VJ jijiijjjiiji ++=×-×+×++= HHHHH SSSLSL 2ll

nmjmin =jj

!+-

+-

+= ååm jmjn ini

ji EEVm

mEE

Vnn

00

0000

000000jj

!+-

+-

+= ååm iimn iin

jiji EEVmmV

EEVnnV

VV00

000000000000jjjj

å

å

-

×-×+××-×+×+

-

×-×+××-×+×+»

m iim

jiijjjiijiijjjii

n iin

jiijjjiijiijjjii

EEJmmJ

EEJnnJ

V

0

0

00200200

00200200

SSSLSLSSSLSL

SSSLSLSSSLSL

llll

llll

jiijjjiijiijjjii nJnnnJ SSSLSLSSSLSL ×-×+×=×-×+× 02000000000200 llll

{ }{ } jiijjzjzjyjyjxjx

iziziyiyixix

nJSnLSnLSnL

SnLSnLSnL

SS ×-+++

++=

0200000000000

000000000

l

l

jiijjjii nJnn SSSLSL ×-×+×= 0200000000 ll

DM interaction：2nd order perturbation theory

!+-

+-

+= ååm jmjn ini

jiji EEVmmV

EEVnnV

VV00

000000000000jjjj

[ ][ ]

[ ][ ]å

å

-

×-×+××-×+×+

-

×-×+××-×+×+»

m jmj

jiijjjiijiijjjii

n ini

jiijjjiijiijjjii

EEJmmmmJmm

EEJnnnnJnn

V

0

0

00020000000002000000

00020000000002000000

SSSLSLSSSLSL

SSSLSLSSSLSL

llll

llll

( ) ( ) ( ) ( ) ( ) 2110212

2

2010 **0,00000 rrrrrr ddrenJnJ jinjiij jjjjòò==

( ) ( )[ ] ( ) ( )[ ] !+××-

-××-

-= ååm

jiijjmjn

jiiiini

mEEmJn

EEnJV SSSLSSSL ,000,02,00,002

00

ll

( )[ ] ( )[ ]

( )[ ] ( )[ ]å

å

-

×-××-×+

-

×-××-×+»

m jmj

jijjjijj

n ini

jiiijiii

EEmJmmJm

EEnJnnJn

V

0

0

00,0200,0020

00,0200,0020

SSSLSSSL

SSSLSSSL

ll

ll

( )[ ] ( ) ( ) ijijiijii SSSSSSSSS ×-×=×,

( ) ( ) ijzizjyiyjxixjzizjyiyjxixi SSSSSSSSSSSS SS ++-++=

[ ] [ ]{ } [ ] [ ]{ } [ ] [ ]{ } zjyiziyjxixizyjxiyixjziziyxjzixizjyiyix SSSSSSSSSSSSSSSSSS eee ,,,,,, -+-+-=

( ) ( ) ( ) zjyixjxiyyjxizjzixxjziyjyiz SSSSiSSSSiSSSSi eee -+-+-=

jii SS ´-=

( ) ( )[ ] ( ) ( )[ ]

( ) ( ) [ ]jim

jjmjn

iini

mjiij

jmjnjiii

ini

mEE

mJnEEnJi

mEEmJn

EEnJ

SSLL

SSSLSSSL

´×úúû

ù

êêë

é

--

-=

××-

-××-

-

åå

åå

000,000,002

,0000,02,00,002

00

00

l

ll

[ ]ji SSD ´×=

Parasitic ferromagnetism of α-Fe2O3（hematite）explained by DM interaction

945 K以下： antiferromagnetism260-945 K： ferromagnetism appears in AF phase

（ parasitic ferromagnetism ）[ ] 02

,,<´×+×-= åå JJ

jiji

jiji SSDSSH

( ) qqq sincos2 22 SSJE D--=

Canting angle of spins

qp -

JD2

tan =q

ji SSD ´- //

４. Molecular field thoery

å å><

-×-=ji i

izBjiij SHgJH,

2 µSSå><

×-=ji

jiijJH,

2 SS

Temperature dependence of magnetization

magnetic susceptibility specific heat

å å><

-×-=ji i

izBjiij SHgJH,

2 µSS

å><

×-=ji

jiijJH,

2 SS

4.1 Heisenberg model

(4.1)

(4.2)

Hamiltonian of magnetic material

( )H,0,0=H

・no magnetic anisotropy, but z-axis is easy axis

HSgg zBB µµ =×=×- HSHμ HSg zBµ-

：external field

：pair of nearest neighbor lattice pointsji,

・sign of spin

Hereafter, S: spin magnetic moment

Equation of (4.1) or (4.2) cannot be solved exactly.

iS：total spin of a atom at i-th lattice point

Heisenberg model Ising model @S=1/2

4.2 Molecular field theory - finite temperature

( ) å -×-= +k

izBkii HSgJiH µSS2 (4.3)

Consider iʼth spin

・Sum of k is from one to the number of the nearest neighbor atoms.・Exchange integrals of the nearest neighbor spins are assumed to be equal.

