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The dynamics involving directed separation of a monopole-antimonopole pair in two- or three-dimensional spin ices is conventionally defined as magnetic current or magnetricity. While the feasibility of magnetricity in artificial spin ices has been shown theoretically and experimentally [1,2], the controllability of magnetic currents is still under debate. Particularly, it has been theoretically shown that maximization of Dirac string’s tension is a necessary condition for stabilization of directed magnetic currents [1]. Experimentally, however, magnetic currents are often of only short duration, because magnetic monopoles created at one sample edge do not reach the other sample edge, but rather stick in the middle of the sample. Here, we explore theoretically the role of the termination of arrays and the higher order magnetic interactions for the directionality of magnetic currents in two-dimensional artificial spin-ice. Our Monte-Carlo calculations show that arrays of point dipoles and identical array with realistic multipole moments give qualitatively similar results with respect to the end states and with respect to the averaged propagation of the monopoles. However, the Monte-Carlo calculations using the higher order contributions reflect much better formation of domains with phase shifted Dirac strings, found in recent experiments [3]. Our calculations permit us to elaborate a phase diagram predicting the probability of directed propagation of monopole current as a function of termination type of a realistic artificial spin-ice array.References: [1] E. Y. Vedmedenko, Dynamics of Bound Monopoles in Artificial Spin Ice: How to Store Energy in Dirac Strings. Physical Review Letters 116, 077202 (2016). [2] A. Farhan et al., Direct Observation of Thermal Relaxation in Artificial Spin Ice. Phys. Rev. Lett. 111, 057204 (2013). [3] A. Hanu, E. Y. Vedmedenko, J. Cuia, J. Vijayakumard, A. Kleibertd, L.J. Heyderman, Control of emergent magnetic monopole currents in artificial spin ice, submitted.

MMM 2020

Role of Higher-Order Magnetostatic Interactions for the Directional Injection of Monopole Currents

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The demand of flexibility in the existing rigid devices has introduced a new field of research ‘Flexible Spintronics’, which has enormous significance from application viewpoint [1-2]. Flexible device requires all the components of rigid spintronics in a flexible i.e. bendable, foldable, twistable and stretchable mode [3]. The ideal next generation device need to be power efficient as well as flexible. To meet such device requirement stress/strain has proven to be a potential candidate [4-5]. As magnetic thin films are prime component of every spintronic device hence the flexible counterpart of those are prerequisite. Cobalt/Platinum (Co/Pt) thin films with perpendicular magnetic anisotropy (PMA) are well studied as it has increased magnetic storage density, thermal stability etc. [6]. In our work we have prepared a PMA Co/Pt film on a flexible polyimide substrate by magnetron sputtering. Strain has been generated on the film by fixing it on semi-circular shaped molds of different radius. Here systematically we have measured the effect of both tensile and compressive stresses on the PMA, domain dynamics and relaxation mechanism by MOKE based microscopy. A minimal in-plane tensile strain has increased the coercivity of the film by 34% of its initial value, while a very small change of coercivity has been found under compressive strain (fig. 1) [7]. The size of ferromagnetic domains also decreases under tensile strain, while no change is observed under compressive stress (fig. 2). Due to stress induced anisotropy, Bloch wall formation energy has increased which hindered the domain wall motion and size of bubble domains reduced. Further, magnetization relaxation measured at sub-coercive field values yields a longer relaxation time in the tensile strained state.References: 1. P. Sheng et al., J. Semicond. 39, 011006 (2018) 2. Z. Bao et al., Adv. Mater. 28, 4177 (2016) 3. K. Harris et al., J. Mater. Sci. 51, 2771–2805 (2016) 4. A. Bukharaev et al., Phys. Uspekhi 61, 1175 (2018) 5. A. K. Biswas et al., Nano letters 17, 3478–3484 (2017) 6. B. Tudu et al., Vacuum 146, 329 (2017) 7. E. Pandey et al., Nano Express 1, 010037 (2020)

Strain Engineered Domain Structure and Their Relaxation in Perpendicularly Magnetized Co/Pt Deposited on Flexible Polyimide

