Abstract
Using antiferromagnets as active elements in spintronics requires the ability to manipulate and readout the Néel vector orientation. Here we demonstrate for Mn_{2}Au, a good conductor with a high ordering temperature suitable for applications, reproducible switching using current pulse generated bulk spinorbit torques and readout by magnetoresistance measurements. Reversible and consistent changes of the longitudinal resistance and planar Hall voltage of starpatterned epitaxial Mn_{2}Au(001) thin films were generated by pulse current densities of ≃10^{7} A/cm^{2}. The symmetry of the torques agrees with theoretical predictions and a large readout magnetoresistance effect of more than ≃6% is reproduced by ab initio transport calculations.
Introduction
Antiferromagnets (AFMs) are magnetically ordered materials which exhibit no net moment and thus are insensitive to magnetic fields. Antiferromagnetic spintronics^{1} aims to take advantage of this insensitivity for enhanced stability, while at the same time active manipulation up to the natural THz dynamic speeds of AFMs^{2} is possible, thus combining exceptional storage density and ultrafast switching. However, the active manipulation and readout of the Néel vector (staggered moment) orientation is challenging. Recent predictions have opened up a path based on a new spinorbit torque^{3}, which couples directly to the Néel order parameter. This Néel spinorbit torque was first experimentally demonstrated in a pioneering work using semimetallic CuMnAs^{4}.
For the key application operations of reading and writing in AFMs, different approaches have been previously put forward. Initial experiments on spinvalve structures with an AFM as the active layer manipulated the Néel vector by an exchangespring effect with a ferromagnet (FM) and readout via tunnelinganisotropic magnetoresistance (TAMR) measurements^{5}. Other related experiments were based on the same effect^{6}, or on a FM to AFM phase transition^{7}. However, the most promising approach is to use currentinduced spinorbit torques for switching the Néel vector. It exhibits superior scaling and its counterpart in FMs is already established and considered among the most efficient switching mechanisms for memory applications^{8,9}.
Only two compounds, CuMnAs and Mn_{2}Au, are known to provide at room temperature the collinear commensurate antiferromagnetic order and specific crystal structure, which is predicted to result in the staggered spin accumulation in the sublattice structure, leading to bulk Néel spinorbit torques allowing for currentinduced switching of the Néel vector^{3}.
Semimetallic CuMnAs was grown previously by molecular beam epitaxy (MBE) with a Néel temperature of ≃500 K^{10} and currentinduced switching of these samples was recently demonstrated for the first time^{4,11}. However, for spintronics applications the compound Mn_{2}Au provides several advantages, as it is a good metallic conductor and does not contain toxic components. Furthermore, its magnetic ordering temperature is well above 1000 K^{12}, providing the necessary thermal stability for applications. Mn_{2}Au shows a simple antiferromagnetic structure with the collinear magnetic moments in the (001) plane^{12,13,14}. Thin film samples were previously grown in (101) orientation by MBE^{15} and Fe/Mn_{2}Au(101) bilayers showed AMR effects of up to 2.5% in a 14 T rotating magnetic field^{16}.
While Mn_{2}Au was the first compound for which currentinduced internal staggered spinorbit torques were predicted^{3}, corresponding experimental evidence has been missing. Here we report currentinduced Néel vector switching in Mn_{2}Au(001) epitaxial thin films, which is easily readout by a large AMR. We compare the experimental results with calculations of the AMR from ab initio theory, from which we conclude an extrinsic disorder related origin of the exceptionally large magnetoresistance.
Results
Mn_{2}Au samples preparation
Our Al_{2}O_{3}(Substrate)/Ta(10 nm)/Mn_{2}Au(75 nm)/Ta(3 nm) samples were prepared by radio frequency sputtering from a single stoichiometric target and structurally and magnetically characterized as described elsewhere^{17}. By xray diffraction, we demonstrated that the Mn_{2}Au thin films grow with the (001) axis perpendicular to the thin film surface. The inplane orientation is given by the epitaxial relation with the Ta(001) buffer layer, which results in the [100] direction of the Mn_{2}Au thin films aligned parallel to the [100] direction of the epitaxial Ta layer. These samples were patterned into a star structure as shown in Fig. 1a. This geometry allows for electric writing of the Néel vector orientation by pulsing currents along the two perpendicular directions \(I_{{\mathrm{pulse}}}^{\mathrm{1}}\) and \(I_{{\mathrm{pulse}}}^{\mathrm{2}}\) and for electric readout by measuring either the transversal resistivity ρ_{ xy }, i.e., the Planar Hall Effect (PHE), or the longitudinal resistivity ρ_{long}, corresponding to the AMR of the samples. Depending on the inplane orientation of the patterned structure, the pulse currents can be sent along different crystallographic directions, i.e., along [100] or along [110] (45° inplane rotation of the star pattern, Fig. 1b–d).
