Introduction

The control of magnetization switching in magnetic structures by an electric current is crucial for the development of spintronics devices1. An established approach is via spin-transfer torque (STT) which involves the transfer of angular momentum from spin-polarized current to the magnetization2. Lately, current-induced spin-orbit torque (SOT) has emerged as an efficient alternative to STT3,4. The SOT is observed in magnetic multilayer structures composed of a ferromagnetic (FM) layer sandwiched by two heavy metal (HM) layers5,6,7. In this structure, the conduction electrons of FM and HMs exhibit strong spin-orbit coupling which leads to two well-known phenomena, i.e., the Rashba effect and the spin Hall effect (SHE)3,4. Due to the Rashba effect, spins accumulate in the FM layer, which exerts both a damping-like and field-like torque on the magnetization of FM layer8. Meanwhile, due to SHE, polarized spins are induced to accumulate at the FM/HM interface and diffuse into FM layer, which gives rise to an STT effect on the magnetization9. The torque from the Rashba effect and SHE is namely the SOT comprising of a field-like torque and damping-like torque. Generally, the two torques are represented by the two corresponding effective fields, field-like term HF and damping-like term HD3,4,5,6,7,9,10.

The dependence of the two effective fields on the orientation of the magnetization has been studied, especially in materials with perpendicular magnetic anisotropy (PMA)11,12,13,14,15,16,17. Reported SOT measurements on Ta/CoFeB/MgO structures have shown that the damping-like term HD changed the direction when the magnetization is reversed12. Recently, the dependence of the field-like term HF which has been considered as a constant on the polar angle of magnetization was observed experimentally in films with PMA14,15. The SOT effective fields also depend on the thicknesses of the FM and HM layers in Ta/CoFeB/MgO structure18. The increase of both field-like and damping-like terms with respect to the thickness of Ta has been reported, which is due to a more significant amount of current in a thicker Ta layer. For the dependence on the thickness of the FM layer, the field-like term decreases with increasing CoFeB thickness, while the damping-like term remains constant. Such dependence was ascribed to the giant magnetoresistance effect18. In the investigations of the referred dependences, sweeping magnetic fields were used. However, the sweeping behavior, i.e., magnetizing process, gives rise to magnetization variation or magnetization non-uniformity in magnetic devices. Hence, to precisely characterize the dependencies, the relationship between the SOT effective fields and the magnetic uniformity is required. Specifically, this relationship in a magnetic system with in-plane magnetic anisotropy (IMA) requires investigation, as the SOT devices based on IMA show promising application in spintronics19.

Here, we demonstrate the dependence of the SOT effective fields on the magnetization uniformity in wires consisting of Ta/Co/Pt layers with IMA. The impact of HM and FM layers thickness dependence of SOT is eliminated. Varying the uniformity and characterizing the SOT effective fields were achieved concurrently in each wire by applying a magnetic field along the long axis of the wire. Experimental results show that the field-like term decreases with respect to the magnetization uniformity, whereas the damping-like term increases. It is proposed that the magnetization uniformity increase leads to an increase of electron diffusion constant to decrease the field-like term and increase damping-like term.

Experiments and Discussion

Harmonic Hall resistance measurement technique, which has been reported earlier20, were employed to characterize the SOT effective fields. In this technique, a constant field, Hx−ext, is applied longitudinally to the wire long axis to ensure a constant magnetization uniformity, while a transverse field to the wire, Hy−ext, is swept in the plane to obtain the SOT effective fields accurately. The transverse field changes the magnetization azimuthal angle φ0. With Hx−ext being constant, the cosine of the angle, X, which can be simultaneously calculated by \(X=\,\cos \,{\phi }_{0}={H}_{x-{\rm{ext}}}/\sqrt{{H}_{x-{\rm{ext}}}^{2}+{H}_{y-{\rm{ext}}}^{2}}\). X is used to apply in the expression of the second harmonic Hall resistance R2ndHall which is:

$${R}_{{\rm{2nd}}{\rm{Hall}}}=\frac{{R}_{{\rm{AHE}}}}{2{H}_{\perp }}{H}_{{\rm{D}}}X+\frac{{R}_{{\rm{PHE}}}}{{H}_{x-{\rm{ext}}}}{H}_{{\rm{F}}}(2{X}^{4}-{X}^{2}),$$
(1)

where HD and HF are the damping-like term and field-like term, respectively, RAHE and RPHE are the amplitudes of anomalous Hall effect and planar Hall effect resistances, and H⊥ is the effective field orientating the magnetization in the film plane. By fitting the experimental second harmonic Hall resistance with Eq. (1), the SOT effective fields can be extracted. In Eq. (1), the parameter RPHE/Hx−ext can be obtained from the first harmonic Hall resistance R1st Hall, which is expressed as \({R}_{1stHall}={R}_{PHE}\,\sin \,2{\phi }_{0}\). Due to the expression of \(\cos \,{\phi }_{0}\), the maximum and minimum values of R1st Hall occur at Hy−ext = ±Hx−ext, which give RPHE.

The measurements were carried out in the wires with stacks of Ta(t nm)/Co(2 nm)/Pt(5 nm), where t = 4, 6, 8 and 1020. The fabrication and patterning processes of the wires are described elsewhere17,20,21,22. The SOT fields are quantified as a function of the longitudinal fields in the sample of Ta(4 nm)/Co(2 nm)/Pt(5 nm)20. In this quantification, the constant longitudinal field Hx−ext was applied in a range of 250 Oe to 650 Oe with a 50 Oe increment. For each value of Hx−ext, the ratio of the maximum value of the sweeping field Hy−ext to Hx−ext was fixed. An AC frequency of 307.1 Hz was used for the low-frequency harmonic Hall resistance measurements. The amplitudes of the AC current were in the range of 3 × 1010~10 × 1010 Am−2 with an increment of 1010 Am−2. The first and second harmonic Hall resistances were measured using a 7265 DSP lock-in amplifier. The obtained harmonic Hall resistances at applied fields Hx−ext = 250 Oe, 450 Oe, and 650 Oe and current density of 1 × 1011 Am−2 are shown in Fig. 1.

Figure 1
figure 1

(a) The measured first harmonic resistances R1st Hall with respect to the azimuthal angle φ0 of magnetization. Inset is the measured R1st Hall with respect to the applied transverse field Hy−ext. (b) The measured RPHE and the calculated M, with respect to the longitudinal field. RPHE is obtained by fitting the sin2φ0 curves of (a). (c) The measured second harmonic Hall resistances R2ndHall with respect to the cosine X of the azimuthal angle. Inset is the measured R2ndHall with respect to applied transverse field Hy−ext.

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The measured first harmonic Hall resistances R1st Hall exhibit typical \(\sin \,2{\phi }_{0}\) behaviors as functions of the azimuthal angle φ0 of the magnetization, and the minimum and maximum values of R1st Hall are at φ0 = ±45 degrees. Correspondingly, for each value of Hx−ext shown in the inset of Fig. 1(a), the minimum and maximum values of R1st Hall occur at Hx−ext = ±Hy−ext, which give the values of RPHE and the ratio of RPHE/Hx−ext shown in Fig. 1(b). In Fig. 1(c), the measured second harmonic Hall resistances, R2nd Hall, are shown to increase with increasing X for each value of Hx−ext. Fitting the experimental R2nd Hall by Eq. (1), where the values of RPHE and RPHE/Hx−ext are recorded in Fig. 1(b), we compute the two effective SOT fields, HF and HD.

As shown in Fig. 2, for each value of Hx−ext, HF and HD increase with respect to the current density at each value of Hx−ext. The values of HF and HD are similar to that reported in the same stack20. We notice that HF and HD vary with the longitudinal field for each value of the applied current densities. However, at the current density of 3 × 1010 Am─2, the field-like term variation is 19.7%, and the damping-like term variation is 16.9% when Hx−ext increases from 250 Oe to 650 Oe. At the current density of 1011 Am─2, the field-like term variation increases to 21.5% and the damping-like term variation decrease to 12.4%. The variations are not only the functions of the magnetization uniformity but also the applied current. The role of the current in the variation indicates that possible thermal effect, such as the anomalous Nernst effect, should exist during the SOT measurement23,24,25,26. In the Ta/Co/Pt stack where our measurements were carried out, there are differences in the resistivity of Ta and Pt. Due to Joule heating which is induced by electric current, the different resistances give rise to a thermal gradient along the normal direction of the stack. The thermal gradient contributes to the measured second harmonic Hall resistances which were used to calculate the SOT effective fields13. Depending on the direction of the thermal gradient; the thermal contribution may lead to an over or underestimation of the SOT effective field when the current density increases. However, in our measurement regime, the thermal effects are considered negligible because the current density in our experiments was in the range of ~107 A/m2, where Joule heating is generally negligible5. Furthermore, both the damping like term and field like term relates monotonically with the applied current density as shown in Fig. 2, implying that the thermal effects insignificantly to the SOT effective field27,28.

Figure 2
figure 2

The measured field-like term (a), and damping-like term (b), with respect to the longitudinal field.

