Tunable Spin Hall Magnetoresistance in All-Antiferromagnetic\\ Heterostructures

Lin Huang(黄琳), Yongjian Zhou(周永健), Tingwen Guo(郭庭温), Feng Pan(潘峰), and Cheng Song(宋成)$^{\hyperlink{s}{*}}$

doi:10.1088/0256-307X/39/4/047502

⇦Chinese Physics Letters, 2022, Vol. 39, No. 4, Article code 047502⇨ Tunable Spin Hall Magnetoresistance in All-Antiferromagnetic Heterostructures Lin Huang (黄琳), Yongjian Zhou (周永健), Tingwen Guo (郭庭温), Feng Pan (潘峰), and Cheng Song (宋成)* Affiliations Key Laboratory of Advanced Materials (MOE), School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Received 1 February 2022; accepted 11 March 2022; published online 28 March 2022 ^*Corresponding author. Email: songcheng@mail.tsinghua.edu.cn Citation Text: Huang L, Zhou Y J, Guo T W et al. 2022 Chin. Phys. Lett. 39 047502 Abstract We investigate the spin Hall magnetoresistance (SMR) in all-antiferromagnetic heterostructures $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ with Pt contacts. When the temperature is ultralow ($ < $ 50 K), the spin current generated in the Pt layer cannot be transmitted through Cr$_{2}$O$_{3}$ ($t = 4$ nm), and the SMR is near zero. Meanwhile, when the temperature is higher than the spin fluctuation temperature $T_{\rm F}$ ($\approx $ 50 K) of Cr$_{2}$O$_{3}$ and lower than its Néel temperature $T_{\rm N}$ ($\approx $ 300 K), the spin current goes through the Cr$_{2}$O$_{3}$ layer and is reflected at the $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ interface; an antiferromagnetic (negative) SMR is observed. As temperature increases higher than $T_{\rm N}$, paramagnetic (positive) SMR mainly arises from the spin current reflection at the Cr$_{2}$O$_{3}$/Pt interface. The transition temperatures from negative to positive SMR are enhanced with increasing Cr$_{2}$O$_{3}$ layer thickness, accompanied by the absence of SMR signals when $t = 10$ nm. Such a tunable SMR builds a bridge between spin transport and structures. It also enriches antiferromagnetic spintronics.

