Abstract
Electric currents carrying a net spin polarization are widely used in spintronics, whereas globally spinneutral currents are expected to play no role in spindependent phenomena. Here we show that, in contrast to this common expectation, spinindependent conductance in compensated antiferromagnets and normal metals can be efficiently exploited in spintronics, provided their magnetic space group symmetry supports a nonspindegenerate Fermi surface. Due to their momentumdependent spin polarization, such antiferromagnets can be used as active elements in antiferromagnetic tunnel junctions (AFMTJs) and produce a giant tunneling magnetoresistance (TMR) effect. Using RuO_{2} as a representative compensated antiferromagnet exhibiting spinindependent conductance along the [001] direction but a nonspindegenerate Fermi surface, we design a RuO_{2}/TiO_{2}/RuO_{2} (001) AFMTJ, where a globally spinneutral charge current is controlled by the relative orientation of the Néel vectors of the two RuO_{2} electrodes, resulting in the TMR effect as large as ~500%. These results are expanded to normal metals which can be used as a counter electrode in AFMTJs with a single antiferromagnetic layer or other elements in spintronic devices. Our work uncovers an unexplored potential of the materials with no global spin polarization for utilizing them in spintronics.
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Introduction
The field of spintronics utilizes the spin degree of freedom in condensed matter for information processing and storage^{1}. Most spintronic applications rely on electric currents with sizable spin polarization for detection or manipulation of the magnetic order parameter in spintronic devices. A typical and widely used spintronic device is the magnetic tunnel junction (MTJ), where a longitudinal charge current spin polarized by one ferromagnetic metal quantummechanically tunnels into another ferromagnetic metal through an insulating barrier layer^{2,3}. Conductance of the MTJ is controlled by the relative magnetization orientation of the two ferromagnetic electrodes, resulting in a tunneling magnetoresistance (TMR) effect^{4}.
Contrary to the spinpolarized currents, spinneutral currents are usually considered impractical for spintronics due to being unable to directly interact with the magnetic order parameter. This fact challenges spintronics based on nonferromagnetic materials, such as compensated antiferromagnets, which normally do not support spinpolarized currents. Due to being robust against magnetic perturbations, the absence of stray fields, and ultrafast spin dynamics, antiferromagnets are considered as outstanding candidates to replace the widely used ferromagnets in the next generation spintronics^{5,6,7,8,9}. This promising route has been recently stimulated by the demonstrated control of the antiferromagnetic Néel vector by spinorbit torques^{10,11}. However, the absence of a net magnetization and hence spinindependent conductance makes the electrical detection of the Néel vector using conventional methods, such as TMR measurements, unfeasible^{5}. So far, the electrical detection of the Néel vector has been performed using anisotropic^{10,11} or spinHall^{12,13,14,15} magnetoresistance. Unfortunately, both methods suffer from relatively small signals easily influenced by perturbations^{16} and require multiple inplane terminals resulting in large device dimensions^{7}. Antiferromagnetic spin valves^{17,18,19,20} and antiferromagnetic tunnel junctions (AFMTJs)^{21,22} have been theoretically proposed, promising, in some cases, sizable magnetoresistance effects. However, these magnetoresistance effects rely on perfect interfaces and switching the interfacial magnetic moment alignment between parallel and antiparallel. This mechanism is not robust against disorder and interface roughness inevitable in experimental conditions. Recent efforts have been aimed at exploring unconventional methods for the Néel vector detection based on topological properties^{9,23,24,25} but require an experimental confirmation.
One promising direction is to create spinpolarized currents in antiferromagnets. Recently, it has been predicted that certain types of compensated antiferromagnets exhibit a momentumdependent spin splitting of the Fermi surface^{26,27,28}, resulting in spinpolarized currents along certain crystallographic orientations^{29,30,31}. These predictions indicate that these antiferromagnets can work as ferromagnets in spintronic devices, which broadens the range of materials useful for spintronics.
