Abstract
The interconversion of charge and spin currents via spinHall effect is essential for spintronics. Energyefficient and deterministic switching of magnetization can be achieved when spin polarizations of these spin currents are collinear with the magnetization. However, symmetry conditions generally restrict spin polarizations to be orthogonal to both the charge and spin flows. Spin polarizations can deviate from such direction in nonmagnetic materials only when the crystalline symmetry is reduced. Here, we show control of the spin polarization direction by using a noncollinear antiferromagnet Mn_{3}GaN, in which the triangular spin structure creates a low magnetic symmetry while maintaining a high crystalline symmetry. We demonstrate that epitaxial Mn_{3}GaN/permalloy heterostructures can generate unconventional spinorbit torques at room temperature corresponding to outofplane and Dresselhauslike spin polarizations which are forbidden in any sample with twofold rotational symmetry. Our results demonstrate an approach based on spinstructure design for controlling spinorbit torque, enabling highefficient antiferromagnetic spintronics.
Introduction
Currentinduced spinorbit torque enables highly efficient manipulation of magnetization for spintronic applications^{1,2,3,4,5,6,7,8}. In the classical picture of currentinduced magnetization dynamics^{9,10} (Fig. 1f), charge currents in a multilayer sample flowing along the inplane direction (x direction) can generate outofplane spin currents (flowing in the z direction) that have spin polarization σ, required by symmetry to be along the y direction corresponding to a Rashbalike spin polarization. This particular spin current can give rise to an antidamping spin torque in an adjacent ferromagnet, which has magnetization vector m, of the form m × (m × y). The antidamping torque is responsible for efficient magnetization manipulation when the torque is collinear with the magnetization leading to a direct change of the effective magnetic damping. But as the antidamping torque is restricted to lie along an inplane direction, it is efficient for manipulating only samples with magnetic anisotropy along the inplane y axis, not along the outofplane (z) direction or collinear with the current (x direction). Such a limitation of controlling the spintorque or spin polarizations makes the switching of magnetic devices with magnetic anisotropy along the x or z axis nondeterministic and much less efficient (than that along y direction)^{11}.
To efficiently and deterministically drive, for example, perpendicularlymagnetized devices that are preferred for highdensity memories, an outofplane antidamping torque is required. Such an outofplane spintorque can originate from spinorbit scattering from ferromagnetic interfaces^{12,13,14}, or can arise at the interface in systems with reduced symmetry, such as in bilayers of a nonmagnetic transitionmetal dichalcogenide and a ferromagnetic metal^{15}. However, these effects based on interface or heterostructure engineering have not been demonstrated to be strong enough for practical antidamping switching. Here we demonstrate an alternative strategy to achieve unconventional spinorbit torques, based on longrange noncollinear magnetic order within the bulk of the spinsource layer. In particular, we use the spinHall effect in epitaxial thin films of Mn_{3}GaN, a metallic antiferromagnet that has a 120° triangular spin texture, which reduces the symmetry sufficiently to allow spin current generation with different spin polarization directions to generate unconventional spintorques. In heterostructures of epitaxial Mn_{3}GaN/permalloy, we observe not only the outofplane antidamping torque, but also the antidamping torque corresponding to a Dresselhauslike spin polarization^{16,17,18}, besides the conventional Rashbalike symmetry. When the noncollinear spin texture is eliminated by heating above the Néel temperature of Mn_{3}GaN (345 K), the unconventional spintorques go to zero. Such a control of the spin polarizations is coincident with our symmetry analysis and theory calculation upon the magnetic space groups across the Néel transition. Although the spinHall effect has been previously demonstrated in antiferromagnetic thin films, only Rashbalike symmetry has been observed^{19,20,21,22}.
