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Threshold logic [1] efficiently implements complex logical functions within a single gate, but its widespread adoption has been impeded by insufficient nonlinearity in conventional transistors. The recent experimental demonstration of cascaded logic gates with current-driven magnetic domain walls [2] operates based on the nonlinearity intrinsic to anisotropic systems, thus naturally enabling threshold logic gates. However, these logic gates are described in [2] as reconfigurable NAND and NOR gates, rather than as three-input minority threshold logic gates. We therefore propose a novel domain wall threshold logic (DWTL) paradigm that enables circuits that are simpler and more efficient than can be achieved with the reconfigurable gates of [2]. Similar to [2], magnetic domains magnetized in the +z (-z) direction represent logic ‘1’ (‘0’), and each in-plane region causes magnetization reversal due to chiral coupling resulting from the interfacial DMI. As demonstrated in [2], DWTL gates produce an output domain that is the inversion of the majority of the inputs. That is, this logic gate performs the minority threshold function, which is significantly more logically expressive than NAND or NOR. By leveraging this minority gate behavior, the one-bit full adder of Fig. 1 can be realized with three logic gates, 80% fewer than the 15 required in [2]. We also propose DWTL functions with logical expressivity increased further by using more than three inputs, and with inputs that have non-uniform impacts on the output magnetization. As shown in Fig. 2a, a four-input DWTL gate can be realized in which one input has twice the width of the other three; therefore, this wider input will have roughly twice the impact on the output magnetization. DWTL gates with greater logical expressivity further reduce the number of devices required to perform logical operations, such as in the three-bit DWTL full adder of Fig. 2b that requires only two gates per bit (87% reduction).References: [1] D. Hampel and R. O. Winder, IEEE Spectrum, Vol. 8, pp. 32-39 (1971) [2] Z. Luo, A. Hrabec, T. P. Dao, et al., Nature, Vol 579, pp. 214-218 (2020)

MMM 2020

Threshold Logic with Current-Driven Magnetic Domain Walls

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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technical paper

The efficiency of manipulation of domain walls and skyrmions in ferromagnetic racetracks with perpendicular anisotropy determines perspectives of development of data storage and logic devices relying on spintronic and spin-orbitronic concepts [1, 2]. The domain wall dynamics is dependent on its orientation with respect to the racetrack axis. In-plane fields [3], edge roughness [4] and current [5] result in the domain wall tilt in samples, possessing Dzyaloshinskii-Moriya interaction (DMI). Here, we show theoretically, that the tilt can appear in equilibrium and describe the domain wall dynamics under the action of external field. We consider a thin biaxial stripe with DMI of interfacial type [6]. The main easy axis of anisotropy is perpendicular to the plane, and the direction of the second easy axis lies in the stripe plane under the angle α to the stripe axis. While the shape anisotropy results in α = 0, a general case α ≠ 0 can appear under the influence of other effects, e.g crystalline structure [7]. While the second easy axis defines the preferable in-plane magnetization within the domain wall, the DMI forces the domain wall being perpendicular to the magnetization gradient. The competition between these two energy contributions and the domain wall tension results in the unidirectional tilt of the whole domain wall. If the DMI is weak enough, there is an additional metastable domain wall state, tilted into the opposite direction. The symmetry break is observed not only for static magnetization texture, but also in the domain wall dynamics under the action of external magnetic field. The domain wall reveals fast and slow motion regimes for the opposite signs of A. The maximum of the Walker field and Walker velicities is determined by the angle A of the second easy axis anisotropy and does not coincide with a shape-induced anisotropy direction A=0. The domain wall possesses the switch of the magnetization direction inside the domain wall in the slow motion regime, which results in the faster motion.References: [1] K.-S. Ryu, L. Thomas, S.-H. Yang et al., Nat. Nanotech., Vol. 8, 527 (2013) [2] O. Pylypovskyi, D. Sheka, V. Kravchuk et al., Sci. Rep. Vol. 6, 23316 (2016) [3] C. Muratov, V. Slastikov, A. Kolesnikov et al., Phys. Rev. B. Vol. 96, 134417 (2017) [4] E. Martinez, S. Emori, N. Perez et al. J. Appl. Phys. Vol. 115, 213909 (2014) [5] O. Boulle, S. Rohart, L. Buda-Prejbeanu et al., Phys. Rev. Lett. Vol. 111, 217203 (2013) [6] O. Pylypovskyi, V. Kravchuk, O. Volkov et al., J. Phys. D. (2020), DOI:10.1088/1361-6463/ab95bd [7] M. Heide, G. Bihlmayer, S. Blügel, Pys. Rev. B, Vol. 78, p. 140403 (2008).