Here, we embark on a different path and argue that globally spinindependent conductance in compensated antiferromagnets can be efficiently used in spintronics, provided their crystal symmetry supports a nonspindegenerate Fermi surface and thus momentumdependent spin polarization. While such a spin polarization is cancelled out in the net conductance due to being antisymmetric with respect to certain symmetry operations, its presence in the momentum space can be functionalized if such an antiferromagnet is combined with another similar antiferromagnet in a spintronic device such as an AFMTJ. In this case, the resistance change of the AFMTJ occurs in response to the orientation of the antiferromagnetic Néel vector due to changing matching conditions between the spinpolarized conduction channels in the two metal electrodes. These considerations can be expanded to normal (nonmagnetic) metals with spinorbit coupling where the combined space inversiontime reversal symmetry is broken, indicating that they can also be utilized in spintronics despite globally spinneutral currents.
Results
Spin polarized conduction channels
To explore the possible use of spinneutral currents in a spintronic device, we first consider ballistic conductance of a material under investigation. Since the ballistic conductance is determined by the number of conduction channels, i.e. propagating Bloch states at the Fermi energy, it can provide an important characteristic of a spintronic device where this material is used as a metal electrode^{32,33}. In the absence of spinorbit coupling, the ballistic conductance g per unit area along the z direction can be obtained in terms of two spin components as follows^{34}.
Here \(\sigma\) denotes the spin component ↑ or ↓, \(\overrightarrow{k}\) is the wave vector in the threedimensional Brillouin zone, \({N}_{\parallel }^{\sigma }({\overrightarrow{k}}_{\parallel })\) is the number of conduction channels (integer) at the transverse wave vector \({\overrightarrow{k}}_{\parallel }=({k}_{x},{k}_{y})\) for spin \(\sigma\), \({E}_{n}^{{{{{{\rm{\sigma }}}}}}}\) is energy for the nth band, \({v}_{nz}^{\sigma }=\frac{\partial {E}_{n}^{\sigma }(\overrightarrow{k})}{\hslash \partial {k}_{z}}\) is the band velocity along the transport z direction, and \(f\) is the Fermi distribution function.
The net transport spin polarization is defined by
and represents an important quantity useful in spintronics. For example, in a crude approximation of \({\overrightarrow{k}}_{\parallel }\)independent transmission between two ferromagnetic electrodes with spin polarizations \({p}_{1}\) and \({p}_{2}\) in an MTJ, the TMR effect is given by the wellknown Julliere’s formula^{2} \(TMR=\frac{2{p}_{1}{p}_{2}}{1{p}_{1}{p}_{2}}\). Clearly, a larger spin polarization of the electrodes favors a larger TMR.
A large spin polarization \(p\) is generally expected for ferromagnets where the finite net magnetization breaks time reversal symmetry \(\hat{T}\) . The latter flips the spin \(\sigma\) and changes sign of \({\overrightarrow{k}}_{\parallel }\), resulting in \(\hat{T}{N}_{\parallel }^{\uparrow }({\overrightarrow{k}}_{\parallel })\,={N}_{\parallel }^{\downarrow }({\overrightarrow{k}}_{\parallel })\). Compensated antiferromagnets do not have net magnetization and hence (with some exceptions^{29,30,31}) do not support the macroscopic spinpolarized current. However, even though macroscopically the net transport spin polarization \(p\) is absent, microscopically the conductance could be spin polarized as reflected in the spin polarization of conduction channels at \({\overrightarrow{k}}_{\parallel }\):
If both electrodes in a twoterminal spintronic device are made of materials with zero net spin polarization but have spinpolarized conduction channels, this momentumdependent spin polarization will be reflected in the device conductance and can be functionalized through the antiferromagnetic Néel vector. Indeed, in the transport regime conserving spin (no spinorbit coupling) and wave vector \({\overrightarrow{k}}_{\parallel }\) (no diffuse scattering), the device conductance is largely affected by the spin matching of the conduction channels \({\overrightarrow{k}}_{\parallel }\) of the electrodes. If their spin polarization changes in response to the Néel vector rotation in an antiferromagnetic electrode, this alters the net conductance of the device. Thus, although the spin polarization of the charge current remains zero, the device conductance reflects the \({\overrightarrow{k}}_{\parallel }\) dependent spin polarization of the conduction channels.