DOI:10.1088/0256-307X/39/4/047502 © 2022 Chinese Physics Society Article Text Spin transport at magnet/heavy metal (HM) bilayers provides a rich platform for spintronics.[1] The spin Hall magnetoresistance (SMR) effect is a phenomenon in which the resistance of the HM layer depends on the angle between the spin polarization $\sigma$ of spin current and the magnetic order $M$ of magnetic layers. When $\sigma$ is perpendicular (parallel) to $M$, the resistance is high (low) because of the absorption (reflection) of spin current at the interface.[2–6] Next, the SMR in antiferromagnet/HM is negative because of the spin-flop transition (Néel order is perpendicular to the magnetic field above a critical field). The introduction of an additional magnetic layer would enrich or complicate the spin transport. A sign change in SMR was observed in ferromagnetic insulators (FMIs) YIG (or $\gamma$-Fe$_{2}$O$_{3}$)/Pt bilayers by inserting an antiferromagnetic insulator (AFI) NiO,[3,7–10] revealing the magnon spin current transmission through AFI and spin current reflection at the FMI/AFI interface. In Y$_3$Fe$_5$O$_{12}$ (YIG)/Cr$_{2}$O$_{3}$/Pt heterostructures,[11] the spin conducting and non-conducting states are modulated by the spin structure of Cr$_{2}$O$_{3}$. Meanwhile, in YIG/NiO/YIG structures, the SMR response depends on the orientation of two YIG magnetizations (parallel or antiparallel).[12] However, whether the sign of SMR can be modulated in all-antiferromagnetic heterostructures has not been reported. In this Letter, we experimentally investigate the temperature-dependent SMR in $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$/Pt heterostructures. Here $\alpha$-Fe$_{2}$O$_{3}$ is an AFI with an easy plane (0001) at room temperature. In bulk, it exhibits a Néel temperature of $T_{\rm N} = 953$ K with the Morin transition at $T \approx 263$ K.[13] The $\alpha$-Fe$_{2}$O$_{3}$/Pt bilayer exhibits a large negative SMR.[14,15] Cr$_{2}$O$_{3}$ is a prototypical AFI with out-of-plane uniaxial anisotropy and $T_{\rm N} \approx 307$ K.[16,17] The experiments investigate the SMR variation as a function of temperature and Cr$_{2}$O$_{3}$ thickness in all-antiferromagnetic heterostructures. The epitaxial $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Cr$_{2}$O$_{3}$ ($t$ nm) bilayers were fabricated on Al$_{2}$O$_{3}$ (0001) substrates through pulsed laser deposition from Fe$_{2}$O$_{3}$ and Cr$_{2}$O$_{3}$ targets. The pulsed laser source employed a KrF laser of 248 nm wavelength. The laser energy density was kept at 3 J/cm$^{2}$ with a repetition rate of 2 Hz and 3 Hz for $\alpha$-Fe$_{2}$O$_{3}$ and Cr$_{2}$O$_{3}$, respectively. The base pressure of the chamber before deposition was less than $3 \times 10^{-5}$ Pa. The deposition temperature was 450 ℃ in an oxygen atmosphere of 2.67 Pa. Additionally, the epitaxial growth is confirmed through in situ reflection high-energy electron diffraction (RHEED). The films were cooled down to room temperature after deposition at the rate of 10 ℃/min under the oxygen pressure of $2 \times 10^{4}$ Pa. Then, a 5-nm-thick Pt layer was grown on the surface of the Cr$_{2}$O$_{3}$ film by magnetron sputtering, where the base pressure of the chamber was less than $6 \times 10^{-7}$ Pa. The film roughness was measured by an atomic force microscope. The qualities of films were characterized by x-ray diffraction (XRD). The cross-section imaging was performed on a high-resolution transmission electron microscopy (HRTEM). Finally, a Hall bar of 500 µm $\times 50$ µm was patterned for electrical measurements by photolithography and Ar ion beam etching. Figure 1(a) shows typical RHEED images of $\alpha$-Fe$_{2}$O$_{3}$ and Cr$_{2}$O$_{3}$ surfaces. The RHEED pattern of the $\alpha$-Fe$_{2}$O$_{3}$ and Cr$_{2}$O$_{3}$ surfaces is streaky, indicating a flat surface of the bilayer. According to the structure factor of the corundum structure, [11$\bar{2}$0] projected reciprocal lattice of the single-crystalline corundum (0001) surface is asymmetric.[18] Figure 1(b) shows an XRD result of $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ on the Al$_{2}$O$_{3}$ (0001) substrate, where peaks (0006) of $\alpha$-Fe$_{2}$O$_{3}$ and Cr$_{2}$O$_{3}$ can be seen at 39.5$^{\circ}$ and 40.1$^{\circ}$, respectively. The latter is next to the Al$_{2}$O$_{3}$ (0001) substrate peak. Figure 1(c) shows the surface topography of the bilayer using atomic force microscopy, exhibiting a flat surface with a small surface roughness of 0.18 nm. Figure 1(d) shows a typical cross-sectional HRTEM image of $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$/Pt cross section. Therefore, the surface of the bilayer is relatively smooth, which is essential to grow a high-quality top layer of heavy-metal Pt on it.

Fig. 1. (a) RHEED patterns of $\alpha$-Fe$_{2}$O$_{3}$ and Cr$_{2}$O$_{3}$ surfaces. (b) XRD spectra of $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ grown on the Al$_{2}$O$_{3}$ (0001) substrate. (c) Atomic force microscopy image of $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ surface with a roughness of 0.18 nm. (d) HRTEM image of the $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$/Pt cross section.

