The domain wall (DW) magnetic tunnel junction (MTJ) is a promising component of next-generation neuromorphic, in-memory, and logical computing hardware due to features including non-volatility, multiple analog resistive states, low power, and ultrafast read-write operation [1-3]. Predicting the DW motion along a magnetic track involves complicated and time-consuming micromagnetic simulations, which greatly impede the development of computing systems designed with interconnected DW-MTJ devices [4]. The neural ordinary differential equations (ODE) is a recently conceptualized deep learning framework that can predict the behavior of dynamic systems [5]. In this approach, the ODE that defines the time derivative of the system output is expressed as a deep neural network; when provided with initial conditions, an ODE solver determines the system response at future times. Chen et al. [6] extended this approach by expressing hidden variables that are difficult to probe with higher-order derivatives of the output, which are then replaced with mathematically equivalent time-delayed versions. External inputs can be applied to the network in a similar time-delayed fashion. In this work, DW motion by spin-transfer torque excitation is modeled utilizing the neural ODE framework. Temporal DW position data along a magnetic track excited by a set of random sinusoidal current inputs are generated by mumax3 [7] and are used to train a neural ODE in a supervised fashion through backpropagation. For a different set of random sinusoidal current inputs, the DW position forecasted by the trained network matches with the results generated by mumax3 with very high accuracy (Fig. 2) while reducing the simulation time by a factor greater than 200x, thereby opening the path to efficiently design integrated systems interconnecting a large number of DWMTJs. 

**References** [1] M. Alamdar, T. Leonard, C. Cui et al., Appl. Phys. Lett., vol. 118, no. 11, p. 112401 (2021).
[2] N. Hassan, X. Hu, L. Jiang-Wei et al., Jour. Appl. Phys., vol. 124, no. 15, p. 152127 (2018).
[3] W. H. Brigner, N. Hassan, L. Jiang-Wei et al., IEEE Trans. Electron Devices, vol. 66, no. 11, pp. 4970-4975 (2019).
[4] X. Hu, A. Timm, W. H. Brigner et al., IEEE Trans. Electron Devices, vol. 66, no. 6, pp. 2817-2821 (2019).
[5] R. T. Q. Chen, Y. Rubanova, J. Bettencourtet al., Proceedings of NeurIPS, 31 (2018).
[6] X. Chen, F. A. Araujo, M. Riou et al, Nat. Comm., vol. 12, no. 1016 (2022).
[7] A. Vansteenkiste, J. Leliaert, M. Dvornik et al., AIP Advances, vol. 4, no. 10, p. 107133 (2014). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/01364715490ab753cd3ac85dcf7e5cc1.png) 
Up (blue) and down (red) polarized magnetic domains along a track separated by a DW (yellow). Current inputs control the position of the DW by applying spin-transfer torque. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/9e4b8c52f3c00844a1d6aa3d22edfa85.png) 

DW position forecasted by a trained neural ODE in comparison with mumax3 simulation.

Modeling Magnetic Domain Wall Motion with Neural Ordinary Differential Equations

Ultrafast and energy-efficient writing by spin-orbit torque (SOT) prevents device degradation within thermally stable magnetic random-access memory (MRAM) with stable perpendicular magnetic anisotropy (PMA) by separating the read and write paths. However, orthogonality between spin polarization and PMA impedes its application at industrial level because of non-deterministic switching. Previously proposed approaches to break this orthogonality includes structural deformation and external magnetic field control. However, all of those approaches increase the fabrication complexity, are highly sensitive to the SOT current duration and magnitude, or increase the switching energy. By exploiting the precessional nature of the field-like SOT component, we propose toggle PMA SOT-MRAM that achieves field-free and energy-efficient switching with a simple structure, and is robust to variations in the SOT current magnitude and duration. 
The proposed three-terminal structure consists of a magnetic tunnel junction (MTJ), heavy metal, and compensating ferromagnet. A unidirectional SOT current applied through the heavy metal stabilizes the MTJ free ferromagnet magnetization at an excited steady state after crossing the hard axis plane. Switching off the excitation relaxes the magnetization to the nearest easy axis, thus completing a toggle cycle. The next excitory pulse toggles the magnetization back to its initial state. The deformation-free structure provides efficient data read-out and can also be leveraged for directional writing through a simple XOR between the stored and incoming bits. Robust MRAM writing with greater than 50% tolerance to variations in the excitory SOT current magnitude has been demonstrated through micromagnetic simulation. Moreover, room temperature simulations indicate that this switching is robust to thermal noise. The proposed device therefore provides a promising solution for compact, highly-efficient, deformation-free and robust SOT-MRAM with PMA.

Field-Free Toggle Mode Spin-Orbit Torque Driven MRAM with Perpendicular Magnetic Anisotropy

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

HOC: New Approaches in Computational Magnetism

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Naimul Hassan

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Modeling Magnetic Domain Wall Motion with Neural Ordinary Differential Equations

Field-Free Toggle Mode Spin-Orbit Torque Driven MRAM with Perpendicular Magnetic Anisotropy

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

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