Mn_{3}GaN is a metallic nitride with the antiperovskite crystal structure^{23,24} (identical to the perovskite structure, but with anion and cation positions interchanged) and a lattice parameter close to that of commonly used perovskite oxide substrates. In the bulk, it is known to exhibit antiferromagnetic ordering with a noncollinear Γ^{5g} Kagomelike structure (magnetic space group: \({\mathrm{R}}\bar 3m\)) stabilized by the magnetic frustration of the Mn atoms in the (111) plane (Fig. 1a)^{23,25}. In the (001) plane of Mn_{3}GaN (Fig. 1b), the (110) plane is the only mirror plane. In this lowsymmetry state, we find that charge currents along x generate unconventional antidamping torque components in the form of \({\mathbf{\tau }}_x \propto {\mathbf{m}} \times ({\mathbf{m}} \times {\mathbf{x}})\) and \({\mathbf{\tau }}_z \propto {\mathbf{m}} \times ({\mathbf{m}} \times {\mathbf{z}})\) in addition to the conventional \({\mathbf{\tau }}_y \propto {\mathbf{m}} \times ({\mathbf{m}} \times {\mathbf{y}})\), which correspond to spin currents with σ along x, z, and y, respectively (Fig. 1c). These spin currents have corresponding spinHall conductivities \(\sigma _{zx}^x\), \(\sigma _{zx}^z\), and \(\sigma _{zx}^y\) (in the form of \(\sigma _{jk}^i\), where i, j and k denote the spin polarization, spin flow and charge flow directions). The symmetry allowed, and experimentally observed, nonzero spinHall conductivities are consistent with our linear theory calculation (Supplementary Note 9). Figure 1d shows that the \(\sigma _{zx}^x\), \(\sigma _{zx}^z\), and \(\sigma _{zx}^y\) calculated by using the bulk Mn_{3}GaN band structure are large within a wide energy window around the charge neutrality point, implying the existence of a sizable spinHall current even in the presence of charge carrier doping by defects. Above the antiferromagnetictoparamagnetic transition temperature (Néel temperature T_{N}), disordered spins give rise to a highsymmetry state (space group: \(Pm\bar 3m\)) having 4 mirror planes in the crystal lattice (Fig. 1e), and consequently only the conventional spinHall conductivity \(\sigma _{zx}^y\) can be nonzero. We list the matrices of the spinHall conductivity tensors, obtained from symmetry analysis and calculations, for Mn_{3}GaN in antiferromagnetic and paramagnetic phases in the Supplementary Table 1.
Results
Epitaxial Mn_{3}GaN thin films were grown on (001) (La_{0.3}Sr_{0.7})(Al_{0.65}Ta_{0.35})O_{3} (LSAT) substrates by reactive magnetron sputtering with insitu reflection highenergy electron diffraction (RHEED, see “Methods”). The outofplane Xray diffraction around the (002) LSAT substrate peak shows an epitaxial Mn_{3}GaN film (Fig. 2a). The distinct Kiessig fringes around the Mn_{3}GaN (002) peak and the streaky RHEED pattern of the Mn_{3}GaN film surface (Fig. 2a inset) indicate a high crystalline quality and a smooth film surface. We also confirmed the cubeoncube epitaxial relationship between the Mn_{3}GaN film and underlying LSAT substrate (Supplementary Note 1). Ferromagnetic permalloy Ni_{81}Fe_{19} (Py) and Cu (as a spacer layer) thin films were then deposited in situ on Mn_{3}GaN to form the Py/Cu/Mn_{3}GaN trilayer, and finally were patterned into device bars for spintorque measurements. In Fig. 2b, we show the crosssectional filtered STEMHAADF image of a 30nm Mn_{3}GaN film on LSAT capped with 10nm Py, which reveals sharp interfaces between both Mn_{3}GaN/LSAT (left) and Py/Mn_{3}GaN (right). Atomic force microscope images of the sample surface indicate an atomicallysmooth surface with a surface roughness of ~0.3 nm. Using neutron diffraction, we determined that our 250nm Mn_{3}GaN films order with the bulk antiferromagnetic triangular Γ^{5g} spin structure below a Néel temperature of T_{N} = ~350 K (Supplementary Note 3), consistent with that for the thinner 20nm films having T_{N} = 345 K, but higher than that for polycrystalline bulk samples (T_{N} ~ 290 K), possibly due to less grain boundaries and slight nitrogen deficiency in thin film samples (Supplementary Note 2). Using Xray magnetic linear and circular dichroism with photoemission electron microscopy, we observe antiferromagnetic domains with size on the order of 200–300 nm (Supplementary Note 12). We note that domains with differing spin configurations can affect the unconventional spin torque, since the unconventional spinHall conductivity terms can be averaged out to zero under certain symmetry operations (Supplementary Note 11). The fact that we observe nonzero unconventional spin torques in Mn_{3}GaN, as described below, suggests that certain antiferromagnetic domain configurations are more favorable, which is inferred to be due to a tetragonal distortion that can induce a small noncompensated magnetic moment in Mn_{3}GaN thin films. This unbalanced antiferromagnetic domain population is also evidenced by the finite Xray magnetic linear dichroism (XMLD) signal from the Mn_{3}GaN films at the Mn edge for a beam area 100’s of microns in scale (Supplementary Note 12).