Next, we identify magnetic space group symmetry requirements for crystals to exhibit the \({\overrightarrow{k}}_{\parallel }\)dependent spin polarization. Obviously, in ferromagnets, the conduction channels are spin polarized due to the spindependent Fermi surface. Thus, passing a spinneutral current through a ferromagnetic material makes it spin polarized (Fig. 1a). On the contrary, most compensated antiferromagnets contain symmetries that not only prevent the net magnetization but also lead to a spindegenerate Fermi surface and thus spinindependent conduction channels. For example, if a compensated antiferromagnet exhibits \(\hat{P}\hat{T}\) symmetry, where \(\hat{P}\) and \(\hat{T}\,\) are space inversion and time reversal symmetries, respectively, \({p}_{\parallel }=0\) due to \(\hat{P}\hat{T}{N}_{\parallel }^{\uparrow }({\overrightarrow{k}}_{\parallel })\,={N}_{\parallel }^{\downarrow }({\overrightarrow{k}}_{\parallel })\). This property follows from the spindegenerate Fermi surface due to \(\hat{P}\hat{T}{E}_{n}^{\uparrow }(\overrightarrow{k})\)=\(\,{E}_{n}^{\downarrow }(\overrightarrow{k})\) (Fig. 1b). The spin degeneracy also appears in compensated antiferromagnets with \(\hat{T}\hat{t}\,\)symmetry (\(\hat{t}\) is half a unit cell translation) in the absence of spinorbit coupling.
The spin degeneracy is however broken in compensated antiferromagnets belonging to magnetic space groups with violated \(\hat{P}\hat{T}\) and \(\hat{T}\hat{t}\) symmetries^{26}. The vanishing net magnetization in such antiferromagnets originates from the combination of some other magnetic space group symmetries of the crystal. For example, Fig. 1c shows a collinear antiferromagnet with the Néel vector pointing along the z direction. The zero net magnetization in this antiferromagnet is guaranteed by two glide symmetries \({\hat{G}}_{x}\) and \({\hat{G}}_{z}\), where \({\hat{G}}_{l}=\{{\hat{M}}_{l}\hat{t}\}\) represents mirror symmetry \({\hat{M}}_{l}\) with a mirror plane normal to vector \(\overrightarrow{l}\) combined with translation \(\hat{t}\). The symmetry transformation \({\hat{G}}_{x}{N}_{\parallel }^{\uparrow }({k}_{x},{k}_{y})={N}_{\parallel }^{\downarrow }({k}_{x},{k}_{y})\) flips the spin and thus according to Eqs. (1) and (2) results in a vanishing net spin polarization \(p\) for the current along the z direction. On the other hand, due to the Néel vector pointing along the z axis, the symmetry transformation \({\hat{G}}_{z}{E}_{n}^{{{{{{\rm{\sigma }}}}}}}({\overrightarrow{k}}_{\parallel },{k}_{z})={E}_{n}^{{{{{{\rm{\sigma }}}}}}}({\overrightarrow{k}}_{\parallel },{k}_{z})\) conserves the spin \({{{{{\rm{\sigma }}}}}}=\uparrow ,\downarrow\) of the conduction modes at \({\overrightarrow{k}}_{\parallel }\) and there is no symmetry operation which would enforce \({p}_{\parallel }=0\). The presence of spinpolarized conduction channels in this type of antiferromagnets can be understood in terms of two congruent (but not identical) up and downspin Fermi surfaces, which are transformed to each other by the symmetry transformation \({\hat{G}}_{x}\) (Fig. 1c). In this case, each conduction channel is spin polarized (except highsymmetry \({\overrightarrow{k}}_{\parallel }\) points invariant to \({\hat{G}}_{x}\)), whereas the net conductance is spin neutral.