Magnetic elastomers are one of the most important types of magnetoresponsive materials. They are characterized by large changes in their shape and rehological properties as a response to external magnetic fields [1]. Similarly to magnetic fluids and gels, these are hybrid materials that consist of magnetic nano- or microparticles embedded in a magnetically passive carrier material. Magnetic elastomers are candidate materials for many technological applications, like adaptive damping devices, vibrational absorbers, stiffness tunable mounts, soft actuators and micromanipulators, force sensors or artificial muscles, see [2] and references therein. Here, we study theoretically a novel microstructural design for a hybrid magnetic elastomer material that has been synthesized for the first time very recently [3-5]. This design is based on the embedding within the polymer matrix of two types of magnetic particles in different amounts: a low volume fraction of magnetically hard (MH) colloidal particles---typically, ferromagnetic particles of 5-100μm of diameter---and a high volume fraction of smaller, magnetically soft (MS) particles---paramagnetic particles with a diameter typically smaller than 5μm. We study the response of this hybrid magnetic material to external magnetic fields by means of two theoretical approaches. First, we perform extensive computer simulations with a bead-spring model of the system that takes into account the magnetic influence of the MH particles on the MS ones, as well as the mechanical coupling imposed by the polymer matrix on the rearrangements of the particles [6]. Finally, we compare the results obtained from this approach with the ones provided by a continuous magnetomechanical model. In both approaches we focus on a minimal representative volume of the material, consisting of a single MH particle surrounded by a cloud of MS ones and the mechanical coupling between all of them. We analyze the deformations of this elementary volume as a response to the external fields.References: [1] S. Odenbach. Arch. Appl. Mech., 86:269–279, (2016). [2] P. Sánchez et al., Soft Matter, 15, 175-189 (2018). [3] J.M. Linke, D.Y. Borin, S. Odenbach, RSC Adv., 6, 100407 (2016). [4] M.V. Vaganov, J. Linke, S. Odenbach, Yu.L. Raikher, J. Mag. Mag. Mat. 431, 130-133 (2017). [5] P. Sánchez et al., Soft Matter, 14, 2170-2183 (2018). [6] P. Sánchez et al., Soft Matter, 15, 7145--7158 (2019).

Theoretical Modelling of Hybrid Magnetic Elastomers

Progress has been made on emulating the neuron and the synapse using magnetic devices [1], [2] especially with three-terminal magnetic tunnel junctions (3T-MTJs) [3], [4]. Increasingly more accurate biological mimics are being developed with the 3T-MTJ like the Leaky-Integrate-Fire (LIF) [5] neuron displaying lateral inhibition, and the 3T-MTJ synapse model showing spike timing dependent plasticity (STDP) [6], [7]. However, more work must be done to achieve real-time machine learning. Using an earlier developed synapse circuit [8] and a 3T-MTJ spice model [9], we show the transient behavior of a 2 by 2 crossbar array (cartoon in Fig 1). There is STDP for different delay conditions between the pre- and post-synaptic neural spikes. The shorter the time difference between the onset of the pre- and post-synaptic neuron spike, the higher the current through the ferromagnet (Fig 2). Higher current leads to higher rate of change of the domain wall (DW) position per neural event and higher DW displacement. At t=0, the initial pair of presynaptic and postsynaptic pulses helps to randomly give the DWs a head start. DW for synapse S2 (red plot) rises faster because it has the least delay of 1 ns between its pre- and post-synaptic pulse despite its pulses lagging that of S1 whose DW had a head start. Scaling this up, the 3T MTJ synapse is robust in more complex systems such as a machine learning clustering task.References: [1] S. Lequeux et al., Sci. Rep., vol. 6, no. August, 2016, doi: 10.1038/srep31510. [2] J. Torrejon et al., Nature, vol. 547, no. 7664, pp. 428–431 (2017) [3] N. K. Upadhyay, H. Jiang, Z. Wang et al, Adv. Mater. Technol., vol. 1800589, pp. 1–13 (2019) [4] N. Hassan et al., J. Appl. Phys., vol. 124, no. 15, p. 152127 (2018) [5] G. Bi and M. Poo, Annu. Rev. Neurosci., vol. 24, no. 1, pp. 139–166 (2002) [6] H. Markram, W. Gerstner, and P. J. Sjöström, Front. Synaptic Neurosci., vol. 4, pp. 2–5 (2012) [7] O. Akinola, X. Hu, C. H. Bennett et al, J. Phys. D. Appl. Phys., vol. 52, no. 49 (2019) [8] X. Hu, A. Timm, W. H. Brigner et al, IEEE Trans. Elect. Dev., vol. 66, no. 6, pp. 2817–2821 (2019)

