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The end of Moore’s Law for complementary metal-oxide semiconductor (CMOS) technology has prompted the search for low-power computing, resulting in several promising proposals based on magnetic logic. One approach aims at tailoring arrays of nanomagnetic islands in which the magnetostatic interactions constrain the equilibrium orientation of the magnetization to embed logical functionalities [1]. Despite the realization of several proofs of concepts of such nanomagnetic logic, it is still unclear what the advantages are compared to the widespread CMOS designs, due to their need for clocking and/or thermal annealing for which fast convergence to the ground state is not guaranteed.&lt;br/&gt;In fact, it seems increasingly evident that “beyond CMOS” technology will require a fundamental rethinking of our computing paradigm. In this respect, digital memcomputing machines have been proposed [2]. These machines consist of &lt;i&gt;self-organizing &lt;/i&gt;&lt;i&gt;logic gates&lt;/i&gt;&lt;i&gt; &lt;/i&gt;(SOLGs)&lt;i&gt; &lt;/i&gt;which are&lt;i&gt; &lt;/i&gt;able to dynamically “self-organize” into their logically correct states irrespective of whether the signal is applied to the traditional input terminals, or the traditional output terminals. This type of terminal-agnostic logic thus allows for reversed computing.&lt;br/&gt;Here [3], we introduce a novel concept of SOLGs that employs stray-field coupled nanomagnetic islands to perform terminal-agnostic logic: nanomagnetic self-organizing logic gates. We thereby focus on the design of a functionally complete NAND gate with which any Boolean circuit can be constructed in a bottom-up approach. Our strategy to build such a nanomagnetic SO-NAND gate relies on two main properties: &lt;i&gt;balancedness&lt;/i&gt; and &lt;i&gt;dynamic error suppression&lt;/i&gt; (DES) – see Figure 1.&lt;br/&gt;For reversible logic to work, the logically correct states should be occupied equally in thermodynamical equilibrium (and not only after a relaxation process into the ground states, as balancedness was originally defined [4]). This contrasts the balanced NAND gate proposed in Ref. [5]. To allow for larger circuits, the probability of being in a logically incorrect state should suppressed. To this end, our DES scheme periodically checks the logical correctness of the gate/coupling and applies additional fields on each island of the gate/coupling if necessary.&lt;br/&gt;Our DES scheme is thus based on &lt;i&gt;local&lt;/i&gt; information, i.e. the magnetization state of a single gate or coupling. This locality is in sharp contrast to the biasing used in p-bit logic inversion, for which the logical behavior of a collection of super-paramagnetic tunnel junctions (sMTJ) is fully imposed by &lt;i&gt;globally&lt;/i&gt;-determined time-varying spin currents [6]. Owing to its locality, our approach does not suffer from scaling issues.&lt;br/&gt;To show the usefulness of our SO-NAND gate, we use it as a building block to construct a self-organizing two-bit multiplier – see Figure 2. Our SO-2BM is capable of decomposing &lt;i&gt;any&lt;/i&gt; output number into its (prime)factors, finding each possible decomposition without clearly favoring nor penalizing any solution. This fact indicates that the balancedness remains largely conserved throughout the whole system, which allows the extension to larger systems in order to solve more complex problems.

**References:**

[1] A. Imre, G. Csaba, L. Ji &lt;i&gt;et al&lt;/i&gt;., Science, Vol. 311, p. 205-208 (2006).&lt;br/&gt;[2] F. Traversa and M. Di Ventra, Chaos, Vol. 27, p. 023107 (2017).&lt;br/&gt;[3] P. Gypens, J. Leliaert, M. Di Ventra &lt;i&gt;et al&lt;/i&gt;., arXiv preprint arXiv:2012.12721 (2020).&lt;br/&gt;[4] J. Lust and B. Dixon, Inf. Sci., Vol. 113, p. 193-204 (1999).&lt;br/&gt;[5] P. Gypens, J. Leliaert and B. Van Waeyenberge, Phys. Rev. Appl., Vol. 9, p. 034004 (2018).&lt;br/&gt;[6] W. Borders, A. Pervaiz, S. Fukami &lt;i&gt;et al&lt;/i&gt;., Nature, Vol. 573, p. 390-393 (2019).&lt;br/&gt;[7] J. Leliaert, A. Vansteenkiste, A. Coene &lt;i&gt;et al&lt;/i&gt;., Med. Biol. Eng. Comput., Vol. 53, p. 309-317 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7d77a99461c449a155c9bf16943ac1b1.jpg)
Fig. 1: Schematic representation of a self-organizing NAND gate consisting of four nanomagnetic islands with perpendicular anisotropy. The probability distribution is obtained with the macrospin simulation tool Vinamax [7] which numerically solves the Landau-Lifshitz-Gilbert equation taking thermal fluctuations into account. Logically incorrect states are suppressed by a dynamic error suppression (DES) scheme, while still preserving the balancedness of the gate. P&lt;sub&gt;rel&lt;/sub&gt; is the probability relative to the most likely logical state.

![](https://assets.underline.io/uploads/markdown_image/1/image/bdd9a7d3218e170d709dc62ca31af92c.jpg)
Fig. 2: Schematic representation of a self-organizing two-bit multiplier and the relative probability distribution obtained with Vinamax for an output fixed to 0. P&lt;sub&gt;rel&lt;/sub&gt; is the probability relative to the most likely logical state. The four output islands of the SO-2BM, labeled from a to d, are fixed such that they represent a certain number (0, 1, 2, 3, 4, 6, or 9) and the system is allowed to explore its state space. The magnetization of the input islands, corresponding to the solution of the computation, are labeled from A to D. All states of which one of the inputs is logically inconsistent are aggregated into the &quot;??&quot;-state.

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 Nanomagnetic Self-Organizing Logic Gates

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The end of Moore’s Law for complementary metal-oxide semiconductor (CMOS) technology has prompted the search for low-power computing, resulting in several promising proposals based on magnetic logic. One approach aims at tailoring arrays of nanomagnetic islands in which the magnetostatic interactions constrain the equilibrium orientation of the magnetization to embed logical functionalities [1]. Despite the realization of several proofs of concepts of such nanomagnetic logic, it is still unclear what the advantages are compared to the widespread CMOS designs, due to their need for clocking and/or thermal annealing for which fast convergence to the ground state is not guaranteed. In fact, it seems increasingly evident that “beyond CMOS” technology will require a fundamental rethinking of our computing paradigm. In this respect, digital memcomputing machines have been proposed [2]. These machines consist of self-organizing logic gates (SOLGs) which are able to dynamically “self-organize” into their logically correct states irrespective of whether the signal is applied to the traditional input terminals, or the traditional output terminals. This type of terminal-agnostic logic thus allows for reversed computing. Here [3], we introduce a novel concept of SOLGs that employs stray-field coupled nanomagnetic islands to perform terminal-agnostic logic: nanomagnetic self-organizing logic gates. We thereby focus on the design of a functionally complete NAND gate with which any Boolean circuit can be constructed in a bottom-up approach. Our strategy to build such a nanomagnetic SO-NAND gate relies on two main properties: balancedness and dynamic error suppression (DES) – see Figure 1. For reversible logic to work, the logically correct states should be occupied equally in thermodynamical equilibrium (and not only after a relaxation process into the ground states, as balancedness was originally defined [4]). This contrasts the balanced NAND gate proposed in Ref. [5]. To allow for larger circuits, the probability of being in a logically incorrect state should suppressed. To this end, our DES scheme periodically checks the logical correctness of the gate/coupling and applies additional fields on each island of the gate/coupling if necessary. Our DES scheme is thus based on local information, i.e. the magnetization state of a single gate or coupling. This locality is in sharp contrast to the biasing used in p-bit logic inversion, for which the logical behavior of a collection of super-paramagnetic tunnel junctions (sMTJ) is fully imposed by globally-determined time-varying spin currents [6]. Owing to its locality, our approach does not suffer from scaling issues. To show the usefulness of our SO-NAND gate, we use it as a building block to construct a self-organizing two-bit multiplier – see Figure 2. Our SO-2BM is capable of decomposing any output number into its (prime)factors, finding each possible decomposition without clearly favoring nor penalizing any solution. This fact indicates that the balancedness remains largely conserved throughout the whole system, which allows the extension to larger systems in order to solve more complex problems.