Due to the nonspindegenerate Fermi surface and spinpolarized conduction channels, the globally spinneutral conduction of the compensated antiferromagnets can be exploited in spintronic devices, such as AFMTJs. Fig 1d shows an AFMTJ which contains two identical antiferromagnetic electrodes separated by a nonmagnetic insulating spacer. The antiferromagnets are assumed to have spinpolarized conduction channels along the outofplane transport direction. The functionality of the AFMTJ is controlled by the relative orientation of the Néel vector of the two antiferromagnetic electrodes. In the parallel state, the spinpolarized conduction channels of the electrodes perfectly match, resulting in a low resistance state. In the antiparallel state, the spin polarized conduction channels are mismatched, resulting in a high resistance state.
Electronic structure of RuO_{2}
To demonstrate this spintronic functionality, we consider the recently discovered roomtemperature antiferromagnetic metal RuO_{2}^{35} suitable for realizing the proposed AFMTJ. RuO_{2} exhibits interesting properties such as spin splitting without spinorbit coupling^{36}, a crystal Hall effect^{37}, and a magnetic spin Hall effect^{31,38}. RuO_{2} has a rutile structure with an outofplane Néel vector (Fig. 2a) and magnetic space group P4_{2}'/mnm', which contains glide \({\hat{G}}_{x}=\{{\hat{M}}_{x}(\frac{1}{2},\frac{1}{2},\frac{1}{2})\}\), \({\hat{G}}_{y}=\{{\hat{M}}_{y}(\frac{1}{2},\frac{1}{2},\frac{1}{2})\}\) and mirror \({\hat{M}}_{z}\) symmetries. In the absence of spinorbit coupling, the energy bands are spin degenerate at the kplanes invariant to \({\hat{G}}_{x}\) and \({\hat{G}}_{y}\), such as \({k}_{x}=0,\frac{\pi }{2}\) or \({k}_{y}=0,\frac{\pi }{2}\). This is evident from our firstprinciples density functional theory (DFT) calculations. As seen from Fig. 2b, the energy bands of RuO_{2} are spin degenerate along the Г–X, Г–Z, X–M, Z–R, and R–A directions lying in these glideinvariant planes. On the other hand, a large spin splitting appears along the directions away from these planes, such as Г–M and Z–A.
The spindependent band structure leads to the momentumdependent spin polarization of the conduction channels along the z direction. We explicitly demonstrate this by calculating the number of conduction channels \({N}_{\parallel }^{\sigma }({\overrightarrow{k}}_{\parallel })\) in the twodimensional (2D) Brillouin zone of RuO_{2}. As seen from Fig. 2c, the distributions of \({N}_{\parallel }^{\uparrow }\) and \({N}_{\parallel }^{\downarrow }\) in the \(({k}_{x},{k}_{y})\) plane have congruent shapes and are symmetric with respect to the \({\hat{G}}_{x}\) and \({\hat{G}}_{y}\) symmetry transformations, which enforce a zero global spin polarization in the conductance along the z direction. On the contrary, as seen from Fig. 2d, the \({\overrightarrow{k}}_{\parallel }\)dependent spin polarization \({p}_{\parallel }\) remains finite across a sizable portion of the 2D Brillouin zone. This demonstrates that RuO_{2} (001) exhibits a globally spinneutral conductance through spinpolarized conduction channels.
TMR in a RuO_{2}/TiO_{2}/RuO_{2} AFMTJ
Next, we design an AFMTJ using RuO_{2} (001) as electrodes and TiO_{2} (001) as an insulating barrier layer. Due to both having a rutile structure and a similar lattice constant, this AFMTJ is feasible in practice. Fig 3a shows the atomic structure of the RuO_{2}/TiO_{2}/RuO_{2} (001) supercell, which is used in our DFT and quantum transport calculations and includes 8 TiO_{2} layers in the center and 10 RuO_{2} layers on each side. We find that a wide band gap of TiO_{2} is well maintained in this heterostructure, and the Fermi energy E_{F} is located deeply inside the band gap (Fig. 3b).