Online Training of Spiking Neural Networks Using Domain Wall Magnetic Tunnel Junction Synapses

Domain wall-magnetic tunnel junction (DW-MTJ) devices intrinsically emulate neuron [1] and synapse [2] functionalities for neuromorphic systems. While these devices efficiently emulate neurobiological components, significant CMOS overhead is required to integrate multiple perceptron layers [3]. This work proposes a four-terminal DW-MTJ neuron [4] that enables purely spintronic multilayer perceptrons that do not require CMOS. The proposed leaky integrate-and-fire neuron [5] in Fig. 1 has a ferromagnetic DW track between terminals T1 and T2 and an MTJ between terminals T3 and T4. The MTJ free layer is coupled to the DW track by an electrically insulated layer. Spin-transfer torque induced by the current through the T1-T2 path moves the DW from right to left to perform integration. DW leaking from left to right intrinsically occurs due to either dipolar magnetic field [1], anisotropy gradient along [6], or shape variation [7]. When the DW passes underneath the electrically-isolated MTJ, the free ferromagnet reorients itself to match the magnetization of the DW track due to dipolar coupling, thereby switching the MTJ between the anti-parallel resistive state and parallel conductive state. A constant voltage applied at the T3 terminal, fires by switching between low and high output currents. Additionally, by exploiting stray magnetic field interactions, these neurons perform lateral inhibition among a group of neighboring neurons [1]. In the multilayer perceptron of Fig. 2, each four-terminal DW-MTJ neuron is connected to the three-terminal analog DW-MTJ input synapses through the T1 terminal and is cascaded to the output synapses through the T4 terminal without any CMOS interface. Finally, an unsupervised learning algorithm results from feedback between the T4 terminals of the neurons and right terminals of the synapses, providing best results of 98.11% accuracy on the Wisconsin breast cancer clustering task [8].References: [1] N. Hassan, X. Hu, L. Jiang-Wei, et al., J. Appl. Phys., Vol. 124, no. 15, p. 152127 (2018) [2] O. Akinola, X. Hu, C. H. Bennett, et al., J. Phys. D: Appl. Phys., Vol. 52, no. 49, LT01 (2019) [3] A. Sengupta, Y. Shim, and K. Roy, IEEE TBioCAS, Vol. 10, no. 6, pp. 1152–1160 (2016) [4] W. H. Brigner, N. Hassan, X. Hu, et al., arXiv preprint arXiv: 2002.00862 (2020) [5] A. N. Burkitt, Biol. Cybern., Vol. 95, pp. 97–112 (2006) [6] W. H. Brigner, X. Hu, N. Hassan, et al., IEEE JXCDC, Vol. 5, no. 1, pp. 19–24 (2019) [7] W. H. Brigner, N. Hassan, L. Jiang-Wei, et al., IEEE TED, Vol. 66, no. 11, pp. 4970-4975 (2019) [8] A. Velasquez, C. H. Bennett, N. Hassan, et al., arXiv preprint arXiv: 2003.11120 (2020)

Purely Spintronic Multilayer Perceptron Enabled by Four-Terminal Domain Wall-Magnetic Tunnel Junction Neuron

Magnetic materials are already widely used for long-term data storage, but ongoing work seeks to use them as both working memory and computing architectures. Particularly, devices based on magnetic domain walls (DWs) can perform logic operations and can readily store information [1]. However, the stochasticity of DW pinning limits the feasibility of creating technologically viable devices. Here, we demonstrate how stochasticity can be changed from a technologically inhibitive behaviour into an integral property of machine learning algorithms for specialised neuromorphic devices. We first present experimental measurements that demonstrate the tunability of stochastic processes by applying external stimuli. Focused magneto-optic Kerr effect measurements were used to probe the pinning of DWs at notch-shaped defect sites in 400 nm wide Permalloy nanowires (Fig. 1). DWs were injected into the nanowires and propagated to the defect sites using a field parallel to the nanowire (Hx). The probability of the DWs being pinned at the defect sites was found to depend sigmoidally on the magnitude of a field transverse to the nanowires (Hy). The stochastic behaviour of the nanowire were then integrated into machine learning models. We have developed feedforward neural networks with the nanowires acted as binary stochastic synapses (BSS) that can be sampled by repeating the network. It is found that for a single cycle of the network it can be trained to recognise handwritten digits with an error rate of 19%. Repeated sampling of the network decreases the error rate to 9% for 128 cycles - close to mean field performance (Fig. 2). Adding a hidden layer improves performance with extra pathways for the stochastic behaviour to be averaged over. Our work illustrates how the stochasticity of DW devices can be harnessed to provide tunable bespoke hardware for machine learning tasks.References: [1] R. L. Stamps et al., J. Phys. D. Appl. Phys. Vol. 47, p. 333001 (2014)

