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Band inversion of electrons due to strong spin-orbit coupling (SOC) is commonly seen in magnetic materials. It characteristically appears in topological insulators (TIs) with inverted bandgaps [1] and Weyl semimetals (WSMs) with pointlike band crossings [2], while metallic systems are also capable of having band inversion. In momentum space around the SOC-induced band inversion, the electron spin is often correlated to its momentum, known as spin-momentum locking (SML). Moreover, the electron system acquires nontrivial band topology characterized by the Berry curvature, which serves as an effective magnetic field in momentum space. It is well understood that the band topology gives a substantial contribution to anomalous transport phenomena, such as the anomalous Hall effect (AHE) and the spin Hall effect (SHE), due to the anomalous velocity of electrons triggered by the Berry curvature. Recent experimental attempts have been successful in synthesizing materials having both nontrivial band topology and magnetism (magnetic TIs, WSMs, etc.), and it is an important question if their magnetic functionalities are also governed by their band topology.&lt;br/&gt;&lt;br/&gt;In this talk, I show from theoretical aspect that the current-induced spin torques in such magnetic topological materials are influenced and occasionally enhanced due to their band topology. After reviewing the current status of theoretical and experimental studies on magnetic topological materials, I will introduce some of my recent works.&lt;br/&gt;&lt;br/&gt;From the general point of view, I first propose that the momentum-space topology gives rise to an &quot;intrinsic&quot; contribution to the spin torques, which is robust against disorder and thermal fluctuation similarly to the intrinsic AHE and SHE [3]. This intrinsic torque originates from electron spin polarization converted from the anomalous velocity by SML, and is distinct from the conventional spin-transfer torque (STT) and the spin-orbit torque (SOT) driven by electron spins in transport current. In particular, I point out the intrinsic torque exerting on magnetic textures, such as domain walls (DWs), which we call the &quot;topological Hall torque (THT)&quot;. The THT is induced by the cooperation of the real-space magnetic texture and the momentum-space Berry curvature, and can emerge dominantly in bulk crystals without building any complex heterostructures.&lt;br/&gt;&lt;br/&gt;As a typical example, I show our microscopic calculation result of current-induced torques with the low-energy effective model of magnetic WSM [4]. The band crossing points, called &quot;Weyl points&quot;, serve as source and sink of the Berry curvature in momentum space, accompanied with SML of electrons around them (see Fig. 1). The Weyl electrons are therefore capable of generating a strong intrinsic torque. From the microscopic calculation in response to an externally applied electric field, we have derived the SOT, the STT, and the THT exerted by the Weyl electrons. The THT gets enhanced at a sharp magnetic DW, due to the localized states of Weyl electrons formed at the DW [5,6].&lt;br/&gt;&lt;br/&gt;For more realistic situations, I also show the phenomenological calculation of the THT in a metallic ferromagnet with band inversion [3]. In response to an external electric field applied to a DW (see Fig. 2), we have obtained a large THT. The THT acts on a DW in the same form as the conventional nonadiabatic STT, but with the unusual size of the nonadiabaticity parameter β≈2, which will enable an efficient control of DW motion by electric field. Experimental measurability of the THT in the metallic ferromagnet SrRuO&lt;sub&gt;3&lt;/sub&gt;, in connection with the temperature dependence in its AHE [7], will also be mentioned.

![](https://assets.underline.io/uploads/markdown_image/1/image/d0716b6a26c3c8035d704ada28777bbc.jpg)

Schematic image of Weyl points emerging from band inversion. The Berry curvature (magenta curves) emerges in the vicinity of the Weyl points.

![](https://assets.underline.io/uploads/markdown_image/1/image/9d1d264c88091f5f7c8512236cd257bf.jpg)

Schematic image of the setup of magnetic DW in our calculations [3]. We consider an external electric field Eext applied across the DW, and calculate the torque in response to Eext.

