Spin Transfer Torque Magnetoresistive Random Access Memory
57
magnetic moments. Such magnetostatic interaction can have a long range effect,
leading to the formation of domains in an effort to lower the energy cost in contrast
to keeping spins aligned parallel as a single domain. As such, it earns its name as the
demagnetization term since it competes with the short range exchange interaction.
Therefore, at an intermediate range where the exchange interaction is no longer
strong enough to hold spins in the parallel configuration, magnetic domains are
formed with transitions between domains known as domain walls. Domain walls
could be an alternative method to store information along with the STT effect as a
writing mechanism to drive domain walls along a nanowire [54, 55, 56].
It is generally difficult to calculate the demagnetization term for a magnet that is
either non-uniform or arbitrarily shaped. However, in the case of an ellipsoid, the
demagnetization field can be expressed as [57, 58]:
H dem = −
=
N M = −
⎛
⎝
N x 0 0
0 N y 0
0 0 N z
⎞
⎠ M,
(13)
where
=
N is the dimensionless demagnetization tensor of rank 2, with its trace N x +
N y + N z = 1 for a coordinate system orientated along the principle axes of the
ellipsoid. The energy of the demagnetizing energy is simply the integral over the
volume of the magnet:
E dem = −
μ 0
2
V
M · H dem dV .
(14)
The demagnetization term can be utilized to induce shape anisotropy, which is
crucial for the development of iMTJ. Shape anisotropy has also been proposed to
create pMTJs smaller than 10 nm (see Sect. 5.6). In the case of infinitely extended
thin films, the demagnetization factors can be set as N x = N y = 0, leaving N z = 1.
3.3.2 Anisotropy Energy
In MRAM applications, magnetic anisotropy is an important effect as it results in an
easy axis where magnetic moments tend to align towards to for energy minimization. The direction of the magnetic moment and subsequently the bit information is
therefore confined to this axis, which can be used to achieve maximum TMR effect.
In addition, the anisotropy energy would constitute as the energy barrier required for
magnetic reversal, influencing the thermal stability of the bit information.
The total effective anisotropy energy is summed up due to the contribution of the
bulk contribution K v (e.g. magnetocrystalline anisotropy), the surface anisotropy
term
2K s
t
and the demagnetization energy (2π M
2
s for PMA), leads to the following
expression [59, 60]:
57
magnetic moments. Such magnetostatic interaction can have a long range effect,
leading to the formation of domains in an effort to lower the energy cost in contrast
to keeping spins aligned parallel as a single domain. As such, it earns its name as the
demagnetization term since it competes with the short range exchange interaction.
Therefore, at an intermediate range where the exchange interaction is no longer
strong enough to hold spins in the parallel configuration, magnetic domains are
formed with transitions between domains known as domain walls. Domain walls
could be an alternative method to store information along with the STT effect as a
writing mechanism to drive domain walls along a nanowire [54, 55, 56].
It is generally difficult to calculate the demagnetization term for a magnet that is
either non-uniform or arbitrarily shaped. However, in the case of an ellipsoid, the
demagnetization field can be expressed as [57, 58]:
H dem = −
=
N M = −
⎛
⎝
N x 0 0
0 N y 0
0 0 N z
⎞
⎠ M,
(13)
where
=
N is the dimensionless demagnetization tensor of rank 2, with its trace N x +
N y + N z = 1 for a coordinate system orientated along the principle axes of the
ellipsoid. The energy of the demagnetizing energy is simply the integral over the
volume of the magnet:
E dem = −
μ 0
2
V
M · H dem dV .
(14)
The demagnetization term can be utilized to induce shape anisotropy, which is
crucial for the development of iMTJ. Shape anisotropy has also been proposed to
create pMTJs smaller than 10 nm (see Sect. 5.6). In the case of infinitely extended
thin films, the demagnetization factors can be set as N x = N y = 0, leaving N z = 1.
3.3.2 Anisotropy Energy
In MRAM applications, magnetic anisotropy is an important effect as it results in an
easy axis where magnetic moments tend to align towards to for energy minimization. The direction of the magnetic moment and subsequently the bit information is
therefore confined to this axis, which can be used to achieve maximum TMR effect.
In addition, the anisotropy energy would constitute as the energy barrier required for
magnetic reversal, influencing the thermal stability of the bit information.
The total effective anisotropy energy is summed up due to the contribution of the
bulk contribution K v (e.g. magnetocrystalline anisotropy), the surface anisotropy
term
2K s
t
and the demagnetization energy (2π M
2
s for PMA), leads to the following
expression [59, 60]: