Field-free spin-orbit torque-induced switching of perpendicular magnetization in a ferrimagnetic layer with vertical composition gradient
Zhenyi Zheng, Yue Zhang, Victor Lopez-Dominguez, Luis Sánchez-Tejerina, Jiacheng Shi, Xueqiang Feng, Lei Chen, Zilu Wang, Zhizhong Zhang, Kun Zhang, Bin Hong, Yong Xu, Youguang Zhang, Mario Carpentieri, Albert Fert, Giovanni Finocchio, Weisheng Zhao, Pedram Khalili Amiri
FField-free spin-orbit torque-induced switching of perpendicular magnetization in a ferrimagnetic layer with vertical composition gradient
Zhenyi Zheng , Yue Zhang , Victor Lopez-Dominguez , Luis Sánchez-Tejerina , Jiacheng Shi , Xueqiang Feng , Lei Chen , Zilu Wang , Zhizhong Zhang , Kun Zhang , Bin Hong , Yong Xu , Youguang Zhang , Mario Carpentieri , Albert Fert , Giovanni Finocchio , Weisheng Zhao , Pedram Khalili Amiri Fert Beijing Research Institute, School of Integrated Circuit Science and Engineering, Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, P. R. China Department of Electrical and Computer Engineering, Northwestern University, Evanston, IL, USA School of Electronics and Information Engineering, Beihang University, Beijing, P. R. China Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Messina, Italy Dipartimento di Ingegneria Elettrica e dell’Informazione, Politecnico di Bari, Bari, Italy Unité Mixte de Physique, CNRS, Thales, Univ. Paris-Sud, University of Paris-Saclay, Palaiseau, France These authors contributed equally: Zhenyi Zheng, Yue Zhang. * e-mail: [email protected], [email protected], [email protected], [email protected]. bstract Current-induced spin-orbit torques (SOTs) are of interest for fast and energy-efficient manipulation of magnetic order in spintronic devices. To be deterministic, however, switching of perpendicularly magnetized materials by SOT requires a mechanism for in-plane symmetry breaking. Existing methods to do so involve the application of an in-plane bias magnetic field, or incorporation of in-plane structural asymmetry in the device, both of which can be difficult to implement in practical applications. Here, we report bias-field-free SOT switching in a single perpendicular CoTb layer with an engineered vertical composition gradient. The vertical structural inversion asymmetry induces strong intrinsic SOTs and a gradient-driven Dzyaloshinskii–Moriya interaction (g-DMI), which breaks the in-plane symmetry during the switching process. Micromagnetic simulations are in agreement with experimental results, and elucidate the role of g-DMI in the deterministic switching. This bias-field-free switching scheme for perpendicular ferrimagnets with g-DMI provides a strategy for efficient and compact SOT device design.
Introduction
Spin-orbit torque (SOT) is a leading contender as a fast and low-power method to manipulate magnetic order in spintronic devices, particularly for artificial intelligence and high-performance computing applications where high-speed on-chip memory is required [1-4]. SOT switching of ferrimagnetic (FIM) materials, in particular, is of great current interest [5-9]. FIMs exhibit the exchange-dominated high-frequency sub-terahertz) dynamics of antiferromagnets, which can result in high switching speed [5-7], while avoiding the difficulties of controlling domain size commonly encountered in antiferromagnets [4]. They can also utilize the relatively straightforward electrical readout methods available in ferromagnetic material systems (due to their non-zero magnetization) [7-9]. Hence, FIMs are promising as near-term candidates for practical high-performance SOT devices. Typically, for memory applications where information is encoded in the direction of the magnetization in magnets with perpendicular magnetic anisotropy (PMA), SOT switching is realized in a non-magnet/ferromagnet (NM/FM) or NM/FIM heterostructure with the assistance of an external in-plane magnetic field H ex along the current direction [1-2,10-12]. SOTs originating from the Spin Hall effect (SHE) in the NM (e.g. heavy metals [2,10] or topological insulators [11-12]) or from the Rashba effect [1,13] at the NM/FM interface can drive the adjacent magnetization to switch, while H ex breaks the in-plane inversion symmetry and leads to a deterministic switching direction for a particular direction of in-plane current. The requirement of external H ex in switching, however, hinders the integration of SOT devices on semiconductor chips. Thus, mechanisms to break the in-plane (structural or magnetic) symmetry, instead of H ex , are being investigated to realize practical SOT memory devices [13-28]. To date, these have included in-plane exchange bias fields [14-16], interlayer exchange coupling [17-18], in-plane structural, composition, or interfacial oxidation asymmetry [19-25], and combining multiple competing sources of spin torque in one device [26-28]. However, with the exception f methods combining more than one source of torque, these schemes all rely on breaking the static in-plane symmetry in the device. In many cases, this makes it difficult to build large arrays with uniform device properties on the same wafer. Here we experimentally show, instead, that a structure featuring in-plane symmetry, along with inversion asymmetry only along the growth direction, can also exhibit deterministic switching due to the dynamic in-plane breaking of symmetry during the switching process, induced by the Dzyaloshinskii–Moriya interaction (DMI). The DMI effect can induce spin textures (e.g. domain walls and skyrmions) with broken chiral symmetry [29-32]. Therefore, DMI has recently been theoretically proposed as a symmetry-breaking mechanism which, in combination with damping-like and field-like SOTs, may enable deterministic switching in the absence of an external field [33-35]. However, this type of field-free combined DMI-SOT switching has not been experimentally observed to date. Our approach to engineer the required DMI is inspired by a recent work reporting a large bulk DMI in a rare-earth (RE)-transition-metal (TM) ferrimagnet (FIM), where a vertical composition gradient was brought about by the natural inhomogeneous elemental composition distribution in alloys with large thickness [36]. Here, in order to maximize the DMI, we instead engineer a vertical composition gradient in the FIM layer by depositing ultrathin individual layers with a varying RE/TM composition ratio, leading to a broken inversion symmetry along the growth direction. This results in a gradient-driven DMI (g-DMI) which has the same symmetry as interfacial DMI n previously studied NM/FIM and NM/FM systems, which, however is present throughout the bulk of the FIM due to the continuous composition gradient. Additionally, as RE elements possess large spin-orbit coupling (SOC) due mainly to their 5d band [37-38], they can generate spin-currents as large as those generated in nonmagnetic 5d metals such as Pt at the opposite end of the 5d metal series [39]. The 3d electrons of cobalt possess a much smaller SOC and the f electrons of Tb do not participate in the conduction. The combination of large spin-orbit-coupling and broken inversion symmetry generates Rashba interactions [40], which give rise to current-induced Rashba spin polarizations and resulting spin currents [41] which, in turn, can produce an SOT on the FIM magnetization, as discussed in the next section. Using the combination of SOT and g-DMI effects, we demonstrate efficient field-free SOT magnetization switching in a single ferrimagnetic CoTb layer with a vertical composition gradient. Experiments and simulations indicate that this scheme can eliminate the need of NM and H ex simultaneously in the previously mentioned SOT switching systems. The field-free switching data are in agreement with micromagnetic simulations. Our work provides a pathway towards field-free SOT switching of perpendicular ferrimagnets which can potentially be scaled to large wafer size with good uniformity. Large spin-orbit torques originating from the Tb composition gradient
As a ferrimagnetic alloy, in which RE elements and TM elements are coupled anti-ferromagnetically, CoTb can exhibit robust PMA in a wide composition range 8-9]. To investigate the exact composition range of PMA, a series of Al (2 nm) / CoTb (6 nm) / Al (3 nm) films (see Fig. 1a) were deposited on thermally oxidized silicon substrates. Here, Al was chosen due to its negligible SOT. CoTb was constructed by a co-sputtering process of Co and Tb at different powers to control the stoichiometry of the final CoTb layer. As shown in Fig. 1b, Co Tb and Co Tb were the most Co-rich and the most Tb-rich samples which provided good PMA properties in this heterostructure. In addition, by fitting of the net saturation magnetization M s dependence on composition, Co Tb was determined as the magnetization compensation point at room temperature. Guided by this information, samples with a vertical composition gradient were deposited next. As illustrated in Fig. 1c, the ferrimagnet consists of six layers of CoTb, each having a thickness of 1 nm. Here, we fixed the bottom layer composition as Co Tb , while the other five deposited layers followed a certain Tb composition step δ, resulting in a nominal gradient of δ per nanometer. Hence, their composition can be expressed as Co Tb , where n = 1, 2, 3, 4, 5. Fig. 1d shows that increasing the composition gradient leads to a decrease of the overall M s of the sample, which corresponds to the Co-rich part tendency in Fig. 1b, i.e., increasing the average Tb composition makes the sample approach the magnetic compensation point [8-9]. When δ arrives at 0.07, the sample is nearly compensated ( M s < 50 emu/cc), while the sample starts to show an in-plane magnetization component when exceeding δ = 0.07. We thus only discuss gradients δ in the range from 0 to 0.07. To verify the existence of and quantify the composition gradient in our samples, high-resolution ross-sectional scanning transmission electron microscopy (STEM) and energy dispersive X-ray spectroscopy (EDS) were performed on the sample with δ = 0.07. Fig. 1e shows the structure of the ordered layers in the sample. No obvious crystal structure is observed, revealing the amorphous nature of the CoTb layer as expected. EDS curves versus the thickness direction in Fig. 1f exhibit a prominent inhomogeneous spatial distribution of Co and Tb elements. With the measured position penetrating to the substrate, Co (Tb) intensity shows an increasing (decreasing) tendency, which corresponds well with the designed structure in Fig. 1c. To perform electrical measurements, the films were fabricated into Hall bar devices with a width of 10 μm by conventional lithography and ion milling. The inset in Fig. 2a displays the device as well as the measurement contact configurations. Based on the anomalous Hall effect resistance ( R AHE ) measurements, all samples show clear PMA consistent with the film-level magnetometry results. The case of δ = 0.06 is shown as an example in Fig. 2a. Note that with δ varying from 0.00 to 0.07, the R AHE loop does not change its polarity, revealing that all samples remain Co-rich overall. Standard harmonic measurements were then carried out in a large field range to determine the SOT efficiency as well as the effective spin Hall angle of each sample [42-43]. When an alternating current j = j sin(ωt) is applied along the x axis, the SOT-induced alternating effective field will generate magnetization oscillations around the equilibrium position, which contributes to the rise of a second-harmonic Hall resistance R . In the regime where the in-plane magnetic field |𝐻 (cid:2934) | exceeds the agnetic anisotropy field H k , R can be described by the equation [42-43] 𝑅 (cid:2870)(cid:3104) = (cid:2869)(cid:2870) (cid:3019) (cid:3250)(cid:3257)(cid:3254) (cid:3009) (cid:3253)(cid:3261) |(cid:3009) (cid:3299) |(cid:2879)(cid:3009) (cid:3286) + 𝑅 (cid:3042)(cid:3033)(cid:3033)(cid:3046)(cid:3032)(cid:3047) , (1) where H DL and R offset are the damping-like SOT effective field and resistance offset, respectively. Fig. 2b and Fig. 2c plot the first harmonic resistance R ω and R of the sample with δ = 0.06. By fitting Fig. 2c with equation (1), the SOT efficiency ξ = H DL / j CoTb , where j CoTb is the corresponding current density in the CoTb layer, is determined to be 3.2 ± 0.3 Oe/(10 A/m ). The calculation of j CoTb can be found in Supplementary Note S1. We then calculated the damping-like effective spin Hall angle by θ DL = 2eM s t CoTb H DL / ℏj CoTb , where t CoTb is the CoTb layer thickness. The θ DL values of samples with different gradients δ are