2022 Joint MMM-INTERMAG

Theory of Spin Torques Emerging from Band Topology

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Band inversion of electrons due to strong spin-orbit coupling (SOC) is commonly seen in magnetic materials. It characteristically appears in topological insulators (TIs) with inverted bandgaps [1] and Weyl semimetals (WSMs) with pointlike band crossings [2], while metallic systems are also capable of having band inversion. In momentum space around the SOC-induced band inversion, the electron spin is often correlated to its momentum, known as spin-momentum locking (SML). Moreover, the electron system acquires nontrivial band topology characterized by the Berry curvature, which serves as an effective magnetic field in momentum space. It is well understood that the band topology gives a substantial contribution to anomalous transport phenomena, such as the anomalous Hall effect (AHE) and the spin Hall effect (SHE), due to the anomalous velocity of electrons triggered by the Berry curvature. Recent experimental attempts have been successful in synthesizing materials having both nontrivial band topology and magnetism (magnetic TIs, WSMs, etc.), and it is an important question if their magnetic functionalities are also governed by their band topology. In this talk, I show from theoretical aspect that the current-induced spin torques in such magnetic topological materials are influenced and occasionally enhanced due to their band topology. After reviewing the current status of theoretical and experimental studies on magnetic topological materials, I will introduce some of my recent works. From the general point of view, I first propose that the momentum-space topology gives rise to an "intrinsic" contribution to the spin torques, which is robust against disorder and thermal fluctuation similarly to the intrinsic AHE and SHE [3]. This intrinsic torque originates from electron spin polarization converted from the anomalous velocity by SML, and is distinct from the conventional spin-transfer torque (STT) and the spin-orbit torque (SOT) driven by electron spins in transport current. In particular, I point out the intrinsic torque exerting on magnetic textures, such as domain walls (DWs), which we call the "topological Hall torque (THT)". The THT is induced by the cooperation of the real-space magnetic texture and the momentum-space Berry curvature, and can emerge dominantly in bulk crystals without building any complex heterostructures. As a typical example, I show our microscopic calculation result of current-induced torques with the low-energy effective model of magnetic WSM [4]. The band crossing points, called "Weyl points", serve as source and sink of the Berry curvature in momentum space, accompanied with SML of electrons around them (see Fig. 1). The Weyl electrons are therefore capable of generating a strong intrinsic torque. From the microscopic calculation in response to an externally applied electric field, we have derived the SOT, the STT, and the THT exerted by the Weyl electrons. The THT gets enhanced at a sharp magnetic DW, due to the localized states of Weyl electrons formed at the DW [5,6]. For more realistic situations, I also show the phenomenological calculation of the THT in a metallic ferromagnet with band inversion [3]. In response to an external electric field applied to a DW (see Fig. 2), we have obtained a large THT. The THT acts on a DW in the same form as the conventional nonadiabatic STT, but with the unusual size of the nonadiabaticity parameter β≈2, which will enable an efficient control of DW motion by electric field. Experimental measurability of the THT in the metallic ferromagnet SrRuO3, in connection with the temperature dependence in its AHE [7], will also be mentioned.

![](https://assets.underline.io/uploads/markdown_image/1/image/d0716b6a26c3c8035d704ada28777bbc.jpg)

Schematic image of Weyl points emerging from band inversion. The Berry curvature (magenta curves) emerges in the vicinity of the Weyl points.

![](https://assets.underline.io/uploads/markdown_image/1/image/9d1d264c88091f5f7c8512236cd257bf.jpg)

Schematic image of the setup of magnetic DW in our calculations [3]. We consider an external electric field Eext applied across the DW, and calculate the torque in response to Eext.

technical paper

Antiferromagnets (AFMs) are magnetically ordered materials without a macroscopic magnetization. There is accelerating interest in using them for memory applications, motivated by the possibility for electrically controlling AFM order via spin-orbit torques, and its read-out via magnetoresistive effects [1-2]. Here we show that micrometer-scale bits of the antiferromagnet PtMn, grown on a Ta or Pt heavy metal (HM) layer, can be switched reversibly between different magnetic states by electric currents, with readout through a second harmonic voltage technique [3]. The device exhibits multiple states, which can be set using different current amplitudes and may find application in neuromorphic computing. Micromagnetic simulations suggest that these metastable states consist of vortex and anti-vortex textures that move in response to current. We also demonstrate, using a modified device layout, direct voltage measurements to read out the electrically-induced switching of noncollinear PtMn3. The modified layout and measurement methodology, which was recently demonstrated using the IrMn3/Pt material system [4], is utilized here to distinguish current-induced magnetic and nonmagnetic switching signals in PtMn3/Pt devices. We also realize switching down to 10 µs pulse width, which shows the potential for fast device applications. The effect of temperature and external magnetic field (up to 15 T) are also studied. Our results pave the way towards practical PtMn and PtMn3 antiferromagnetic memories integrated on silicon, and indicate the importance of micromagnetic textures in the static and dynamic behavior of antiferromagnetic devices.