We now investigate the SMR responses in $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Cr$_{2}$O$_{3}$ (4 nm)/Pt heterostructure at temperatures ranging from 10 K to 350 K. Figure 2(a) shows the schematics of the Hall bar with in-plane rotation angle $\alpha$ defined between the external magnetic field ${\boldsymbol H}$ and $x$-axis in the $xy$ plane. The external magnetic field ${H}$ is 50 kOe, which is sufficient for the spin-flop transition (several thousands of Oe) of $\alpha$-Fe$_{2}$O$_{3}$.[14,15,19,20] The $xy$ plane is the film plane, where the current $J_{\rm e}$ is applied along the $x$-axis. The magnetoresistance (MR) is calculated as the change in resistance $\Delta R_{xx}=R_{xx}-R_{\min}$ and normalized to the resistance $\Delta R_{xx}/R_{\min}$, where $R_{\min}$ is the minimum resistance during the magnetic field rotation from 0$^{\circ}$ to 360$^{\circ}$. Figure 2(b) shows the SMR responses in $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Cr$_{2}$O$_{3}$ (4 nm)/Pt heterostructure as a function of angle $\alpha$ with different temperatures. At ultralow temperature ($T = 10$ K), the Néel order of Cr$_{2}$O$_{3}$ is aligned out-of-plane, the spin current is blocked, and SMR is nearly zero. Next, when the temperature ranges from 50 K to 200 K, the SMR curve shows [$1-\cos(2\alpha)$] angle dependence, featured as M-shape, and the maximum resistance appears at $\alpha = 90^{\circ}$ and 270$^{\circ}$. The M-shape SMR arises from the spin current reflected at $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ interface, which is quite characteristic of the antiferromagnetic $\alpha$-Fe$_{2}$O$_{3}$. The spin current can be transmitted through the Cr$_{2}$O$_{3}$ layer when the Cr$_{2}$O$_{3}$ magnetic moment is in the magnetic fluctuation state with temperatures ranging from 50 K to 200 K. When the temperature is higher ($T \ge 300$ K), SMR shows a $\cos (2\alpha)$ angular-dependent behavior, and the maximum resistance appears at $\alpha = 0^{\circ}$ and 180$^{\circ}$, which is considered as W-shape. The W-shape SMR arises from the spin current reflected at Cr$_{2}$O$_{3}$/Pt interface, where Cr$_{2}$O$_{3}$ is in the paramagnetic phase.[21–23]

Fig. 2. (a) Schematic of in-plane $\alpha$ angular dependence of magnetoresistance in $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Cr$_{2}$O$_{3}$ (4 nm)/Pt. The magnetic field ${\boldsymbol H}$ is rotated in the $xy$ plane with angles $\alpha$ relative to the $x$-axis, and charge current $J_{\rm e}$ is applied along the $x$ direction. (b) In-plane $\alpha$ angular-dependent magnetoresistance measured with magnetic field ${H} = 50$ kOe for various temperatures (marked in the figure) for the $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Cr$_{2}$O$_{3}$ (4 nm)/Pt sample.

For comparison, the SMR responses of $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Pt [Fig. 3(a)] and Cr$_{2}$O$_{3}$ (4 nm)/Pt bilayer [Fig. 3(b)] were measured with a temperature ranging from 10 K to 350 K. The measuring method and device geometry are identical with Fig. 2(a). As shown in Fig. 3(a), the SMR signal of $\alpha$-Fe$_{2}$O$_{3}$/Pt bilayer exhibits an M-shape. The maximum resistance appears at 90$^{\circ}$ and 270$^{\circ}$, where the Néel vector ${\boldsymbol n}$ is coplanar with charge current $J_{\rm e}$ and ${\boldsymbol H}$, and perpendicular to the in-plane magnetic field ${\boldsymbol H}$.[24–26] As the temperature increases from 10 K to 350 K, the magnitude of the SMR signal increases because the spin-orbit coupling of Pt is stronger at higher temperatures.[27] As shown in Fig. 3(b), the SMR signal shows a W-shape when $T > T_{\rm N}$ ($\approx 300$ K), originating from the paramagnetic state of Cr$_{2}$O$_{3}$. This paramagnetic polarization of Cr$_{2}$O$_{3}$ will always follow the large external magnetic field, producing a positive SMR.[21,28,29] The situation differs dramatically when the sample is cooled below the Néel temperature; the signals become almost constant without a typical SMR shape ($T = 200,\, 50$, and 10 K). In this scenario, the Néel order of Cr$_{2}$O$_{3}$ is aligned out of plane, producing no SMR signal.