Harnessing Stochastic Domain Wall Pinning for Machine Learning

Lateral inhibition (LI) is an important neuromorphic network functionality, modeled after the biological neuron behavior that a firing neuron deactivates its neighbors and prevents their firing. In most neuromorphic hardware platforms LI is implemented by external circuitry, thereby decreasing the energy efficiency and increasing the area overhead. Recently, the domain wall – magnetic tunnel junction (DW-MTJ) neuron is shown in modeling to be intrinsically inhibitory1. LI in DW-MTJ neurons results from magnetostatic interaction between adjacent neurons (Fig.1): the two DWs, DWN and DWI, are driven by electrical current of densities JeI<JeN, so that the DW velocities vDWI<vDWN and the active Neuron N fires first. DWI falls behind DWN and is subjected to the stray field of Neuron N in the −z direction. It is demonstrated that such stray field can decrease DWI velocity and prevent Neuron I from firing, thus emulating LI. However, the LI mechanism in DW-MTJ neurons has not been studied thoroughly, leading to weak inhibition only in very closely spaced devices. In our work2 we model the current- and field-driven DW motion in two adjacent DW-MTJ neurons to approach these problems. LI is maximized by an optimal magnetic interaction strength, determined by tuning the spacing between the two neurons (Fig.2). The results are explained by the current- and field-driven DW velocity characteristics and are quantified by an analytical model. We further study the influence of device parameters and demonstrate the LI behavior in an array of 1000 neurons, which renders our proposed DW-MTJ array promising for implementing a k-winner-take-all layer. With strong LI achieved, a path towards competitive learning algorithms are made possible on the DW-MTJ neurons. This work is funded by Sandia National Lab. Sandia National Lab (SNL) is managed and operated by NTESS under DOE NNSA contract DE-NA0003525References: 1. N. Hassan et al. Journal of Applied Physics, Vol.124, p.152127 (2018) 2. C. Cui et al. Nanotechnology, Vol.31(29), p.294001 (2020)

Maximized Lateral Inhibition in Paired Magnetic Domain Wall Racetracks for Neuromorphic Computing

Neuromorphic computing attempts to mimic the neurons and synapses in a human brain in order to provide significant improvements in the computation of unstructured, real-world data. In past research, we have proposed three separate leaky integrate-and-fire (LIF) neurons that provide the leaking, integrating, and firing characteristics without the use of any additional circuitry [1]-[3]. These neurons are therefore able to significantly reduce the area and energy overhead of neuromorphic circuits. To increase the biological accuracy of these neurons and to better match the neurons to different neuromorphic schema and algorithms, it is desirable for neuron leaking to implement specific mathematical functions in addition to exhibiting the three basic LIF neuronal functionalities. By varying the shape of the devices, it is possible to implement a variety of leaking characteristics. In this work, we discuss the implementation of linear leaking characteristics. In the neurons of [1]-[3], the leaking DWs accelerate towards the ends of the tracks. To counter this, it is necessary to alter these devices. In fact, by simply introducing slight exponential variations in the sides of the neuron from [3], as in Figure 1, we are capable of constructing neurons with considerably more linear leaking characteristics than those discussed in [1]-[3]. As the exponential variations become more pronounced, the linearity of the neuron’s leaking increases as well. Beyond a certain point, however, continued increases in the curvature causes a regression in the leaking functionality’s linearity. Figure 2 demonstrates the effect of the curvature of the DW track on the device’s leaking functionality, starting with no curvature and progressing to more pronounced curvature.References: [1] N. Hassan, X. Hu, L. Jiang-Wei et al., “Magnetic domain wall neuron with lateral inhibition,” J. Appl. Phys., vol. 124, no. 15, p. 152127, 2018. [2] W. H. Brigner, X. Hu, N. Hassan, et al., “Graded-Anisotropy-Induced Magnetic Domain Wall Drift for an Artificial Spintronic Leaky Integrate-and-Fire Neuron,” IEEE Journal on Exploratory Solid-State Computational Devices and Circuits, vol. 5, no. 1, pp. 19–24, 2019. [3] W. H. Brigner, N. Hassan, L. Jiang-Wei, et al., “Shape-Based Magnetic Domain Wall Drift for an Artificial Spintronic Leaky Integrate-and-Fire Neuron,” IEEE Transactions on Electron Devices, vol. 66, no. 11, pp. 4970-4975, 2019.