To measure the symmetry of the spin torques, we use the spintorque ferromagnetic resonance (STFMR) technique (Fig. 3a)^{15,26}. During the STFMR measurement, a microwave current applied to Mn_{3}GaN produces alternating torques on the Py, and excites the Py magnetic moment into precession, generating a corresponding alternating sinusoidal change of the resistance R due to the anisotropic magnetoresistance (AMR) of Py. We measure a dc voltage signal V_{mix} across the device bar that arises from the mixing between the alternating current and changes in the device resistance. The resonance in V_{mix} is obtained by sweeping the external inplane magnetic field through the Py resonance condition (see “Methods”). Both inplane and outofplane torque components can then be determined individually, as the symmetric and antisymmetric part of the line shape are proportional to the amplitude of the inplane and outofplane torque components, respectively. Considering only the conventional spinHall effect (or the Rashba–Edelstein effect and Oersted field), the inplane and outofplane torque components would only have the form of m × (m × y) and m × y, respectively^{8,27}. This corresponds to the case of samples containing materials with 2fold rotational symmetry, in which case if m is inverted by rotating the inplane magnetic field angle φ (with respect to x) by 180^{°}, V_{mix} must retain the same amplitude but change sign, giving \(V_{{\mathrm{mix}}}\left( \varphi \right) =  V_{{\mathrm{mix}}}\left( {\varphi + 180^ \circ } \right)\). Any difference in the resonance line shape between \(V_{{\mathrm{mix}}}\left( \varphi \right)\) and \( V_{{\mathrm{mix}}}\left( {\varphi + 180^ \circ } \right)\) indicates the presence of an additional, unconventional torque component.
Figure 3b shows resonance spectra of a 10 nm Py/2 nm Cu/20 nm Mn_{3}GaN sample with the current flow along the [100] direction for the magnetic field angle φ equal to 40^{°} and 220^{°}, measured at room temperature when the Mn_{3}GaN is in the antiferromagnetic state. The Cu insertion layer breaks the exchange coupling at the Py/Mn_{3}GaN interface, but it allows the transmission of the spin current since Cu has a long spin diffusion length. We find that the V_{mix} (40°) and −V_{mix} (220°) scans are notably different in the antiferromagnetic phase, indicating the presence of unconventional torque components^{15}.