**References:**

[1] A. Imre, G. Csaba, L. Ji et al., Science, Vol. 311, p. 205-208 (2006). [2] F. Traversa and M. Di Ventra, Chaos, Vol. 27, p. 023107 (2017). [3] P. Gypens, J. Leliaert, M. Di Ventra et al., arXiv preprint arXiv:2012.12721 (2020). [4] J. Lust and B. Dixon, Inf. Sci., Vol. 113, p. 193-204 (1999). [5] P. Gypens, J. Leliaert and B. Van Waeyenberge, Phys. Rev. Appl., Vol. 9, p. 034004 (2018). [6] W. Borders, A. Pervaiz, S. Fukami et al., Nature, Vol. 573, p. 390-393 (2019). [7] J. Leliaert, A. Vansteenkiste, A. Coene et al., Med. Biol. Eng. Comput., Vol. 53, p. 309-317 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7d77a99461c449a155c9bf16943ac1b1.jpg)
Fig. 1: Schematic representation of a self-organizing NAND gate consisting of four nanomagnetic islands with perpendicular anisotropy. The probability distribution is obtained with the macrospin simulation tool Vinamax [7] which numerically solves the Landau-Lifshitz-Gilbert equation taking thermal fluctuations into account. Logically incorrect states are suppressed by a dynamic error suppression (DES) scheme, while still preserving the balancedness of the gate. Prel is the probability relative to the most likely logical state.

![](https://assets.underline.io/uploads/markdown_image/1/image/bdd9a7d3218e170d709dc62ca31af92c.jpg)
Fig. 2: Schematic representation of a self-organizing two-bit multiplier and the relative probability distribution obtained with Vinamax for an output fixed to 0. Prel is the probability relative to the most likely logical state. The four output islands of the SO-2BM, labeled from a to d, are fixed such that they represent a certain number (0, 1, 2, 3, 4, 6, or 9) and the system is allowed to explore its state space. The magnetization of the input islands, corresponding to the solution of the computation, are labeled from A to D. All states of which one of the inputs is logically inconsistent are aggregated into the "??"-state.

technical paper

There are many potential applications for spintronics including in-memory, neuromorphic, and radiation hard computing. In- memory computing offers a solution to the memory wall bottleneck that hinders traditional architectures. Neuromorphic computing has the potential to revolutionize how data-intensive tasks, such as image recognition, are computed by emulating how a biological brain operates. Radiation hard computing is useful for certain situations, mainly space applications, where the Earth’s atmosphere does not protect sensitive CMOS hardware. All of these applications can be achieved using domain wall motion-based devices. The domain wall magnetic tunnel junction (DW-MTJ) device architecture is explored in this work as shown in figure 1. DW-MTJ devices operate by sending a current pulse along a ferromagnetic wire to propagate a domain wall from the input to the output of the device. As the domain wall passes under the MTJ, the magnetization state of the bottom layer of the CoFeB/MgO/CoFeB junction changes. By controlling where the domain wall is, the MTJ can have programmable resistance. We use current driven domain wall motion to electrically switch the devices. It has been shown that spin logic requires high tunneling magnetoresistance, low switching voltages, and high stability. These hurdles have been overcome in this work by utilizing perpendicular anisotropy, spin orbit torque switching, and optimized fabrication. Incorvia et al. in 2015 demonstrated that information could be encoded in a magnetic wire by moving a domain wall, but the TMR of those devices was 12% [1]. Here, we develop a process that maintains as much TMR as possible, achieving TMR &gt; 150%, as well as no reduction in resistance-area (RA) product after device fabrication. There were several improvements made to the previous fabrication process. The main change was to remove the reactive ion etch step to contact the MTJ. Incorvia et al. opened a via in polymethylmethacrylate (PMMA) and then used a CF4 reactive ion etch through a hydrogen silsesquioxane (HSQ) layer to expose the MTJ and wire contacts. In our new process, we replaced this step with a negative tone resist and encapsulation step to achieve the same result of an open via to the MTJ and domain wall track without exposing the sensitive layers to a reactive etch. This way the MTJ is never directly exposed: during all etches and encapsulation, the MgO is protected by negative resist. Furthermore, the encapsulation was performed at relatively low temperatures of 100 °C to prevent any additional loss of TMR, and silicon nitride was used instead of silicon oxide to reduce oxidation. These factors allowed our devices to maintain their high TMR post-processing. The TMR is shown in the field loops of four devices in figure 2. The average achieved was 164±17%, compared to the unpatterned film TMR of 168±6%. Before fabrication, the film had RA = 35±2 Ω-μm2 and the devices have RA = 31±3 Ω-µm2. This indicates that our fabrication process had little to no negative impact on the magnetic integrity of our material. These devices utilize electrical initialization of domain walls rather than an external magnetic field, so they can be fabricated as straight wires rather than curved. This allows for increased density if this technology becomes industrialized. Previous iterations of these devices were curved to allow an external magnetic field to nucleate a domain wall at an initial position. Here, we opted for an electrical approach by implementing an oersted field line near the left contact of each device. A short pulse is sent through this line which nucleates a domain wall in the ferromagnetic track [2]. This is one step closer towards all electric operation of the memory-in-logic devices. These oersted field lines increased stability and repeatability by nucleating the domain wall in the same position each time, rather than an unpredictable location via external magnetic field nucleation. Calculating the variability based on the switching voltage range over 10 cycles with both nucleation methods showed an improvement from external magnetic field (105%) variability to the oersted field line (7%) variability. We show in simulation that these improvements bring the variability of the technology within the regime for large circuit function such as a 32-bit full adder [3]. Partial funding from Sandia National Lab (SNL). SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525.

**References:**

1J. A. Currivan-Incorvia, S. Siddiqui, S. Dutta, E. R. Evarts, C. A. Ross & M. A. Baldo. 2015 IEEE Int. Electron Devices Meet. 32–36 (2015) 2G. Zahnd, V. T. Pham, A. Marty, M. Jamet, C. Beigné, L. Notin, C. Vergnaud, F. Rortais, L. Vila & J. P. Attané. J. Magn. Magn. Mater. Vol. 406, p. 166–170 (2016) 3T. Patrick Xiao, M. J. Marinella, C. H. Bennett, X. Hu, B. Feinberg, R. Jacobs-Gedrim, S. Agarwal, J. S. Brunhaver, J. S. Friedman & J. A. C. Incorvia. IEEE J. Explor. Solid-State Comput. Devices Circuits., Vol. 5, p. 188-196 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bee6a10b4751acb8eb32a297ccf3365a.jpg)
Fig.1: Top-down SEM image of a DW-MTJ device. Patterned domain wall track (labeled DW track) and magnetic tunnel junction (labeled MTJ) identified. Device width (labeled w) is 450 nm. Widths down to 250 nm were patterned and tested.

![](https://assets.underline.io/uploads/markdown_image/1/image/694813823627de4f60ea1974a8edb598.jpg)
Fig. 2: Field loop for devices in this work. RMTJ is the measured resistance through the tunnel junction. Spin-orbit torque current switching was also performed and will be presented.