Relation of AMR and PHE
The AMR of a single domain sample is given by
where ρ_{long} is longitudinal resistivity, ϕ is the angle between the Néel vector and current direction and [hkl] is the Néel vector orientation in the basis of the tetragonal conventional unit cell (Fig. 1b). The PHE usually observed in ferromagnetic materials scales with the AMR and shows a dependence on the angle ϕ given by^{18,19}:
Thus also in AFMs, ρ_{ xy } has its maximum value and changes sign if ϕ switches from +45° to −45°.
Switching and readout
For technical reasons the applicable pulse length for switching was limited to a minimum of 1 ms. This allows us using an oscilloscope to monitor the timedependent sample resistivity during the application of the current pulses (Fig. 2). By comparison with the temperature dependence of the resistivity of our Mn_{2}Au thin films obtained from separate measurements, this allows to study the current pulse induced heating effects. As shown in Fig. 2, significant heating with local temperatures up to 300 °C is obtained for the highest possible current densities not resulting in sample destruction.
Currentinduced Néel vector switching was only observed applying pulse current densities associated with notable heating. As single pulses resulted in very small changes of the readout signals only, trains of 100 current pulses with a pulse length of 1 ms and a delay between the pulses of 10 ms were applied. This relatively long delay between the pulses was chosen to avoid the destruction of the sample by excessive accumulated heating. After a pulse train thermal relaxation on a time scale of 1 s was observed. Thus the readout was performed with a delay of 10 s. The magnetic state of a sample probed this way is longterm stable, i.e., no changes of the readout signals were observed within the probed time scales of up to 1 h.
Figure 3 shows the transversal resistivity ρ_{ xy } versus the number of applied pulse trains. First, a pulse current density of 1.4 × 10^{7} A/cm^{2} was applied along the [1\(\bar 1\)0] direction, resulting in a small change of the corresponding Hall voltage. Without reaching saturation after 50 pulse trains the pulse current direction was switched to [110], resulting in a reversal of the corresponding change of the transversal resistivity. This sequence could be reproduced several times. Increased pulse current densities of 1.7 × 10^{7} and 1.8 × 10^{7} A/cm^{2} resulted in larger changes of the corresponding Hall voltages. Those current densities are about 1 order of magnitude smaller than predicted by Železný et al.^{3,20}, which could be related to thermal activation processes enabled by the above mentioned current pulse induced heating. By increasing the number of pulse trains applied along the [110] direction to 500, a trend towards saturation of the Hall voltage was obtained.
Internal field like spinorbit torques are expected to generate reversible switching between distinct stable states if the current is injected along biaxial easy directions^{3,21}. However, we observed reversible switching to stable states for pulse currents along both the crystallographic [110] and [100] axes (rotated star pattern). Thus we conclude that the inplane magnetic anisotropy of our Mn_{2}Au thin films is weak. This is consistent with our calculations of the magnetocrystalline anisotropy energy (MAE), which is almost negligible within the (001) plane (Methods section).
An example of the resulting changes of the transversal and longitudinal resistivities generated by pulse currents along the [100] and [010] directions is displayed in Fig. 4. The inset of Fig. 4a shows that it is possible to apply current pulses until a large degree of saturation of the induced longitudinal resistivity changes is obtained. However, this is achieved for current densities that result for prolonged injection of pulses in the destruction of the samples. Thus to allow for an increased number of current pulse cycles with different polarities this regime has to be avoided. The current density at which eventually the sample is destroyed varies by a factor of ≃2 from sample to sample. The main upper panel (Fig. 4a) shows the longitudinal resistivity probed after each of the first 1600 pulse trains consisting of 100 pulses each with a current density 1.0 × 10^{7} A/cm^{2}. For the first sequences only small variations of the longitudinal resistivity were observed. However, with the application of subsequent pulse trains the magnitude of the effect increased. This traininglike behaviour may be associated with the motion and pinning of AFM domain walls in the sample. After 1600 pulse trains a constant resistance change of \({\mathrm{\Delta }}\rho _{{\mathrm{long}}}{\mathrm{/}}\bar \rho = 2.5\)% induced by 100 pulse trains was reached, which is an order of magnitude higher than what is observed for CuMnAs^{4}.
