France



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 Ni&lt;sub&gt;80&lt;/sub&gt;Fe&lt;sub&gt;20&lt;/sub&gt; [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.&lt;br/&gt;&lt;br/&gt;Fig. 1 (A) shows an SEM micrograph of a representative rectangular Ni&lt;sub&gt;80&lt;/sub&gt;Fe&lt;sub&gt;20&lt;/sub&gt; 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].&lt;br/&gt;&lt;br/&gt;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 Ni&lt;sub&gt;80&lt;/sub&gt;Fe&lt;sub&gt;20&lt;/sub&gt; 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-L&lt;sub&gt;3&lt;/sub&gt; 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.&lt;br/&gt;&lt;br/&gt;&lt;br/&gt;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.&lt;br/&gt;&lt;br/&gt;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)&lt;br/&gt;[2] Branford et al., Science 335, 1597 (2012)&lt;br/&gt;[3] Jensen et al., ALIFE 2018: The 2018 Conference on Artificial Life (July 2018) pp.15-22&lt;br/&gt;[4] Negoita et al., J. Appl. Phys. 114, 013904 (2013)&lt;br/&gt;[5] R. Dawidek et al., Adv. Funct. Mater. 2021, 2008389&lt;br/&gt;[6] I. Vidamour et al. presented at MMM-2020&lt;br/&gt;[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.

INTERMAG 2021

 Imaging the emergent behaviour in nanoring assemblies for reservoir computing applications

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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.

technical paper

Compensated ferrimagnets show both ferromagnetic and antiferromagnetic dynamics, i.e., fast magnetization dynamics [1-5] and vanishing Skyrmion Hall effect [6] with finite magnetization. As the dynamics of ferrimagnets can be driven with current-induced spin torques [7, 8], it is important to find a way of quantifying spin torques in ferrimagnetic materials. One of the main methods to quantify the spin torque in ferromagnets is the spin torque ferromagnetic resonance (ST-FMR) technique [9, 10]. For ferromagnets, the damping-like (DL) and field-like (FL) torque components are separately determined through ST-FMR line-shape analysis. However, it is unclear if this line-shape analysis is valid even for ferrimagnets. In this work, we report a theory of spin torque ferrimagnetic resonance (ST-FiMR) in ferrimagnet/heavy metal bilayer structures. We investigated ST-FiMR for two cases. (i) Identical DL and FL torque efficiencies of two sublattices in a ferrimagnet, and (ii) Different DL and FL torque efficiencies of two sublattices, considering different spin polarization between rare-earth and transition metal elements [11]. For both cases, the mixing voltage originating from the DL torque is linearly proportional to the effective net spin density of the system, resulting in sign reversal at the angular momentum compensation condition. In addition, at the compensation condition, the line-shape can be anti-symmetric, in contrast to entirely symmetric signals for ferromagnets. Our result suggests that the line-shape analysis established for ferromagnets is invalid for ferrimagnets especially near the compensation condition so that the proposed theory must be implemented to analyze the spin torque ferrimagnetic resonance.

**References:**

[1] K.-J. Kim et al., Nat. Mater. 16, 1187 (2017). [2] S.-H. Oh et al., Phys. Rev. B 96, 100407(R) (2017). [3] L. Caretta et al., Nat. Nanotechnol. 13, 1154 (2018). [4] S. A. Siddiqui et al., Phys. Rev. Lett. 121, 057701 (2018). [5] S.-H. Oh and K.-J. Lee, J. Magn. 23, 196 (2018). [6] Y. Hirata et al., Nat. Nanotechnol. 14, 232 (2019). [7] J. Yu et al., Nat. Mater. 18, 29 (2019). [8] T. Okuno et al., Nat. Electron. 2, 389 (2019). [9] A. A. Tulapurkar et al., Nature 438, 339 (2005). [10] J. C. Sankey et al., Phys. Rev. Lett. 96, 227601 (2006). [11] C. Kaiser et al., Phys. Rev. Lett. 95, 047202 (2005).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f8bb2c0202f39a1849a2dd00cdb94a4f.jpg)
Fig. 1. Spin torque ferrimagnetic resonance signal only with damping-like torque (DLT).