 Domain Wall-Magnetic Tunnel Junction Spin Orbit Torque Devices for In-Memory Computing

Stochastic behaviour has traditionally been a limiting factor in the development of nanomagnetic technology. However, it can also give rise to rich physical behaviours. For example, in artificial spin ice (ASI) arrays [1,2] stochastic effects give rise to emergent behaviour that can help us to understand phenomena in frustrated systems. In particular, this emergent behaviour is useful for a novel form of neuromorphic computing called ‘reservoir computing’ which is highly efficient for time domain processing of signals [3]. One such system that exhibits probabilistic, emergent behaviour with complexity is an array of interconnected nanowire rings of Ni80Fe20 [4-5]. Here we characterise the emergent behaviour of domain wall (DW) states in a nanoring array using X-ray imaging under the application of rotating fields of different amplitudes in order to better understand the array’s magnetic response and ascertain its suitability for reservoir computing. Fig. 1 (A) shows an SEM micrograph of a representative rectangular Ni80Fe20 ring array of which we have obtained the magnetic response by applying an in-plane rotating magnetic field. Magneto-optic Kerr effect (MOKE) measurements show a strongly nonlinear magnetisation variation with field amplitude (Fig. 1 (B)) - an essential response for reservoir computing. We have shown this system to be a promising candidate for reservoir computing using various macroscopic measurement techniques [5-6]. It is clear that these complex dynamics are due to the interplay of pinned and propagating DWs (Fig. 1 (B)) and different magnetic states in the array. We used X-ray Photo Emission Electron Microscopy (X-PEEM) to image the magnetic states across a ring array with rings of radius 2 microns and width of 500 nm with an Ni80Fe20 thickness of 10 nm and with a 2nm Al layer to prevent oxidation. X-PEEM images were obtained with an Elmitec SPELEEM-III microscope on the I06 beamline at the Diamond Light Source using XMCD at the Fe-L3 edge. Samples were mounted on cartridges with a quadrupole magnet, to provide an in-plane magnetic field with arbitrary direction, designed and built at the CIRCE XPEEM beamline at the ALBA Synchrotron. Samples were subject to an initialisation field pulse of 160 Oe followed by 30 rotations of an in-plane magnetic field of various strengths. A representative X-PEEM image of the array (Fig. 2 (A)) obtained with a rotating field strength of 30 Oe at a frequency of 8 Hz shows multiple magnetisation states. The allowed magnetic configurations are shown schematically in Fig. 2 (B). There are four types of ‘onion’ states (two DWs separated by 180°), two ‘vortex’ states (zero DWs) and eight ‘three-quarter’ states (two DWs separated by 90°) possible. A dedicated Python code was used to analyse the magnetisation states in the X-PEEM image (Fig. 2 (A)). The rings were mapped to circles using the Hough transform available in the OpenCV library [7] and the color of each quarter was recorded. Then the color scheme of an entire circle was assigned to a magnetic configuration. The population of various states at different amplitudes of the rotating field was extracted and is shown in Fig. 2 (C). We can see that ‘onion’ states (at lower fields) give way to ‘vortex’ and ‘three-quarter’ states and at higher fields ‘onion’ states dominate again. This is consistent with the previous MOKE measurements that were attributed to static onion states at low field magnitudes, due to DW pinning, and dynamic onion states at high fields (as shown in Fig. 1 (B)) when DWs readily overcome the pinning potentials presented by ring junctions. Note that this behaviour leads to the emergent dynamics which the ring assembly shows under the influence of a rotating magnetic field. The variation in the magnetic states population also indicates ‘fading memory’ as the state population at a given field is independent of the population at a previous field. This is another desirable property for a system to function as a reservoir. The normalised magnetisation response of the array can be extracted from the state population (Fig. 2 (D)) and in order to measure the array’s response to field strength. This is similar to that obtained from previous MOKE measurements but highlights the significance of dynamic local variation in magnetic structure. We have thus imaged the emergent DW dynamics in nanoring arrays using X-PEEM and have obtained their magnetic response to show their suitability for reservoir computing applications. The different complex magnetic states in the array, which give rise to their emergent magnetic behaviour, provide insights into the evolution of these states. We hope that this study will pave the way for further investigations into the suitability and implementation of these systems for advanced computing purposes.

**References:**

[1] Wang et al., Nature 439, 303 (2006) [2] Branford et al., Science 335, 1597 (2012) [3] Jensen et al., ALIFE 2018: The 2018 Conference on Artificial Life (July 2018) pp.15-22 [4] Negoita et al., J. Appl. Phys. 114, 013904 (2013) [5] R. Dawidek et al., Adv. Funct. Mater. 2021, 2008389 [6] I. Vidamour et al. presented at MMM-2020 [7] Bradski, Dr. Dobb’s Journal of Software Tools 2236121 (2000)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4c1d76f0e5846822be2b0bb1d2560a9a.jpg)
Fig. 1: (A) An SEM image of a fabricated ring array. (B) MOKE measurements of the array showing the ‘nonlinear response’ of its magnetization and the different dynamic states in the system.

![](https://assets.underline.io/uploads/markdown_image/1/image/0b16e317875081de61e6fa8d49103c5f.jpg)
Fig. 2: (A) A representative X-PEEM image of a ring array. (B) A schematic showing the different types of magnetic states possible in the array. The arrows show the magnetisation direction which the measurement was sensitive to. (C) The variation of the magnetisation states in the array with rotating field amplitude and (D) the normalised magnetisation derived from the magnetisation states.

 Imaging the emergent behaviour in nanoring assemblies for reservoir computing applications

Reservoir computing has been gaining traction as a bio-inspired, machine-learning approach for solving computationally hard problems, such as speech recognition and prediction of chaotic time series, that are becoming fundamental to modern technology [1]. By making use of a complex, fixed reservoir with a single output layer, the approach minimises expensive training steps. Magnetic systems are well positioned to fulfill the requirements of the reservoir, providing non-linearity, fading memory, and reproducible responses. In particular, magnetic systems subject to a temporal input sequence can perform the entire role of the reservoir, replacing complex networks of transistors whilst only requiring a single input and output [2]. This offers the potential for compact devices, with reduced energy costs for computation. Here, we propose strain-mediated, voltage controlled superparamagnetic ensembles as an ideal candidate for an ultra-low-energy reservoir. Single superparamagnetic nano-dots with uniaxial anisotropy exhibit stochastic behavior, switching between states with a characteristic timescale set by their energy barriers. Micromagnetic simulations (Fig. 1a) demonstrate this telegraphing behavior for CoFeB nano-dots, where the auto-correlation time has been verified to be below 5 ns and the strain-induced anisotropy can compete with the intrinsic anisotropy [3-6]. Borders et al. [7] have shown how tuning such telegraph noise in magnetic tunnel junctions can be used for integer factorization. Here, however, we make use of the collective behavior of extended ensembles of nano-dots such that the average magnetisation becomes predictable; thus, showing the reproducible response required for reservoir computing. Input is provided by addition of a strain-mediated anisotropy (Fig. 1b)—a technique that is predicted to achieve ultra-low-power consumption [8]. This acts to rotate the intrinsic anisotropy axis, biasing dwell time in one state over the other [9]. Whilst the internal system dynamics are driven by thermal noise, the average magnetisation obeys simple rules that, when subject to a temporally varying input, produce a complex non-linear response with fading memory (Fig. 1c). Using an analytical model of the response of the superparamagnetic ensemble [10], we simulate the physical reservoir. By inputting a temporal sequence and training a single layer of weights from the output, we can perform standard machine learning benchmarks with competitive performance over a range of input timescales. Fig. 2 presents the performance for spoken digit recognition on the TI-46 dataset and chaotic time series prediction on the NARMA10 task. In both cases the performance approaches that of competing reservoirs [1,2], despite the relative simplicity of the system. The intrinsic timescale of the system (τ) is controlled by the ratio of energy barrier (anisotropy, KV) to the thermal energy (KBT at 300 K). Fig. 2a demonstrates that by tuning the input rate and strength (relative to the intrinsic timescale and anisotropy respectively), optimal performance of 95 % recognition of the spoken digits 0-9 with five female speakers can be achieved. This is realised for τ ~ 50 ns (KV/KBT = 20) and τ ~ 1 ms (KV/KBT = 50). For the NARMA10 task (Fig. 2b), the highest performance is achieved across this entire timescale range by tuning the strength of a feedback term (i.e. the output is fed back into the input stream with a given weight). Robust performance on timescales ranging from hundreds of nanoseconds up to seconds would allow a physical realisation of such a reservoir to be tuned to provide computation in real time for a wide range of possible physical inputs: from decision-making in driverless cars (fast) to speech recognition (slow). The simplicity of the system and the ease of manufacture, coupled with the low energy consumption expected for such a, thermally driven, device makes it an ideal candidate for edge computing where high performance is needed at very low latency and power.