Fig. 3. The $\alpha$-dependent magnetoresistance measured with magnetic field ${H} = 50$ kOe for various temperatures in (a) $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Pt and (b) Cr$_{2}$O$_{3}$ (4 nm)/Pt bilayers.

Figure 4(a) shows the temperature-dependent SMR ratio of $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Cr$_{2}$O$_{3}$ ($t$ nm)/Pt with different Cr$_{2}$O$_{3}$ thicknesses $t$ ($t = 0,\, 2,\, 4,\, 6$, and 10 nm). The SMR ratio is defined as SMR ratio = ($R_{0}-R_{90}$)/$R_{0}$; here, $R_{0}$ and $R_{90}$ denote the resistance value at $\alpha = 0^{\circ}$ and 90$^{\circ}$, respectively. When the temperature ($T = 10$ K) is lower than $T_{\rm N}$, the Néel vector of Cr$_{2}$O$_{3}$ is along the easy axis (out-of-plane), the spin current cannot be transmitted through Cr$_{2}$O$_{3}$, and no SMR signal is observed when $t \ge 4$ nm. Moreover, when the temperature is between 50 K and 200 K ($0 < t \le 4$ nm), the negative M-shape SMR is observed, and the Néel vector of Cr$_{2}$O$_{3}$ is in a magnetic fluctuation state. The uniaxial anisotropy of Cr$_{2}$O$_{3}$ reduces as temperature increases, resulting in the inclination of the Néel vector when magnons are excited and enabling the spin current transportation with an in-plane component of the spin polarization. Phenomenologically, we define the fluctuation temperature $T_{\rm F}$ based on the negative SMR that appears. The spin current transmitted through the Cr$_{2}$O$_{3}$ layer is reflected at the $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ interface, as shown in the schematic in the inset (i) of Fig. 4(a). As the temperature increases higher than $T_{\rm N}$ and Cr$_{2}$O$_{3}$ is in the paramagnetic phase, a positive W-shape SMR is observed with the thickness $t = 2,\, 4$, and 6 nm. The paramagnetic state of Cr$_{2}$O$_{3}$ follows the external magnetic field, and positive SMR mainly arises from the spin current reflection at the Cr$_{2}$O$_{3}$/Pt interface, instead of the $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ interface shown in the inset (ii) of Fig. 4(a). Figure 4(b) shows the temperature-dependent SMR ratio of Cr$_{2}$O$_{3}$/Pt bilayers with different thicknesses. Positive SMR is observed in the paramagnetic phase of Cr$_{2}$O$_{3}$ with $t = 2$ and 4 nm; however, it vanishes when cooling below the Néel temperature. Meanwhile, the SMR signal is absent in the bilayers when $t$ is thicker ($t = 6$ and 10 nm). The results show that the spin current can be transmitted through Cr$_{2}$O$_{3}$ with thinner thicknesses, and the transmittance of the spin current decreases as the thickness of Cr$_{2}$O$_{3}$ increases.

Fig. 4. (a) SMR ratio of $\alpha$-Fe$_{2}$O$_{3}$ (4 nm)/Cr$_{2}$O$_{3}$ ($t$ nm)/Pt as a function of temperature with $t = 0,\, 2,\, 4,\, 6$, and 10 nm. (b) SMR ratio of Cr$_{2}$O$_{3}$ ($t$ nm)/Pt as a function of temperature with $t = 2,\, 4,\, 6$, and 10 nm.