![](https://assets.underline.io/uploads/markdown_image/1/image/293aa0f64f1b4c6580837ff6a06e31d7.jpg)
Fig. 2. Sign reversal of spin torque ferrimagnetic resonance near angular momentum compensation condition.

 Theory of Spin-Torque Ferrimagnetic Resonance

The orthogonal magnetization multilayer has the in-plane and perpendicular anisotropy magnetic layers, which has been proposed for the fast magnetization switching by spin transfer torque [1]. On the other hand, it is difficult to obtain the stable spin transfer torque oscillation (STO) in the orthogonal magnetization since the switching occurs too easily. In this study, we introduced the 90 degree magnetic coupling to obtain the STO in the orthogonal configuration. In typical, the 90 degree coupling between two magnetic layers is described by the bilinear coefficient A12 and biquadratic coefficient B12, and obtained under the conditions of A12&lt;|B12| and B12&lt;0 by using suitable layer as a spacer. [2] In our model, the multilayer consists of three layers with thickness of 2 nm; a top layer with the in-plane magnetic anisotropy, a nonmagnetic spacer, and a bottom layer with the perpendicular magnetic anisotropy sequentially. Figure 1 shows the calculated magnetization map in the top layer (a) and the side views of the magnetization configuration of the top and bottom layers (b) when the top layer is CoFe with Ms of 1450 emu/cc, Hk of 35 Oe, and a damping constant of 0.01, and the bottom layer is Fe/Pt with Ms of 800 emu/cc and Hk of 15000 Oe. The top and bottom layer has the vortex magnetization and the perpendicular magnetization, respectively, when a current density is zero. By increasing the current density, the magnetization of the top layer oscillates with frequency 35 GHz while the bottom layer maintains the perpendicular magnetization. This stable STO is due to the 90 degree coupling. Figure 2 shows the STO frequency as a function of the current density for various conditions. First, let us focus on the relation between the STO frequency and magnitude of 90 degree coupling. The STO frequency of the CoFe top layer was increased by decreasing B12 from −0.3 to −0.6, that is, increasing the 90 degree coupling. Next, we compared the STO performances by changing the top layer material from CoFe to NiFe and Ni with small Ms and small damping constant. As a result, the frequency achieved 45 GHz and the necessary current density was reduced to 6x107 A/cm2. In summary, it was found that the STO was stably realized in the orthogonal configuration by introducing 90 degree magnetic coupling, and the STO frequency was increased by enhancement of 90 degree coupling. Additionally, our calculation indicates that it is possible to increase the frequency and necessary current density by material selection.

**References:**

[1] H. Liu et al., Appl. Phys. Lett. 97, 242510 (2010). [2] G. Nagashima., et al, J. Appl. Phys. 126, 093901 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0ab8e84ec749f770d055d306dfad3181.jpg)
Fig. 1 (a) Magnetization configuration in the top layer. (b) Side views of magnetization configuration in the top and bottom layers. The shown magnetization corresponds to the enclosed area in (a).

![](https://assets.underline.io/uploads/markdown_image/1/image/0d156a12ee9515515270d71c3b90fa7a.jpg)
Fig. 2 STO frequency as a function of the current density for various conditions; B12=−0.3 and B12=−0.6 for CoFe top layer, and B12=−0.6 for CoFe, NiFe, and Ni top layers.