To examine the torque components quantitatively, we perform STFMR measurements as a function of the inplane magnetic field angle φ. Figure 3c and d show the angular dependence of symmetric V_{S} and antisymmetric V_{A} part for the 10 nm Py/2 nm Cu/20 nm Mn_{3}GaN sample, measured at room temperature. The angular dependence of STFMR can be understood as the product of the AMR in Py \([dR/d\varphi \propto \sin (2\varphi )]\), with the inplane τ_{} or outofplane torque τ_{⊥} components, as \(V_{\mathrm{S}} \propto \sin \left( {2\varphi } \right)\tau _\parallel\) and \(V_{\mathrm{A}} \propto \sin \left( {2\varphi } \right)\tau _ \bot\). For ferromagnetic metal/normal metal bilayers (i.e., Py/Pt), the conventional antidamping torque \({\mathbf{\tau }}_{{\mathrm{y}},{\mathrm{AD}}} \propto {\mathbf{m}} \times ({\mathbf{m}} \times {\mathbf{y}})\) and fieldlike torque \({\mathbf{\tau }}_{{\mathrm{y}},{\mathrm{FL}}} \propto {\mathbf{m}} \times {\mathbf{y}}\) both have a cos(φ) dependence, giving rise to an overall angular dependence of the form \(\sin \left( {2\varphi } \right){\mathrm{cos}}(\varphi )\) for both V_{S} and V_{A}. We find the angular dependence of both V_{S} and V_{A} for the Mn_{3}GaN clearly deviate from this simple model (Fig. 3c and d, gray line), but can be well fitted by adding additional, unconventional torque terms with the presence of spin currents with spin polarizations oriented away from y. The spin currents that are polarized along x would generate torque [\({\mathbf{\tau }}_{{\mathrm{x}},{\mathrm{AD}}} \propto {\mathbf{m}} \times ({\mathbf{m}} \times {\mathbf{x}})\) and \({\mathbf{\tau }}_{{\mathrm{x}},{\mathrm{FL}}} \propto {\mathbf{m}} \times {\mathbf{x}}\)] with a sin(φ) dependence; while the torques with spin polarization along z [\({\mathbf{\tau }}_{{\mathrm{z}},{\mathrm{AD}}} \propto {\mathbf{m}} \times ({\mathbf{m}} \times {\mathbf{z}})\) and \({\mathbf{\tau }}_{{\mathrm{z}},{\mathrm{FL}}} \propto {\mathbf{m}} \times {\mathbf{z}}\)], since \({\mathbf{m}}\) is oriented in the plane, are independent of φ. We thus fit \(V_{mix,\,S}(\varphi )\) and \(V_{mix,\,A}(\varphi )\) to more general forms to take all possible torque terms into account:
We can then find nonzero antidamping torque terms \(\tau _{{\mathrm{x}},{\mathrm{AD}}}\), \(\tau _{{\mathrm{y}},{\mathrm{AD}}}\), and \(\tau _{{\mathrm{z}},{\mathrm{AD}}}\) (see “Methods” for calculation details) demonstrating the existence of unconventional torque originated from spin polarizations along x and z. This is consistent with the symmetryallowed spin currents derived from the noncollinear antiferromagnetic Mn_{3}GaN magnetic space group through the bulk spinHall effect. This mechanism is distinct from those previously reported in noncentrosymmetric systems, and in magnetic trilayers^{13,14}. The generation of the spin torque \({\mathbf{\tau }}_{{\upnu }},_{\mathrm{AD}}\) relative to the charge current density can be parameterized into the spintorque ratio \(\theta _\nu = \frac{\hbar }{{2{\mathrm{e}}}}\frac{{j_{s,\upsilon }}}{{j_c}}\), where \(j_{s,\upsilon }\) is the spin current density with the spin polarization along ν that is absorbed by the Py, and j_{c} is the charge current density in Mn_{3}GaN estimated from a parallelconduction model. We find at room temperature that \(\theta _x =  0.013 \pm 0.0002\), \(\theta _y = 0.025 \pm 0.0002\) and \(\theta _z = 0.019 \pm 0.0005\), and the spin torque conductivity (\(\sigma _{zx}^\nu = \frac{{\theta _\nu }}{\rho }\frac{\hbar }{{2e}}\), where ρ is the charge resistivity of Mn_{3}GaN) to be around \(\sigma _{zx}^x =  5.9 \times 10^3\,(\hbar /2e)({\mathrm{{\Omega} }}\,m)^{  1}\), \(\sigma _{zx}^y = 1.1 \times 10^4\,(\hbar /2e)({\mathrm{{\Omega} }}\,m)^{  1}\) and \(\sigma _{zx}^z = 8.6 \times 10^3\,(\hbar /2e)({\mathrm{{\Omega} }}\,m)^{  1}\) with \(\rho = 220\,\mu {\mathrm{{\Omega} }}\,{\mathrm{cm}}\). The outofplane fieldlike torque has the form \({\mathbf{\tau }}_{{\mathrm{y}},_{\mathrm{FL}}}\), dominated by the contribution from the currentinduced Oersted field (see “Methods”) with no detectable \({\mathbf{\tau }}_{{\mathrm{x}},{\mathrm{FL}}}\) torque. In addition, we observe an inplane fieldlike torque \({\mathbf{\tau }}_{{\mathrm{z}},{\mathrm{FL}}}\) with a large torque ratio of \(\theta _{{\mathrm{FL}},{\mathrm{z}}} =  0.15 \pm 0.0002\), which could be generated along with \({\mathbf{\tau }}_{{\mathrm{z}},{\mathrm{AD}}}\) by the spin currents polarized along z. The observed unconventional spin torques could also originate from spinorbit precession^{13}, in which a longitudinal spin polarized current from a ferromagnet can scatter from the ferromagnetic/normal metal interface. We note that the longitudinal spin polarized current in the noncollinear antiferromagnet and the transverse spin current that is reported in this paper could share the same physical origin^{28}.