◆Weissʼs molecular field theorySpin around is replaced by the averaged value.iSki+S

zzkiykixki SSSS === +++ ,0

( ) izBizz HSgSSzJiH µ--= 2 ( ) izBz SHgSzJ µ+-= 2

( ) åå -==i

izzi

SSzJiHH 2

(4.4)

( ) ( )åå +-==i

izBzi

SHgSzJiHH µ2

Z(i) of iʼth spin,

( ) ( )å-=

úû

ùêë

é+=

S

Si B

izBz Tk

SHgSzJiZ µ2exp

[ ] ( ) ( ) ASSASAASS

Siiz eeeeAS ----

-=

+++== å 11exp !Tk

HgSzJA

B

Bz µ+=2

( )

TkHgSzJ

STk

HgSzJ

iZ

B

Bz

B

Bz

22

sinh

212

sinh

µ

µ

+

úû

ùêë

é÷øö

çèæ +

+

=

Total magnetic momentN：total number of magnetic atomszB SNgM µ=

{ }ASAAAS eeee 221 ++++= -!

( )

A

SAAS

eee-

-=

+-

11 12

( ) ( )

2/2/

2/12/1

AA

SASA

eeee-

+-+

--

=A

AS

21sinh

21sinh ÷øö

çèæ +

=

(4.5)

(4.6)

(4.7)

◆Temperature dependence of magnetization( )

( )iZ

eSS

S

Si

TkiH

iz

z

Bå-=

-

=

TkHgSzJ

AB

Bz µ+=2

( )( )

ååå-=-=

+

-=

-

===S

Si

ASS

Si

STk

HgSzJS

Si

TkiH

iziz

B

Bz

B eeeiZµ2

( ) å-=

=¶¶ S

Si

ASiz

izeSAiZ

( )( )AiZ

iZSz ¶

¶=1

( )22

21

21

AA

SASA

ee

eeiZ-

÷øö

çèæ +-÷

øö

çèæ +

-

-=

úúú

û

ù

êêê

ë

é

-

-¶¶

-

-=

-

÷øö

çèæ +-÷

øö

çèæ +

÷øö

çèæ +-÷

øö

çèæ +

-

22

21

21

21

21

22

AA

SASA

SASA

AA

ee

eeA

ee

ee

2

22

21

21

222221

21

21

21

22 21

21

÷÷ø

öççè

æ-

÷÷ø

öççè

æ-÷÷

ø

öççè

æ+-÷÷

ø

öççè

æ-÷

÷ø

öççè

æ+÷

øö

çèæ +

-

-=

-

÷øö

çèæ +-÷

øö

çèæ +--÷

øö

çèæ +-÷

øö

çèæ +

÷øö

çèæ +-÷

øö

çèæ +

-

AA

SASAAAAASASA

SASA

AA

ee

eeeeeeeeS

ee

ee

22

22

21

21

21

21

21

212

AA

AA

SASA

SASA

ee

ee

ee

eeS-

-

÷øö

çèæ +-÷

øö

çèæ +

÷øö

çèæ +-÷

øö

çèæ +

-

+-

-

++=

2coth21

21coth

212 ASAS

-úû

ùêë

é÷øö

çèæ +

+=

TkSHgSzJS

ASxB

Bz µ+==2

úû

ùêë

é-÷øö

çèæ ++

=Sx

Sx

SS

SSSSz 2

coth21

212coth

212

( )xSBS=

( )Sx

Sx

SS

SSxBS 2

coth21

212coth

212

-÷øö

çèæ ++

= Brillouin function

[ ] ( ) xxxxB tanhcoth2coth221 =-=

[ ] ( ) SxSx

SxSx

SS eeee

SxxB 2/2/

2/2/

21limcoth -

-

¥®¥® -+

-= ( )11

21limcoth /

/

-+

-=¥® Sx

Sx

S ee

Sx

( ) ( )( ) !

!

+++++

-=¥® 2///

2///221limcoth 2

2

SxSxSxSx

Sx

S

( ) ( )!

!