 Spin Transfer Torque Oscillation in Orthogonal Magnetization Disks

Spin-torque diode is one of the emerging novel technology in spintronics. Based on the spin-torque diode effect, it could be used for effective RF detection and harvesting, as its sensitivity significantly overcomes the sensitivity of commonly used Schottky diodes. Usage of perpendicular magnetic anisotropy (PMA) in the free layer appears to be one of the most promising designs for spin-torque diodes. We report an efficient design rule to develop spin-torque diodes with PMA with record microwave sensitivity. We study the impact of the free layer magnetization tilt on the sensitivity of the spin-torque diode. This magnetization tilt can be effectively engineered by choosing the ratio of the shape and surface anisotropy and interlayer magnetostatic interactions. With the help of micromagnetic modeling over a broad range of input powers and free layer thicknesses, we demonstrate a significant role of non-linearity and inhomogeneous magnetization dynamics in achieving high sensitivity levels (see Fig.1). Our modeling for typical state-of-the-art parameters of magnetic tunneling junction demonstrates a possibility to reach a record-breaking unbiased microwave sensitivity of 3500 mV/mW (see Fig.2), which is far beyond the parameters of commercial Schottky diodes. We also analyze the impact of direct current bias on sensitivity both in case of current is lower than the critical one and in case of current is higher than the critical one and injection locking occurs. These results can be very important for the development of efficient spintronic ambient energy harvesters. Financial support by the Russian Science Foundation (Project No. 19-12-00432) is gratefully acknowledged.

**References:**



**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7698082bae26856f7ff2cb3d64de7fa5.jpg)
Unbiased spin-torque diode sensitivity as a function of free layer thickness derived from micromagnetic modeling. Color refer to input power: red – 2 nW, blue – 20 nW, green – 100 nW, brown – 1000 nW.

![](https://assets.underline.io/uploads/markdown_image/1/image/f2f39922dd26de7259c1364e356d5aef.jpg)
Unbiased diode sensitivity dependences on the thickness for Rp = 600Ω, Rap = 1200Ω (red) and Rp = 100Ω, Rap = 200Ω (blue) in case of 20 nW incident power.

 Sensitivity optimization of spin-torque diode with perpendicular anisotropy through free layer thickness tuning

We present our recent realisation of a nanomagnetic Galton board [1], a nanoscale recreation of the paradigmatic experiment in which balls travel through an array of pegs, designed by Sir Francis Galton to demonstrate the emergence of order from randomness. In our nanoscale experiment, balls are substituted by magnetic domain-walls, a honeycomb nanowire lattice recreates the peg array, and an externally applied magnetic field takes the role of gravity (Figure 1). Using high-resolution immersion Kerr microscopy, we image domain wall propagation through this nanoscale system a very large number of times (Figure 2), and demonstrate that complex domain wall dynamics can be employed to generate high-quality stochasticity suitable for computing applications. By varying the direction of the externally applied field, we were also able to demonstrate the continuous tuneability of the stochastic domain-wall decision process, a critical step to the realisation of functional computing systems. To conclude, we will discuss the technical advances that enabled this study, in which millions of domain-wall trajectories were imaged and analysed. These will be of great interest to those performing stochasticity studies in spintronics, as being able to acquire and deal with large datasets is an important experimental challenge in the field. The results demonstrated open promising paths in the design and development of spintronic devices in which the intrinsic stochasticity of domain-wall dynamics is exploited using new computing architectures.

**References:**

[1] Sanz-Hernández, Dédalo, et al. "Tunable Stochasticity in an Artificial Spin Network", DOI: 10.1002/adma.2020081

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9e8bc476a41cbcab0159a897cf862dd2.jpg)
Figure 1. Nanoscale magnetic Galton board formed by an elliptical domain-wall injection pad (top) and a bifurcating nanowire network.

![](https://assets.underline.io/uploads/markdown_image/1/image/705a04356594395b96ae40d6c0dfa320.jpg)
Figure 2. Oil-immersion longitudinal Kerr microscopy image of the board presented in Figure 1, after the injection of a domain-wall in the system. The trajectory followed by the wall is observed as a dark trace.