Electrical manipulation of antiferromagnetic order in PtMn and PtMn3 pillars

In antiferromagnetic topological insulator MnBi2Te4, the interplay between topological electrons and magnetization dynamics in the form of spin torques and charge pumping remains an open question. Using the Berry phase theory, we develop a universal formalism to quantify the spin-orbit torques (SOTs) and their reciprocal effects in layered MnBi2Te4, where the symmetry of SOTs exhibits an even-odd layer-number pattern. The dynamical consequences of our finding are demonstrated by the SOT-driven magnetic resonances. In particular, we identify an anomalous exchange (AE) mode in the tri-septuple layer (SL) case, where the magnetic moments from the top and bottom SL rotate with a pi phase difference while those in the middle SL stay static. Unlike the ordinary exchange mode and the acoustic mode, the AE mode does not react to a microwave electromagnetic field. Therefore, the AE resonance can only be excited by the peculiar SOTs we find, providing a unique way to verify the SOTs in MnBi2Te4. Our study lays the theoretical foundation of current-induced spin dynamics in intrinsic antiferromagnetic topological insulators.

![](https://assets.underline.io/uploads/markdown_image/1/image/4045560a7cd6d072b78a10061469d722.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/d50337c92d0bcff8e7d7da23d0825a66.jpg)

Anomalous Exchange Resonance Driven by Spin-Orbit Torques in Intrinsic Antiferromagnetic Topological Insulator

Spin pumping has been an efficient way of spin generation, which converts coherent spin precessions in a magnet into pure spin currents in an adjacent metal. Recently, spin pumping has been demonstrated in easy-axis antiferromagnets (AFs) that operate at the terahertz frequency range, opening the exciting possibility of signal generation and processing at ultrafast speed. However, easy-plane AFs which are experimentally easier to prepare than their easy-axis counterparts, were considered to be bad candidates for spin pumping because the N\'{e}el vector is linearly polarized due to the easy-plane anisotropy, placing a major restriction on the choice of materials for practical applications. Here, using NiO and α-Fe2O3 (hematite) as two archetypal easy-plane AFs, we theoretically challenge this seemingly established conclusion by showing that easy-plane AFs can pump DC spin currents with the help of either the Dzyaloshinskii-Moriya interaction or an applied magnetic field without inducing the spin-flop transition. The magnitude of spin pumping turns out to be as strong as that in easy-axis AFs, which significantly broadens the materials pool for antiferromagnetic spintronics. Furthermore, we find that in the presence of a magnetic field, NiO can function as a sub-terahertz spin-torque oscillator at a highly reduced threshold current density.

![](https://assets.underline.io/uploads/markdown_image/1/image/0487d1f128d8b2c5f538c85b9f9d5307.jpg)

DC output as a function of the applied magnetic field for different values of DM interaction in α-Fe2O3

![](https://assets.underline.io/uploads/markdown_image/1/image/c322020ac034c68f560001032918d42e.jpg)

DC output as a function of the applied magnetic field along the in-plane easy axis direction below the spin-flop transition for different driving frequencies in NiO

Coherent Spin Pumping in Easy-Plane Antiferromagnets

The spin-1/2 of an electron makes it an archetypal two-level system and inspires the description of other two-level systems using an analogous pseudospin. The quantized spin excitations of an ordered antiferromagnet can be viewed as pairs of spin-up and -down magnons and can be characterized by a magnonic pseudospin. The similarity between electronic spin and magnonic pseudospin has triggered the prediction of exciting phenomena like emergent spin-orbit coupling and topological states in antiferromagnetic magnonics. In the last year, first experimental observations of the associated dynamics of antiferromagnetic pseudospin have been reported [1,2]. Based on these findings, we will expand the concept of magnon pseudospin and the description of magnon pseudospin dynamics [3,4] and discuss the influence of dimensionality on the magnon Hanle effect. Additionally, we show our recent experiments demonstrating control of magnon spin transport and pseudospin dynamics in thin films with varying thickness of the antiferromagnetic insulator hematite (α-Fe2O3) utilizing two Pt strips for all-electrical magnon injection and detection [2] . We observe an oscillation in polarity of the magnon spin signal at the detector as a function of the applied magnetic field, which we quantitatively explain in terms of diffusive magnon transport. In particular, we observe a coherent precession of the magnon pseudospin caused by the easy-plane anisotropy and the Dzyaloshinskii-Moriya interaction. Moreover, we find peculiar changes in the magnon spin signal for thicker hematite layers. Our results are paramount in unnlocking the high potential of antiferromagnetic magnonics towards the realization of electronics-inspired phenomena. Financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC-2111 - 390814868 and project AL2110/2-1, and the Research Council of Norway through its Centers of Excellence funding scheme, project 262633, "QuSpin" is gratefully acknowledged.