summarized in Fig. 2d. Interestingly, θ DL shows a clear increasing tendency as a function of δ. At δ = 0.07, the maximum value of θ DL is determined to be 0.061 ± M ) and SHE symmetry (spin polarized along the y axis for current along x and emission along z), see Note S1 in [39] for Rashba-induced spin currents and the more general theory by Stiles and coworkers [45-46]. The ASHE term, polarized along M , cannot generate a torque on M . The SHE-like spin current in a magnetic layer can generate a Damping-Like (DL) spin transfer torque on M if there is a possible transfer of the spins carried by this current, either outside the magnetic layer [41,47] or into a different part of the magnetic layer. Spin transfer to outside cannot work in our samples because the CoTb layer is inserted between two thin layers of the light element and weak absorber Al. In contrast, the asymmetric distribution of CoTb layers of different compositions introduces spin transfers to the bottom part of the sample from the top where the higher Tb concentration induces stronger Rashba interactions, higher spin polarizations and downward emissions of a spin current j CoTb θ DL,top , where θ DL,top is the dampling-like effective spin Hall angle of the top CoTb. The final torque is difficult to predict precisely because it will depend on how the gradient of Tb concentration is reflected in the gradient of Rashba interaction and the variation of the emitted spin current from j CoTb θ DL,top to j CoTb θ DL,bottom , where θ DL,bottom is the dampling-like effective spin Hall angle of the bottom CoTb. In rough approximation, the DL torque should correspond to the transfer of a spin current of the order of j CoTb θ DL,top = j CoTb ( θ DL,top - θ DL,bottom ) (which, as expected, becomes zero if the layer is homogeneous). ield-free switching of perpendicular CoTb by SOT and gradient-driven DMI
We next performed SOT-induced magnetization switching experiments on the CoTb samples with a vertical composition gradient. Following the measurement strategy shown in Fig. 3a, successive write current pulses with a duration of 0.1 ms (orange bars, current from positive to negative and then back to positive) were injected in the x axis of the Hall bar, while a small sensing current of 0.1 mA (blue triangles) was applied after each pulse to detect the magnetization state via R AHE . Fig. 3b shows R AHE as a function of the injected write current density j CoTb in the sample with δ = 0.07, under a bias field H ex along the x axis varying from 100 Oe to -100 Oe. The successful magnetization switching verifies the existence of intrinsic SOTs in the CoTb with composition gradient. Meanwhile, the current-driven R AHE shows a similar amplitude to the field-driven R AHE , revealing that a nearly complete switching of the CoTb is achieved by current. We define the critical switching current j c as the value of j CoTb where half of the maximum resistance change is achieved. From Fig. 3b, the j c of this 6 nm thick sample was determined to be around 9×10 A/m . An analysis of the H ex dependence of j c can be found in Supplementary Note S2. Supplementary Note S8 compares the SOT switching efficiency of these CoTb gradient samples – defined as j c / (t∙K eff ) , where t is the magnetic film thickness and K eff is its effective perpendicular anisotropy energy density – with: (i) previous reports of SOT switching of ferrimagnetic CoTb films (in the presence of a bias magnetic field) [9,48-50], and (ii) field-free deterministic SOT switching of perpendicular ferromagnetic layers [15,19,21,27]. In all cases, the CoTb samples with vertical omposition gradient show the best SOT switching efficiency. As expected in the SOT framework, the switching loops show opposite switching polarities under ±100 Oe. More interestingly, a full SOT switching loop in the absence of H ex is observed as well. This field-free switching loop has the same polarity as the cases with a positive H ex . In addition, almost no switching signal is detected in this sample when H ex = -20 Oe. These results indicate the existence of an effective internal field which participates in the SOT switching process. We attribute this effective field to the