**References:**

[1] L. Appeltant et al., Nat Commun. 2, 468 (2011). [2] J. Torrejon et al., Nature. 547, 428–431 (2017). [3] A. Vansteenkiste et al., AIP Advances. 107133 (2014). [4] J. Leliaert et al., AIP Advances. 7, 125010 (2017). [5] S. Zhang et al., Scientific Reports. 4, 3727 (2014). [6] C. Safranski et al., arXiv:2010.14393[cond-mat] (2020). [7] W. A. Borders et al., Nature. 573, 390–393 (2019). [8] K. Roy, S. Bandyopadhyay and J. Atulasimha, Applied Physics Letters. 99 (2011). [9] Z. Guo et al., Appl. Phys. Lett. 112, 052904 (2018). [10] O. Hovorka et al., Appl. Phys. Lett. 97, 062504 (2010).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/2ab92e894162fe72ba365c3a5186c89a.jpg)
Fig. 1: a. Telegraphing behaviour of a single superparamagnetic nano-dot. Micromagnetic simulation for a 40 nm diameter, 4 nm thick CoFeB cylinder with 8.2 KJ/m-3 uniaxial anisotropy at 300 K. b. Rotation of the intrinsic anisotropy axis in a nano-dot via strain anisotropy. The rotation biases the dwell time in one magnetic state (left/right pointing) versus the other. c. Analytical response of an ensemble of superparamagnetic nano-dots to a global input via strain. A temporal input sequence produces the non-linear response with fading memory required for reservoir computing.

![](https://assets.underline.io/uploads/markdown_image/1/image/7c8ac57f39d07c9f25ded428f97ed89f.png)
Fig 2: a. Performance on a spoken-digit recognition test using the TI-46 dataset. Performance is tuned by the input rate (relative to the intrinsic timescale) and strength. Optimal performance (95 % recognition accuracy) is achieved for intrinsic timescales of 50 ns (KV/KBT = 20, left) and 1 ms (KV/KBT = 50, right) and approaches that of competing reservoirs [2]. b. Performance on a chaotic time series prediction task (NARMA10). Performance is maintained across the entire timescale range by tuning a feedback parameter.

 Voltage-Controlled, Thermally Driven Superparamagnetic Ensembles for Tuneable Timescale Reservoir Computing

Stochastic computing has recently gain attention mainly due to its low cost, fast and energy efficient circuit level implementations. In addition, it is rather fault-tolerant and noise resilient. The success of stochastic computing is due to the fact that complex functions are extremely difficult to be implemented by conventional binary logic design while they can be effectively approximated by stochastic computing. Stochastic computing uses probabilistic representation of real-valued numbers. It represents and processes numbers thanks to bitstreams. Such a stream represents the number p if the proportion of digit 1 in the stream is equal to p [1]. One of the major requirements for accurate stochastic computation is to use non-repeating and uncorrelated random number generators to produce the bitstreams. In pure digital conventional circuits, such randomness arises from Linear Feedback Shift Register (LFSR) based Random Number Generator (RNG). They output a bitstream that represents the quantity 0.5, and is then processed to represent whatever number [1]. However, LFSRs are made of CMOS logic gates and flip-flops. The produced bitstream is totally predictable, requires an important amount of energy along with a high silicon area. Recently, nanoscale devices such as Magnetic Tunnel Junctions (MTJ) have been considered for bistream generators due to their inner randomness [2]. The so-designed circuits produce pure random bitstreams (the sequence is not deterministic), and they operate in a very energy efficient way. The MTJ is a structure in which a binary information is stored as the relative direction of the magnetization of a magnet with respect to a reference one. The behaviour of this magnetization depends in part on thermal fluctuations [2], which can be leveraged as a source of stochasticity. Two types of schemes have been proposed for an MTJ based bitstream generator. First, a MTJ based RNG [3-5]: a sequence of bits ‘0’ and ‘1’ with a proportion of 50% each is generated, then processed to produce the bitstream. Second, a programmable circuit tunes the MTJ in such way that it generates directly the required bitstream, without the need of any post-processing [6]. Its implementation implies less hardware cost. However, full circuit implementation of the second scheme has not been proposed so far. In this work, we designed a bitstream generator composed of a single MTJ, a writing circuit, a reading circuit and a feedback loop. The operating principle, schematically represented in Figure 1, is the following: a current passes through the writing path of the MTJ (that has been previously set in the logic state “0”) during a short period with a specific amplitude. This brings about a probability (denoted p) that the magnetization experiences a switching. After repeating the process N times, the resulting bitstream is composed of an average of Np bit seen as “1”. The obtained bitstream then represents the number p in its stochastic representation. The reading circuit, similar to the latch-type voltage-mode sense amplifier published in [7], designed and optimized to allow asynchronous operations for low power operation, has the advantage that the static currents required to measure the resistance of the MTJ last a very short time, allowing for important energy saving. The inner CMOS based latch decorrelates the output of the reading circuit to the MTJ so that the state of the read MTJ is stored and remains available after the reading process, while the MTJ can be used again. The feedback loop, connected to the output of the reading circuit extracts the number that is represented by determining the ratio of bits at “1” and “0”. It determines the gap between the number that is computed with respect to the one that is to be represented. In the case of a large gap, the writing current is modified accordingly. This current is increased if the there are too many “0”s, and reversely it is decreased when there are too many “1”s. The writing unit is connected in series to the writing path of the MTJ. It aims at imposing a current through the writing path that is determined by the feedback loop. It is made of several transistors in parallel, acting like independent current sources. This configuration allows the circuit to find the proper writing current required for a given bitstream. This also allows handling temperature dependence of the writing current. Indeed, with temperature variation the writing current may vary in time. In our proposed circuit, the real-time tracking of the optimal writing current by the proposed feedback loop handles this problem. In conclusion, we propose an innovative single MTJ based bitstream generator leveraging the unique stochastic properties of the magnetization of the storage layer. The critical issue of PVT variations and noise are solved by our system thanks to the feedback loop and to the versatility of the writing circuit that allow for the generation of a relevant writing current that can be generalized to any bitstream computation. This study opens the way to the design of fast and energy efficient stochastic computing based circuit.

