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

ronicsronics

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

ronicsronics

• 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 ferromagetism

Dilute electrons

Local moments

RKKY indirect interaction

Kondo Lattice model

i

ii

ji

ji SJcctH

,,

,,

With Zeeman energies

i

ZiB

i

ZiB hgShg

1)()();()( nj

njiij SStStG

HtAdt

tdAi ),(

)(

Arbitrary S local moment Green’s function

Equation of motion

1, )()(;,)( n

jn

jinji SSHStGdt

di

The time derivative of local spin greens function

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

)()(;)(

11

1,

nj

njiz

nj

nji

Z

nj

nji

Znji

SSSSSSJ

SSSJtGdt

di

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

SSSJtGdt

di

1)()(; nj

nji SSIncluding spin flip

Greens function of conducting electrons

equal to

1)()(; nj

njii SScc

),(1

2

1)(

)(,

qGdee

NtG 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

k

n

ZZ

kqk

kkqkqk

ZZ

Z

qGSJ

cccc

N

SJJ

);()1

22(

2

k ZZ

kqk

kkqkqk

Z

iSJ

cccc

N

SJ

q

1

2),(

2

Self-energy

Dyson’s general formula of magnetization

1212

1212

)()(1

)()(1)(1)(

SS

SSZ

SS

SSSSSSS

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

2m

kk

0

2 sin21

ddkkN k

The summation becomes

dk

m

kq

m

qSJ

m

kq

m

qSJ

nq

fkm

VS

J

dk

m

kq

m

qSJ

m

kq

m

qSJ

nq

fkm

VS

JJ

Zq

Zq

k

C

Z

Zq

Zq

k

C

ZZq

)

2

2( 2

2

)

2

2( 2

2

*

2

*

22

*

2

*

22

02

*2

*

2

*

22

*

2

*

22

02

*2

Spinwave Spectrum

where

k

BZZ

k

kk

f

TKSJ

cc

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

12

2

2

12

2

2

Imaginary part of self energy will cause the spin waves spread

)()(

cos2

2

)()(1

2),(Im

0

1

1

22

2

kqkZ

kqk

C

Z

kqkZ

kkqk

Z

SJff

ddkkV

SJ

SJffN

SJ

q

02

cos *

22

*

2

m

q

m

kqSJ 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

SJ

SJ

SJ

SJ

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

net

izat

ion

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

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

Su

scep

tib

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,

h

SSS

dh

dstatic

Zh

Z

hZ 0)(

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

hZS

0 ZSand

Where

• 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