g-DMI in the CoTb layer [36]. To quantify the g-DMI-induced effective field H DMI in our sample, a method based on the magnetic droplet nucleation model was used [36,51]. As depicted in the inset of Fig. 3c, the hysteresis loop was measured by sweeping the magnetic field at an angle θ with respect to the z axis. Fig. 3c illustrates the magnetization switching curves in the negative field range with different angles. If we denote the magnetic field where the magnetization is switched from the “up” state to the “down” state by H sw , then, the coercive field H c and the accompanying in-plane field H n are determined to be H sw sinθ and H sw cosθ , respectively. Fig. 3d shows the measured H c as a function of H n . As described in earlier works, the curve shows a clear plateau, and H c starts to decrease after H n passes a threshold value. This threshold H n corresponds to H DMI [36,51]. In this manner, H DMI was determined to be around 23 Oe in the 6 nm CoTb layer with δ = 0.07. We note that this value is very close to the absolute value of applied in-plane H ex (-20 Oe, see Fig. 3b) where magnetization switching vanishes, which is consistent with the hypothesis hat the g-DMI-induced effective field is responsible for the observed field-free switching [33-35]. To investigate the role of the composition gradient on the switching behavior in more detail, SOT switching experiments were also performed in the following samples: (A) 6 nm thick CoTb with δ = 0.06 per 1 nm; (B) 9 nm thick CoTb with δ = 0.07 per 1.5 nm; (C) 4.2 nm thick CoTb with δ = 0.07 per 0.7 nm. Note that samples A and B have a smaller slope of the composition gradient compared to the previously discussed samples (where we had a 6 nm thick CoTb film with δ = 0.07 per 1 nm), due to the smaller value of δ and the larger overall CoTb thickness, respectively. On the other hand, the slope of the composition gradient is increased in sample C, due to the reduced overall CoTb thickness for the same value of δ. The results are shown in Supplementary Notes S3, S4 and S5. All the samples exhibited clear deterministic switching loops in the absence of H ex , and the value of H ex where switching vanished corresponded well to the measured value of H DMI . This is in agreement with the fact that the g-DMI is responsible for the in-plane symmetry-breaking and zero-field switching observed in these samples. As expected, the j c values of samples A and B (> 10×10 A/m ) were larger than that in the 6 nm thick CoTb layer with δ = 0.07. On the contrary, a smaller j c (~ 7.5×10 A/m ) as well as a higher H DMI (~ 40 Oe) were achieved in sample C, as expected based on its steeper composition gradient. witching behavior of a CoTb film with an inverse composition gradient
To further verify the origin of SOTs and g-DMI, we studied the SOT switching behavior in a 6 nm thick CoTb layer with an inverse composition gradient (δ = -0.07), i.e. Tb concentration decreasing from bottom to the top of the layers. The details of this sample are described in Fig. 4a, while Fig. 4b shows the obtained SOT switching curves of this sample under different H ex . The switching loops always show opposite polarity compared to those of the sample with δ = 0.07 (comparison of Fig. 4b and Fig. 3b). This indicates that the vertical broken symmetry directions in these two samples are opposite, confirming again that the SOTs in the CoTb layer truly originate from the vertical composition gradient. SOT switching in the absence of H ex was also observed in this sample, and the zero-field loop exhibits a polarity similar to switching loops under negative H ex . In addition, the deterministic switching vanishes at a positive H ex of ~15 Oe. Both of these characteristics, which are determined by g-DMI in our field-free switching explanation, are opposite to the ones observed in the sample with δ = 0.07. Thus, we conclude that the inverse composition gradient also induces an H DMI with opposite sign compared to the previous case. The combination of the reversed sign of SOT compared to our reference device, along with the opposite sign of H DMI explains the field-free switching data of Fig. 4b and is consistent with the observations in Fig. 3b.