Domain Wall Tilt and Enhancement of the Walker Limit in Stripes with Dzyaloshinskii-Moriya Interaction and Perpendicular Anisotropy

Chiral magnetic domain walls stabilised by the interfacial Dzyaloshinskii-Moriya interaction are of great technological interest since the latter favors a given Néel wall with fast current-driven domain wall propagation [1]. We study experimentally by Kerr microscopy, numerically and analytically chiral magnetic domain wall motion in the creep regime under easy-axis driving fields in materials with a perpendicular magnetic anisotropy with an in plane magnetic field. We study two samples: one asymmetric [Pt(3nm)/Co(0.6nm)/Ta(3nm)] and one symmetric [Pt(3nm)/Co(0.6nm)/Pt(3nm)]. Brillouin light scattering has been used to determine the DMi constant: D = -1.2mJ/m2 and D = 0 mJ/m2 in the asymmetric and symmetric case respectively. As expected from the DMi constant measured by BLS, the DW velocity curves as a function of the in-plane magnetic field are qualitatively different. In the asymmetric system, we could not reach a minimum of the velocity v within our experimental in plane field range (Fig 1a), consistently with the DM field = 220 mT estimated from the BLS measurements. For the symmetric system, a slight asymmetry is observed with a minimum around µ0Hip = 0 mT (Fig 1c). We detail the creep regime analysis by studying the evolution of the depinning field from the creep regime to the flow regime for different µ0Hip by studying the linear variation of ln(v) vs (μ0Hz)−1/4. In the asymmetric case the slope is constant (Fig 1b) unlike in the symmetric case (Fig 1d): when |µ0Hip| < 50 mT the DW goes from left-handed Néel to Bloch to right-handed Néel wall and the slope varies while for larger fields the slope is constant. We can extrapolate the evolution of the depinning field which is strongly influenced by the in-plane biasing field. In the asymmetric system, we observe an excess of velocity with respect to the creep law close to the depinning field (Fig 1d, explained by the increase of the size of the deterministic relaxation close to the depinning transition [2]. The stronger the DMi field, the more this excess of velocity occurs at larger out-of-plane field values.References: [1] A. Thiaville, et al. EPL 100, 57002 (2012) [2] N. B. Caballero, et al., Phys. Rev. B 96, 224422 (2017).

Chiral magnetic domain walls dynamics in creep regime

Spintronics has developed into a growing field with the fabrication of magnetic memories and logic devices. Domain wall motion is a major part of the research in spintronics where spin polarized current is used to move the domain walls by two mechanisms spin tranfer torque(STT) and spin orbit torque(SOT). Compensated ferrimagnets have been in focus recently for their ability to reach the magnetic and angular compensation points. At the angular compensation point, the precessional motion of the local magnetic moment becomes negligible which massively enhances the domain wall velocity near this point[1,2]. In our previous studies, we had shown epitaxially grown ferrimagnet Manganese Nitride(Mn4N) having an anti-perovskite crystal structure with Mn atoms at the corner and the face centre sites while Nitrogen atom in the centre[3]. It has a very low magnetization (71kA/m) and a high perpendicular magnetic anisotropy(PMA) with mm sized domains[3]. Such small magnetization and high PMA is necessary for faster domain wall motion. Such properties led us to observe a very high domain wall velocity of 900 m/s at 1.2*1012 A/m2 [4] using pure STT. Here we report an even larger domain wall velocity of more than 2000 m/s at 1.16*1012 A/m2 using pure adiabatic STT near the magnetic and angular compensation point in epitaxially grown Ni substituted Mn4N nanowires (MnNiN). We also show a very small thresold current density of 1*1011 A/m2. This high speed is comparable to the best velocities obtained by SOT with the assitance of an external in-plane field. In our case, such domain wall velocity is achieved without any external in-plane field, with the sole action of pure current in this single layer system. Addtionally, we also show a reversal in the direction of the domain wall motion after the magnetic compensation point, where the domain walls move in the same direction as the current. We discuss this reversal of the domain wall motion phenomena with the help of DFT calculations.References: 1. K.J. Kim, S.K. Kim, Y Hirata, Nature Materials., Vol 16, p1187 (2017) 2. L. Caretta, M. Mann, F. Buttner, Nature Nanotechnology., Vol 13, p1154 (2018) 3. T. Gushi, L. Vila, O. Fruchart., Jpn J. Appl. Phys. Vol 57, p120310 (2018) 4. T. Gushi, M. J. Klug, J.P. Garcia, Nano Letters, Vol 19, p8716 (2019)