Antiferromagnetic Magnon Pseudospin Dynamics and the Magnon Hanle Effect in Hematite Thin Films

Antiferromagnetic materials (AFMs) provide a pathway to overcome the limitations of ferromagnet-based spintronics, enabling fast and high density memory devices, as well as terahertz detectors and sources. This has fueled intensive research on the electrical manipulation of metallic AFMs using interfacial spin-orbit torque (SOT) from a heavy metal (HM) layer [1-4]. However, the threshold current density in HM/AFM bilayers is typically large, thus creating non-magnetic contributions to resistive switching signals, complicating their interpretation. Here, we present a new device structure and measurement protocol based on differential voltage measurements, allowing in situ reference measurements and to systematically separate the switching signals originating in the AFM and HM layers of a HM/AFM heterostructure [5]. The proposed device consists of a double-cross HM structure with six terminals, where the AFM, patterned in a micropillar shape, is placed on one of the two crosses. We successfully use this method in devices formed by Pt and the non-collinear AFM IrMn3. The results provide clear evidence of electrical switching of the AFM order parameter in the IrMn3 pillar by SOT in the range of writing current densities from 36 to 50 MA/cm2 with no appreciable switching signal from the Pt layer. We discuss the role of different switching configurations (180° and 90° writing currents), as well as the role of thermal activation on the switching characteristics. Control experiments in Pt-only double-cross samples without an AFM pillar reveal HM-dominated switching at higher current densities ~60 MA/cm2. We discuss active readout of the switching signals using a differential amplifier, with large peak-to-peak output voltage ~100 mV. The structure allows for reliable electrical measurement of resistive signatures of AFM switching induced by SOT, which will be important not only for the fundamental understanding of current-induced AFM domain dynamics, but also as an electrical readout method for practical applications in memory and computing devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/9f7b7e5713bbb79e6a1b7f3436e814c5.jpg)

Fig. 1. Design of double-cross device for differential voltage measurements and the measured differential voltage for the 180° switching configuration with 20 mA write pulses.

Observation of current-induced switching in non-collinear antiferromagnetic IrMn3 by differential voltage measurements

Antiferromagnetic materials have recently attracted much interest due to a number of advantages including high resonance frequencies up to the terahertz. Further, the possibility to electrically modulate the magnetic properties such as the anisotropy makes them an excellent candidate for energy-efficient devices at ultra-high frequencies. In this study, we theoretically explored one such application in the dynamic manipulation of electromagnetic responses with a broad tunable spectral range. The materials considered were dielectric antiferromagnets with strong magneto-electric or –elastic coupling which can achieve robust control in the perpendicular magnetic anisotropy. The metallic counterparts were not suitable due to the anticipated large loss. A theoretical analysis based on the Landau-Lifshitz equation was used to derive the frequency-dependent permeability and the resonant frequency of the antiferromagnet for different biaxial anisotropy profiles and incident electromagnetic wave polarizations. Numerical calculations based on typical parameters available in the literature (such as NiO) strongly indicated that the resonance frequency modulation as much as 10’s of percent may be possible in the sub-mm wavelength range, along with the corresponding changes in the permeability. Application to specific devices such as a bandpass filter further illustrated the validity of the concept. Funding Acknowledgement This work was supported, in part, by the U.S. Army Research Office (No. W911NF-16-1-0472).

![](https://assets.underline.io/uploads/markdown_image/1/image/6ffd93f402fd8b4c21123e86d0bbb698.jpg)

Fig.1 Schematic of the magnetic-piezoelectric composite. Electric field along out-of-plane direction causes magneto-elastic effects and influences eigen-frequency by modifying in-plane magnetic anisotropy. EM wave is incident along x-axis.