We further confirmed the correlation between the observed unconventional spin polarization and the noncollinear spin structures in Mn_{3}GaN by performing angulardependent STFMR measurements across its antiferromagnetictoparamagnetic phase transition. The Néel temperature of the 20 nm Mn_{3}GaN thin film is determined to be 345 K by tracking the temperature dependence of the outofplane lattice parameter (Fig. 4a) because the magnetic phase transition of Mn_{3}GaN produces a region of strong negative thermal expansion^{29}. Figure 4b–d show the temperature dependence (300 K–380 K) of the ratios between antidamping torque components and the Oersted torque, \(\tau _{{\mathrm{y}},{\mathrm{AD}}}/\tau _{{\mathrm{y}},{\mathrm{FL}}}\), \(\tau _{{\mathrm{x}},{\mathrm{AD}}}/\tau _{{\mathrm{y}},{\mathrm{FL}}}\) and \(\tau _{{\mathrm{z}},{\mathrm{AD}}}/\tau _{{\mathrm{y}},{\mathrm{FL}}}\) (extracted from the full angulardependent STFMR measured at each temperature, see Supplementary Note 7). The unconventional torque ratios \(\tau _{{\mathrm{x}},{\mathrm{AD}}}/\tau _{{\mathrm{y}},{\mathrm{FL}}}\) and \(\tau _{{\mathrm{z}},{\mathrm{AD}}}/\tau _{{\mathrm{y}},{\mathrm{FL}}}\) vanish when the sample temperature is above the Néel temperature, while the conventional component \(\tau _{{\mathrm{y}},{\mathrm{AD}}}/\tau _{{\mathrm{y}},{\mathrm{FL}}}\) remains nonzero, with a weak peak near the transition temperature (a similar peak in \(\tau _{{\mathrm{y}},{\mathrm{AD}}}\) has been observed near the Curie temperature of Fe_{x}Pt_{1x} alloys^{30}). The vanishing of \(\tau _{{\mathrm{x}},{\mathrm{AD}}}/\tau _{{\mathrm{y}},{\mathrm{FL}}}\) and \(\tau _{{\mathrm{z}},{\mathrm{AD}}}/\tau _{{\mathrm{y}},{\mathrm{FL}}}\) directly demonstrates the strong correlation between the noncollinear spin structure and the existence of the unconventional spin torques. We also find that the unconventional torques persist at temperatures well below the Néel temperature (Supplementary Note 8), but decrease gradually at lower temperature with the increase of the uncompensated moment and the onset of the anomalous Hall effect in Mn_{3}GaN (Supplementary Note 4 and 5). The correlation between the observed unconventional spinorbit torques, the uncompensated magnetic moment and the anomalous Hall effect in Mn_{3}GaN at low temperatures requires further study.
Discussion
In summary, we have demonstrated the generation of unconventional spinorbit torque based on lowsymmetry noncollinear spin ordering present in the bulk of an epitaxial antiferromagnetic thin film with an antiperovskite structure. Such unconventional torques can be robustly manipulated by controlling the antiferromagnetic ordering across the Néel temperature. This work provides essential insight into understanding how unconventional spinorbit torques can arise in systems with lower crystalline or magnetic symmetry. This strategy of controlling the spin polarization direction by the design of the spin structure will lead to a much more efficient manipulation and deterministic switching of nanomagnets with arbitrary magnetization, as well as antiferromagnetic tunnel junctions^{28}. In addition, our finding offers the possibility to design and control spin currents through manipulating the noncollinear spin order via strain, temperature, chemical doping, and possibly external excitation, opening new areas of research opportunities in antiferromagnetic spintronics^{31,32,33,34,35}.