 Tuneable Stochastic Domain-Wall Trajectories in a Magnetic Galton Board

Domain walls (DWs) in magnetic nanowires are a topic of intense interest due to proposals to use them as data carriers in non-volatile logic and memory devices. However, despite their apparent technological potential such devices are challenging to realise, in part because DWs pinning and propagation is highly stochastic, making digital devices unreliable [1]. While materials engineering approaches can be used to supress stochasticity [2], it is also interesting to consider whether alternative computer paradigms could prove more resistant to, or even benefit from, stochasticity. In this talk we will present three examples of how embracing DW stochasticity as a functional, rather than technologically inhibiting feature can facilitate novel computational devices. Figure 1(a) presents results that illustrate typical stochastic DW behaviours in a 400 nm wide Ni80Fe20 nanowire device, as characterised by focused magneto-optic Kerr effect (FMOKE) magnetometry. DWs were injected from the pad to a patterned defect in the nanowire’s middle and the distributions of magnetic fields required to propagate them through the defect characterised. The depinning field histogram shows both fractional pinning probabilities and multimode depinning field distributions, both characteristic features of stochasticity induced by complex DW dynamics [3]. However, while individual reversals of such a device are highly stochastic, average properties are well-defined, and more importantly, highly tuneable. Figure 1(b) illustrates how applying modest magnetic fields orthogonal to a similar nanowire/defect combination allows DWs pinning probabilities to be tuned between 0 and 1. Thus, stochastic pinning can be used to create random bit-streams with tuneable average values. In our first example of the utility of these effects we use stochastic DW pinning to realise the stochastic computing paradigm in DW logic networks. In stochastic computing, floating point numbers are encoded in the average value of random binary bit-streams, and single conventional logic gates perform complex mathematical operations that would usually require extended logic networks to perform. For example, an AND gate performs a multiplication operation. Figure 2(a) presents the results of FMOKE magnetometry, illustrating how the output of a DW AND gate with stochastic input produces output bitstreams consistent with those expected. In our second example we use the stochastic pinning of DWs to facilitate two neuromorphic computing approaches: a feed-forward neural network and a Boltzman machine. In the feed-forward network, nanowires act as binary stochastic synapses, with the probability of DW propagation taking the role of a synaptic weight. Simulations of network behaviour allow us to demonstrate that such a device can perform machine learning tasks including written digit recognition. In the Boltzman machine, where stochasticity is used to sample a defined, or trained, probability distribution, the nanowires take the role of stochastic neurons. We apply this approach to an integer factorisation problem where network is defined so that the binary representation of the integer factors are the most probable states for the neurons. In our final example, we show how DWs driven by rotating magnetic fields within an interconnected nanoring array, can be used to perform reservoir computing [4]. We characterise a ring array’s behaviour using a combination of MOKE, polarised neutron reflectivity and photoemission electron microscopy, and show how emergent behaviour arises from stochastic interactions between DWs at the array’s junctions (Figure 2(b)). Using the magnitude of the rotating applied field as an input we then demonstrate how this allows the array to be used as a reservoir and perform a range of time series analysis tasks including speech recognition and chaotic series prediction.

**References:**

[1] T.J. Hayward and K.A. Omari, Beyond the quasi-particle: stochastic domain wall dynamics in soft ferromagnetic nanowires, Journal of Physics D: Applied Physics 50, 8, 084006 (2017). [2] T.J. Broomhall, A.W. Rushforth, M.C. Rosamond, E.H. Linfield, and T.J. Hayward, Physical Review Applied 13, 024039 (2020). [3] T.J. Hayward, Scientific Reports 5, 13279 (2015). [4] R. W. Dawidek, T. J. Hayward, I. T. Vidamour, T. J. Broomhall, M. Al Mamoori, A. Mullen, S. J. Kyle, P. W. Fry, N. J. Steinke, J. F. K. Cooper, F. Maccherozzi, S. S.Dhesi, L. Aballe, M. Foerster, J. Prat, E. Vasilaki, M. O. A. Ellis, D. A. Allwood, submitted to Advanced Functional Materials (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/158b0ac6376b52704edb84a394fcac23.jpg)
Figure 1: (a) Characterisation of stochastic DW pinning at a dN = 80 nm notch in in a 400 nm wide, 25 nm thick Ni80Fe20 nanowire. The experimental geometry is shown at the top of the figure with green circles representing where injection and depinning field histograms were measured. (b) Illustration of how the pinning probability of DWs in similar system were tuned by applying magnetic fields (Hy) orthogonal to the nanowire.