Current induced domain wall motion at the magnetic compensation point in Manganese Nickel Nitride (MnNiN) thin films

Bringing nanomagnetism into the third dimension is of growing interest due to the many advantages that 3D structures provide. The possibility for larger volumes and higher surface-to-volume ratios, as well as the introduction of new topologies, chirality and curvature into the structure, lead to numerous opportunities for new physics and applications.[1] Combining this promising field of 3D nanomagnetism with the well-established field of spintronics offers huge opportunities, a key example being the proposal to use 3D nanomagnets in a 3D racetrack architecture that promises the delivery of non-volatile and ultra-high-density information storage[2]. However, the integration of 3D nanomagnets into 2D microelectronics is challenging and has so far impeded the realisation of technological applications such as sensing, computing or memory. Now, with 3D Nano-printing capabilities, we have developed a method to integrate complex 3D magnetic nanostructures [3] into 2D microelectronic circuits and to measure the magneto-transport properties of our 3D nanomagnetic structures. Here, we demonstrate the direct integration of a complex 3D magnetic nanostructure in a microelectronic circuit (Figure 1). In particular, we exploit the recent advances of Focused Electron Beam Induced Deposition, a 3D nano-printing technique, to directly print a high-quality 3D Cobalt nanobridge on pre-synthesized electrical contacts. This 3D nanomagnetic circuit allows for the determination of the bridge’s magnetic and transport properties by performing magnetotransport measurements for different geometries of the field with respect to the 3D nanostructure (Figure 2). In addition, we infer the field-induced switching process of the 3D nanostructure from the transport measurement, that includes the nucleation, propagation and pinning of domain walls. [4]. This methodology can be extended to more complicated geometries and materials, and so this work represents the first step towards the realisation of 3D nanomagnetic devices and will facilitate further exploration of 3D nanomagnetism for both fundamental and device-based studies.References: 1 A. Fernandez-Pacheco, R.Streubel, O. Fruchart, Nature Communications., Vol. 8, p.15756 (2017) 2 S. Parkin, M. Hayashi, L.Thomas, Science., Vol. 320, p.190 (2008) 3 L.Skoric, D. Sanz-hernandez, F. Meng., Vol. 20, p.184 (2020) 4 F.Meng, C. Donnelly, C. Abert (In preparation).