The RuO_{2}/TiO_{2}/RuO_{2} (001) structure in Fig. 3a is then used as the scattering region of the AFMTJ connected to two semiinfinite RuO_{2} (001) electrodes for calculating transmission. The transmission is obtained for parallel (Fig. 4a) and antiparallel (Fig. 4b) alignments of the Néel vectors of the electrodes. For the parallelaligned AFMTJ, the \({\overrightarrow{k}}_{\parallel }\)resolved transmission \({T}_{P}^{\sigma }({\overrightarrow{k}}_{\parallel })\) is shown in Fig. 4c for spin up (left panel) and spin down (right panel). The clearly seen spin asymmetry between \({T}_{P}^{\uparrow }({\overrightarrow{k}}_{\parallel })\) and \({T}_{P}^{\downarrow }({\overrightarrow{k}}_{\parallel })\) reflects the related asymmetry in the distribution of the spinpolarized conduction channels in RuO_{2} (Fig. 2c). The suppressed transmission near the Brillouin zone corners is due to a larger decay rate of the evanescent states for \({\overrightarrow{k}}_{\parallel }\) away from the zone center.
For the antiparallelaligned AFMTJ, the transmission \({T}_{AP}^{\sigma }({\overrightarrow{k}}_{\parallel })\) is blocked for the wave vectors \({\overrightarrow{k}}_{\parallel }\) with no conduction channels in one of the spin states, i.e. \({N}_{\parallel }^{\sigma }({\overrightarrow{k}}_{\parallel })=0\) for \(\sigma =\uparrow\) or \(\sigma =\downarrow\). These are the regions where \({p}_{\parallel }=\pm 1\) or \({N}_{\parallel }^{\uparrow }={N}_{\parallel }^{\downarrow }=0\) in Fig. 2c and d. As a result, only the states near the zone center where the spin channels are degenerate (enforced by \({\hat{G}}_{x}\) and \({\hat{G}}_{y}\)) contribute to the transmission (Fig. 4d). This leads to the total transmission being much smaller for the antiparallel state (T_{AP}) than for the parallel state (T_{P}) (Fig. 4e). At the Fermi energy \({E}_{F}\), we find the TMR ratio \(({T}_{P}{T}_{AP})/{T}_{AP}\) as large as ~500%. This value is comparable to the values obtained for the wellknown Fe/MgO/Fe (001) MTJs^{39,40} which are currently used in magnetic randomaccess memories. The giant TMR appears not only for \(E={E}_{F}\) but also for the energies around the Fermi level, with the smallest value of ~50% at \(E={E}_{F}0.25\,\)eV (Fig. 4f). This fact indicates that the large TMR will be sustained under an applied bias voltage.
The predicted TMR is largely independent of the interface terminations and the relative alignment of the interface magnetic moments, as follows from our explicit DFT calculations (Supplementary Figs. S1 and S2). This distinguishes our results from the previous findings^{21,22}, where the interface termination controls TMR and implies that the predicted TMR effect is likely less sensitive to the interface roughness than that in the previous studies. The bulk origin of TMR in the proposed AFMTJs makes it also more robust against other types of disorder, as long as the crystallinity of the tunnel junction and the direct tunneling transport mechanism are maintained (see Section C of Supplemental Material).
Discussion
The above properties are sustained in the presence of spinorbit coupling. This is due to the momentumdependent spin polarization in rutile antiferromagnets being inherited from the antiferromagnetic order rather than spinorbit coupling^{27,31,37}. In RuO_{2}, the \({\hat{M}}_{z}\) symmetry transformation reverses the wave vector component \({k}_{z}\), conserves the spin component \({\sigma }_{z}\) but flips \({\sigma }_{x}\) and \({\sigma }_{y}\). As a result, the conduction modes at \({\overrightarrow{k}}_{\parallel }\) are spin polarized purely along the z axis and the x and yspin components vanish (see Supplementary Section D). Supplementary Fig. S5e shows the results of the DFT calculation for RuO_{2} in the presence of spinorbit coupling. It is seen that the spin polarization of the most conduction channels is well preserved, indicating that the giant TMR is robust against spinorbit coupling in the RuO_{2} based AFMTJ.