Modeling and simulations o shed light on the underlying mechanism of this field-free switching phenomenon, we performed micromagnetic simulations. Our micromagnetic framework is based on a two-sublattice model strongly coupled through an exchange interaction [41,52-55]. Along with the exchange interaction, we also take into account the anisotropy, the DMI, and the torque exerted by SOT. Supplementary Note S6 provides details of the micromagnetic model. The g-DMI originates from the composition gradient along the z -axis [36] and therefore the symmetry breaking is, from a modeling point of view, similar to the case of interfacial DMI observed in other multilayer systems [56-58]. To mimic the experimental setup (see Fig. 2a) we consider a cross geometry initially saturated along the +z direction. From this initial state we apply an electric current for 20 ns (from t on,1 to t off,1 ), let the system relax for 12 ns more and repeat the process with the opposite polarization of the current (from t on,2 to t off,2 ). Fig. 5b shows the time evolution of the average z component of the magnetization for the first sublattice (black line) together with the timing of the two current pulses. As observed in previous numerical studies [33-35,59], the deterministic switching starts with the nucleation of a domain at the edge of the sample because of the magnetization tilting toward an in-plane direction imposed by the DMI boundary conditions (see Fig. 5a). In other words, the nucleation of a new domain at the edge is driven by the SOT applied to this in-plane component at the edge (Fig. 5c and d). The switching occurs via a strongly nonuniform magnetization pattern, given by the higher DW velocity in the edges as compared to the central region due to the combined ffects of the DMI boundary conditions, the DMI, and exchange fields (Fig. 5e-g). After the current is switched off, the magnetization relaxes to the reversed uniform ground state (Fig. 5h and i). The same process is observed for the current pulses of opposite polarity (See Supplementary Note S7). Summary and Conclusions
We have demonstrated a field-free DMI-SOT switching scheme in a thick perpendicular CoTb layer with vertical composition gradient. The broken structural inversion symmetry along the growth direction induces both strong SOTs and g-DMI. Notably, this gradient-driven switching mechanism is promising to realize field-free switching under a relatively low current density and has the potential to be scalable to large wafer size. The SOT switching efficiency is significantly better than in previous field-free SOT switching experiments in ferromagnets, as well as previous (field-assisted) SOT switching experiments in conventional CoTb layers without composition gradient. Given that the SOT in our structure originates within the bulk of the CoTb gradient film, it could in principle be further enhanced by interfacing with an appropriate heavy metal layer to provide additional interfacial SOT. Notably, since the g-DMI in our structure also originates from the composition gradient within the bulk, it does not impose any constraints on the choice of the heavy metal layer for interfacial SOT, thus providing freedom in device design. Finally, we note that large tunneling magnetoresistance (TMR) ratios have already been demonstrated in magnetic tunnel junctions based on FIMs [60-62]. Therefore, it is possible to integrate he CoTb gradient films presented here into SOT memory devices with TMR readout. Our findings go beyond the conventional paradigm of SOT-induced perpendicular magnetization switching with the assistance of external field and SOT sources, and being applicable to ferrimagnets, pave the way towards the development of ultrafast SOT memory arrays.
Methods Sample growth and characterization:
A series of Al (2) / CoTb (6) / Al (3) (thickness in nm) stacks were deposited on thermally oxidized silicon substrates by DC and RF sputtering under a base pressure lower than 10 -8 Torr. The composition of CoTb was controlled by setting the deposition power of Co and Tb targets. Vibrating sample magnetometry was used to quantify the magnetic properties of each sample. STEM and EDS characterizations were performed by a JEM-ARM200F microscope.
Device fabrication and electrical measurement:
The films were fabricated into Hall bar devices of 10 μm width by optical lithography and ion milling techniques. For AHE and DMI measurements, DC current was applied along the x axis, while for SOT switching measurements, current pulses (0.1 ms width) were applied. A Hall probe was inserted in the probe station to detect the actual magnetic field values during measurement, thus guaranteeing the absence of any 𝐻 (cid:3032)(cid:3051) contributions due to remanent fields. For harmonics measurements, two SR830 lock-in amplifiers were used to detect the first and second harmonic Hall voltage induced by an AC current with a frequency of 133.33 Hz. Author contributions
P.K.A., W.S.Z. and Y.Z. planned and supervised the project. Z.Y.Z. and Y.Z. designed the experiments and fabricated the devices with help from V.L.-D. and J.S. Z.Y.Z., V.L.-D., X.F., L.C., Z.W., Z.Z. and Y.X performed the measurements of the samples and analyzed the data. B.H., K.Z. and Y.G.Z. performed the STEM imaging. L.S.-T., M.C., and G.F. implemented the micromagnetic code and performed the simulations. A.F. contributed to the theoretical explanation and data interpretation. Z.Y.Z., P.K.A., G.F. and Y.Z. co-wrote the manuscript. All the authors read and commented on the manuscript.