To check the origin of these changes, the transversal resistivity of the sample was measured and also showed reproducible pulse currentinduced changes (Fig. 4b). The increase of the transversal resistivity induced by 100 pulse trains amounted to ρ_{ xy } = 0.5 μΩ cm.
Based on these numbers the identification of the longitudinal and transversal resistivities with the AMR and PHE can be verified: If both effects originate from the same anisotropic electron scattering, they have to be related by Eq. (2). We assume a switching of the Néel vector in parts of the sample corresponding to a change of ϕ in Eq. (2) from +45° to −45°, i.e.,
Thus we find that ρ_{ xy } = 0.5 μΩ cm corresponds again to \({\mathrm{\Delta }}\rho _{{\mathrm{long}}}{\mathrm{/}}\bar \rho = 2.5\)%. This consistency of the longitudinal and transversal resistivities provides strong evidence for an intrinsic electronic origin of the pulse currentinduced changes of the magnetoresistance signals.
After two more pulse current direction reversals reproducing the previous behaviour of the sample, the pulse current direction was kept along [100] for 800 additional pulse trains. This resulted in a sign reversal of the PHE, which according to Eq. (2) corresponds also to a sign change of the angle ϕ between the Néel vector and the current direction. Although a small offset of the transversal voltage due to e. g. imperfections of the patterned structure is possible, the magnitude of the transversal voltage measured with both signs can only be explained by a switching of the Néel vector. As no saturation of the readout signals without destroying the samples could be reached, we can conclude that ϕ was switched from +45° to −45° for the most part of the sample, but not everywhere. After about 300 pulse trains along the [100] direction a beginning saturation of the PHE resistivity appeared, but was not completed when after 500 additional pulse trains the sample broke. A maximum transversal resistivity of ρ_{ xy } = 1.25 μΩ cm was reached, which based on Eq. (3) corresponds to an AMR of 6.25%. This is one of the largest AMR ratios found in metallic magnetic thin films, and its size bodes well for easy readout of the antiferromagnetic state as necessary for device applications. While small variations exist between samples, we observe consistently larger AMR effects for pulse currents along the [100] than for the [110] directions.
Origin of the AMR
To understand the origin of the magnetoresistance effects, we calculated the AMR of single domain Mn_{2}Au assuming a complete 90° switching of the Néel vector. In general, the AMR originates from effects of spinorbit coupling on the band structure^{22} and from scattering from an extrinsic disorder potential^{23}. Incorporating the effects of realistic disorder in the calculations, two types of disorder were considered: offstoichiometry and intersite swapping between Mn and Au atoms. Experimentally, the former was analyzed by energy dispersive xray spectroscopy (EDX) of 500 nm thick Mn_{2}Au films resulting in a stoichiometry of 66.2 ± 0.3% Mn and 33.8 ± 0.3% Au, which indicates a slight Au excess. In addition, a small degree of intersite disorder is to be expected, but its quantification is experimentally not accessible. Thus we simulated a slight excess of Au randomly distributed over the Mn sites and random Mn–Au swapping.
We calculated the AMR for two crystal directions of the Néel vector, AMR_{100} and AMR_{110} from ab initio theory. We employ the fully relativistic Dirac tightbinding linearmuffin tin orbital (FRDTBLMTO) density functional theory (DFT) combined with the KuboGreenwood formula for the longitudinal conductivity (see refs. ^{24,25,26}):
where V_{x,k} is the velocity operator labeled by the Cartesian coordinate x, and wavenumber k, \({\mathrm{Im}}\overline g _{\bf{k}}\) is the auxiliary Green function within the TBLMTO formalism^{26}, all evaluated at the Fermi level, \(\overline g\) denotes the configuration averaging in the presence of disorder, and v.c. are vertex corrections^{27}. We treat the scattering in the chemicaly disordered Mn_{2}Au within the coherent potential approximation (CPA). To reveal the role of the AMR contribution from spinorbit coupling effects in the band structure, we also calculated the AMR of Mn_{2}Au without chemical disorder in the constant relaxation time approximation (RTA) (see Methods section). The theoretical AMR is obtained by calculating the resistivity tensor ρ = σ^{−1} from Eq. (4) and substituting it into Eq. (1).