The spinpolarized conduction channels are not limited to RuO_{2}, but typical for a wide group of materials with violated \(\hat{P}\hat{T}\) symmetry, including those with a noncollinear antiferromagnetic order^{29}. For the antiferromagnets with magnetization being compensated by combined mirror and/or rotation symmetries, the spinpolarized conduction channels can exist purely due to the antiferromagnetic order, even in the absence of spinorbit coupling^{26,27,28,29,30,31}. On the other hand, in compensated antiferromagnets with \(\hat{T}\hat{t}\) symmetry, the spinpolarized conduction channels appear due to the spin degeneracy lifted by spinorbit coupling. Such antiferromagnets can also be used in AFMTJs if they have sizable spinorbit splitting.
These considerations can be expanded to normal metals with broken space inversion symmetry, where the band spin degeneracy is lifted by spinorbit interaction. In these materials, the two conduction channels with opposite spin polarizations are linked by the time reversal symmetry operation. For example, in topological metal TaN^{41,42}, the conduction channels along the [001] direction carry the spin polarization pointing along the same [001] direction (Supplementary Fig. S6). This property is enforced by the \({\hat{M}}_{z}\) mirror symmetry. While the momentumdependent spin polarization in noncentrosymmetric normal metals is fixed by their crystal symmetry and band structure, they can be used in spintronics in conjunction with antiferromagnets. For example, the antiferromagnetic reference layer in the AFMTJ in Fig. 1d can be replaced by a normal metal layer. Alternatively, one can create an antiferromagnet/normal metal interface. In such systems with a single antiferromagnetic layer, the spintronic functionality is controlled by the Néel vector orientation that regulates a matching of the conduction channels in the antiferromagnetic and normal metal layers.
Note that in junctions with a single antiferromagnetic layer, reversal of the Néel vector is equivalent to the timereversal transformation which does not change the resistance. However, the resistance changes with rotation of the Néel vector, resulting in a tunnelling anisotropic magnetoresistance (TAMR) effect^{43}.
Another possibility is to utilize noncentrosymmetric insulators as a tunneling barrier layer in an AFMTJ. Due to the broken space inversion symmetry and spinorbit coupling, the evanescent states in these insulators are spinpolarized^{44,45}. Therefore, the Néel vector of the free antiferromagnetic layer can be used to control the matching between the propagating Bloch states in the antiferromagnetic electrode and the evanescent gap states in the barrier resulting in a TAMR effect. An additional useful functionality of this kind of tunnel junctions may be provided by a switchable polarization of the noncentrosymmetric insulating barrier layer if it is ferroelectric^{45}.
The proposed use of spinneutral currents in spintronics is feasible from the experimental perspective. For example, the proposed RuO_{2}/TiO_{2}/RuO_{2} (001) AFMTJ has all rutile structure with a good match of the RuO_{2} and TiO_{2} lattice constants and thus can be grown epitaxially preserving crystallinity of the overall heterostructure. The Néel vector of the antiferromagnetic free layer can be switched by a spinorbit torque via the spin current from an adjacent heavy metal layer generated by an inplane charge current^{8,46}. With the inplane writing path and outofplane reading path, only two inplane terminals and one outofplane terminals are required for such an AFMTJ, which is desirable for nanoscale spintronic applications. In addition, the large magnitude of TMR indicates a possibility of a strong spin transfer torque in the AFMTJs, which may be robust against disorder^{47} and may offer an alternative way to switch the Néel vector.
In conclusion, we have proposed that globally spinneutral currents flowing through oppositely spinpolarized conduction channels can be efficiently used in spintronics. Such currents exist in compensated antiferromagnets and normal metals with the magnetic space group symmetries which lift the spindegeneracy of the Fermi surface. In the heterostructures, such as antiferromagnetic tunnel junctions or antiferromagnet/normal metal interfaces, these currents can be controlled by the Néel vector orientation providing a useful functionality for spintronics. Based on firstprinciples density functional theory combined with quantum transport calculations, we have demonstrated such functionality using a roomtemperature antiferromagnetic metal RuO_{2} as electrodes in a RuO_{2}/TiO_{2}/RuO_{2} AFMTJ and predicted a giant TMR effect of ~500%. Our work uncovers an unexplored potential of the materials with no global spin polarization for utilizing them in spintronics. We hope therefore that our predictions will stimulate experimental investigations of these materials and the associated phenomena.