Creating 3D nanomagnetic circuits for Spintronics applications

The interfacial Dzyaloshinskii-Moriya Interaction (DMI), an anti-symmetric exchange originating from strong spin-orbit coupling (SOC) in systems with broken inversion symmetry, plays a critical role in stabilizing chiral spin textures such as magnetic skyrmions. Magnetic skyrmions with their non-trivial properties have recently emerged as a potential candidate for next-generation spintronics applications such as race-track memory and logic devices [1, 2]. Recently, ferrimagnetic materials have been found to be more attractive for hosting stable ultra-small skyrmions at room temperature than conventional ferromagnets, due to their low saturation magnetization, low stray field, and fast spin dynamics [3, 4]. However, to manipulate skyrmions properties, controlling the DMI is essential. Herein, we present a systematic analysis of the DMI variation in compensated ferrimagnetic Pt/GdCo/Pt1-xWx multilayer as a function of Tungsten (W) composition (x) using Density Functional Theory (DFT). We find that a small amount of W (~10%) is sufficient to give rise to a non-zero DMI by breaking the inversion symmetry of Pt/GdCo/Pt. We also find that our calculated DMI increases as the W composition increases, but saturates at higher W composition, in agreement with experiment [5]. We show that the vanishing of spin-orbit coupling (SOC) energy to the adjacent metal layers of the top interface, and the simultaneous constancy of the bottom interface are responsible for such a saturating behavior of the DMI. Additionally, we investigate Pt/GdCo/X, where X=Ta, W, Ir to demonstrate the effect of capping layer heavy metals on the DMI. Our results predict that W in the capping layer favors a higher value of the DMI than Ta and Ir. Our results open up exciting combinatorial possibilities for controlling the DMI in ferrimagnets to nucleate and manipulate ultrasmall high-speed skyrmions.References: [1] A. Fert, N. Reyren and V. Cros, Nat. Rev. Mater., 2, 1-15 (2017) [2] H. Vakili, Y. Xie and A. W. Ghosh, arXiv (2020), 2001.00729 [3] L. Caretta, M. Mann and F. Büttner, Nat. Nanotechnol., 13, 1154-1160 (2018) [4] S. J. Poon and C. T. Ma, J. Supercond. Novel Magn., 33, 269-273 (2020) [5] Y. Quessab, J. Xu, C. T. Ma, Sci. Rep., 10, 1-8 (2020)

Interface Engineering to Control the Dzyaloshinskii-Moriya Interaction in Ferrimagnetic GdCo: ab initio Calculations

In ferrimagnetic films spin-orbit torque (SOT) has been used for current-induced magnetization switching1 but also domain wall2 and skyrmion motion3. Ferrimagnets with interfacial Dzyaloshinskii-Moriya interaction (DMI) are attractive materials due to their low stray field and fast spin dynamics which can lead to ultrasmall2 and fast skyrmions. Yet, so far, the reported skyrmion velocities3 are much lower than of domain walls2. The current-induced dynamics of DMI skyrmions is intrinsically related to the charge-spin conversion efficiency. Here we investigate the DMI and SOT in CoGd ferrimagnetic thin films, in which we previously showed that we can tune the DMI in a range that allows sub-100 nm skyrmion formation4. DMI and SOT both arise from interfacial effects. It is thus important to understand the interplay between these two effects to achieve efficient and fast skyrmion motion. For this purpose, we studied Pt/CoxGd1-x/(Ir, W or Ta) films as a function of the cap layer to systematically vary the DMI and spin-Hall efficiency (SHE). We performed SOT magnetization switching experiments that we correlate to the measurements of DMI energy and SHE. Figure 1 shows an example of SOT switching phase diagram. The switching current density for different field values and cap layer is summarized in Fig. 2(a). W and Ta capping, which induce the largest net DMI, provide the lowest switching current density, thus the most efficient switching. However, Pt/CoGd/Ir requires a smaller in-plane field to achieve full magnetization switching. In the low field region, the films capped by W and Ta exhibit a metastable state [Fig. 2(b)] composed of skyrmion bubbles as suggested by micromagnetic simulations. Our results provide insight into achieving energy-efficient and fast skyrmion devices.References: 1S.-G. Je et al., APL 112, 062401 (2018) 2L. Caretta et al., Nature Nanotechnology 13, 1154 – 1160 (2018) 3S. Woo et al., Nature Communications 9, 959 (2018). 4Y. Quessab et al., Scientific Reports 10, 7447 (2020)

Interplay between spin-orbit torque and Dzyaloshinskii-Moriya interactions in thin amorphous ferrimagnetic films