![](https://assets.underline.io/uploads/markdown_image/1/image/c816108946b00cb61778bad36bbb92b6.jpg)
Figure 2: (a) Stochastic computing with a DW AND gate. An SEM of the measured device is shown at the top of the figure, with green circles indicating where FMOKE measurements were made to measure the probability of DWs passing through the input nanowires (PA and PB) and output nanowire (Pout). The plot shows PA x PB and Pout as a function of applied field; the close tracking of these curves shows that AND gate performed a multiplication operation on the input bit-streams. (b) Behaviour of an interconnected nanoring array as a function of rotating applied field amplitude. The number of rings with propagating DWs, pinned DWs and in vortex states are shown at each applied field value.

 From Stochasticity to Functionality: Harnessing Magnetic Domain Walls for Probabilistic and Neuromorphic Computing

Magnetic Skyrmion is a particle-like magnetic excitation in ferromagnetic material similar to the Dirac-monopole in the spin-ice systems. Since the magnetic Skyrmion is topologically protected [1], it can survive under thermal agitation and shows Brownian motion. The skyrmion system can be an ideal platform to design stochastic computer [2] and/or in Brownian computer [3], since magnetic Skyrmion is a special particle that may show Brownian motion in all-solid-state devices. Therefore, investigation of the Brownian motion of the magnetic Skyrmion is now the subject of scientific and technical interest. The dynamics of the Skyrmions in a two-dimensionally extended film at finite temperature is described by the Thiele equation [4]. According to the equation, the Skyrmion may pose a velocity perpendicular to the force's direction and tend to turn right or left depending on its chirality determined by the Skyrmion number. This gyro-motion reduces the (diagonal element of the) diffusion constant [5]. Naïvely thinking, we may suppose that the diffusion flow may pose a component perpendicular to the gradient of the skyrmion density because of the gyro-motion. The component should be proportional to the gradient of the Skyrmion density and newly defined “gyro-diffusion constant” (off-diagonal element of the diffusion constant) [6]. However, such flow is divergence-less and does not appear in the equation of diffusion. According to this context, we mention the gyro-diffusion constant as “the hidden diffusion constant”. In this talk, we show evidence of gyro-motion in diffusion and discuss the observation of the gyro-diffusion constant. Then, we mention recent progress to implement “Brownian computer” by use of Skyrmions. In our experiments, the magnetic Skyrmions are bubble-shaped magnetic domains generated in a ferromagnetic thin film having spontaneous magnetization perpendicular to the film surface. The Skyrmions have their diameter adjusted to 1 μm to allow the observation by an optical microscope with polarization analysis (MOKE microscope). To create the Skyrmions, Ta/CoFeB /Ta sandwiched films [7] possessing perpendicular magnetic anisotropy and the interfacial- Dzyaloshinskii-Moriya interaction (i-DMI) were fabricated. Changing the upper and lower Ta film thicknesses independently, the ratio of the i-DMI energy to the magnetic anisotropy energy was adjusted. Besides, by eliminating the annealing process, the generation of grain boundaries was suppressed. The annealing free process enhanced the diffusion coefficient of skyrmions by a factor of more than 10 compared to the previous study [2] and comparable to the recent observations [7,8]. Firstly, we have developed the Skyrmion channel by a slight modification of the SiO2 passivation layer thickness. By this method we could make degradation and dipole-field free channels. With this method the Skyrmions show free Brownian motion even in a complicated Skyrmion circuit like Hub (Y-cross) [7]. Secondly, we put a Skyrmion in one-dimensional channels to interfere with its gyro-motion. As a result, a significant increase in the diagonal diffusion constant was observed. This is clear evidence of the role of gyro-motion in the diffusion process. Thirdly, we have observed the diffusion of a Skyrmion in uniformly extended ferromagnetic films. In the uniform film, Skyrmions show free diffusion as it is shown in Fig. 1. The analysis of the simultaneous position-position correlation function &lt;X(t)*X(t)&gt;, which is a half of the mean square displacement (MSD), results that &lt;X(t)*X(t)&gt; is proportional to the elapsed time providing the diagonal diffusion constant as its slope. On the other hand &lt;X(t)*Y(t)&gt; is zero as it is expected form the symmetry. While, we have found that simultaneous velocity-position correlation functions (&lt;Vx(t)*X(t)&gt; and &lt;Vx(t)*Y(t)&gt;) are non-zero. From the theory, &lt;Vx(t)*Y(t)&gt; corresponds to the gyro-diffusion constant. The size of the observed gyro-diffusion constant is about 1/10 of the diagonal diffusion constant, and non-negligible. However, the sign of the observed gyro-diffusion constant is opposite to that expected from a sense of gyro-motion. Since the sense of the gyro-motion is determined by the sign of the Skyrmion number, it should not be changed easily. The change of the sign in the gyro-diffusion constant will be discussed in terms of the diffusion in a potential. Finally, essential devices to construct a Brownian computing circuit, i.e. hub, ratchet, and c-join [3], were tested. By now, ratchet and c-join need a control using MOKE observation and manual feedback. To overcome this difficulty, we are now developing autonomous devices using voltage control [9, 10] of the Skyrmion (Fig. 2). Acknowledgments The authors acknowledge Mr. Y. Jibiki of Osaka University, Dr. S. Auffret, Dr. C. Baraduc, and Dr. Hélèn Béa of Spintech, France for their initial contributions. The authors also acknowledge Dr. M. Oogane for his FMR measurements. This research and development work was supported by the ULVAC, Inc., the Ministry of Internal Affairs and Communications, Basic research S (20H05666) of JSPS, and CREST (Non-classical Spin project) of JST.

