We use calibration magnetic fields to quantitatively determine the strength of the SOTs from the measured MOKE signals. For the DL-SOT calculation, we use an out-of-plane calibration field, while an in-plane calibration field is utilized for extracting the FL-SOT.
The out-of plane calibration field is generated by a specially designed calibration wire, shown in Fig. 3.5(a). Given all the dimensions of the calibration wire, the generated cal- ibration field hCal due to a certain calibration current can be calculated. Recalling the
magnetization reorientation expression in Eq. 3.2, we can write down the MOKE signal due to the out-of-plane magnetization tilt caused by hCal as
Figure 3.5: (a) Diagram of the designed calibration wire for out-of-plane calibration field. Calibration current is applied from the left to the right. (b) Generated out-of-plane Oersted field distribution for a calibration current of 400 mA. The 50 µm × 50 µm area is indicated by the red dot in (a). The values on the legend bar are in Oe.
∆VCal = αPolar∆θM =
hCal
|Hex| + Meff
. (3.8)
Since the DL-SOT effective field, hDL ∝ ~m × ~σ, linearly depends on the magnetization, it
changes sign when the external magnetic field Hex reverses. Given hz = hDL+ hOerstedOut and
the Oersted field is independent of Hex, the hDL-caused magnetization tilt can be calculated
as: ∆θhDL =
∆θhz(+Hex)−∆θhz(−Hex)
2 . Therefore, the MOKE signal generated by the DL-SOT
can be expressed as ∆VDL= αPolar∆θhDL = ∆V (+Hex) − ∆V (−Hex) 2 = hDL |Hex| + Meff . (3.9)
From Eqs. 3.8 and 3.9, we derive the final equation to calculate the effective magnetic field due to the DL-SOT as
hDL=
∆VDL
∆VCal
Figure 3.6: Illustration of the DL-SOT calibration process. (a) The hDL-induced MOKE
signal. The signal has a step-like shape because of the change of sign of hDLon magnetization
reversal, as described by Eq. 3.9. (b) the MOKE signal due to the calibration field. The symmetric tilted shape is described by Eq. 3.8. (c) The signals used to extract hDL via a
line-scan method, for comparison.
We verified the accuracy of this method by comparing its result with the result of a line-scan method [81]. The line-scan method utilizes the different symmetries of hDL and
hOerstedOut. By measuring the overall MOKE signal across the sample width under positive
and negative external magnetic fields, signals due to hDLand hOerstedOutare extracted through
simple addition and subtraction. Since the current-induced Oersted field can be readily calculated with Ampere’s law, we can calibrate the amplitude of hDL through fittings of
the SOT and Oersted curves, shown in Fig. 3.6(c). Although the line-scan method is self- calibrated, which means no extra calibration field is needed, it is also more time-consuming. Therefore, we replace the line-scan method in our experiments. A sample with the structure,
substrate/Py(20)/Pt(3), was used in the comparison experiment, where the values in the parenthesis are in nanometers. Fig. 3.6(a) and (b) show the polar MOKE signal of the DL-SOT measurement and the calibration measurement, respectively. The MOKE signal due to the DL-SOT, ∆VDL ≈ −30 µV, is extracted from Fig. 3.6(a) by taking half the
difference between the positive field and the negative field signals. And, the calibration signal, ∆VCal ≈ −220 µV, is the zero-field reading in Fig. 3.6(b). Since the calibration
current is 400 mA, which corresponds to a calibration field hCal = 9.6 Oe, the DL-SOT-
induced effective field can then be computed using Eq. 3.10 to be hDL= 1.3 ± 0.2 Oe. This
result is in good agreement with the self-calibrated line-scan result, hDL = 1.1 ± 0.2 Oe, that
is extracted from Fig. 3.6(c).
Applying the same methodology, we use an in-plane calibration field for the FL-SOT measurements. The in-plane field is generated by a 2 mm wide straight wire on a PCB board, that we attach underneath the sample with the wire parallel with the current direction (x- direction). The wide width of the wire ensures a relatively uniform in-plane (y-direction) Oersted field across the sample area. Since both the FL-SOT effective field hFL and in-plane
calibration field hCalIn rotate the magnetization in the xy-plane, from Eqs. 3.1 and 3.7, we
can write the quadratic MOKE signal due to hFL and hCalIn as
∆VFL = βQuadratic∆φhFL =
hFL+ hOerstedIn
Hex
, (3.11)
∆VhCalIn = βQuadratic∆φhCalIn =
hCalIn
Hex
, (3.12)
where hOerstedIn is the Oersted field generated by the current that generates the SOTs.
As shown in Fig. 3.7(a) and (b), both the FL-SOT and calibration quadratic MOKE signals yield nice 1/Hex dependence as predicted by Eqs. 3.11 and 3.12. By replotting the
FL-SOT result as a function of the calibration values and performing a linear regression fitting (Fig. 3.7(c)), we extract a slope value, which is equal to the ratio of hFL+hOerstedIn
Since hCalInand hOerstedIn can be calculated with Ampere’s law, the strength of the FL-SOT
effective field can be quantitatively determined as
hFL = slope × hCalIn− hOerstedIn. (3.13)
Figure 3.7: Illustration of the FL-SOT calibration process. (a) Quadratic MOKE signal of the FL-SOT. The signal includes contributions from the current-induced FL-SOT and the in-plane Oersted field generated by the same current. (b) The quadratic MOKE signal of the in-plane calibration field. Both (a) and (b) follow a clear 1/Hex dependence. (c) Plot of the
FL-SOT MOKE signal as a function of the calibration-field-induced signal. The slope of the linear fit can be used to extract the amplitude of the effective field of FL-SOT. The negative sign before the field value indicates the direction of the hFL is opposite to the calibration