Methods
Sample growth, fabrication, and characterization
Epitaxial Mn_{3}GaN thin films were grown on (001)oriented LSAT substrates by DC reactive magnetron sputtering using a stoichiometric Mn_{3}Ga target in a vacuum chamber with a base pressure of 1 × 10^{−8} Torr. During the growth, the Mn_{3}GaN growth mode and surface crystalline structure were monitored by in situ reflection high energy electron diffraction (RHEED). The growth undergoes a 3D to 2D growth mode transition. The streaky RHEED pattern after the deposition implies a smooth film surface (Fig. 2a inset). The growth was performed at a substrate temperature of 550 °C and an Ar (62 sccm)/N_{2} (8 sccm) atmosphere of 10 mTorr. After the Mn_{3}GaN growth, the sample was cooled down in vacuum. The Cu and Py thin films were subsequently sputter deposited at an Ar pressure of 3 mTorr. The atomically flat sample surface was verified using atomic force microscopy (Supplementary Fig. 1). We confirmed the thickness, epitaxial arrangement, and coherence of the Mn_{3}GaN films using Xray reflectivity, Xray diffraction, and reciprocal space mappings. The growth rate of Py and Cu films were calibrated using Xray reflectivity.
We patterned the samples by using photolithography followed by ion beam milling. Then 200 nm Pt/5 nm Ti electrodes were sputter deposited and defined by a liftoff procedure. Devices for STFMR were patterned into microstrips (20–50 μm wide and 40–100 μm long) with groundsignalground electrodes. Devices for electrical transport measurements were patterned into 100 μm wide and 500 μm long Hall bars.
STEM measurements
The STEM sample was prepared through mechanical polishing down to a thickness of ~10 μm by using the precise polishing system (EM TXP, Leica). The polished specimen was then ionmilled using a 1–3 keV Ar ion beam (PIPS II, Gatan) to make the hole for the STEM observation. Afterwards, a low energy milling was performed using 0.1 keV Ar beam to minimize the surface damage from the prior ionmilling process.
The atomic structures were observed using a STEM (JEMARM200F, JEOL) at 200 kV equipped with an aberration corrector (ASCOR, CEOS GmbH). The optimum size of the electron probe was ~0.8 Å. The collection semiangles of the HAADF detector were adjusted from 68 to 280 mrad in order to collect largeangle elastic scattering electrons for clear Zsensitive images. The obtained raw images were processed with a bandpass Wiener filter with a local window to reduce a background noise (HREM research Inc.).
STFMR measurements
During STFMR measurements, a microwave current at a fixed frequency was applied with the inplane magnetic field swept from 0 to 0.2 T for driving the ferromagnetic layer Py through its resonance condition. The amplitude of the microwave current is modulated at a low frequency (1.713 kHz), and the mixing voltage is detected through a lockin amplifier. For the low temperature STFMR measurements (including the room temperature results shown in Fig. 3), the device was wire bonded to a coplanar waveguide and then transferred into a liquid helium flow cryostat. For the high temperature measurements (Fig. 4), the sample is placed on a resistive heater with the device probed by the groundsignalground rf probe. The STFMR resonance line shape can be fitted to a sum of symmetric V_{S} and antisymmetric V_{A} Lorentzian components in the form \(V_{{\mathrm{mix}}} = V_{{\mathrm{mix}},{\mathrm{S}}}\frac{{W^2}}{{(\mu _0H_{{\mathrm{ext}}}  \mu _0H_{{\mathrm{FMR}}})^2 + W^2}} + V_{{\mathrm{mix}},{\mathrm{A}}}\frac{{W(\mu _0H_{{\mathrm{ext}}}  \mu _0H_{{\mathrm{FMR}}})}}{{(\mu _0H_{{\mathrm{ext}}}  \mu _0H_{{\mathrm{FMR}}})^2 + W^2}}\), where W is the halfwidthathalfmaximum resonance linewidth, μ_{0} is the permeability in vacuum and H_{FMR} is the resonance field. The inplane τ_{} and outofplane τ_{⊥} components are proportional to V_{mix,S} and V_{mix,A} components, which can be expressed as,
where I_{rf} is the microwave current, R is the device resistance as a function of inplane magnetic field angle φ due to the AMR of Py, α is the Gilbert damping coefficient, and M_{eff} is the effective magnetization. The AMR of Py is determined by measuring the device resistance as a function of magnetic field angle with a field magnitude of 0.1 T. We calibrate the microwave current I_{rf} by measuring the microwave currentinduced device resistance change due to Joule heating^{36,37} (Supplementary Note 6). The inplane and outofplane torques can be expressed as the angular dependence of the torque components with different spin polarization directions,
The strength of the torque components can then be determined from Eqs. (3–6) with the calibrated I_{rf} values, from which we noticed that the primary contribution to \({\mathbf{\tau }}_{{\mathrm{y}},{\mathrm{FL}}}\) is the currentinduced Oersted field. The spin torque ratios can be expressed as,
where M_{s} and t_{Py} are the saturation magnetization and the thickness of Py; t_{MGN} is the thickness of Mn_{3}GaN. \(\hbar\) is the reduced Planck’s constant and e is the electron charge. The saturation magnetization of Py was measured with SQUID magnetometry, and is indistinguishable from the effective magnetization determined by STFMR.
