theory of diluted ferromagnetic iii-v compound semiconductor materials of spintronics
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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
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