**References:**

[1] A. Alaghi and J. P. Hayes, ACM Trans. Embed. Comput. Syst [2] W. Brown, Phys. Rev. 130 (1963) 1677-1685 [3] Akio Fukushima et al 2014 Appl. Phys. Express 7 083001 [4] F. Ouattara, A. Nejat and L. Torres, IEEE Access, vol. 7, pp. 59271-59277, 2019 [5] S. Oosawa, T. Konishi and N. Onizawa, IEEE 13th International New Circuits and Systems Conference (NEWCAS), Grenoble, 2015, pp. 1-4 [6] LA de Barros Naviner et al., 13th IEEE International New Circuits and Systems Conference (NEWCAS), 2015 [7] H. Zhang, W. Kang and, IEEE Access, vol. 6, pp. 64250–64260

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6719b321db54b403c7f441a57764f6a8.jpg)
Fig.1 : Schematic of the bitstream generator

 Hardware Implementation of a Magnetic Tunnel Junction based Bitstream Generator for Stochastic Computing

The spintronic devices such as the domain wall and skyrmion based devices are being explored for their applications in neuromorphic computing [1-3], due to their non-volatile behaviour, stochastic switching and energy efficiency. The electric field control of these devices has attracted extensive attention and has further gained popularity both in memory and logic applications as it provides an efficient way to improve the data storage density [4-8]. Neuromorphic computing has been shown to be a superior alternative in terms of speed and energy, when compared to training on a conventional computing unit. A magnetic tunnel junction (MTJ), which constitutes the building block of a magnetic random-access memory (MRAM), is a competitive candidate for the next generation memory devices because it is non-volatile and exhibits high endurance, low power consumption, high operation speed and integration capability [8-9]. The present work discusses the proposal of a spintronic neuromorphic system with spin orbit torque driven domain wall motion-based neuron and synapse. The domain wall motion in the neuron is controlled by the electric field at the gate generated by voltage signals from pre-neurons as shown in Fig. 1(a). The application of voltage as input and output variable helps in reducing power consumption, as pinning site can be turned ON/OFF only when required. Further it provides better fanout as post-neuron output from first stage can act as infinite number of pre-neurons to next stage, which is going to play an important role in realization of system level neuromorphic architectures. For the implementation of the synaptic weight, we propose 3- terminal MTJ with stochastic domain wall motion in the free layer. The edge roughness results in intrinsic pinning of domain wall and this adds to the stochastic behaviour of domain wall motion in presence of spin transfer torque and/or spin orbit torque [10]. We have incorporated these intrinsic pinning effects by creating triangular notches on the sides of free layer as shown in Fig. 1(b). The complete neuromorphic circuit containing neurons, synapses and control circuitry implementation of spike time dependent plasticity is shown in Fig. 1(a). The sensing unit consists of a voltage comparator which provides high positive or zero voltage feedback pulse to the weight-updating unit. The weight updating unit takes this signal and the pre-neuron signal to decide the sign of the synaptic weight update pulse based upon spike timing correlation between the pre- and post-neuron. For modelling the micromagnetics and spin transport in both synapse and neuron, we developed a micromagnetic Non-Equilibrium Green’s Function (MuMag-NEGF) coupled model. In Fig. 2(a), we show that in the presence of the gate voltage the uniaxial anisotropy for a 20 nm-wide region is varied slightly which results in the pinning of domain wall at 40 nm from the centre. In the absence of gate voltage uniaxial anisotropy is uniform and the domain wall moves more than 100 nm right in 10 ns, where it is sensed by the MTJ as post neuron spike. Also, the reduction of domain wall velocity is shown in Fig. 2(b). The thermal effects result in the stochastic domain wall motion but stochasticity is tuned by the external current in the form of spin orbit torque. Fig. 2 (c) shows the stochastic magnetization switching in MTJ synapse and stochasticity is tuned in either direction with application of positive and negative current pulses. In the presence of more positive pulses, the MTJ gradually moves into low resistance state, while in the presence of more negative pulses it moves into high resistance state, as shown in Fig. 2(d). When a current pulse is applied, the probability of domain wall moving increases. The dynamic pattern recognition capabilities of the circuit are tested on 9x9 cross-bar architecture, consisting of 9 pre-neurons, 9 post-neurons and 81 synapses. In summary, here we have successfully designed and demonstrated the on-chip stochastic dynamic learning by adopting the spike time dependent plasticity as the learning algorithm. At the end of training phase, we demonstrated that all pattern synapses stochastically evolve into low resistance state (high weight) while the remaining background synapses are in high resistance state (low weight).

**References:**

[1]. Deming Zhang, Weisheng Zhao et. al, “All spin neural networks based on compound spintronic synapses and neurons,” IEEE Advances in Biomedical Circuits and Systems, vol. 10, no.4, August-2016. [2]. Otitoaleke Akinola, Jean Anne C Incorvia et.al, “Three terminal magnetic tunnel junction circuits showing spike timing-dependent plasticity,” J. Phys. D: Appl. Phys. 52 (2019) 49LT01 (11pp). [3]. Yangqi Huang, Weisheng Zhao et.al, “Magnetic-skyrmion based synaptic device,” IOP Nanotechnology, 28 08LT02, 2017 [4]. Jiang Nan, Weisheng Zhao et.al, “Efficient Magnetic Domain Nucleation and Domain Wall Motion with Voltage Control Magnetic Anisotropy Effect and Antiferromagnetic/Ferromagnetic Coupling,” IEEE TRANSACTIONS ON MAGNETICS, Vol. 55, No. 7, JULY 2019. [5]. Jialin Cai et.al, “Voltage-Controlled Spintronic Stochastic Neuron Based on a Magnetic Tunnel Junction,” Phys. Rev. Applied 11, 034015, 6 March 2019.[6]. M. Bauer, R. Alexis, G. Atwood, B. Baltar, A. Fazio, K. Frary, M. Hensel, M. Ishac, J. Javanifard, M. Landgraf, D. Leak, K. Loe, D. Mills, P. Ruby, R. Rozman, S. Sweha, S. Talreja, and K. Wojciechowski, in IEEE International Solid-State Circuits Conference Digest of Technical Papers (1995), p. 132. [7]. T. S. Jung, Y. J. Choi, K. D. Suh, B. H. Suh, J. K. Kim, Y. H. Lim, Y. N. Koh, J. W. Park, K. J. Lee, J. H. Park, K. T. Park, J. R. Kim, J. H. Yi, and H. K. Lim, IEEE J. Solid-State Circuits 31, 1575 (1996). [8]. X. H. Lou, Z. Gao, D. V. Dimitrov, and M. X. Tang, Appl. Phys. Lett. 93, 242502 (2008). [9]. N. Locatelli, V. Cros, and J. Grollier, Nat. Mater. 13, 11 (2014). [10]. F. N. Tan et.al, “High Velocity Domain wall propagation using voltage control magnetic anisotropy,” Scientific Reports, 2019) 9:7369.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/11865266d897f1c54b8d59f2476fef1b.jpg)
Fig. 1: (a) Neuromorphic circuit implementation of spike time dependent plasticity-based pattern recognition. 
The circuit consists of voltage-controlled magnetic anisotropy post neuron and stochastic 3T-MTJ synapse with domain wall driven by SOT. (b) 3T-MTJ stochastic synapse schematic with small notches. (c) Cross-bar architecture implementation of the circuit.

![](https://assets.underline.io/uploads/markdown_image/1/image/568d7875203ba7028c8a7d9c8698bfa0.jpg)
Fig. 2: (a) Domain wall position in neuron for pinning ON and pinning OFF controlled by gate voltage. (b) Domain wall velocity for the pinning and non-pinning case. (c) Magnetization in the free layer MTJ synapse showing stochastic behavior with a positive current pulse. (d) MuMAG – NEGF coupled computed resistance evolution of the synapse.

