Deterministic Spin-Orbit Torque Induced Magnetization Reversal In Pt/[Co/Ni]n/Co/Ta Multilayer Hall Bars

Spin-orbit torque (SOT) induced by electric current has attracted extensive attention as an efficient method of controlling the magnetization in nanomagnetic structures. SOT-induced magnetization reversal is usually achieved with the aid of an in-plane bias magnetic field. In this paper, we show that by selecting a film stack with weak out-of-plane magnetic anisotropy, field-free SOT-induced switching can be achieved in micron sized multilayers. Using direct current, deterministic bipolar magnetization reversal is obtained in Pt/[Co/Ni]2/Co/Ta structures. Kerr imaging reveals that the SOT-induced magnetization switching process is completed via the nucleation of reverse domain and propagation of domain wall in the system.


S1. Harmonic Hall voltage measurement technique
For the purpose of characterizing the magnitude and direction of the effective fields induced by SOT, the illustration of the harmonic Hall voltage measurement scheme for a ferromagnetic heterostructure system with perpendicular magnetic anisotropy (PMA) is shown in Fig. S1. The magnetic energy density of the system can be expressed as [1][2][3] • , where the effective perpendicular anisotropy energy is . is the uniaxial perpendicular magnetic anisotropy energy and is a positive value for out of plane magnetic easy axis (z-axis). Ms is the saturation magnetization, and the in-plane anisotropy energy is from the shape anisotropy of the nanowire. In this system, the demagnetization factors along basic directions satisfy ≫ due to the geometry of the wire. The in-plane easy axis in our system is along the x-axis (the wire direction).
Hence, we neglect and , and set 1 for the convenience of computation. Figure S1. The schematic illustration of the coordinate system. Ms and H represent the magnetization and the external magnetic field respectively. and are the polar and azimuthal angles of the magnetization respectively. and are the polar and azimuthal angles of the external magnetic field respectively.
Using the extraordinary Hall effect (EHE), the Hall voltage Vxy is a product of the Hall resistance Rxy and the applied current Ix, expressed as 4 Considering an AC current applied, the polar and azimuthal angles of the magnetization will oscillate as , expressed as ∆ and ∆ . ∆ and ∆ are the modulation amplitudes of the magnetization angle from its equilibrium position ( , ). The Hall voltage Vxy can be expressed in harmonic form as Where, and are the first and second harmonic voltage, respectively.
For our system, the initial magnetization M lies perpendicular to the film plane due to a high Keff. As we apply an external out-of-plane magnetic field along the z direction, we obtain ∆ .
The ± sign corresponds to the case for up (Ms along +z direction) and down (Ms along -z direction) magnetization states, respectively.
For the case of an external in-plane magnetic field, , both the first and second harmonic Hall voltage can be obtained by , , , , Here, is defined by . Hx and Hy represent the longitudinal and transverse external magnetic field, respectively. The ± sign corresponds to the case for up and down magnetization states, respectively. where the perpendicular external magnetic field Hz sweeping field from ~-750 Oe to ~+750 Oe was applied. The change of AHE is about 0.9 mV. ∆ is calculated to be ∆ . 0.18 , according to eq. (4). Figure S2 Table S1, which shows the negative correlation between the effective fields and the Co/Ni layer numbers.

S5. AHE measurements using DC bias current for n = 2 structures
For all the AC current measurements, the frequency was set to 333 Hz while the AC current values refer to the amplitude. Given that the AHE measurements were conducted for field induced reversal, the AC current effect is only to aid in the reversal process, although the AC current induces an oscillating SOT within the system. AHE measurements have also been performed using DC bias current on n = 2 structures.
Both positive (along +x orientation) and negative (along -x orientation) current were used.
The measured AHE loops by various current values are shown in Figs. S4(a) and (b), -8 -respectively. The switching field is decreased as current value increases. The coercivity values extracted from corresponding AHE loops including DC and AC measurements were plotted in the Fig. S4(c). In the same method, the value of coercivity Hc is given by , where Hc+ and Hc-are positive and negative switching fields, respectively.
The positive and negative DC bias currents lead to different coercivity trends for current smaller than 3mA, as seen in Fig. S4(c). This change in coercivity cannot be explained by just Joule heating effect, which is dependent on the magnitude of the current but independent of the current directions.
Interestingly, for small current, irrespective of the DC current direction, the obtained coercivity is much lower that for AC current. This is consistent with the lower RMS value (IAC/2) of the AC current. However, for current larger that 3mA, we note that the current induced magnetization reversal dominates irrespective of the type of current bias used.   Néel DW chirality. This is consistent with Pt being the dominant source of DMI.

S7. Joule heating effect on the coercivity trend
To assess the Joule heating effect, the resistance of the wire was monitored by a DC current flow for 10 minutes. Figure S6(a) shows the measured wire resistance for n = 2 structures by 1mA and 3 mA, respectively. Though the variation of resistance at a fixed DC bias is ~0.1 Ω, a change of ~1 Ω was observed in the resistance of n = 2 sample as the current is increased from 1 mA to 3 mA, where the coercivity is tuned by current. Therefore, the Joule heating assists on the reversal process. No similar change was observed for the n = 4 sample, as shown in Fig. S6(b). As such, the coercivity trend observed for n = 2 structures can be attributed to a combination of joule heating and SOT-induced switching. For the n = 4 sample, no change in the wire resistance was observed as the DC bias was increased, as shown in Fig. S6(b). As such, the coercivity trend observed for n = 2 structures can be attributed to a combination of joule heating and SOT-induced switching.

S8. Effect of Oersted field on reversed domain nucleation
To show that the edge-defects do not play a role in the switching, the reversed process was repeated for a +z and z initial magnetization. The reversed domains are nucleated at opposite current directions for +z and z initial magnetizations, as shown in Figs. S7(a) and (b). Figure S7 shows that though the reversal occours from the edge, in both configurations, they occur at different locations, which is close to the current applied, implying that the defects do not play a role in the reversal domain nucleation. Figure S7. The reversal occurs from the edge for initial magnetization along (a) +z and (b)z directions. The two configurations occurs at opposite current directions.