Current-Driven Domain Wall Dynamics in Magnetic …
125
coupled to each other via exchange interaction J ex as shown in Fig. 18. We assume
that M L and M U are single spins. The energy of the system (E ex ) can be written as:
E ex = −2J ex
M L .
M U = −2J ex M L M U cos θ,
(14)
where ‘θ’ is the angle between the two DWs and ‘J ex ’ is the interlayer exchange
coupling constant. If J ex > 0, the two DWs are aligned parallel to each other, i.e. θ =
0, whereas if J ex < 0, the two DWs are aligned antiparallel to each other, i.e. θ = π. In
both the cases, energy of the system is minimum according to Eq. (14), thus does not
experience any torque. However, when this equilibrium position is perturbed through
an external source such as current and its effect, the two collinear DWs are deviated
away from θ = 0 (J ex > 0) and θ = π (J ex < 0) equilibrium states, generating a torque.
From the exchange theory of two spins, the magnetic moment of upper DW spin can
be written as
M U = −gμ U
S U (For lower layer
M L = −gμ L
S L ). The magnetic field
experienced (H
U
ex )) by the top DW can then be expressed as
5 :
H
U
ex = −
∂ E ex
∂ M U
= −
2J
gμ U
S L ,
(15)
Hence, the torque on the top DW due to the exchange field is given by:
τ
U
ex = −
M U ×
H
U
ex = −2J
S U ×
S L
.
(16)
In the similar way, the exchange torque for a DW in lower ferromagnetic layer
can be given by
τ
L
ex = −
M L ×
H
L
ex = −2J
S L ×
S U
.
(17)
For antiferromagnetic coupling, ‘J ex ’ is negative and constant, therefore, the direction of the torque depends on the projection of
S U ×
S L
, which is always out-ofplane in the case of antiferromagnetically coupled Néel DWs in the PMA SAF wires
[89]. From Eqs. (16) and (17), one can conclude that (1) the exchange torque is in
out-of-plane direction, (2) the magnitude of τ
U
ex is same to that of τ
L
ex (3) τ
U
ex and τ
L
ex
are opposite in direction and (4) they become maximum when the DWs in upper and
lower ferromagnetic layers are perpendicular to each other.
5 Introduction to Solid State Physics. Edited by Charles Kittel. 2004, Wiley, ISBN: 978–0-471–41,
526-8.
125
coupled to each other via exchange interaction J ex as shown in Fig. 18. We assume
that M L and M U are single spins. The energy of the system (E ex ) can be written as:
E ex = −2J ex
M L .
M U = −2J ex M L M U cos θ,
(14)
where ‘θ’ is the angle between the two DWs and ‘J ex ’ is the interlayer exchange
coupling constant. If J ex > 0, the two DWs are aligned parallel to each other, i.e. θ =
0, whereas if J ex < 0, the two DWs are aligned antiparallel to each other, i.e. θ = π. In
both the cases, energy of the system is minimum according to Eq. (14), thus does not
experience any torque. However, when this equilibrium position is perturbed through
an external source such as current and its effect, the two collinear DWs are deviated
away from θ = 0 (J ex > 0) and θ = π (J ex < 0) equilibrium states, generating a torque.
From the exchange theory of two spins, the magnetic moment of upper DW spin can
be written as
M U = −gμ U
S U (For lower layer
M L = −gμ L
S L ). The magnetic field
experienced (H
U
ex )) by the top DW can then be expressed as
5 :
H
U
ex = −
∂ E ex
∂ M U
= −
2J
gμ U
S L ,
(15)
Hence, the torque on the top DW due to the exchange field is given by:
τ
U
ex = −
M U ×
H
U
ex = −2J
S U ×
S L
.
(16)
In the similar way, the exchange torque for a DW in lower ferromagnetic layer
can be given by
τ
L
ex = −
M L ×
H
L
ex = −2J
S L ×
S U
.
(17)
For antiferromagnetic coupling, ‘J ex ’ is negative and constant, therefore, the direction of the torque depends on the projection of
S U ×
S L
, which is always out-ofplane in the case of antiferromagnetically coupled Néel DWs in the PMA SAF wires
[89]. From Eqs. (16) and (17), one can conclude that (1) the exchange torque is in
out-of-plane direction, (2) the magnitude of τ
U
ex is same to that of τ
L
ex (3) τ
U
ex and τ
L
ex
are opposite in direction and (4) they become maximum when the DWs in upper and
lower ferromagnetic layers are perpendicular to each other.
5 Introduction to Solid State Physics. Edited by Charles Kittel. 2004, Wiley, ISBN: 978–0-471–41,
526-8.