCq

eqN

titRtRtitRR

ii

i

20

,,,

,,,,,,

,,,,

,,

)0,0(,)0,0(,)(),,(

kqkp

qkkppkqkpkkqk

kqkkqks

CCCCCCHCC

HCCCCttqkdt

di

p kkpkkpqkpkqkkpk

p k kkqkkpkkpkpkkkqkkpk

p k kpkkkqkkpkkqkkpkkpk

p k kkpkkqkkpkkpkkpkkqk

p k kkpkkpkkqk

Ukqk

kpkp k k

kpkU

CCCCCCCCN

U

CCCCCCCCN

U

CCCCCCCCCCCCN

U

CCCCCCCCCCCCN

U

CCCCCCN

U

HCC

CCCCN

UH

',',',,,,,','

' ''',,'','',,''',,'',,','

' '','',,'',,',',',,'','',,'

' '','','',,,',','','',',',,

' '','','',',',,

,,

,'',''' ''

,','

,

,,

,

,

21

,,,,,,

,,'

,,',','

,,

',,,',,,',,

',,',,,,

RPA In

,,

,,

,)0,0(,

kqkkkqkqk

pqkp p

kppppkp p

pqk

p pkpppqkpppkqk

p ppppkqkkqk

ffCCCC

CCCCCCCC

CCCCCCCC

CCCCCC

)1.........()0,0(;)0,0(;

)0,0(;)0,0(;

)0,0(;

)0,0(;

)0,0(;)(),,(

,,,'

,,,'

,,''

,',,,'

,','

',,,','

',',',,,,,','

,,,,

ppkpqkqk

kkqkk

pkpkp k

kqkpkqkp k

kpk

p kpkqkkpk

p kkpkkpqkpkqkkpk

kqkqkkkqks

CCfN

UCCf

N

U

CCCCN

UCCCC

N

U

CCCCN

U

CCCCCCCCN

U

CCffttqkt

i

22

)2.........()0,0(;)0,0(;

)0,0(;)0,0(;

)0,0(;

,,,'

,,,'

,','

,,',,'

,','

',',',,

ppkpqkk

kkqkk

kpqkp k

kpkkpqkp k

kpk

p kkpkkpqk

CCfN

UCCf

N

U

CCCCN

UCCCC

N

U

CCCCN

U

)0,0(;)0,0(;)2()1( ,,'

,,,,'

,','

pkpqkk

qkkkqkk

kk CCffN

UCCff

N

U

),,(

),,()(),,(

)0,0(;)0,0(;

)0,0(;)(),,(

,,

~

','

~

',',,

,,,,'

,,,','

,,,,

,,

tqpkffN

U

tqkfN

Uf

N

Uffttqk

ti

CCffN

UCCff

N

U

CCffttqkt

i

s

pqkk

s

kkqk

kkkkqk

s

ppkpqkqkk

kkqkkk

kqkqkkkqks

qkk

23

);();();();(1);();(

~~1

);(

~~);(11

);,(1

~~);(1

);,(

,,

,,

,,

,,

,,

,,

qqUqqUqq

ff

Nq

qUff

Nqk

N

qUffqk

sss

k kqk

kqk

k kqk

skqk

k

s

kqk

skqks

);(1

);();(

qU

qqs

(spin dependent in PI)

PI to AFM

In CO state

Mn+4

Mn+3iRiQ

iiCO enH

Induced

),,( type-Gfor Q

24

Qkk CC

k

k

Qk

kk c

cH

:state COIn

22

22

k

kk

);(1

);();(

qU

qqs

CO to AFM

x TC TN0.55 222 1560.60 260 1430.65 265 1300.70 250 1250.75 215 1130.80 180 1060.85 130 102

Substituting to

Experimental data

);(1

);();(

qU

qqs

);()(21

);();(

qqVU

qqc

To determine interaction relationsCheong and Hwang (1999)

25

Results and discussion

U

1 2 3 4 5 6 7 8 9 10

Vco

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.55 TC=222

x=0.60 TC=260

0.65 TC=265

0.70 TC=250

0.75 TC=215

0.80 TC=180

x=0.85 TC=130

CO

AFM

CO/AFM

31 MnOSrLa xx

Reflection different transitions

26

U=5.1

x

0.55 0.60 0.65 0.70 0.75 0.80 0.85

Vco

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Consistent with John Teller distortion

non-symmetry symmetry

More distortion

27

U

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

0

1

2

3

4

5

6

7

8

9

10

x=0.55x=0.60x=0.65x=0.70x=0.75x=0.80x=0.85

Charge gaps are depressed by U

28

U=5.1

x

0.55 0.60 0.65 0.70 0.75 0.80 0.85

0

1

2

3

4

5

6

7

Charge gap fluctuation

The competition between HV and HU

29

Conclusions Substituting experimental critical transition temperatures of TCOs and TNs to charge-charg

e and spin-spin susceptibility functions offer the determination of the inter-Coulomb repu

lsions and charge gaps for x > 0.5, respectively.

These Inter-Coulomb repulsions increase with x increasing but not in linear.

In small on-site repulsion U the phase transitions only occur pare-insulator to charge-ord

ering transitions and in large U only occur para-insulator to antiferromagnetic transitions.

The consequential phase transitions for para-insulator to charge-ordering following char

ge-ordering to antiferromagnetic transitions occur in a moderate U. In charge ordering st

ates the charge gaps are opened and are depressed by U.

The scale of the charge gap increases linearly with x increasing excluding a small range

of deviation. This deviation comes from the charge gap fluctuation according to the comp

etition between inter-Coulomb and on-site Coulomb interactions.

30

Thanks for your attendance!!