Linear Intrinsic Leaking in a Domain-Wall Magnetic Tunnel Junction Neuron

The relaxation kinetics and emerging correlations of arrays of interacting Ising-like nanomagnets are governed by the switching rates of the individual magnets. These rates depend, via the Arrhenius law, on the energy barrier for moment reversal in the nanomagnets. In this work, we consider how the switching behaviour of a nanomagnet in archetypical artificial square ice is modified by the interaction with its six nearest neighbours. We find that due to two effects the transition barriers can be significantly lowered compared to an approximation commonly used for kinetic Monte Carlo simulations: (1) Based on a simplified point-dipole model, we find that the barriers for clockwise and counter-clockwise uniform rotation can differ significantly due to the interactions with neighbours. For a subset of environmental microstates this leads to a significant enhancement of the switching rate, which is especially marked in the limit of rare events. (2) Micromagnetic simulations employing the string method to model the moment reversal in an extended nanomagnets reveal that, in addition to the clockwise/counter-clockwise splitting, the environments promote non-uniform reversal leading to further reductions of the switching barriers. These results highlight that local microstates can significantly enhance moment reversals already in ideal systems, without defects or additional anisotropies, and are particularly pronounced for fully-magnetised configurations. Thus, we expect that the initial stages of demagnetisation will proceed faster than assumed from previous models. These findings are of integral importance to achieve realistic kinetic Monte Carlo simulations of emergent correlations in extended artificial spin systems or the evolution of small-scale nanomagnetic logic circuits.

Switching barriers in artificial square ice

Modeling computer architecture after the human brain is a promising approach to the ever-increasing demands for real-time processing of massive amounts of data. But, the building blocks of the brain, i.e. neurons and synapses, have more detailed behavior than simply how they are connected, and this behavior is central to their computing efficiency. If we want to capture the energy efficiency of such a neuromorphic architecture, we need devices themselves that can capture the behavior of the biological elements. Magnetic materials and devices have many properties similar to the brain that make them attractive candidates as artificial neurons and synapses. We will present our recent work on designing and modeling three-terminal domain wall-magnetic tunnel junctions (DW-MTJs) that can act as both leaky, integrate, and fire neurons with inherent lateral inhibition [1-2] and as synapses with online learning and spike-timing dependent plasticity behavior [3]. The DW-MTJ neuron consists of a perpendicularly magnetized wire containing a single DW and an MTJ on top of the wire. When current is applied to the wire, the DW propagates through spin transfer torque (STT) or spin orbit torque (SOT). The MTJ defines the firing point of the neuron: when the DW moves past the junction, the MTJ resistance switches to generate a spiking current at the MTJ output terminal. We will show results of integration, firing, and inherent leaking of the device. When assembled into a crossbar array, 1-10 digit identification is simulated by measuring which of ten neurons reach the firing point first. Lateral inhibition (LI) is an important function in neuromorphic computing, modeled after the biological neuron behavior that a neuron prevents its less active neighbors in the same layer from firing. In most neuromorphic hardware platforms, LI is implemented with external circuitry, thereby decreasing the energy efficiency and increasing the area overhead of such systems. We show that the DW-MTJ neuron has inherent LI via magnetic interactions. We will show results from modeling current- and field-driven DW motion in adjacent DW-MTJ neurons. Figure 1 shows that with 90 nm spacing between two 50 nm wide DW-MTJs, the slower device’s DW velocity is drastically reduced from 75 m/s to 20 m/s and a LI of 75% is achieved. The results are explained by current-driven DW velocity characteristics in response to external magnetic fields and are quantified by an analytical model based on the Landau-Lifshitz-Gilbert (LLG) equation. We will show results of arrays of DW-MTJs implementing a soft winner-take-all function and application of the effect to image recognition. We will furthermore present simulation results of the DW-MTJ showing synaptic behavior when the MTJ is made long. By constructing a small circuit around the device we implement spike-timing-dependent plasticity (STDP), where the current flowing through the synapse, which will determine its resistive synaptic weight, is determined by the delay between input and output neurons (Fig. 2). We will show that the STDP synapses can process time-dependent input data. The results will be compared with other emerging nonvolatile devices for artificial neural networks. The work presented shows that DW-MTJs can provide a monolithic platform for neuromorphic computing circuits, where both synapses and neurons can be made from the same thin film stack, and motivate further study of magnetic devices for neuromorphic computing. SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525.References: 1. N. Hassan, X. Hu, L. Jiang-Wei, W. H. Brigner, O. G. Akinola, F. Garcia-Sanchez, M. Pasquale, C. H. Bennett, J. A. C. Incorvia & J. S. Friedman. Magnetic domain wall neuron with lateral inhibition. J. Appl. Phys. 124, (2018). 2. C. Cui, O. G. Akinola, N. Hassan, H. Christopher, M. J. Marinella, J. S. Friedman & J. A. C. Incorvia. Maximized Lateral Inhibition in Paired Magnetic Domain Wall Racetracks for Neuromorphic Computing. IOP Nanotechnology 31, 29 (2020). 3. O. G. Akinola, X. Hu, C. H. Bennett, M. Marinella, J. S. Friedman & J. A. C. Incorvia. Three-terminal magnetic tunnel junction synapse circuits showing spike-timing-dependent plasticity. J. Phys. D. Appl. Phys. 52, 49LT01 (2019).

