Spin–orbit torques in normal metal/Nb/ferromagnet heterostructures

Quantifying the spin–orbit torque (SOT) efficiency with changing the layer thickness is crucial for understanding the physical background of SOT. This study investigates the Nb-thickness-dependent SOT efficiency of two types of layered heterostructures: Ta/Nb/CoFeB and Pt/Nb/CoFeB. We find that the Nb thickness dependence of the SOT efficiency in the two samples is quite different. In the Pt/Nb series, the SOT sign changes according to the thickness variation because Pt and Nb have different spin–orbit coupling signs. We observe the resulting reversal in switching polarity through current-induced SOT switching experiments. However, due to the same spin–orbit coupling signs of Ta and Nb, no such polarity reversal was observed in Ta/Nb series. Further, we extract the spin diffusion length of Nb in each heterostructure. These results provide a systematic understanding of the material- and thickness-dependent SOT characteristics.


Supplementary Note 2. The 1 st and 2 nd harmonic curves for Nb based SOT structure
We measured all PMA films for their SOT efficiency using harmonics measurement. We induced the external magnetic field varying up to 18 kOe and used a 1-3 mA alternating current with a fixed frequency of 13.7 Hz.  and 2 nd harmonic (f) signals for Pt (3)/Nb ( )/CoFeB 1.1/MgO 1/Ta 2 films.

Supplementary Note 3. SOT measurement of the films with in-plane magnetic anisotropy
We patterned films into a Hall bar structure with a 5 μm width and 35 μm length using the photolithography and lift-off method to measure the spin Hall properties. The electrode consists of Ti (10)/Au (100) (in nm) was evaporated by e-beam. The Hall bars were wire-bonded and placed on the stage with motors that rotate the device in both angles; polar (θ) and azimuthal (φ) angles. The external magnetic field induced to the device varies up to 18 kOe. We used 1-3 mA of amplitude a.c current during the measurement, and its frequency was fixed as 13.7 Hz.
The magnetization of the films stays at an equilibrium position just before the inducing current. When current flows through the device, the magnetization starts to oscillate due to the effective field exerted by the spin current. We can obtain the spin Hall properties that reflect this oscillation from the θ and φ angle dependence between the measured Hall resistance and the magnetization direction. The first ( 1 ) and second-harmonic Hall resistance ( 2 ) is expressed as follows: 1 = AHE cos + PHE sin 2 sin 2 .
The second-harmonic Hall resistance ( 2 ) contains the degree of the magnetization oscillations in terms of the total effective field = + + . represents a SOT induced damping-like field, is a field-like field, and is an electrical current induced-Oersted field. The Hall resistance occurred from a thermo-electric process expressed as .
In specific measurement geometry, = /2, we can get the simplified expression of secondharmonic Hall resistance.
where and are the effective anisotropy field and the applied external field, respectively S1, S2 . When an applied external field is large enough to regard the direction of magnetization as parallel to that of , we can assume the contribution of thermo-electric Hall resistance is constant. Also, we can vanish the contribution of the + by rotating azimuthal angle into = /4. In this case, we can separate the effect of with other effective fields. By measuring a relation between Hall resistance and azimuthal angle, the effect of + also could be obtained.

Supplementary Note 4. The resistivity of Nb films in (NM)/Nb/CoFeB/MgO structures
We measured in three systems: (ⅰ) Nb/CoFeB/MgO/Ta, (ii) Ta/Nb/CoFeB/MgO/Ta, and (iii) Pt/CoFeB/MgO/Ta. We fabricated two reference films consisting of Ta (3)  proportional to each other. However, such a relationship between these parameters was not observed in the Pt/Nb series; this means that the Nb films above the Pt layer are of low quality.

Supplementary Note 5. Drift-diffusion model in the FM/NM1/NM2 structure
Here, we derived the analytical expression of the SOTs from the spin Hall effects in the NM1 and NM2 layers. A schematic of the NM1/FM and NM2/NM1/FM structures is provided in Fig. S4.
From the spin current at the NM1/FM interface, we obtain the spin-orbit torque on the FM layer: The expressions for the SOT efficiencies are described as: Therefore, we have Note that is the charge current density in the NM1/NM2 layer, which is obtained by the parallel circuit model

Supplementary Note 6. Spin diffusion length considering spin current generation at the normal metal/ferromagnet interface
We fitted the data using an extended spin diffusion model containing an interfacial source of spin current. We assumed that spin current is generated only at Nb/CoFeB interface for simplicity. Therefore, two conductivity parameters corresponding with spin-filtering and spinprecession were additionally considered for the Nb/CoFeB boundary conditions modified by interface generated spin current 27 (see Supplementary Note 5). , = 3.64 ± 0.72 nm was obtained for the Nb/CoFeB bilayer system, as shown in Fig. S5a. For reducing the ambiguity of the tri-layer system fitting, we assumed that the parameters describing interfacial conditions were the same as those of the Nb/CoFeB bilayer system. Fig. S5b and S5c showed that the extracted values were , = 8.61 ± 1.83 and , = 4.48 ± 1.00 nm in Ta/Nb and Pt/Nb series, respectively. , extracted by using an extended model slightly increased compared to those when the conventional bulk model was applied, but the enhancement was negligible in all cases.

Supplementary Note 7. NM 1/NM 2 interface analysis
To characterize the structural properties of these systems, we deposited an NM/Nb bi-layer and then conducted a TEM analysis. Fig. S6a and S6b show the side-views of Ta 3/Nb 5 nm and Pt 3/Nb 5 nm, respectively. Note that annealing at 300 °C for 1 h was conducted for both samples. The situations at the NM layer and the NM/Nb interface were completely different.
The 3 nm Ta layer maintains an amorphous structure, despite heat treatment. It is well-known that films deposited over the amorphous buffer layer show well-defined interface quality and crystallinity S8-S10 , as demonstrated by Fig. S6a. When very thin layers of Pt were grown on a substrate lacking a buffer layer, Pt exhibited an amorphous structure. However, in the Pt/Nb structure, the Pt layer after annealing at 300 °C featured polycrystallinity, as shown in Fig. S6b.
Besides, we observed a low-quality interface that appears to have formed during the crystallization process. To verify this in duplicate, a TEM energy dispersive spectroscopy (EDS) analysis was also conducted. Each inset of Fig. S6a and S6b depicts the atomic ratio between NM and Nb for the regions represented as Regions 1 and 2, respectively. Nb atoms were almost absent in the Ta region (~0.40%), whilst some Nb atoms diffused into the Pt region (~17.8%) for Pt/Nb. The atoms diffused into the NM layer, and the resulting interfaces inevitably affected the transport of current through scattering; the drift-diffusion model does not consider this.

Supplementary Note 8. Surface roughness of Nb thin films on different underlayers
We measured the roughness of Nb thin films using atomic force microscopy (AFM) to confirm the effect of the underlayer type on the interfacial roughness of Nb films. We scanned 5 × 5 areas of each sample, and the root-mean-square (RMS) of the area was indicated as a data point. We indicated the error range using the standard deviation. The overall trend of roughness increases as increases regardless of the type of underlayers, as shown in Fig. S7.
The Nb thin films grown on Ta underlayer were less rough than the other series grown on SiO2 and Pt underlayer. The films grown on SiO2 and Pt showed higher roughness than the Ta series but a similar roughness level. Although the overall tendency was similar in all series, there were the difference in Nb roughness between Ta/Nb and Pt/Nb series that may induce different transport behavior at the Nb/CoFeB interface.