 Domain Wall Motion based Neuromorphic Computing with Voltage Controlled Spin Neuron and Stochastic Magnetic Tunnel Junction Synapse

Physical reservoir computing using spin-torque oscillator (STO) has been attracting much attention after the demonstration of highly accurate human voice recognition [1]. To enhance the computational ability of STOs, there are two approaches: an array of STOs, and delayed feedback system. In the STO array, theoretical study was performed on 10-1000 STOs with interaction to obtain highly nonlinear dynamics, resulting in enhancement of computational ability that is competitive to quantum reservoir [4]. A similar highly nonlinear dynamics can be generated by using the feedback system, where outputted power from STO was reinjected in the form of electrical current or magnetic field to induce additional torque. By tuning the additional torque, complex magnetization dynamics such as amplitude modulation [5] and chaos [6,7] can be induced. These nonlinear dynamics are expected to enhance the computational ability even without interactions between STOs. Recent study showed that tunable nonlinear dynamics and delayed response by the feedback effect are adaptable for specific temporal pattern recognition task [7]. However, quantitative comparison of computational ability between the array system and the feedback system has not been investigated. In this work, we quantitatively evaluate the computational ability of STO with delayed-feedback circuit by performing a general task. The computational ability was evaluated by performing second order nonlinear autoregressive moving average (NARMA2) task [3,4]. The NARMA2 task aims to reproduce target outputs, defined by nonlinear transformation of the random inputs, from the amplitude of the STO. Figure 1 is a schematic illustration of vortex-type STO [2] with a delayed feedback circuit. To excite an auto-oscillation of the vortex core, perpendicular magnetic field of about 0.5 T and bias voltage of 350 mV were applied. In addition, we superposed random-pulse voltage with the width of 25 ns on the bias voltage. The random pulse was based on the uniform random number in the range of [-1,1] with the magnitude of 75 mV; therefore, the total input voltage varied from 275 to 425 mV. The outputted signal of the STO was amplified and converted to the RF magnetic field, which was applied to the STO with time delay of about 30 ns. The strength of the feedback signal was controlled by a tunable attenuator. Figure 2(a) shows power spectral density of the STO with delayed feedback as a function of attenuation. In the large attenuation region, the power spectra show a broad peak. With decreasing the attenuation, sharp multiple peaks appeared in the spectra below 30 dB. Such a split of spectrum is often observed when the feedback effect induces the amplitude modulation [5,6]. Figure 2(b) summarizes the dependence of normalized mean square error (NMSE) of NARMA2 task on the attenuation. It is clearly shown that NMSE can be minimized at an attenuation of 34 dB, below which the sharp multi peak structure appeared in the spectra. The observed minimum NMSE (6.4×10-6) is comparable to that theoretically obtained in the array of 16-coupled STOs [3]. This result indicates that single STO with delayed-feedback circuit can compete with the STO array. This work is supported by NEDO.

**References:**

[1] J. Torrejon et al., Nature 547, 428 (2017). [2] S. Tsunegi et al., Jpn. J. Appl. Phys. 57, 120307 (2018). [3] T. Kanao et al., Phys. Rev. Applied 12, 024052 (2019). [4] K. Fujii et al., Phys. Rev. Applied 8, 024030 (2017). [5] D. Kumar et al., Sci. Rep. 6, 30747 (2016). [6] T. Taniguchi et al., Phys. Rev. B 100, 174425 (2019). [7] J. Williame et al., Appl. Phys. Lett. 114, 232405 (2019). [8] M. Riou et al., Phys. Rev. Applied 12, 024049 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d49dc6b3239e7837d4a4db73ab12d4ba.jpg)
Schematic illustration of spin-torque oscillator (STO) with delayed-feedback circuit. Auto-oscillation is driven by random pulse voltage Vin(t). Power of the feedback is controlled by the rate of attenuator rAtt. Reservoir computing is performed using time-series amplitude Aout(t) obtained by the Hilbert transformation of output voltage Vout(t).

![](https://assets.underline.io/uploads/markdown_image/1/image/ae2ed6bb6522935b9ce2976e571c4ff4.jpg)
(a) Power spectral density (PSD) of STO with the delayed feedback circuit as a function of attenuation rate rAtt. (b) Normalized mean-squared error (NMSE) of second order nonlinear autoregressive moving average (NARMA2) task against attenuation rate. NMSE shows a minimum value (6.4×10-6) at rAtt = 34 dB.

 Enhanced Computational Ability of Spin Torque Oscillator with Delayed-Feedback Circuit for Physical Reservoir Computing

Recent research on deep learning shows that Binary Neural Network (BNN) is able to significantly reduce computation and memory cost with small precision loss [1][2]. By binarizing inputs and weights to +1 or -1 in BNNs, multiply-and-accumulate (MAC) operations can be replaced by XNOR operation in memory. Nonvolatile memory for BNN such as resistive random access memory (RRAM) and magnetic random access memory (MRAM) was reported in [3]-[6]. In this paper, we use spin-toque transfer magnetic random access memory (STT-MRAM) to complete XNOR and single-step convolution in memory. The main contributions include: (1) A STT-MRAM based structure with four transistors and two magnetic tunnel junctions (MTJ) (4T-2M) is proposed for binary convolution and fully connection operations using current mode sense amplifier. (2) The impact of different tunnel magnetoresistance ratio (TMR) and different size of MTJ on accumulating in memory has been explored. (3) By quantizing sensing yield into model, the trade-off between energy and performance is investigated. As shown in Fig. 1 (a), a 4T-2M bit-cell structure represents one synaptic weight, ‘-1’ is represented by two MTJs as parallel (P) and anti-parallel state (AP). The opposite combination is used to represent ‘1’. Two encoded word lines (WLs) represent one input value. To complete the XNOR operation, an offset-canceling current-sampling sense amplifier (OCCS SA) [7] is used (see Fig. 1(b)) to compare the current difference between source line (SL) and SLB. Fig.1 (c) shows the structure of the proposed STT-MRAM block. For convolution operation, the row decoder gated by appropriate timing decodes the multiple WLs to input parallel, and output through path &lt;1&gt; as the result of convolution. For fully connected operation, the row decoder activates the WLs row by row, and output through path &lt;2&gt; with bit-counting as the result of one fully connected. We implement 9 bits XNOR and accumulation to perform 3 × 3 convolution. Fig. 1(d) lists all the combinations in the convolution. The worst case is four ‘-1’ and five ‘1’ accumulation or its opposite combination, which causes the minimum current difference between “5IAP+4IP” and “5IP+4IAP”. Fig. 1(e) illustrates the circuit realization of BNN. When one group of WLs is simultaneously activated, ISL and ISLB will be different depending on the resistance of MTJs on two SLs. A CSA is implemented to generate high level when weighted sum is positive, and low level when weighted sum is negative. The output of the CSA represents the output of the neuron. SPICE-level simulation is performed using a Verilog-A model of STT MTJ and an industrial 28-nm CMOS process. Fig.2 (a) shows the simulation results that the sensing yield in different TMR and MTJ critical dimension (CD) can up to 99.8%. The sensing yield is positively correlated with TMR and negatively correlated with MTJ CD. As Fig.2(b) indicated, smaller MTJ CD leads to higher sensing margin [8]. As shown in Fig. 2(c), a BNN model consisting of two convolutional stages and one fully connected layer as the classifier is designed. Validation results on MNIST handwritten digit dataset is shown in Fig. 2(d). Sensing yield is quantized into model when sensing processes is failed, the results of convolution will be flipped. Lower TMR and larger MTJ CD result in a lower sensing yield and a dramatic decrease in accuracy. When TMR is greater than 2 with 40nm MTJ or TMR is greater than 1.5 with MTJ smaller than 35nm, the accuracy could achieve &gt;95% (&lt;1.8% loss). Fig. 2(e) illustrates the energy saving of single-step convolution can achieve 51.2% compared with that of row by row with a slight decrease in accuracy.