Electrical transport measurements of Mn_{3}GaN
Electrical transport measurements of Mn_{3}GaN films were performed directly on asgrown films wirebonded in a fourcorner van der Pauw geometry. Both sheet resistance and Hall resistance were measured as a function of temperature and magnetic induction in a Quantum Design Physical Property Measurement System. Film resistivity was computed by solving the van der Pauw equation in conjunction with film thickness as measured with xray reflectivity, while Hall resistance was calculated by summing two orthogonal Hall configurations. The longitudinal resistivity and the Hall resistance of Mn_{3}GaN films vs. temperature are reported in Supplementary Fig. 4.
Temperature dependence of neutron diffraction
Single crystal neutron diffraction measurements were performed on the WISH timeofflight diffractometer^{38} at ISIS, the UK neutron and muon source. A stack of eight, approximately 250 nm thick (001) Mn_{3}GaN film samples with lateral dimensions 10 × 8 mm, were coaligned and oriented for the measurement of nuclear and magnetic diffraction intensities in the (HK0) reciprocal lattice plane. The sample was first mounted within a ^{4}He cryostat, and diffraction patterns were collected from a base temperature of 1.5 K up to 300 K, in 25 K steps. The sample was transferred to a mediumrange furnace, and diffraction patterns were then collected at 320, 340, 360, and 390 K.
Temperature dependence of Xray diffraction
The Xray diffraction data were acquired at beamline 6IDB at the Advanced Photon Source with 12 keV incident Xray energy. The sample temperature was controlled employing an ARS high temperature cryostat. Data were collected with 5 K steps; and at each temperature the sample position was realigned with respect to the basetemperature reciprocal space matrix. The sample was mounted on a standard PSI Huber diffractometer. The representative temperature dependence of xray diffraction spectra around the LSAT (003) reflection can be found in Supplementary Fig. 2.
Theoretical calculations
The electronic band structure of Mn_{3}GaN was calculated by using firstprinciples density functional theory (DFT) with Quantum ESPRESSO^{39} and fully relativistic ultrasoft pseudopotentials^{40}. The exchange and correlation effects were treated within the generalized gradient approximation (GGA)^{41}. The planewave cutoff energy of 57 Ry and a 16 × 16 × 16 kpoint mesh in the irreducible Brillouin zone were used in the calculations. Spinorbit coupling and noncollinear Γ^{5g} antiferromagnetism were included in all electronic structure calculations. We note that even though the spinorbit coupling in Mn_{3}GaN is relatively small, it still plays an important role to couple the spin and the lattice, which lifts the spin rotation symmetry and allows the existence of the nonvanishing spinHall conductivity (Supplementary Note 10). The calculated band structures for Mn_{3}GaN in antiferromagnetic and paramagnetic phases are shown in Supplementary Fig. 9.