**References:**

[1] N. Nagaosa et al., Nat. Nanotech. 8, 899-911 (2013). [2] J. Zazvorka et al., Nat. Nanotech. 14, 658 (2019). [3] F. Peper et al., ACM J. Emerg. Technol. Comput. Syst, 9, 1, Article 3 (2013). [4] A. Thiele et al., Phys. Rev. Lett. 30, 230 (1973). [5] C. Schuette et al., Phys. Rev. B 90, 174434 (2014) [6] E. Tamura et al. https://arxiv.org/abs/2005.04860. [7] Y. Jibiki et al., Appl. Phys. Lett. 117, 082402 (2020). [8] L. Zhao et al., Phys. Rev. Lett. 125, 027206 (2020). [9] T. Nozaki et al., Appl. Phys. Lett. 114, 012402 (2019). [10] M. Scott et al., Nano Lett. 17, 3006 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6533c8637225b7f1b2cad390443cbfe3.jpg)
Fig. 1 Trajectories of magnetic skyrmions observed at 316 K using a MOKE microscope.

![](https://assets.underline.io/uploads/markdown_image/1/image/4c8fd416dc80b83ea49919378f59a037.jpg)
Fig. 2 Voltage/current controlled Skyrmion ratchet (test device).

 Hidden skyrmion diffusion and Brownian computing

Magnetic tunnel junctions offer significant and near-term improvements to new computer chips, most immediately from non-volatile memory applications [1]. These devices consist of two magnetic layers whose relative alignment – parallel or antiparallel – determine the device's two-terminal resistance. When the energy barrier between these two states is sufficiently small, thermal fluctuations can cause the device to exhibit random telegraph noise in its resistive state. In this case, the device is called a superparamagnetic tunnel junction [2]. Computing with random telegraph noise can offer favorable properties such as noise-tolerance, massive parallelism, and – for certain operations – low power and area footprints. The field of stochastic computing addresses questions of engineering such systems but has mostly been restricted to the use of low-quality pseudorandom number generators. The low periodicity and high cross-correlations between these devices have restricted traditional stochastic computing by forcing frequent regeneration of stochastic bitstreams. Crossing in and out of the stochastic domain can incur significant energy penalties [3]. In our work [4], we introduce a low-energy circuit for extracting programmable randomness from superparamagnetic tunnel junctions. The required probability is determined via a digital input signal, rather than an analog current, so ohmic losses are minimized (Figure 1). Using detailed analog circuit simulation, we find that our device requires only half the energy of the traditional all-CMOS solution, while also offering infinite-period truly random bitstreams. To demonstrate the value of truly random bitstreams, we designed a deep neural network modeled after LeNet5 [5], a famous convolutional neural network used for handwriting recognition. Traditional neural networks multiply neuron outputs by a synaptic weight, add these products together at the input of another neuron, and then apply a nonlinear operation to that input sum. In our implementation, the values passed between neurons are encoded in the expected value of the voltage on any given wire. To multiply two signals together, we need only use a Boolean AND gate to construct the joint distribution of the wires. Feeding many wires into a Boolean OR gate generates a polynomial function which looks like linear addition near the