Modeling biological behavior in domain wall-magnetic tunnel junction artificial neurons and synapses for energy-efficient neuromorphic computing INVITED

Novel computational paradigms in combination with proper hardware solutions are required to overcome the limitations of our state-of-the-art computer technology, in particular regarding energy consumption. Due to the inherent complex and non-linear nature, spintronics offers the possibility towards energy-efficient, non-volatile hardware solutions for various unconventional computing schemes.[1-3] In this talk, I will address the potential of topologically stabilized magnetic whirls – so-called skyrmions – for in particular two unconventional computing schemes – reservoir computing and stochastic computing. Reservoir computing is a computational scheme that allows to drastically simplify spatial-temporal recognition tasks. We have shown that random skyrmion fabrics provide a suitable physical implementation of the reservoir [4,5] and allow to classify patterns via their complex resistance responses either by tracing the signal over time or by a single spatially resolved measurement. [6] Stochastic computing is a computational paradigm that allows to speed up a calculation while trading for numerical precision. Information is encoded in terms of bit-streams as a probability. A key requirement and simultaneously a challenge is that the incoming bitstreams are uncorrelated. The Brownian motion of magnetic skyrmions allows creating a device that reshuffles the bit-streams. [7,8]References: [1] J. Grollier, D. Querlioz, K. Y. Camsari, KES, S. Fukami, M. D. Stiles, Nature Electronics (2020) [2] E. Vedmedenko, R. Kawakami, D. Sheka, ..., KES, et al., J. of Phys. D (2020) [3] G. Finocchio, M. Di Ventra, K.Y. Camsari, KES, P. K. Amiri and Z. Zeng, arXiv:1907.04601 [4] D. Prychynenko, M. Sitte, et al, KES, Phys. Rev. Appl. 9, 014034 (2018) [5] G. Bourianoff, D. Pinna, M. Sitte and KES, AIP Adv. 8, 055602 (2018) [6] D. Pinna, G. Bourianoff and KES, arXiv:1811.12623 [7] D. Pinna, F. Abreu Araujo, J.-V. Kim, et al, Phys. Rev. Appl. 9, 064018 (2018) [8] J. Zazvorka, F. Jakobs, D. Heinze, ..., KES, et al., Nat. Nanotech. 14, 658 (2019)

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Role of Higher-Order Magnetostatic Interactions for the Directional Injection of Monopole Currents

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