Theory of Diluted FerromagTheory of Diluted Ferromagnetic III-V compound semicnetic III-V compound semiconductor materials of Spintonductor materials of Spintronicsronics

• Spintronics = Spin + Electronics• The most interesting material is Diluted Ferromagnetic semiconductor III-V based with Mn impurity i.e. (In,Mn)As, (Ga,Mn)As

• III-V DMSs : S = 5/2 (Mn 2+) hole concs. ~ 10% impurities concs. (compensated doping) hole spins couple with Mn AF (p-d coupling)

Compensated doping

Carrier mediated ferromagetismDilute electrons

Local moments

RKKY indirect interaction

Kondo Lattice model

iii

jiji SJcctH

,,,,

With Zeeman energies i

ZiB

i

ZiB hgShg

1)()();()( nj

njiij SStStG

HtAdttdAi ),()(

Arbitrary S local moment Green’s function

Equation of motion

1, )()(;,)( n

jn

jinji SSHStGdtdi

The time derivative of local spin greens function

.)()(;)()(;2

)()(;)(

11

1,

nj

njiz

nj

nji

Z

nj

nji

Znji

SSSSSSJ

SSSJtGdtdi

izZiii

Zii SJSJSHS , Where hg Bz

Then

Through RPA mean field

1

1

1,

)()(;

)()(;2

)()(;)(

nj

njiz

nj

nji

Z

nj

nji

Znji

SSS

SSSJ

SSSJtGdtdi

1)()(; nj

nji SSIncluding spin flip

Greens function of conducting electrons

equal to

1)()(; nj

njii SScc

),(121)( )(

,

qGdeeN

tG ti

q

RRiqji

ji

)(

,2

)(

,

1,,2

1

);,(1

)()(;1

)()(;

ji

ji

RRiq

qk

RRiq

qK

nj

njkqk

nj

njii

ekqkN

eSSccN

SScc

Through the Fourier transformation

Local spin Greens function

spin flip Greens function

);()2

(

);,(1);(

qGJ

kqkN

SJqG

zZ

k

Zn

);,()(

);()(2

);,()22();,(

kqkSJ

qGccccJ

kqkkqk

ZZ

qkqkkk

kqk

Combined together

kn

ZZ

kqk

kkqkqk

ZZ

Z

qGSJ

cccc

N

SJJ

);()1

22(

2

k ZZ

kqk

kkqkqk

Z

iSJ

cccc

N

SJq

1

2),(

2Self-energy

Dyson’s general formula of magnetization

1212

1212

)()(1)()(1)(1)(

SS

SSZ

SSSSSSSS

S

where 1)1(1

)( q

qeN

S

22ZZ

k SJ

22ZZ

k SJ

RPA first order approx. for electrons

k

take the dilute limit by conversing the kinetic energy to free electrons like

*

22

2mk

k

0

2 sin21 ddkkN k

The summation becomes

dk

mkq

mqSJ

mkq

mqSJ

nqfkm

VSJ

dk

mkq

mqSJ

mkq

mqSJ

nqfkm

VSJJ

Zq

Zq

k

C

Z

Zq

Zq

k

C

ZZq

)

2

2( 22

)

2

2( 22

*

2

*

22

*

2

*

22

0 2

*2

*

2

*

22

*

2

*

22

0 2

*2

Spinwave Spectrum

where

k

BZZ

k

kk

f

TKSJcc

1/)22

(exp

1

ak

BZZ

qk

qkqk

f

TKSJ

cc

1/)22

(exp

1

0qfor

By L’Hospital rule

dkSJ

fkV

SJ

dkSJ

fkV

SJ

J

Zq

kC

Z

Zq

kC

ZZq

0

2

2

0

2

2

122

2

122

2

Imaginary part of self energy will cause the spin waves spread

)()(

cos22

)()(12

),(Im

0

1

1

22

2

kqkZ

kqk

C

Z

kqkZ

kkqk

Z

SJff

ddkkV

SJ

SJffN

SJq

02

cos *

22

*

2

mq

mkq

SJ Z

The delta function made a constraint

the existing region for the imaginary part

kppZ

kppZ

SJ

SJ

2

2

q

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0

2e-4

4e-4

6e-4

8e-4

1e-3

)2

(2

)2

(2

Zpp

Z

Zpp

Z

SJSJ

SJSJ

Considering the zero temperature situation

the existing region for the imaginary part

Temperature(K)

0 10 20 30 40 50 60 70 80 90 100

Mag

netiz

atio

n

-0.6-0.4-0.20.00.20.40.60.81.01.21.41.61.82.02.22.42.62.83.0

SZ (C*=1.0E-3)

(C*=1.0E-3)

SZ (C*=1.0E-2)Z (C*=1.0E-2)

From Dyson’s general formula of magnetization Magnetization profile is comparable for Monte Carlo result for Ising interaction(Osamu Sakai, Physica E 10,148(2001)

20.1 En

30.1 En

Temperature

0 10 20 30 40 50 60 70 80

Susc

eptib

ility

0.0

1.0e-5

2.0e-5

3.0e-5

4.0e-5

5.0e-5

6.0e-5

7.0e-5

8.0e-5

9.0e-5

1.0e-4

To evaluate the temperature dependence of static susceptibility,

hSS

Sdhdstatic

Zh

Z

hZ 0)(

are expectation values of local spin with magnetic field turned on and off

hZS 0 ZSan

dWhere

• Kondo lattice model utilizes the equation of motion method with RPA approximation in dilute limitation to obtain a local spin greens function of self consistent solution can well describe the magnetic properties of diluted ferromagnetic semiconductors

Conclusions:

• From examining the imaginary part of self energy reveals that the spin excitations are well established in this model • The temperature dependence of magnetization is qualitatively consistent with Monte Carlo result • the significant peak of susceptibility appearing before Tc agrees with experimental result