![](https://assets.underline.io/uploads/markdown_image/1/image/e677976b7cbcf549791f319205ea030e.jpg)

Fig.2 Data showing a high electric field tunable AFMR frequency shift with external voltage at 0, -V, and +V=8kV/cm (Not the final results)

Voltage-Controlled Dynamic Modulation of Electromagnetic Response in Antiferromagnetic Materials at Terahertz Frequencies

Recently antiferromagnetic materials have been gathering increasing attention from the spintronics community, as detecting and manipulating antiferromagnetic (AFM) order electrically is an important milestone for realizing antiferromagnetic devices. Characterizing the type and amplitude of the spin-orbit torques (SOT) is crucial for understanding and predicting AFM dynamics. One important technique to identify and measure SOTs is to perform harmonic Hall measurements in an AFM/heavy metal (HM) layer. In this study, we report the full angular dependence of 1st and 2nd harmonic voltage measurements in epitaxial c-axis oriented α-Fe2O3/Pt bilayer heterostructures (Fig. 1a). By modeling the harmonic resistance signals together with known material parameters such as the exchange field, anisotropy and DMI constant (Fig. 1b), we extract the amplitudes of field-like and anti-damping-like spin-orbit torques from field-dependent measurements in XY, XZ and YZ plane scans (Fig. 1a). Our model suggests a small angle tilt of the easy-plane and indicates that the amplitude of the field-like torques are significantly larger than anti-damping-like torques. Additionally, we show that there is an anomalous Hall-effect contribution in out-of-plane scans and that the Oersted fields are negligible compared to field-like torques. Our work provides a straight-forward way of characterization of spin-orbit torques in AFM/HM heterostructures.

![](https://assets.underline.io/uploads/markdown_image/1/image/87a94b08c94b21f9137b16231524bc27.jpg)

a) Transverse measurement configuration showing the Pt Hall cross with 5 µm channel width on AFM α-Fe2O3 layer. An AC current and the transverse voltage is measured with a lock-in amplifier. The thicknesses of α-Fe2O layer and Pt layers are 30 nm and 5 nm respectively. Next to it, geometry of magnetic field scans in the 3 principal planes (XY, XZ and YZ) is shown. b) Model geometry: D represents both the DMI vector and the hard axis direction. n is the Néel vector which lies in the easy plane. B is the external magnetic field. Alternating current is in x-direction and the spin accumulation in the y-direction. The Néel vector is perpendicular to both D and B.

Quantifying Spin-Orbit Torques in AFM/HM Heterostructures

Antiferromagnetic materials with non-collinear spin structures (NC-AFMs) offer significant potential for spintronic applications[1]. They have vanishingly small magnetization and ultrafast spin dynamics. In addition, they allow for the presence of important counterparts to the effects considered to be exclusive to materials with net magnetization, such as the anomalous Hall effect (AHE)[2], magneto-optical Kerr effect[3] or anomalous Nernst effect (ANE)[4]. One of the promising candidates of the NC-AFM class are Mn antiperovskites, with a typical representative being Mn3NiN. In this contribution we present ANE measurements on epitaxial thin films of Mn3NiN prepared on SrTiO3 substrate with spin structure lying in the (111) plane. Previous theoretical predictions suggest that despite the modest spin-orbit coupling, the magnitude of the ANE in Mn3NiN ought to exceed that of Mn3Sn NC-AFM due to the Berry phase boost[5]. We implement two complementary methods to detect the ANE response of the Mn3NiN films. First, the global Nernst response is measured using a micro-patterned Hall bar device with platinum resistive heaters generating an in-plane thermal gradient. The measured anomalous Nernst data (Fig.1(a)) are in good agreement with the AHE measurements[5], indicating a similar physical origin of the two effects. Second, to add a spatial resolution, scanning thermal gradient microscopy[4] is used. A laser beam incident on a micro-patterned device induces thermal gradients. The method reveals the sample’s domain structure. We find a signature of an ANE component that we can control with the external field due to the unsaturated moment present lying in the (111) plane, different from the previously measured Mn3Sn. This small moment provides a means for manipulating magnetic order, and even writing magnetic domain-like structures (Fig.1(b)). These two methods provide fruitful insight into the magnetic behaviour of NC-AFMs.