The experimental results show that the spin current could only be transmitted if its polarization direction is parallel or antiparallel to the Néel order of Cr$_{2}$O$_{3}$. When the temperature is ultralow ($T < T_{\rm F}$), the polarization of spin current is perpendicular to the Néel vector of Cr$_{2}$O$_{3}$, and no MR response is observed where the spin current cannot be transmitted through Cr$_{2}$O$_{3}$ [Fig. 5(a)]. Meanwhile, when $T_{\rm F} < T < T_{\rm N}$, a negative M-shape SMR is observed, in which the spin current transmits through Cr$_{2}$O$_{3}$ and reflects from the bottom $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ interface [Fig. 5(b)]. As the temperature increases higher than the Néel temperature ($T > T_{\rm N}$), due to the spin current reflection from the Cr$_{2}$O$_{3}$/Pt interface, a positive W-shape SMR is observed [Fig. 5(c)].

Fig. 5. Illustrations for the magnetic structures and spin current transportation in $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$/Pt with temperature variation. (a) $T < T_{\rm F}$, the two Néel vectors (Cr$_{2}$O$_{3}$ and $\alpha$-Fe$_{2}$O$_{3}$) are perpendicular. (b) $T_{\rm F} < T < T_{\rm N}$, the Néel vector of Cr$_{2}$O$_{3}$ is in the magnetic fluctuation state, and spin current can be transmitted through the in-plane components of Cr$_{2}$O$_{3}$. (c) $T > T_{\rm N}$, magnetic order of Cr$_{2}$O$_{3}$, and Néel vector of $\alpha$-Fe$_{2}$O$_{3}$ are perpendicular.

In summary, we have investigated the SMR response in $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$/Pt heterostructures with varying temperatures and Cr$_{2}$O$_{3}$ thicknesses $t$. As the temperature changes for $t = 2$ and 4 nm, we observed a sign change in SMR, which can be explained by the spin current reflection at $\alpha$-Fe$_{2}$O$_{3}$/Cr$_{2}$O$_{3}$ interface when the temperature ranges from $T_{\rm F}$ to $T_{\rm N}$ (negative SMR) and spin current reflection at the Cr$_{2}$O$_{3}$/Pt interface when the temperature is higher than $T_{\rm N}$ (positive SMR). The spin current cannot be transmitted when Cr$_{2}$O$_{3}$ is thick ($t = 6$ and 10 nm), and only a positive SMR signal is observed at high temperature ($T > T_{\rm N}$). Our results provide a potential advantage for distinguishing the source of spin flow in the signal in multilayer heterostructures and highlight the importance of magnetic structure (Néel vector) in AFI, which opens the possible way to manipulate the spin current transportation and AFI memory device.[30] Acknowledgments. This work was supported by the National Key R&D Program of China (Grant No. 2021YFB3601301), the National Natural Science Foundation of China (Grant No. 51871130), the Natural Science Foundation of Beijing Municipality (Grant No. JQ20010). We acknowledge the support of Beijing Innovation Center for Future Chip, Tsinghua University. References Interface-induced phenomena in magnetismSpin Hall Magnetoresistance Induced by a Nonequilibrium Proximity EffectTunable Sign Change of Spin Hall Magnetoresistance in

Pt / NiO / YIG

StructuresMagnetic properties and anomalous Hall effect of Mn3Sn thin films controlled by defects and ferroelectric 0.7Pb(Mg1/3Nb2/3)O3–0.3PbTiO3 substrateTheory of spin Hall magnetoresistanceSuperconductivity and Fermi Surface Anisotropy in Transition Metal Dichalcogenide NbTe₂Effect of NiO inserted layer on spin-Hall magnetoresistance in Pt/NiO/YIG heterostructuresFull angular dependence of the spin Hall and ordinary magnetoresistance in epitaxial antiferromagnetic NiO(001)/Pt thin filmsElectrical Detection of Spin Backflow from an Antiferromagnetic Insulator/