Note added: During the review of this manuscript, we became aware of the relevant work by Šmejkal et al. posted recently^{48}.
Methods
The atomic and electronic structures shown in Figs. 2a, b, 3, S5a, and S6a, b of the systems are calculated using the projector augmented wave (PAW) method^{49} implemented in the VASP code^{50}. A planewave cutoff energy of 500 eV and a 16 × 16 × 16 \(\overrightarrow{k}\)point mesh in the irreducible Brillouin zone are used in the calculations. The exchange and correlation effects are treated within the generalized gradient approximation (GGA) developed by PerdewBurkeErnzerhof (PBE)^{51}. The GGA+U functional^{52,53} with U_{eff} = 2 eV on Ru 4d orbitals and U_{eff} = 5 eV on Ti 3d orbitals is included in the calculations.
The transport properties shown in Fig. 4 and S1–S3 are calculated using the nonequilibrium Green’s function formalism (DFT+NEGF approach)^{54,55}, as implemented the Atomistic Simulation Toolkit (ATK) distributed in the QuantumWise package (Version 2015.1) (ATOMISTIX TOOLKIT version 2015.1 Synopsys QuantumWise (www.quantumwise.com). QuantumWise A/S is now part of Synopsys, and from the upcoming version ATK will be part of the QuantumATK suite)^{56}. The atomic structures are relaxed by VASP and the nonrelativistic FritzHaberInstitute (FHI) pseudopotentials using a singlezetapolarized basis. The spin polarized GGA+U functional^{51,52} with U_{eff} = 2.3 eV on Ru 4d orbitals and U_{eff} = 5 eV on Ti 3d orbitals is included in the calculations. A cutoff energy of 75 Ry and a 11×11×101 \(\overrightarrow{k}\)point mesh are used for the selfconsistent calculations to eliminate the mismatch of the Fermi level between the electrodes and the central region. Unless mentioned in the text, the transmission is calculated using an adaptive \(\overrightarrow{k}\)point mesh. These parameters are confirmed to yield a good balance between the computational time and accuracy.
The tightbinding Hamiltonians of RuO_{2} and TaN are obtained using Wannier90 code^{57} utilizing the maximally localized Wannier functions^{58}. A 500 × 500 × 500 \(\overrightarrow{k}\)point mesh and the adaptive smearing method^{59} are used to calculate the \({\overrightarrow{k}}_{\parallel }\)resolved ballistic conductance shown in Fig. 2c, d, S5e and S6c, d. The spinprojected Fermi surfaces of RuO_{2} with spinorbit coupling shown in Figs. S5b–d are calculated using WannierBerri code^{60,61}.
Figures are plotted using VESTA^{62}, FermiSurfer^{63}, gnuplot^{64}, and the SciDraw scientific figure preparation system^{65}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank Bo Li for helpful discussions. This work was supported by the Vannevar Bush Faculty Fellowship (ONR grant N000142012844) (C.B.E) and by the National Science Foundation (NSF) through the MRSEC (NSF Award DMR1420645) and EPSCoR RII Track1 (NSF Award OIA2044049) programs (E.Y.T.). S.H.Z. thanks the support of National Science Foundation of China (NSFC Grant No. 12174019). Computations were performed at the University of Nebraska Holland Computing Center.
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D.F.S. and E.Y.T. conceived this project. D.F.S. performed symmetry analysis and calculated electronic properties for RuO_{2} and TaN. S.H.Z calculated the ballistic conductance of RuO_{2} and TaN. D.F.S. and M.L. calculated the transport properties of RuO_{2}/TiO_{2}/RuO_{2} AFMTJ. D.F.S., S.H.Z., M.L. and E.Y.T. analyzed the results and discussed with C.B.E. D.F.S. and E.Y.T. wrote the manuscript. All authors contributed to the final version of the manuscript.
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Shao, DF., Zhang, SH., Li, M. et al. Spinneutral currents for spintronics. Nat Commun 12, 7061 (2021). https://doi.org/10.1038/s41467021269153
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DOI: https://doi.org/10.1038/s41467021269153
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