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The magnetization uniformity is equivalent to magnetization amplitude M for the magnetic wire. This magnetization amplitude can be characterized with respect to the applied longitudinal field by measuring R1st Hall, thereby obtaining the RPHE. As shown in Fig. 1(b), RPHE increases with respect to Hx−ext, where Hx−ext is extended to 100 Oe. This increase is attributed to the polycrystalline structure of the Co layer in the sputtered Ta/Co/Pt film. Without applying Hx−ext, the magnetic moment of Co crystalline grains orientates randomly in the film. This is due to the random orientation of the effective field Hcrys generated by the crystalline magnetic anisotropy of each grain. Hence, the M of the wire equals the value of remanence magnetization Mr, which is determined by the intrinsic demagnetizing field transverse to the wire.

However, when Hx−ext is applied, the magnetic moment m of the grains re-orientates towards the x-axis, as schematically shown in Fig. 3(a). Consequently, M increases starting from Mr. The increase of M leads to the increase of RPHE, as M is related to RPHE by the expression RPHE = kM2 29,30,31, where k is a material related coefficient. The maximum of M is the saturation magnetization Ms. Therefore, MH, defined as MH = Ms−Mr, is the maximum of the magnetization component, which can be manipulated by the external field Hx−ext. We consider the magnetization component as the resultant of two vectors expressed by MH/2, instead of evaluating the contribution from each magnetic grain to the magnetization component. As shown in Fig. 3(b), Hx−ext orientates both vectors along the x-axis, while the nonzero y component HT,crys of Hcrys orientates each vector along the ±y direction, respectively. Hence, each MH/2 orientates at their balanced direction determined by Hx−ext and HT,crys, as shown in Fig. 3(b). The y components of the two vectors cancel each other, while the x component of each vector contributes to the magnetization component as \(({H}_{x-ext}/\sqrt{{H}_{x-ext}^{2}+{H}_{T,crys}^{2}})({M}_{H}/2)\). Consequently, the expression of the magnetization component is obtained as \(({H}_{x-ext}/\sqrt{{H}_{x-ext}^{2}+{H}_{T,crys}^{2}})({M}_{H}/2)\). Therefore, the total magnetization M can be expressed as \(M={M}_{R}+({H}_{x-ext}/\sqrt{{H}_{x-ext}^{2}+{H}_{T,crys}^{2}}){M}_{H}\), and Mr + MH equals to the saturation magnetization Ms of the wires. Considering Mr + MH = 466 emu/cc for the sample of Ta (4 nm)/Co (2 nm)/Pt (5 nm)20, and substituting the M expression into the RPHE expression, we fit the measured RPHE as shown in Fig. 1(b). The fitting plot of RPHE matches the experimental RPHE, which verifies the analytical expression of M and gives \(M=405+({H}_{x-ext}/\sqrt{{H}_{x-ext}^{2}+{165}^{2}})61\) emu/cc.

Figure 3
figure 3

(a) The schematic of a polycrystalline magnetic structure and the orientation of magnetic moment for each of crystalline grain under Hx−ext and transverse demagnetizing field. (b) The schematic of the magnetization composition for the magnetic structure.

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The SOT effective fields per 1011 Am−2 at each value of Hx−ext were obtained from Fig. 2, for comparison. Replacing Hx−ext with the corresponding value of M shown in Fig. 1(b), the SOT effective fields per 1011 Am−2 with respect to M are plotted for sample Ta (4 nm)/Co (2 nm)/Pt (5 nm) in Fig. 4. Similarly, the SOT effective fields per 1011 Am−2 and the magnetization were quantified for samples Ta (t nm)/Co (2 nm)/Pt (5 nm), where t = 6, 8 and 10. As reported previously20, similar saturation magnetization values for the samples tTa = 4 nm and tTa = 8 nm leads to similar tendencies of the field like and damping like fields due to their dependence on magnetization. While substantial differences in the saturation magnetization values for samples tTa = 6 nm and tTa = 8 nm leads to different tendencies of the field-like and damping like SOT fields. As shown in Fig. 4(a,b), the field-like term decreases with respect to the magnetization magnitude while the damping-like term increases in each sample.

Figure 4
figure 4

The measured field-like term per current density (a), and damping-like term per current density (b), with respect to the calculated magnetization M for Ta(t nm)/Co(2 nm)/Pt(5 nm), where t = 4, 6, 8 and 10. (c) The plots of sinξe−ξ and 1−cosξe−ξ with respect to ξ.