Y_{3} {Fe}_{5} O_{12}

InterfaceAntiferromagnetic NiO thickness dependent sign of the spin Hall magnetoresistance in γ-Fe₂ O₃ /NiO/Pt epitaxial stacksSpin colossal magnetoresistance in an antiferromagnetic insulatorA nonlocal spin Hall magnetoresistance in a platinum layer deposited on a magnon junctionMagnetic Susceptibility of

α {Fe}_{2} O_{3}

and

α {Fe}_{2} O_{3}

with Added TitaniumA comparative study of spin Hall magnetoresistance in Fe₂ O₃ -based systemsLarge Spin Hall Magnetoresistance in Antiferromagnetic

α − {Fe}_{2} O_{3} / Pt

HeterostructuresAll-Electric Access to the Magnetic-Field-Invariant Magnetization of AntiferromagnetsPurely antiferromagnetic magnetoelectric random access memoryGiant Anomalous Hall Conductivity at the

Pt / Cr

₂ O₃ InterfaceStructural sensitivity of the spin Hall magnetoresistance in antiferromagnetic thin filmsEvolution of the spin hall magnetoresistance in Cr₂ O₃ /Pt bilayers close to the Néel temperatureMagnetic Structure of Cr₂ O₃Optical study of the antiferromagnetic-paramagnetic phase transition in chromium oxide Cr₂ O₃Terahertz pulse-induced Néel vector switching in α -Fe₂ O₃ /Pt heterostructuresAnisotropic magnetoresistance and nontrivial spin Hall magnetoresistance in

Pt / α − F e_{2} O_{3}

bilayersObservation of the antiferromagnetic spin Hall effectTemperature dependence of spin–orbit torques in Pt/Co/Pt multilayersQuantitative study of the spin Hall magnetoresistance in ferromagnetic insulator/normal metal hybridsSpin-Hall magnetoresistance and spin Seebeck effect in spin-spiral and paramagnetic phases of multiferroic

{CoCr}_{2} O_{4}

filmsMagnetization switching by magnon-mediated spin torque through an antiferromagnetic insulator