Figure 5a shows the AMR results obtained within the CPA for different degrees of disorder in Mn_{2}Au. Large AMR values between 5 and 6%, which are consistent with our experiments, were obtained for Au excess generated scattering. Large AMR values consistent with our experiments were obtained for small degrees of disorder reaching a maximum value of 6.3% for 0.5% excess of Au. Moreover, we obtain AMR_{100} > AMR_{110}, which reproduces the experimentally observed trend for the two crystalline directions. The corresponding calculated residual resistivities as shown in Fig. 5b are consistent with the experimentally obtained values (≃8 μΩ cm)^{17}, corroborating the relevance of the simulated type of disorder.
The two types of disorder have different effects on the density of states (DOS) calculated within the FRDTBLMTO+CPA^{24,26}. In the case of the Mn–Au swapping disorder, Mn located at the Au sites creates a virtual bound states at the Fermi level as shown in Fig. 6a, which increases the resistivity (Fig. 5b). However, such states are not created by Au excess as shown in Fig. 6b. Thus these virtual bound states (VBS) are the dominating origin of the disorder induced resistivity increase but not of the large AMR. We note that the VBS are observed for the all simulated concentrations of Mn–Au swapping and for the sake of clarity we choose in Fig. 6 the largest simulated disorder strength since it leads to the better visibility of the VBS peak in the total density of states.
Instead of considering a specific type of disorder, scattering can also be treated within the RTA by broadening the imaginary part of the complex energy Imz, which introduces unspecific lifetime effects on the electronic states, but otherwise keeps the band structure unperturbed by the disorder. With Imz ~ 13 meV we achieved a residual resistivity ≃10 μΩ cm, which corresponds to the experimental low temperature values. Within this approximation we obtained AMR_{100}~1% and AMR_{110}~0.1%. As these RTA values are significantly smaller than the CPA values, they indicate an extrinsic origin of the experimentally observed large AMR related to the Au excess on Mn sites as described by the CPA.
To further elucidate the effect of this Au excess on the electronic structure and related origin of the large scattering contribution to the AMR, we calculated the Bloch spectral function \(A_{\bf{k}} =  \frac{1}{\pi }{\mathrm{Im}}\overline G _{\bf{k}}\). Here the physical Green function \(\overline G\) is related to the auxiliary Green function \(\overline g\) in Eq. (4)^{26} and is calculated within the CPA along the high symmetry lines in the Brillouin zone of Mn_{2}Au with 0.5% excess of Au. Since the vertex correction turned out to be negligible in Mn_{2}Au with excess of Au, the Bloch spectral function represents an appropriate visualization of the disordered effects. As shown in Fig. 7 the spectral weight at the Fermi level along MΓ line increases when rotating the Néel vector from  [100] (Fig. 7a) to  [010] (Fig. 7b). This suggests that the sharper spectral weight in the direction parallel as compared to perpendicular to the current direction effectively increases the corresponding relaxation time and thus reduces the resistivity. Please note that this picture remains qualitatively unchanged also for higher degrees of disorder, which we explicitly verified for 5% Au excess. This together with the kspace anisotropy of the MΓ band around the Fermi level can explain the high value of AMR_{100} experimentally observed for Aurich Mn_{2}Au.
Discussion
Inplane switching of the Néel vector in the antiferromagnetic metal Mn_{2}Au by current pulses was realized using intrinsic spinorbit torques. Consistent measurements of the AMR and PHE showed pulse current direction dependent reversible changes, providing direct evidence for Néel vector switching. Easy readout of the switching is provided by a large amplitude of the AMR of more than 6%, which is more than an order of magnitude higher than previously observed for other antiferromagnetic systems and one of the highest AMR amplitudes found for metallic magnetic thin films. We can reproduce the magnitude of the effect theoretically by including realistic disorder and, in particular, find the same dependence of the amplitude on the crystallographic directions in the experiment as in the calculation. With the basic principles of writing and readout demonstrated, combined with a theoretical understanding of the underlying spinorbit torques, and the large magnetoresistive effects, the metallic compound Mn_{2}Au is a prime candidate to enable future AFM spintronics.