The spinHall effect is given by^{42}
where \(f_{n\vec k}\) is the FermiDirac distribution for the nth band, \(J_i^k = \frac{1}{2}\{ v_i,s_k\}\) is the spin current operator with spin operator S_{k}, \(v_j = \frac{1}{\hbar }\,\frac{{\partial H}}{{\partial k_j}}\) is the velocity operator, and \(i,j,k = x,y,z\). \({\mathrm{{\Omega} }}_{n,ij}^k\left( {\vec k} \right)\) is referred to as the spin Berry curvature in analogy to the ordinary Berry curvature. In order to calculate the spin Hall conductivities, we construct the tightbinding Hamiltonians using PAOFLOW code^{43} based on the projection of the pseudoatomic orbitals (PAO)^{44,45} from the nonselfconsistent calculations with a 16 × 16 × 16 kpoint mesh. The spinHall conductivities were calculated using the tightbinding Hamiltonians with a 48 × 48 × 48 kpoint mesh by the adaptive broadening method to get the converged values.
Synchrotron spectroscopy and microscopy
Xray magnetic circular dichroism (XMCD) and Xray magnetic linear dichroism (XMLD) spectroscopy were measured at beamline 4.0.2, and Xray microscopy at beamline 11.0.1.1 (PEEM3) at the Advanced Light Source (ALS). In spectroscopy, totalelectronyield mode was employed by monitoring the sample drain current, and a grazing incidence angle of 30° to the sample surface along the [110] direction to probe the magnetic state. The obtained dichroism energies giving information on the magnetic nature of the Mn_{3}GaN were then used to image the domain texture by Xray photoemission electron microscopy (XPEEM), also taken with Xrays at a 30° grazing incidence along the [110] direction. Images taken at the maximum dichroism energies as a function of polarization were normalized by preedge energy images in order to minimize any topographic and work function contrast while emphasizing the magnetic contrast of the Mn_{3}GaN films.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by the National Science Foundation (NSF) under DMREF Grant DMR1629270, the University of Wisconsin Materials Research Science and Engineering Center (DMR1720415), the Army Research Office through Grant W911NF1710462 and AFOSR FA95501510334. Transport and magnetization measurement at the University of WisconsinMadison was supported by the US Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences under Award DEFG0206ER46327. Theoretical research at University of NebraskaLincoln (UNL) was partly supported by the NSF Materials Research Science and Engineering Center (MRSEC) program under Grant DMR1420645. Computations were performed at the UNL’s Holland Computing Center. Work at Cornell was supported by the NSF (DMR1708499), and was performed in part at the Cornell NanoScale Facility, an NNCI member supported by NSF Grant NNCI1542081 and also in part in the shared facilities of the Cornell Center for Materials Research which are supported by the NSF MRSEC program (DMR1719875). This research used resources of the Advanced Photon Source, a U.S. DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DEAC0206CH11357. This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DEAC0205CH11231.
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T.N. and C.B.E. conceived the research. C.B.E., M.S.R., E.Y.T., D.C.R, P.G.R., T.T., and S.Y.C. supervised the experiments. T.N., C.X.Q., and L.G. performed the sample growth and surface/structural characterizations. T.N. performed the device fabrication. T.N. and J.G performed STFMR measurements and analysis. N.C. and J.I. performed magnetic and electrical transport characterizations. G.G. and D.F.S. performed the theoretical calculations. K.S. and S.Y.C. carried out the highresolution TEM experiments. R.D.J., P.M., and P.G.R. conceived and performed the neutron diffraction experiment. P.R., J.W.K., and Y.S.C. carried out the synchrotron diffraction measurements. R.V.C. and I.H. performed synchrotron spectroscopy measurements, and R.V.C. performed and analyzed synchrotron microscopy measurements. T.N., J.I., D.F.S., D.C.R., and C.B.E. wrote the paper. All authors discussed the results and commented on the paper. C.B.E. directed the research.
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T.N. and C.B.E. have filed a patent on the technology in this paper. The remaining authors declare no competing interests.
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Nan, T., Quintela, C.X., Irwin, J. et al. Controlling spin current polarization through noncollinear antiferromagnetism. Nat Commun 11, 4671 (2020). https://doi.org/10.1038/s41467020179994
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DOI: https://doi.org/10.1038/s41467020179994
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