Antiskyrmions and other unconventional topological excitations can potentially be supported by low-symmetry ferromagnet--heavy metal (FM-HM) thin films by the Dresselhaus-type anisotropic Dzyaloshinskii-Moriya interaction (aDMI) arising at the FM-HM interfaces. This work presents the potential of C2V Ag/Pt/Co-based films to exhibit aDMI and stabilize new chiral magnetic textures. X-ray diffraction 2θ-ω scans of Ag/Pt/Co/Pt films grown epitaxially on Si(110) single crystal substrates show that Co grows predominantly with (10.0) hexagonal closed-packed (hcp) orientation, deviating from the (110)-oriented face-centered cubic (fcc) structure of Ag and Pt. These findings were corroborated by selected area electron diffraction patterns and further supported by a strong in-plane uniaxial anisotropy measured along the hcp c-axis by alternating gradient field magnetometry. Crosstie domain walls in these films were directly observed by Lorentz transmission electron microscopy, which are also predicted from micromagnetic simulations. The materials system introduced above creates the previously unexplored situation where an in-plane easy axis is coupled with a strong (potentially anisotropic) DMI, which may lead to interesting spin configurations. However, to break the z-mirror plane [v1] and create a non-zero DMI, we fabricate thinner Ag/Pt/[Co/Ni]x/Pt films. X-ray diffraction analysis suggest that the hcp structure is preserved throughout the Ni layers. Structural and magnetic observations of this lower symmetry film stack are compared to micromagnetic simulations and discussed in the context of C2V films potentially exhibiting Dresselhaus-type aDMI and hosting antiskyrmions.References: 1. U. Güngördü et al., Physical Review B., 93, 064428 (2016) 2. M. Hoffman et al., Nature Communications., 8, 308 (2017)

In Pursuit of the Anisotropic Dzyaloshinskii-Moriya Interaction: Structural and Magnetic Characterization of Uniaxial Pt/Co-based C2V Thin Films

We investigate the effect of the distribution of pinning sites on the magnetization behavior in systems with Dzyaloshinskii–Moriya interaction (DMI). We consider the standard classical spin Hamiltonian with Heisenberg exchange, DMI, uniaxial anisotropy statistically varying across the lattice sites, and the Zeeman energy. The statistical variation of the anisotropy constants follows from a spatial Gaussian distribution with non-zero mean and standard deviation σ, and serves as a way to model the structural disorder in the sample. First, we discuss the reduction of the model to a lattice resolved mean-field theory and the development of energy minimisation algorithm for solving it. This approach allows to compute systematically and efficiently the magnetisation versus field, M(H), dependence for variable σ and temperature, and can be used for computing qualitative thermodynamic phase diagrams to explore material behaviour and to guide the computationally costly Monte-Carlo simulations. We show that as the σ increases, relative to the strength of exchange interaction and DMI, the nature of the reversal modes observed along a typical M(H) hysteresis loop changes in a certain temperature window. Namely, in ‘clean’ systems with small σ, the reversal proceeds first through the appearance of skyrmion lattices at low external fields, while in ‘dirty’ systems with large σ the reversal is through the nucleation of individual or small groups of skyrmions. We systematically quantify this effect and discuss its broader implications for applications.