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We demonstrate that the dependence of the SOT effective fields on the magnetization magnitude or uniformity is attributed to SHE in the Ta/Co/Pt structure. SHE-induced spin accumulation, s, which is at the interfaces of Ta/Co and Co/Pt, diffuses into the FM layer to cause STT on the magnetization9. In the STT model proposed by S. Zhang32, the spin current, which is from a reference layer, leads to transverse spin accumulation in the free layer. Consequently, the transverse spin accumulation induces two effective fields: bmr and amf × mr, where mr and mf are unit vectors of the local magnetization of the reference layer and the free layer, respectively. When mr and mf are in the planes of the magnetic layers, b and a are expressed as \(b=(h{j}_{e}/e{M}_{s}{t}_{F})\sin \,\xi .{e}^{-\xi }\) and \(a=(h{j}_{e}/e{M}_{s}{t}_{F})(1-\,\cos \,\xi .{e}^{-\xi })\), respectively, where h is the Planck constant, je is the electric current density perpendicular to the plane of magnetic layers, tF is the thickness of the free layer, and e is the electron charge. In the expressions of a and b, ξ equals to \({t}_{F}/\sqrt{2}\lambda \) with a spin diffusion length \({\lambda }_{f}=\sqrt{2h{D}_{0}/J}\), where J is a coefficient of the contact interaction between the spin accumulation and the local magnetization of the free layer, and D0 is the electron diffusion constant. Analogously in the Ta/Co/Pt structure, the Ta or Pt layer is used to generate spin current normal to the magnetic Co layer. Hence, the Ta or Pt layer is similar to the reference layer, as such, s can be considered as mr. The spins generated by the Ta and Pt layers are accumulated at the Co layer, which allows us to take the Co layer as analogous to the free layer. Similarly, m is to mf. Consequently, the field-like term, HF is equivalent to bmr, and the damping-like term, HD is equivalent to amf × mr. Thus, we obtain \({H}_{F}=(h{j}_{e}/e{M}_{s}{t}_{f})(\sin \,\xi .{e}^{-\xi })\) and \({H}_{D}=(h{J}_{e}/e{M}_{s}{t}_{F})(1-\,\cos \,\xi .\,{e}^{-\xi })\) for the Ta/Co/Pt samples, where Je is the charge current.

We propose that the magnetization magnitude of Co layer manipulate the SOT effective fields via the coefficient ξ, considering the above expressions of HF and HD. The damping-like term is related to spin Hall angle θSH via the expression HD = θSHhJe/eMstF, where θSH is defined as the ratio of spin current js to charge current je19,33,34,35. Comparing the two expressions of HD, we obtain \({\theta }_{SH}=1-\,\cos \,\xi {e}^{-\xi }\) As such, ξ is ≤ 1.6, since the sum of Pt and Ta spin Hall angles is ≤17,36. As \(\xi ={t}_{F}/(\sqrt{2}{\lambda }_{J})\) and λJ is about 1.2~2.4 nm for Co25, we obtain ξ ≥ 0.6, using the Co layer thickness tF = 2 nm. Therefore, our samples have values of 0.6 ≤ ξ ≤ 1.6. ξ can be rewritten as \(\xi =({t}_{F}/2)\sqrt{J/h{D}_{0}}\), where D0 is related to the magnetization of the wire30,37. Ustinov created a superlattice model to explain the correlation of D0-related magnetoresistance (MR) and magnetization37. In this model, the superlattice comprises of several magnetic layers, and for any of two adjacent layers, magnetizations are initially antiparallel to each other. A transverse magnetic field, which is perpendicular to the initial magnetization in the plane of the magnetic layers, is applied to change the magnetization amplitude of the superlattice. The model concludes that the MR increases with respect to the magnetization of the superlattice. Hence, D0 decreases with increasing magnetization in our samples, as it is inverse proportional to MR. Therefore, ξ increases with respect to the magnetization magnitude, due to \(\xi =({t}_{F}/2)(/)(\sqrt{J/h{D}_{0}})\). In the range of 0.6~1.6 for our samples, the increase of ξ leads to the decrease of the term \(\sin \,\xi {e}^{-\xi }\) and increase of the term 1−cosξe−ξ, as shown in Fig. 4(c). Therefore, HF decreases and HD increases with respect to the magnetization, respectively, as the term hje/eMstF is a constant for each sample.

Conclusion

In conclusion, our measurement results show that the SOT effective fields depend on the magnetization uniformity in Ta/Co/Pt structure. The dependence indicates that the SOT effective fields can be manipulated by varying the magnetization uniformity. The change of magnetization uniformity was achieved in each sample by applying magnetic fields along the long axis of the wire. As the SOT effective fields are concurrently characterized, our characterization method eliminates influences from other SOT dependence effects. As an analogy to the STT effect from a reference layer, the SOT dependence on the magnetization uniformity is attributed to the electron diffusion properties. This dependence suggests that SHE plays a significant role in the dependence of SOT effective fields on magnetization. It also indicates that the SOT effective fields cannot be considered as constant parameters when analyzing domain wall dynamics via SOT. Moreover, we conclude that magnetization enhances the damping-like torque while suppressing the field-like torque.