Current-Driven Domain Wall Dynamics in Magnetic …
141
B θ =
1
2
(−R A sin θ 0 + R P sin 2θ 0 sin 2ϕ 0 ))θ
B ϕ = R P sin
2
θ 0 cos 2ϕ 0 ϕ
The angular term variations θ and ϕ depend on the current induced effective
fields, H SL along the longitudinal direction and H FL along the transverse direction.
The second harmonic Hall voltage, V 2ω shows the dependency on θ and ϕ and thus
contains information about the effective fields through θ and ϕ terms. Following
the derivation in Ref. [83], the respective curvature and slope of V ω and V 2ω versus
the external field are calculated to obtain the ratios H SL and H FL , defined as H SL =
(
∂ V 2ω
∂ H X
/
∂
2 V ω
∂ H
2
X
) and H FL = (
∂ V 2ω
∂ H Y
/
∂
2 V ω
∂ H
2
Y
). As discussed earlier, the measured Hall voltage
also contains contributions from planar Hall effect (PHE) that leads to mixing of the
H SL and H FL . We define the field generated in longitudinal and transverse direction
as Corr_H SL and Corr_H FL , respectively. By defining ξ =
R P
R A
, finally it can be
shown that.
Corr_H SL = −2
H SL ± 2ξ H F L
1 − 4ξ 2
,
(28)
Corr_H F L = −2
H F L ± 2ξ H SL
1 − 4ξ 2
.
(29)
The ± sign corresponds to M pointing along the ± z.
The harmonic measurements are performed on the same SAF structures discussed
in Sect. 4.2. The in-plane magnetic field was swept in the longitudinal (H L ) and transverse (H T ) directions to the current of frequency 133 Hz, to quantify Slonczewski-like
(SL), and Field-like (FL) effective fields, respectively. Variation in V ω and V 2ω with
H L for at a current density 1.04 × 10
11 A/m
2 is shown in Fig. 36. The red and
black curves are corresponding to net M Z > 0 (net ‘up’) and M Z < 0 (net ‘down’)
magnetizations, respectively. The first harmonic of the Hall voltage can be fitted to
an equationV ω = AH x + B H
2
x ,
(30)
where, A and B are the polynomial coefficients of the equation. Similarly, linear
fitting can be performed on the second harmonic Hall voltage to get an equation of
the form
V 2ω = C H x + D,
(31)
where, C and D are the slope and intercept of the linear curve, respectively. Now, the
longitudinal effective fields can be obtained using following equation -
141
B θ =
1
2
(−R A sin θ 0 + R P sin 2θ 0 sin 2ϕ 0 ))θ
B ϕ = R P sin
2
θ 0 cos 2ϕ 0 ϕ
The angular term variations θ and ϕ depend on the current induced effective
fields, H SL along the longitudinal direction and H FL along the transverse direction.
The second harmonic Hall voltage, V 2ω shows the dependency on θ and ϕ and thus
contains information about the effective fields through θ and ϕ terms. Following
the derivation in Ref. [83], the respective curvature and slope of V ω and V 2ω versus
the external field are calculated to obtain the ratios H SL and H FL , defined as H SL =
(
∂ V 2ω
∂ H X
/
∂
2 V ω
∂ H
2
X
) and H FL = (
∂ V 2ω
∂ H Y
/
∂
2 V ω
∂ H
2
Y
). As discussed earlier, the measured Hall voltage
also contains contributions from planar Hall effect (PHE) that leads to mixing of the
H SL and H FL . We define the field generated in longitudinal and transverse direction
as Corr_H SL and Corr_H FL , respectively. By defining ξ =
R P
R A
, finally it can be
shown that.
Corr_H SL = −2
H SL ± 2ξ H F L
1 − 4ξ 2
,
(28)
Corr_H F L = −2
H F L ± 2ξ H SL
1 − 4ξ 2
.
(29)
The ± sign corresponds to M pointing along the ± z.
The harmonic measurements are performed on the same SAF structures discussed
in Sect. 4.2. The in-plane magnetic field was swept in the longitudinal (H L ) and transverse (H T ) directions to the current of frequency 133 Hz, to quantify Slonczewski-like
(SL), and Field-like (FL) effective fields, respectively. Variation in V ω and V 2ω with
H L for at a current density 1.04 × 10
11 A/m
2 is shown in Fig. 36. The red and
black curves are corresponding to net M Z > 0 (net ‘up’) and M Z < 0 (net ‘down’)
magnetizations, respectively. The first harmonic of the Hall voltage can be fitted to
an equationV ω = AH x + B H
2
x ,
(30)
where, A and B are the polynomial coefficients of the equation. Similarly, linear
fitting can be performed on the second harmonic Hall voltage to get an equation of
the form
V 2ω = C H x + D,
(31)
where, C and D are the slope and intercept of the linear curve, respectively. Now, the
longitudinal effective fields can be obtained using following equation -