**References:**

[1] M. Courbariaux, et al., arXiv: 1602.02830, 2016. [2] M. Rastegari, et al., arXiv: 1603.05279, 2016. [3] Z. Dong et al., IEEE Transactions on Electron Devices, vol. 66, no. 1, pp. 793-801(2019) [4] L. Chang et al., IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 27, no. 11, pp. 2668-2679(2019) [5] Y. Pan et al., IEEE Transactions on Magnetics, vol. 54, no. 11, pp. 1-5(2018) [6] Y. Zhang et al., IEEE Transactions on Electron Devices, vol. 67, no. 2, pp. 469-473, Feb. 2020. [7] T. Na et al., IEEE Journal of Solid-State Circuits, vol. 52, no. 2, pp. 496-504, Feb. 2017. [8] Zhao, W. S., et al. Microelectronics Reliability, vol. 52, no. 9–10, Sept. 2012.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/946bef43ce91a98b12b19a0fcba53e6c.jpg)
Fig.1 (a) Cell configurations to represent one weight using two MTJs and four transistors. (b) Schematic of OCCS SA. (c) 4T2M based array to complete convolution and fully connection, <1> indicates path 1 and <2> indicates path 2. (d)The list of all the combinations in the convolution. (e) Schematic of the XNOR and accumulate operation as convolution in memory.

![](https://assets.underline.io/uploads/markdown_image/1/image/f8614fa534b410c9fbcbfa4ee258bd1b.jpg)
Fig.2 (a) The sensing yield in different TMR with different MTJ CDs. (b) The resistance distribution of different MTJ CDs. (c) Diagram of the BNN topology used for evaluation with MNIST. (d) The performance of binary neural network with different MTJ CDs and different TMR. (e) Trade-off between energy saving and accuracy loss.

 In-MRAM Processing Elements with Single-Step Convolution for Binary Neural Network

Classifying or predicting complex time-dependent signals (e.g. speech, financial data, the weather) is a challenging computational task. Reservoir computing (RC) is an efficient neuromorphic computing approach that is ideally suited to such tasks and is typically implemented in software using a recurrent neural network (RNN) with fixed synaptic weights (the reservoir) connected to a single, trainable readout layer. However, more efficient implementations of RC are possible if the software RNN reservoir is substituted with a physical system with the correct properties, such as non-linear response to input signals and inherent memory, leading to a readily deployable, hardware-based neuromorphic computing platform [1]. In this work, we propose a novel approach to RC where the dynamics of a single magnetic domain wall (DW) trapped between two defect sites in a nanostrip acts as a hardware-based reservoir. We demonstrate how such a device, with dimensions smaller than 1 µm, is capable of performing complex data analysis tasks, such as speech recognition. We have modelled a Ni nanowire with two anti-notches (shown in the inset of Fig. 1(a)) using both a simple 1D model[2] and micromagnetic simulations. Both models show the DW exhibit complex oscillatory dynamics similar to the Duffing oscillator, thus giving highly non-linear responses to applied magnetic fields (Fig. 1(a)). We exploit the DW dynamics for RC by using an applied field to inject time-multiplexed input signals into the reservoir and show how this approach allows the device to perform classification tasks. We have explored how the regime of applied fields affects the accuracy of sine and square wave classification, showing that the best recognition rate is obtained at the edge of a chaotic regime of oscillation (Fig. 1(b)). We have also demonstrated that the same approach can be used for more complex tasks, such as spoken digit recognition and handwritten digits recognition. Our work opens a new perspective for neuromorphic computing in nanomagnetic hardware.

**References:**

[1] M. Riou et al., Phys. Review Applied,2019 [2] A. Pivano et al., Phys Review B, 2016

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/947f0ad21dbe4e988fab9e7c037ab764.jpg)
Fig. 1: Simulations of Ni nanostrip at 500 MHz. Figure (a) shows a typical bifurcation diagram computed with 1D model at 500MHz and the inset of Fig. 1(a) illustrates the typical geometry of a Ni nano-strip with two anti-nothces separated by 350 nm. Figure (b) shows the recognition rate for the sine and square task computed with 1D model of a domain wall in a Ni nanostrip. The input field is oscillatory comprising two terms: H-the constant offset of the field and ΔH-the amplitude of the stimulus. The inset shows a sequence of inputs and its corresponding output calculated during the classification process using 1D model.

 Neuromorphic computation with a single magnetic domain wall

Unconventional computing utilizing stochastic behavior to effectively solve complex problems, i.e., "probabilistic computing," has gathered much attention [1]. Stochastic magnetic tunnel junction (MTJ) is a promising device to compose the probabilistic bit, a fundamental unit of the probabilistic computer, with efficient as well as individual control of its stochasticity. The average duration of MTJ dwelling in each state, i.e., relaxation time τ, determines the critical indices of the computer's performance, e.g., accuracy and computing time. As a proof of concept, integer factorization has been demonstrated with stochastic MTJs with a perpendicular easy axis (p-MTJs) [2], whose typical τ is on the order of milliseconds. Meanwhile, τ of MTJs with an in-plane magnetic easy axis (i-MTJs) is reported down to sub-microseconds [3]. Here we experimentally and theoretically address the τ of nanomagnets in low-thermal-stability MTJs and elucidate the physical mechanism causing the different fluctuation timescale between p- and i-MTJs. A stack structure, Ta (5.0)/ PtMn (20)/ Co (2.6)/ Ru (0.90)/ CoFeB (2.4)/ MgO/ CoFeB (2.1)/ Ta (5.0)/ Ru (5.0) (thickness in nm), is deposited on a thermally oxidized Si substrate by dc/rf magnetron sputtering. Both bottom (reference) and top (free) CoFeB layers have an in-plane easy axis. Effective perpendicular anisotropy field μ0HKeff of the top CoFeB layer is -0.46 T (negative sign indicates in-plane easy axis). The stack is processed into elliptic MTJs with geometry averaged diameter D of 55 nm. Typical TMR ratio is 110% and RA = 32 Ωμm2. Figure 1(a) shows the measurement circuit and Fig. 1(b) shows time-averaged 〈R〉 as a function of in-plane magnetic field H along a long axis of MTJs, indicating superparamagnetic behavior with zero coercivity. Figure 1(c) shows a measured random telegraph noise (RTN) at μ0H = -7.9 mT. τ averaged for P and AP states τave = (τPτAP)0.5, where τP (τAP) is the τ to stay at P (AP) state, is 19 ns, which is about two orders shorter than that ever reported in MTJs [3]. We then numerically calculate τ of p- and i-MTJs based on a stochastic Landau-Lifshitz-Gilbert (LLG) equation including a thermal field [4]. We define magnetic energy with HKeff and in-plane anisotropy field HK,in as E = MSv(HKeff+HK,insin2φ)sin2θ/2, with MS, v, θ, and φ being the spontaneous magnetization, the volume of nanomagnet, the polar angle, and the azimuthal angle, respectively. We set the same Δ for both systems; μ0HKeff = 10 mT, α = 0.02, and thermal stability factor Δ = HKeffMSv/2kBT = 3.8 (kBT is thermal energy) are assumed for the p-MTJ, and μ0HKeff = -0.46 T, μ0HK,in = 10 mT, α = 0.02, and Δ = HK,inMSv/2kBT = 3.8 for the i-MTJ. Figures 2(a) and 2(d) show the directions of magnetization, a torque from perpendicular magnetic anisotropy, that by damping, and that by the thermal fluctuation calculated by E and LLG equation for p- and i-MTJs, respectively. In the p-MTJs, the torque from HKeff directs in the azimuthal direction, and the only torque directing in the polar direction to induce magnetization switching is the thermal fluctuation field. Accordingly, the magnetization precesses with perpendicular anisotropy field and switches from one energy stable direction to the other with a random fluctuation field, as shown in the magnetization trajectory in Fig. 2(b). In this case, τ is found to be well described by the random-walk under damping [4,5] as τ = π0.5Δ-0.5(2αγμ0HKeff)-1exp(Δ). τ in this condition is calculated as ≈ 0.8 ns, which is consistent with the calculated RTN shown in Fig. 2(c). On the other hand, as shown in Figs. 2(d) and 2(e), the directions of the torques in i-MTJs are different from those in the perpendicular system due to the difference in the landscape of energy potential. τ is calculated to be 20 ns [Fig. 2(f)], which is close to the experimentally obtained τ, and several tens of times shorter than that calculated for the perpendicular system even with the same Δ. We numerically simulate τ as a function of Δ and clarify that the values are well described by τ = π(αγμ0Δ)-1[HKeff(HKeff+HK,in)]-0.5exp(Δ), which is analytically derived with Fokker-Planck equation [4,6] under Δ » 1 and -HKeff » HK,in. In conventional understanding, a decrease of HKeff decreases Δ, leading to shorter τ for p-MTJs. However, our analysis indicates that it increases the prefactor of exp(Δ), in proportion to HKeff -1.5. This dilemma determines a lower limit of τ as τmin = MSvln2(αγμ0kBT)-1, at which magnetization dynamics is dictated by the thermal fluctuation. For example, τmin is 151 ns for a p-MTJ with D = 60 nm, thickness of 1 nm, MS = 1.4 T, and α = 0.02 at room temperature. For i-MTJs, one can be free from the dilemma because the prefactor is mainly determined by HKeff whereas exp(Δ) is determined by HK,in, leading to (πΔ)-0.5MS|HKeff|v/4kBT times shorter τ than that with p-MTJs for the same Δ. This ratio reasonably explains the different τ observed for p-MTJs and i-MTJs, indicating i-MTJs with μ0|HKeff| ~ 1 T can achieve τ down to nanoseconds. This work is in part supported by JST-CREST No. JP1082016, and JSPS Nos. 19J12206, 19J12926, 19H05622, 20H02178, and 19KK0130.

