186
K. Lee et al.
V H ∝ R AH E I 0 sin(ωt)(m z + I 0 sin(ωt)) = R AH E
I 0 sin(ωt)m z + (I 0 sin(ωt))
2
= R AH E (I 0 sin(ωt)m z + (I 0 cos(2ωt)))
(4)
Therefore, by measuring the first and second harmonics of the ac Hall voltage
one can measure the effective fields generated by the SOT. In order to measure the
damping-like and field-like effective fields one must measure the second harmonics
while sweeping the magnetic field in the x and y direction. There are different analysis
methods depending on which angle of magnetization tilt regime is needed to be
analyzed [36, 40–43]. Here, we will concentrate only on the regime where the out of
plane tilt of magnetization is very small. As shown in Fig. 7c, by applying a magnetic
field in either x or y direction of the Hall cross the First and Second harmonic Hall
voltage can be measured. The effective SOT fields can be described as follows,
B x,y =
∂ V 2ω
∂ H x,y
/
∂
2 V ω
∂ H 2
x,y
(5)
where B x and B y will be the effective fields in the x and y direction, respectively
and H x and H y is the applied magnetic field also in the x and y direction. When
considering the planar Hall effect, a correction in the effective fields need to be
calculated. The corrected effective fields would be,
H DL = −
2
(BX ±2ε B Y )
1−4ε 2
j
, H F L = −
2
(BY ±2ε B X )
1−4ε 2
j
,
(6)
where H DL and H FL is the damping-like effective field and the field-like effective
field, respectively and ξ = R PHE /R AHE . In order to apply the effective fields to the
domain walls there is a conversion factor that needs to be considered [44, 45].
2.1.3 Current-Field Equivalence Measurement and Domain Wall
Motion
Measuring the spin–orbit torques by the current-field equivalence method is also a
well-known method [46–48]. The current-field equivalence method is measuring the
effect of an applied current on the depinning measurements [47, 49]. As shown in
Fig. 8, the depinning measurements are done in the Hall cross. First, the Hall cross
is saturated in a certain magnetization and by passing a current through the Oersted
line (yellow strip in Fig. 8a) a domain wall is formed. By applying a magnetic field
(H p in Fig. 8b), the DW is driven into the Hall cross so it gets pinned in the Hall cross.
Here, one has to measure two different fields, one is the depinning field without a
current, i.e. H c
* , the second is the depinning field with a current applied (H dep ). The
difference between the two values will give the field equivalent to the effective field by
the current. In systems where SOTs are present, the effective field due to the current
K. Lee et al.
V H ∝ R AH E I 0 sin(ωt)(m z + I 0 sin(ωt)) = R AH E
I 0 sin(ωt)m z + (I 0 sin(ωt))
2
= R AH E (I 0 sin(ωt)m z + (I 0 cos(2ωt)))
(4)
Therefore, by measuring the first and second harmonics of the ac Hall voltage
one can measure the effective fields generated by the SOT. In order to measure the
damping-like and field-like effective fields one must measure the second harmonics
while sweeping the magnetic field in the x and y direction. There are different analysis
methods depending on which angle of magnetization tilt regime is needed to be
analyzed [36, 40–43]. Here, we will concentrate only on the regime where the out of
plane tilt of magnetization is very small. As shown in Fig. 7c, by applying a magnetic
field in either x or y direction of the Hall cross the First and Second harmonic Hall
voltage can be measured. The effective SOT fields can be described as follows,
B x,y =
∂ V 2ω
∂ H x,y
/
∂
2 V ω
∂ H 2
x,y
(5)
where B x and B y will be the effective fields in the x and y direction, respectively
and H x and H y is the applied magnetic field also in the x and y direction. When
considering the planar Hall effect, a correction in the effective fields need to be
calculated. The corrected effective fields would be,
H DL = −
2
(BX ±2ε B Y )
1−4ε 2
j
, H F L = −
2
(BY ±2ε B X )
1−4ε 2
j
,
(6)
where H DL and H FL is the damping-like effective field and the field-like effective
field, respectively and ξ = R PHE /R AHE . In order to apply the effective fields to the
domain walls there is a conversion factor that needs to be considered [44, 45].
2.1.3 Current-Field Equivalence Measurement and Domain Wall
Motion
Measuring the spin–orbit torques by the current-field equivalence method is also a
well-known method [46–48]. The current-field equivalence method is measuring the
effect of an applied current on the depinning measurements [47, 49]. As shown in
Fig. 8, the depinning measurements are done in the Hall cross. First, the Hall cross
is saturated in a certain magnetization and by passing a current through the Oersted
line (yellow strip in Fig. 8a) a domain wall is formed. By applying a magnetic field
(H p in Fig. 8b), the DW is driven into the Hall cross so it gets pinned in the Hall cross.
Here, one has to measure two different fields, one is the depinning field without a
current, i.e. H c
* , the second is the depinning field with a current applied (H dep ). The
difference between the two values will give the field equivalent to the effective field by
the current. In systems where SOTs are present, the effective field due to the current