Acknowledgements
This work was supported by a grant from the U.S. National Science Foundation, Division of Electrical, Communications and Cyber Systems (NSF ECCS-1853879), the National Natural Science Foundation of China (Grant No. 61971024 and 51901008), the International Mobility Project (Grant No. B16001) and National Key Technology Program of China (Grant No. 2017ZX01032101). This work made use of the NUFAB facility of Northwestern University’s NUANCE Center, which has received support from the SHyNE Resource (NSF ECCS-1542205), the IIN, and Northwestern’s MRSEC program (NSF DMR-1720139). Z.Y.Z also acknowledges the support from the China Scholarship Council (No. 201906020022)
Competing financial interests
The authors declare no competing financial interest.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request. eferences Miron, I. M. et al. Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection.
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Structure of the homogenous CoTb samples. b, Net magnetization dependence on the Tb concentration in homogenous CoTb samples. A clear magnetization compensation point is found at the point where Tb concentration equals 0.29. c, Structure of the CoTb samples with composition gradient δ. d, Net magnetization dependence on δ, showing a linear decrease. e, STEM images of the sample with δ = 0.07. f, EDS-measured Co and Tb intensity in the sample as a function of vertical position. Co and Tb concentrations show opposite slopes, which verifies the existence as well as the direction of the composition gradient.
Fig. 2 Electrical measurement setup and SOT characterization. a,
Measured AHE loop for the sample with δ = 0.06. The square shape reveals that the samples are perpendicularly magnetized. Inset shows a photograph of a representative Hall bar device. b, R ω versus magnetic field along the x direction, for the sample with δ = 0.06. c, R and the fitting line versus magnetic field along the x direction for the sample with δ = 0.06 under j CoTb ≈ 1×10 A/m . d, Increasing tendency of θ DL versus composition gradient δ. Fig. 3 SOT switching experiments and DMI measurement. a,
Measurement scheme of SOT switching. Writing pulses with a width of 0.1 ms were applied from positive to negative, and then back to positive. After each writing pulse, a small read current was applied to detect the magnetic state of the CoTb layer by AHE. b, SOT switching curves of the sample with δ = 0.07 under different in-plane magnetic fields (including zero-field). c, AHE curves when the magnetization switches from up to down under different magnetic field angles θ with respect to the film normal direction. The inset shows the definition of θ in the xz plane. d, H c as a function of H n . There is a threshold value of H n ( ≈
23 Oe), above which H c starts to decease. The inset shows the increasing relationship between H DMI and δ in 6 nm CoTb films.
Fig. 4 Comparison experiments on a CoTb sample with inverse gradient. a,
Structure of the sample with δ = -0.07. b, SOT switching curves of the sample (δ = -0.07) under different in-plane magnetic fields (including zero-field). This sample shows opposite switching polarity as well as an opposite DMI-induced field compared to the sample in Fig. 3b (δ = 0.07), verifying the vertical gradient-induced origin of SOT and DMI in our samples.
Fig. 5 Micromagnetic simulations. a,
A sketch of the magnetization tilting driven by
DMI boundary conditions allowing domain nucleation at the edges. b, Average z component of the first sublattice magnetization in the switching of the system from 0 ns to 32 ns, and from 32 ns to 64 ns, respectively (under opposite current directions). c-i,c-i,