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The magnetization of the ferromagnetic layer (1) can be excited with the help
of external magnetic fields or spin transfer torque generated by current through
the device. The layer (1) is called the free layer. The magnetodynamics of the
free layer under spin transfer torque is governed by the well known Landau-
Lifshitz-Gilbert (LLG) equation given below (Eq.
1
).
d
m
1
dt
α
γM
S
d
m
1
dt
J
e
G
J
p
=
−γM
S
m
1
×
(
h
eff
−
−
m
2
×
m
1
)
(1)
where,
G
=
−
1
4+(1+
P
)
3
(3 +
s
1
. s
2
)
4
P
3
/
2
−
(2)
and
|
e
|
t
F
J
p
=
μ
0
· M
2
(3)
s
Table
1
defines the symbols. The first term on the right hand side of Eq.
1
relates
to the precessional dynamics in the free layer resulting under the net magnetic
field acting on it. The contributors to this field are the dipolar coupling from the
fixed layer, shape anisotropy and demagnetization field. The second term relates
to damping in the layer. The third term relates to the effect of spin transfer
torque on the layer. Below are some of the key properties of MTJs that are
crucial to understanding the operations of STT-MRAM.
Table 1.
Symbol definitions.
Symbol
Description
m
1
,
m
2
Unit vectors in direction of magnetization of free and fixed layer
γ
Gyromagnetic ratio
M
s
Saturation magnetization
h
eff
Unit vector along effective magnetic field on the free layer arising from
crystalline and shape anisotropy, demagnetization field, exchange field
and external field which also includes coupling from the fixed layer
α
Gilbert damping constant
P
Spin polarizing factor
s
1
, s
2
Unit vectors along the global spin orientation of the free and fixed layers
t
F
Thickness of free layer
e
Electron charge
Reduced Planck's constant
μ
0
Permeability of free space
H
K
Effective anisotropy field including magnetocrystalline anisotropy and
shape anisotropy
H
ext
External magnetic field on free layer including coupling from underneath
fixed layer
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