![](https://assets.underline.io/uploads/markdown_image/1/image/e3391f086f9d4953ae3f5db220e7247b.jpg)

Fig.1. Anomalous Nernst effect in Mn3NiN. (a) Global ANE as a function of the out-of-plane magnetic field at different base temperatures. (b) Scanning thermal gradient microscopy at T = 100 K. The domain-like structure was written by high-intensity laser beam and 500 mT in-plane magnetic field.

Anomalous Nernst effect in thin films of non-collinear antiferromagnet Mn3NiN

We combined XMLD, XMCD and MOKE measurements in order to follow the magnetic properties of epitaxial CoO(111)/Fe(110) [1] and NiO(111)/Fe(110) [2] bilayers. We find that in both studied cases FM sublayer plays a dominant role and determines the magnetic state of the neighboring AFM, however completely different interaction mechanisms are involved. In CoO/Fe bilayers the AFM spins are frozen and their orientation is imprinted by magnetization of Fe layer when the system passes the Neel temperature of CoO. Once the Fe layer grafts the particular magnetic anisotropy (MA) into the CoO overlayer, it later remains frozen and insensitive to external factors like external magnetic field or Fe magnetization direction [3]. Specifically, choice of particular magnetic state of Fe sublayer, when passing Néel temperature of CoO, determines both the axis and direction of interfacial antiferromagnetic spins after the sample is cooled and allows for imprinting their +/- 90° and 0/180° alignment within the sample plane. For example, particular direction of frozen AFM spins determines the corresponding sign of the shift field of exchange biased magnetic hysteresis loop (Fig.1). For NiO/Fe bilayers, due to the weak intrinsic MA of NiO, the AFM spins are rotatable and always follow the reorientation of Fe magnetization that can be controlled by external magnetic field or via the temperature and thickness driven SRT of Fe(110). Recently we showed that in a uniform thickness NiO(111)/Fe(110) system, two magnetic states with orthogonal spin orientations can be stabilized in antiferromagnetic NiO [2]. Field-free, reversible switching between these AFM states was demonstrated. Here we present results of systematic investigations of such field-free, temperature induced switching of AFM NiO spins. Fig. 2 shows temperature dependence of RL2 ratio for various thicknesses of Fe. Critical temperature of SRT in AFM NiO can be tuned in wide temperature window ~ (250 – 380 K) for field-free, reversible switching of AFM spins. XML(C)D-PEEM results obtained on individual self-organized NiO/Fe nanostripes will be also presented.

![](https://assets.underline.io/uploads/markdown_image/1/image/ea6cc84785bb0485bc069d8c09366d06.jpg)

Fig.1. Memory of frozen AFM spins in CoO(111)/Fe bilayer.

![](https://assets.underline.io/uploads/markdown_image/1/image/46723238dbaaa7708138289f718efd3d.jpg)

Fig.2. Temperature dependence of XMLD (RL2 ratio) for NiO/Fe [4].

Memory of frozen and rotatable antiferromagnetic spins in epitaxial CoO(111)/Fe and NiO(111)/Fe bilayers.

Synchronization of large spin Hall nano-oscillators (SHNO) arrays is an appealing approach toward ultra-fast non-conventional computing based on nanoscale coupled oscillator networks [1]. However, for large arrays, interfacing to the network, tuning its individual oscillators, their coupling, and providing built-in memory units for training purposes, remain substantial challenges. Here, we address all these challenges using memristive gating of W/CoFeB/MgO/AlOx based SHNOs. We use this type of stack as their substantial perpendicular magnetic anisotropy (PMA) can generate both localized and propagating spin waves [2]. In its high resistance state (HRS), the memristor modulates the perpendicular magnetic anisotropy (PMA) at the CoFeB/MgO interface purely by the applied electric field, which leads to both a voltage-controlled SHNO frequency and a voltage-controlled threshold current [3]. In its low resistance state (LRS), and depending on the voltage polarity, the memristor adds/subtracts a current Im to/from the SHNO drive. The operation in this LRS also affects the SHNO auto-oscillation mode and frequency, which can be tuned up to 28~MHz/V. This tuning allows us to reversibly turn on/off mutual synchronization in chains of four SHNOs and turn the chain into different partially synchronized states [4]. Memristor gating is, therefore, an efficient approach to input, tune, and store the state of the SHNO array for any non-conventional computing paradigm, all in one platform. Examples include SHNO based Ising Machines [5,6] and pattern recognition using memristive-controlled SHNO chains [4], which will both be discussed in detail in the presentation.

Theory of Spin Torques Emerging from Band Topology

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