+++++

-=¥® Sxx

SxSxxS 2/

2///221limcoth 2

2

( )x

x 1coth -= Langevin function( )xL=

(4.8)

→ Ising model

(4.9)

(4.10)

(4.11)

Brillouin function

●H=0，<Sz> is sufficiently small, Magnetization around transition T

!+-+=®<<453

1coth13xx

xxx

[ ]úúú

û

ù

êêê

ë

é

÷øö

çèæ-+-

úúú

û

ù

êêê

ë

é

÷øö

çèæ +

-+

++

+»

33

2451

231

2

121

212

451

212

31

2121

212

Sx

Sx

SxS

xSSx

SS

xSSS

SxBS

34

3422

21

451

212

451

21

31

212

31 x

Sx

SSx

Sx

SS

÷øö

çèæ+÷

øö

çèæ +

-÷øö

çèæ-÷

øö

çèæ +

=

úúû

ù

êêë

é÷øö

çèæ+÷

øö

çèæ +

úû

ùêë

é÷øö

çèæ+÷

øö

çèæ +

úû

ùêë

é÷øö

çèæ-÷

øö

çèæ +

-+

=223

21

212

21

212

21

212

4531

SSS

SSS

SSSxx

SS

2

23

21221

4531

SSS

SSxx

SS +++

-+

= ( ) ( )[ ] 33

22

9011

31 x

SSSSx

SS +++

-+

=

122

<<=+

=TkSzJS

TkSHgSzJS

xB

z

B

Bz µ

(4.12)

( ) ( )[ ] 33

22

9011

31 x

SSSSx

SS +++

-+

=( )xSBS Sz =

( ) ( )[ ] 3

3

22 290112

31

÷÷ø

öççè

æ+++-

+=

TkSzJS

SSSS

TkSzJS

SS

B

z

B

z

( ) ( )[ ]( )

zB

BBz ST

kJSzS

zJk

zJTk

SSSSS ÷÷

ø

öççè

æ-

+÷øö

çèæ

+++=

312

21145 2

223 (4.13)

0=zS

( )B

C kJSzST

312 +

= (4.14) Curie temperature

From eq(4.13)

◆CTT ³

CTT £◆ ( )( )[ ] C

C

Cz T

TTTT

SSSSS -

÷÷ø

öççè

æ

+++

=2

22

222

11

3100=zS

( )( )

( )( ) C

C

C

C

Cz T

TT

SS

SSTTT

TT

SS

SSS -

++

+±»

-÷÷ø

öççè

æ

++

+±=

2222 1

1310

1

1310

● T 〜TC

(4.15)

zB SNgM µ=Magnetization vanishes at Tc

( ) !+-=®>> - SxS e

SxBx /111

CTT <<●

(4.16)

0»T ( ) !+-==-

TkzJS

SzBeSxSBS2

(4.17)

0@ == TSNgM Bµ Magnetization is saturated.

◆ at arbitrary TzS

( )xBSS

Sz =

TkSzJS

xB

z2= x

zJSTk

SS Bz

22=

( )xBSS

Sz =

xzJSTk

SS Bz

22=

xzJSTk

SS CBz

22=

Temperature dependence of magnetization

zB SNgM µ=

P：free energy minimum pointO：free energy maximum point

◆ Temperature dependence of susceptibility

TkSHgSzJS

xB

Bz µ+=2

0=÷øö

çè涶

=HH

Mc

0=÷÷ø

öççè

涶

=H

zB H

SNgµ

(4.18) susceptibility

( )Hx

dxxdBSNg S

B ¶¶

= µc

TkSg

HS

TkzJS

Hx

B

B

H

z

B

µ+÷÷

ø

öççè

涶

=¶¶

=0

2

(4.19)

( )÷÷ø

öççè

æ+=

TkSg

NgTkzJS

dxxdBSNg

B

B

BB

SB

µµcµ 2

( ) ( )

( )dxxdBzJSTk

dxxdBSgN

SB

SB

2

2

2-=

µc (4.20)

0=zS◆CTT ³

CTT £◆

( )( )

SSzJSTkS

SSgNTT

B

B

C

312

31

2

2

+-

+

=³µ

c

From (4.12) ( ) ( ) ( )[ ] 33

22

9011

31 x

SSSSx

SSxBS

+++-

+=

( )S

SBS 310' +

=

( ) ( )CB

B

TTkSSgN

-+

=1

312µ (4.21)

( ) ( ) ( )[ ] 23

22

3011

31' x

SSSS

SSxBS

+++-

+=

( ) ( )[ ] 2

3

22 23011

31

÷÷ø

öççè

æ+++-

+=

TkSzJS

SSSS

SS

B

z

( )( ) C

Cz T

TT

SS

SSS -

++

+»

221

1310Around TC

From (4.15)

( ) ( ) ( )[ ] ( )( ) ÷÷

ø

öççè

æ-

+++

÷÷ø

öççè

æ+++-

+=

CBS T

TSS

SSTkzJS

SSSS

SSxB 1

11

3102

3011

31'

22

222

3

22

÷÷ø

öççè

æ -÷øö

çèæ+

-+

=C

CC

TTT

TT

SS

SS 2131

( )( ) ( )

( )dxxdBzJSTk

dxxdBSgN

TTS

B

SB

C2

2

2-=£

µc

( ) ( )TTk

SSgN

CB

B

-+

=1

612µ

CT

(4.22)

12 2qq =