The effects of disorder on hysteresis loops in chiral magnets

A promising approach is to encode bits of information for next-generation memory and logic is by using solitons, such as chiral domain walls (DW) or topological skyrmions, which can be translated by currents across racetrack-like wire devices1,2. One technological and scientific challenge is to stabilize small spin textures and to move them efficiently with high velocities, which is critical for dense, fast memory. However, despite over a decade of research on ferromagnetic materials, current-driven spin texture dynamics faced a “speed limit” of a few hundred m/s, and room-temperature-stable magnetic skyrmions were an order of magnitude too large to be useful in any competitive technologies. These problems were rooted in two fundamental characteristics of ferromagnets: large stray fields, which limit spin texture size3 (packing density), and precessional dynamics, which limit speed4. By using a broader class of multi-sublattice magnetic materials, namely compensated metallic and insulating ferrimagnets, fundamental limits plaguing ferromagnets can be overcome. Here, we engineer compensated chiral ferrimagnets with reduced magnetisation and angular momentum , realizing order-of-magnitude improvements in both bit size and speed5. By using ferrimagnetic Pt/Gd44Co56/TaOx films with a sizeable Dzyaloshinskii–Moriya interaction (DMI), we realize a current-driven DW motion with a speed of 1.3 km s–1 near the angular momentum compensation temperature and room-temperature-stable skyrmions with minimum diameters close to 10 nm near the magnetic compensation temperature (Figure 1). Both the size and dynamics of the ferrimagnet are in excellent agreement with a simplified effective ferromagnet theory. In addition to metallic systems, a broader, ubiquitous class of materials – ferrimagnetic insulating garnets – have been extensively studied for their technologically-desired magnetooptical and spintronic properties. Their low damping and ferrimagnetic dynamics also make them ripe candidates for ultrafast DW motion. Although spin-orbit effects and spin-transport phenomenon from adjacent heavy metal layers have been used to manipulate the magnetisation in this class of materials6, the DMI and chiral spin textures had not been discovered. Here, we discover chiral magnetism that allows for pure spin-current-driven chiral DW motion in iron garnet films and elucidate the origins of the chiral exchange interaction in these films. Moreover, by exploiting reduced angular momentum and low-dissipation in ferrimagnetic insulators, we drive DWs to their relativistic limit using pure spin currents from the spin Hall effect of Pt, achieving record velocities in excess of 4300 m/s, within ~10% of the relativistic limit (Figure 2). We observe key signatures of relativistic motion including Lorentz contraction and velocity saturation. The experimental results are well-explained through analytical and atomistic modeling. More broadly, these observations provide critical insight into the fundamental limits of the dynamics of magnetic solitons and establish a readily-accessible experimental framework to study relativistic solitonic physics. Technologically, this work shows that high-speed, high-density spintronic devices based on current-driven spin textures can be realized using materials in which magnetisation and angular momentum are compensated.References: 1. Parkin, S. S. P., Hayashi, M. & Thomas, L. Science 320, 190–4 (2008). 2. Fert, A., Cros, V. & Sampaio, J. Nat. Nanotechnol. 8, 152–156 (2013). 3. Büttner, F., Lemesh, I. & Beach, G. S. D. Sci. Rep. 8, 4464 (2018). 4. Kim, K.-J. J. et al. Nat. Mater. 16, 1187–1192 (2017). 5. Caretta, L. et al. Nat. Nanotechnol. 13, 1154–1160 (2018). 6. Avci, C. O. et al. Nat. Mater. 16, 309–314 (2017).

Ultrafast Domain Wall Dynamics in Metallic and Insulating Ferrimagnets INVITED

Development of complementary metal–oxide–semiconductor (CMOS) logic is expected to approach its fundamental limit as the scaling down of the CMOS technology is reaching an end. As a route to extend the technology roadmap beyond traditional CMOS logic, novel spin-based logic architectures are being developed to provide nonvolatile data retention, near-zero leakage, and scalability. In particular, architectures based on magnetic domain walls (DWs) take the advantage of the fast motion, high density, non-volatility and flexible design of DWs to process and store information [1,2]. Such schemes have so far relied on DW manipulation and clocking using an external magnetic field, which hinders their implementation in dense, large-scale chips. Here we demonstrate a method for performing all-electric logic operations and cascading using DW racetracks [3]. Our concept for the magnetic DW logic is based on efficient DW motion induced by spin-orbit torques (SOTs) [4] and recently developed chiral coupling between adjacent magnets where the magnetic anisotropy competes with the interfacial Dzyaloshinskii–Moriya interaction (DMI) [5,6]. First we created a DW inverter, the essential basic building block for all implementations of Boolean logic, by embedding a narrow in-plane magnetized region in a racetrack with out-of-plane magnetization and demonstrate the operation of the DW inversion on application of an electric current (Fig. 1a). Building on the principles used to construct the DW inverter, we then fabricated reconfigurable NAND and NOR logic gates, making our concept for current-driven DW logic functionally complete. Finally, we cascaded several NAND gates to build XOR and full adder gates, demonstrating electrical control of magnetic data and device interconnection in logic circuits (Fig. 1b). Our work provides a viable platform for scalable all-electric magnetic logic, paving the way for memory-in-logic applications.References: [1] Allwood, D. A. et al. Science 309, 1688–1692 (2005). [2] Parkin, S. & Yang, S.-H. Nature Nanotechnol. 10, 195–198 (2015). [3] Luo, Z. et al. Nature 579, 214-218 (2020). [4] Manchon, A. et al. Rev. Mod. Phys. 91, 035004 (2019). [5] Luo, Z. et al. Science 363, 1435 (2019). [6] Dao, P. D. et al. Nano Lett. 19, 5930-5937 (2019).

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