[1]	Hellman F et al. 2017 Rev. Mod. Phys. 89 025006
[2]	Nakayama H, Althammer M, Chen Y T, Uchida K, Kajiwara Y, Kikuchi D, Ohtani T, Geprägs S, Opel M, Takahashi S, Gross R, Bauer G E W, Goennenwein S T B, and Saitoh E 2013 Phys. Rev. Lett. 110 206601
[3]	Hou D, Qiu Z, Barker J, Sato K, Yamamoto K, Vélez S, Gomez-Perez J M, Hueso L E, Casanova F, and Saitoh E 2017 Phys. Rev. Lett. 118 147202
[4]	Zhao Z P, Guo Q, Chen F H, Zhang K W, and Jiang Y 2021 Rare Met. 40 2862
[5]	Chen Y, Takahashi S, Nakayama H, Althammer M, Goennenwein S T B, Saitoh E, and Bauer G E W 2013 Phys. Rev. B 87 144411
[6]	Zhang X, Luo T, Hu X, Guo J, Lin G, Li Y, Liu Y, Zhang X, Luo T, Hu X, Guo J, Lin G, Li Y, Liu Y, Li X, Ge J, Xing Y, Zhu Z, Gao P, Sun L, and Wang J 2019 Chin. Phys. Lett. 36 057402
[7]	Shang T, Zhan Q F, Yang H L, Zuo Z H, Xie Y L, Liu L P, Zhang S L, Zhang Y, Li H H, Wang B M, Wu Y H, Zhang S, and Li R W 2016 Appl. Phys. Lett. 109 032410
[8]	Baldrati L, Ross A, Niizeki T, Schneider C, Ramos R, Cramer J, Gomonay O, Filianina M, Savchenko T, Heinze D, Kleibert A, Saitoh E, Sinova J, and Kläui M 2018 Phys. Rev. B 98 024422
[9]	Lin W and Chien C L 2017 Phys. Rev. Lett. 118 067202
[10]	Dong B, Baldrati L, Schneider C, Niizeki T, Ramos R, Ross A, Cramer J, Saitoh E, and Kläui M 2019 Appl. Phys. Lett. 114 102405
[11]	Qiu Z, Hou D, Barker J, Yamamoto K, Gomonay O, and Saitoh E 2018 Nat. Mater. 17 577
[12]	Guo C Y, Wan C H, He W Q, Zhao M K, Yan Z R, Xing Y W, Wang X, Tang P, Liu Y Z, Zhang S, Liu Y W, and Han X F 2020 Nat. Electron. 3 304
[13]	Morin F J 1950 Phys. Rev. 78 819
[14]	Zhou Y J, Chen X Z, Zhou X F, Bai H, Chen R Y, Pan F, and Song C 2020 J. Appl. Phys. 127 163904
[15]	Fischer J, Althammer M, Vlietstra N, Huebl H, Goennenwein S T B, Gross R, Geprägs S, and Opel M 2020 Phys. Rev. Appl. 13 014019
[16]	Kosub T, Kopte M, Radu F, Schmidt O G, and Makarov D 2015 Phys. Rev. Lett. 115 097201
[17]	Kosub T, Kopte M, H€U R, Appel P, Shields B, Maletinsky P, Hübner R, Liedke M O, Fassbender J, Schmidt O G, and Makarov D 2017 Nat. Commun. 8 13985
[18]	Moriyama T, Shiratsuchi Y, Iino T, Aono H, Suzuki M, Nakamura T, Kotani Y, Nakatani R, Nakamura K, and Ono T 2020 Phys. Rev. Appl. 13 034052
[19]	Morrish A H 1994 Canted Antiferromagnetism: Hematite (Singapore: World Scientific)
[20]	Ross A, Lebrun R, Ulloa C, Grave D A, Kay A, Baldrati L, Kronast F, Valencia S, Rothschild A, and Kläui M 2020 Phys. Rev. B 102 094415
[21]	Schlitz R, Kosub T, Thomas A, Fabretti S, Nielsch K, Makarovand D, and Goennenwein S T B 2018 Appl. Phys. Lett. 112 132401
[22]	Corliss L M, Hastings J M, Nathans R, and Shirane G 1965 J. Appl. Phys. 36 1099
[23]	Pisarev R V, Krichevtsov B B, and Pavlov V V 1991 Phase Transit. 37 63
[24]	Huang L, Zhou Y, Qiu H S, Guo T, Pan F, Jin B, and Song C 2021 Appl. Phys. Lett. 119 212401
[25]	Cheng Y, Yu S S, Ahmed A S, Zhu M L, Rao Y, Ghazisaeidi M, Hwang J, and Yang F Y 2019 Phys. Rev. B 100 220408
[26]	Chen X, Shi S, Shi G, Fan X, Song C, Zhou X, Bai H, Liao L, Zhou Y, Zhang H, Li A, Chen Y, Han X, Jiang S, Zhu Z, Wu H, Wang X, Xue D, Yang H, and Pan F 2021 Nat. Mater. 20 800
[27]	Chen S, Li D, Cui B, Xi L, Si M, Yang D, and Xue D 2018 J. Phys. D 51 095001
[28]	Althammer M, Meyer S, Nakayama H, Schreier M, Altmannshofer S, Weiler M, Huebl H, Geprägs S, Opel M, Gross R, Meier D, Klewe C, Kuschel T, Schmalhorst J M, Reiss G, Shen L, Gupta A, Chen Y T, Bauer G E W, Saitoh E, and Goennenwein S T B 2013 Phys. Rev. B 87 224401
[29]	Aqeel A, Vlietstra N, Heuver J A, Bauer G E W, Noheda B, van Wees B J, and Palstra T T M 2015 Phys. Rev. B 92 224410
[30]	Wang Y, Zhu D, Yang Y, Lee K, Mishra R, Go G, Oh S H, Kim D H, Cai K, Liu E, Pollard S D, Shi S, Lee J, Teo K L, Wu Y, Lee K J, and Yang H 2019 Science 366 1125