2
2. M ETHODS
Influence of magnetic material properties to the reversal mechanism of magnetization was investigated by modeling ferromagnetic material with perpendicular magnetic anisotropy
Co
x
Si
y
B
z
as a perpendicular magnetized nano-dot having 50
50
20 nm
3
in dimension.
Figure 1
: Nano-dot models which the volume is 50 50 20 nm
3
with appropnate coordinate. Chosen magnetic material parameters for Co
x
Si
y
B
z
are 4
M
s
= 4116,6 G with
K
= 2
10
6
ergcm
3
[4]. Magnetization reversal simulation is done by accomplishing the Landau - Liftshitz Gilbert LLG Equation. It contains the time derivative of the magnetization on one of side only
which is shown in Eq.1. This equation describes the magnetic material response which is characterized by the magnetization direction if it is induced by a current field [5].
M M
H M
M H
2 2
1 1
i i
i i
i i
eff ef f
s
d dt
M
1
where
M
as the magnetization,
M
s
as the saturation magnetization,
as the gilbert damping constant 0.3,
as the gyro-magnetic ratio 1.7 10
7
Oe
-1
.s
-1
and
H
eff
as the effective field. The
H
eff
is composed by anisotropy field
H
k
, magnetostatic field
H
M
, the exchange interaction field
H
ex
, the external field
H
ext
and a random stochastic field
H
T
if a thermal field activated as seen in Eq.2 [6].
H H
H H
H H
i i
i i
i i
eff k
M ex
ext T
2
H
ex
as an exchange stiffness constant function which described in Eq.3[7].
H M
2
ex s
A M
3
where
A
as exchange stiffness constant 1×10
7
ergcm,
as permeability of vacuum and function of
M
as shown in Eq.[8].
M M
M M
2 2
2 2
2 2
x y
z
x y
z
4
The exchange energy between the magnetization
i
and
j
in a system of
N
spin is defined as [9] :
3
M r
M r
M r
2 2
2 , ,
1
N ex
x y
z i j k
E A
5 where
M
, ,
x y z
is the spatial gradient of the magnetization normalized components
corresponding to the
x
,
y
and
z
axis. The temperature dependence of exchange stiffness can be formulated in Eq. 6[10].
2
s s
M T
A T A
M
6 where
T
as actual room temperature 298 K. Relation between
H
k
with anisotropy constant is expressed as a function of the unitary
vector m can be seen at Eq.7[9].
H u m u
2 .
k s
K M
7
where
u
is the unit vector, along the direction of the uniaxial easy axis
1 0
and
M m
s
M
.
The effect of temperature toward the anisotropy and saturation magnetization constant is shown in Eq.8 and 9[10].
2
s s
M T
K T
K M
8
0,5
1
s s
c
M T
M T T
9 where
K
= anisotropy constant,
M
s
= saturation magnetization and
c
T
= curie temperature with the assumed value 373 K. The thermal fluctuation field has zero mean and is assumed to
be Gaussian distributed with a variance
given by the fluctuation-dissipation theorem as shown in Eq. 10, Eq.11 and Eq.12[11].
0
i T
H t
10
2
i j
T T
ij
H t H
t t
t
11
2
B s
k T VM
t
12 with
t
is the delta dirac function,
ij
is the Kronecker delta, the indices
i
and
j
label the unit cell or the vector component,
B
k
= boltzman constant,
V
= volume of nano-dot 50
50
20 nm
3
and
t
= time increment
12
0, 25 10
s.
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