Methods
MAE calculation
The MAE of chemically ordered Mn_{2}Au was calculated using the FLAPW (full potential linearized augmented plane wave)+GGA (Generalized Gradiend Approximation) method in combination with the magnetic force theorem^{28}. We found the inplane MAE: \(E_{[100]}  E_{[110]}\) ≲ \(\left { \pm 10} \right\) μeV per formula unit. However, this method is suitable only for fully ordered crystal structures. Thus to realistically describe our experiments we calculated the MAE within the TBLMTO+CPA also for disordered Mn_{2}Au. For all concentrations of the simulated types of disorder we obtained the outofplane MAE \(E_{[001]}  E_{[100]}\) ≃ 2.9 meV. The inplane MAE calculation gives E_{[100]} − E_{[110]} from −1 to −3 μeV per formula unit at the resolution limit of our methods. While the outofplane MAE is similar to the reported value for chemically ordered Mn_{2}Au^{29}, the inplane MAE of the disordered material has opposite sign. Independent from the sign the tiny value of the inplane MAE at the resolution limit of the calculations is consistent with the experimental observation that by current switching the [100] as well as the [110] Néel vector orientation could be stabilized in disordered Mn_{2}Au.
AMR calculation
To calculate the AMR of Mn_{2}Au ab initio we employed the FRDTBLMTO+CPA method in combination with the Kubo formula^{24,25,26}. stype, ptype, and dtype orbitals were included in the basis and the LSDA (local spin density approximation) and the VoskoWilkNusair exchangecorrelation potential parametrization^{30} were used. The groundstate magnetization and density of states was reproduced consistently with a previous report^{31}. In the transport calculations we used up to 10^{10} k points in the Brillouin zone and for the CPA residual resistivity calculations we set the imaginary part of the complex energy to 0.13 meV. In the RTA, the imaginary part of the complex energy Imz in the Bloch spectral function A_{n,k}(z) was approximated by a finite isotropic kindependent relaxation time. In the CPA, the effective medium was constructed corresponding to a random averaging of the occupancies of the disordered sites. The CPA Bloch spectral function is anisotropic and kdependent and determined by the effective medium potential.
We compared the RTA and the CPA derived calculated resistivities with the corresponding experimental values, from which we concluded that the level of disorder in our samples is at least 0.5%. Resistivity calculations for lower disorder values around, e.g., 0.1% correspond to 10–20 times smaller residual resistivities than observed in experiment and are thus not relevant in the framework of our manuscript. Nevertheless, we also calculated AMR_{100} for 0.1% of Au excess resulting in a value of (3.7 ± 1.5)%. The large error of this calculation is due to the increased number of iterations and k points needed for calculations with such a tiny degree of disorder.
Data availability
The relevant data are available within the article or from the authors on reasonable request.
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Acknowledgements
This work is supported by the German Research Foundation (DFG) through the Transregional Collaborative Research Center SFB/TRR173 Spin+X, Projects A03 and A05. J.S., L.S., O.G., and T.J. acknowledge the support of the Alexander von Humboldt Foundation, the ERC Synergy Grant SC2 (No. 610115), the Ministry of Education of the Czech Republic Grant No. LM2015087, and the Grant Agency of the Czech Republic grant no. 1437427G, L.S. acknowledges support from the Grant Agency of the Charles University, no. 280815. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the program Projects of Large Research, Development, and Innovations Infrastructures (CESNET LM2015042), is greatly appreciated. This work was supported by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project IT4 Innovations National Supercomputing Center LM2015070. The work of I.T. was supported by the Czech Science Foundation (Grant No. 1437427G).
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S.Y.B. and M.J. mainly wrote the paper and performed the transport measurements; L.S. performed the AMR and anisotropy calculations and wrote the corresponding part of the manuscript, I.T. developed the codes for the transport calculations, S.Y.B. and A.A.S. prepared the samples, H.J.E., M.K., O.G., T.J., I.T., and J.S. discussed the results and contributed to the writing of the manuscript; M.J. coordinated the project.
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Bodnar, S.Y., Šmejkal, L., Turek, I. et al. Writing and reading antiferromagnetic Mn_{2}Au by Néel spinorbit torques and large anisotropic magnetoresistance. Nat Commun 9, 348 (2018). https://doi.org/10.1038/s4146701702780x
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DOI: https://doi.org/10.1038/s4146701702780x
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