**References:**

[1] P. R. Feynman, Int. J. Theor. Phys. Vol. 21, p.467-488 (1982) [2] W. A. Borders, A. Z. Pervaiz, S. Fukami, et al., Nature Vol. 573, p.390-393 (2019) [3] B. Parks, A. Abdelgawad, T. Wong, et al., Phys. Rev. Appl. Vol. 13, p.014063 (2020) [4] W. F. Brown Jr., Phys. Rev. Vol. 130, p.1677-1686 (1963) [5] T. Taniguchi, and H. Imamura, Phys. Rev. B Vol. 85, p.184403 (2012) [6] T. Taniguchi, Y. Utsumi, and H. Imamura, Phys. Rev. B Vol. 88, p.214414 (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d931fe36020e94e2cb208537fe219b0d.jpg)
Fig. 1 (a) Measurement circuit for time-averaged resistance 〈R〉 and random telegraph noise (RTN). (b) In-plane magnetic field μ0H vs. 〈R〉. (c) RTN signal measured as transmitted voltage VT.

![](https://assets.underline.io/uploads/markdown_image/1/image/3dbfad6da1f9511ba98d94f34ea33a69.jpg)
Fig. 2 (a)(d) Directions of the torques. (b)(e) Calculated magnetization trajectories and (c)(f) time vs. their easy axis component. The conditions for (a)-(c) perpendicular and (d)-(f) in-plane easy axes are shown.

 Superiority of In-Plane Easy-Axis Stochastic Nanomagnet for Shorter Relaxation Time

The bottleneck between data storage and processing on conventional computing hardware is seen as a significant limitation to many forms of machine learning algorithms [1]. One route to solve this issue is to employ physical devices that can naturally mimic brain-like behaviour, known as neuromorphic computing. Magnetic materials are already widely used for long-term data storage, but ongoing work seeks to make use of their potential as both working memory and computing architectures. Particularly, devices based on magnetic domain walls (DWs) can perform logic operations [2] and can readily store information [3]. 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 that could be used in specialised neuromorphic devices. We first present experimental measurements that demonstrate the feasibility of tuning 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(a)). 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 nanowire length (Hy), (Fig. 1(b)). We apply this stochastic behaviour in two possible contexts, as a stochastic neuron for a binary network or as a stochastic synapse for feed forward network, where we model the pinning behaviour based on the experimental results. In the first case, we have modelled these DW wires as neurons in a Boltzmann machine, which exploits their randomness in order to sample a defined, or trained, probability distribution. We apply this to the task of integer factorisation [4], where the connectivity of the neurons is defined so that the binary representation of the integer factors are the most probable states. Such a device inherently exploits the tunable stochasticity of the system to solve complex problems. In the second case, we have explored the properties of feedforward artificial neural networks where the nanowires acted as binary stochastic synapses (BSS). These synapses are individually random but the output can be improved by repeatedly sampling the network to include redundancy. We have developed both mean-field and stochastic based learning methods, where for low repeats of the synapse the stochastic learning is seen to outperform the mean-field based method. It is found that for a single cycle of the network it can be trained to recognise handwritten digits with error rate of 17%. Repeated sampling of the network decreased the error rate to 8% for 128 cycles - close to mean field performance (Fig. 2). Following this we have expanded our network to become a multi-layer perceptron with both inhibitory and excitatory synapses. The additional layer increases the pathways through the network thus reducing the impact of the stochasticity on the classification. For the same classification test the error rate is seen to reduce to 5.5% with a hidden layer of 500 neurons. In conclusion, the pinning of DWs in nanowires can be tuned by applying a magnetic field transverse to the wire and the resultant pinning probability has been used as both stochastic neurons and synapses. As a neuron, we have demonstrated its ability to perform integer factorisation as part of a Boltzmann machine. While as a stochastic synapse, we have been able to train single and multi-layer networks to classify handwritten digits with close to digital network performance. Our work illustrates how the intrinsic stochasticity of DW devices can be harnessed to provide tunable bespoke hardware for machine learning tasks.

**References:**

[1] M. M. Sabry Aly et al., Proceedings of the IEEE, Vol. 107, no. 1, p. 19 (2019) [2] D. A. Allwood et al., Science, Vol. 309, Issue 5741, p. 1688 (2005) [3] R. L. Stamps et al., J. Phys. D. Appl. Phys., Vol. 47, p. 333001 (2014) [4] W. A. Borders et al., Nature, Vo. 573, p. 390 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/488de22bf5135fdfd14a1e4eeaa1d155.jpg)
Fig. 1: (a) Schematic of a notched magnetic nanowire for use as a synapse with a parallel driving field (Hx) and a tunable transverse depinning field (Hy). (b) Pinning probability as a function of transverse field, showing a sigmoidal dependence.

![](https://assets.underline.io/uploads/markdown_image/1/image/0c39f92c401e21de2e6218da03982401.jpg)
Fig. 2: The validation error rate of handwritten digits using a single layer network with binary stochastic synapses (BSS) for varying number of cycles of the network. Lines and points show the training using the mean-field or stochastic methods respectively.

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