origin, but acquires high-order nonlinear corrections when the inputs are strong. However, the OR gate only functions in this way because the inputs can be regarded as statistically independent, which would be impossible with pseudorandom number generators absent an unacceptable increase in energy consumption. We implement LeNet5 using superparamagnetic tunnel junctions together with these simple logic gates and train a simulation of this deep neural network. We are able to achieve 97 % accuracy on handwritten digit recognition using only 150 nJ per inference. This energy efficiency represents a factor of 1.4 to 7.7 improvement over comparable proposals that do not use tunnel junctions. However, after systematic optimization, the best CMOS-only solution is able to achieve 99.1 % accuracy [6]. Questions about the best way to train OR-gate neurons for optimal performance is a question for future research. Nevertheless, our demonstration shows that taking a digital approach to utilizing SMTJ fluctuations can unlock extremely energy-efficient circuit designs with the potential for offering stochastic neural networks computing in near-term application specific integrated circuits.

**References:**

[1] Apalkov, Dieny, and Slaughter, Proc. IEEE 104, 1796 (2016) [2] Camsari, Faria, Sutton, and Datta, Phys. Rev. X 7, 031014 (2017) [3] Manohar, IEEE Comp. Arch. Lett. 14, 119 (2015) [4] Daniels, Madhavan, Talatchian, et al., Phys. Rev. App. 13, 034016 (2020) [5] LeCun, Bottou, Bengio, et al., Proc. IEEE 86, 2278 (1998) [6] Li, Ji, Ren, et al., IEEE Trans. Comp. Aid. Des. Int. Circ. Sys. 38, 1543 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a7dfe1770e892cca4d62d45c58f83e71.jpg)
Neurons in the brain have analog outputs in analog time; superparamagnetic tunnel junctions have digital states in analog time. To use digital circuitry to discretize the time variable, removing the need for steady state currents and the associated ohmic losses.

 Running Deep Neural Networks on Superparamagnetic Fluctuations

Despite attempts to design correlated systems to influence the properties of a physical system, it is often those inspired by nature that yield the best results. Artificial spin ice (ASI) are lithographically defined 2D projections of geometrically frustrated materials that are often used for experimentally model more complex atomistic systems [1-2]. In addition, ASI exhibit interesting physics, including collective dynamics [3-4] and emergent monopole stabilisation [5], that can be applied to probabilistic computing, signal propagation, and logic devices [6-7]. The capacity to design an almost infinite number of topologies to induce novel effects, and the ability to probe these characteristics optically, electrically and by scanning probe techniques have made ASI an exciting candidate for probabilistic computing [8]. Here, two custom ASI systems are shown to exhibit interesting defect behaviours from the inherent array design. The first design models the influence of a ferromagnetic defect in the centre of the structured array. Defects in crystal systems can have bountiful effects on the local physical properties. These effects in-turn can often be exploited for novel applications. For example, defects in transition-metal dichalcogenides have applications in quantum information processing as single photon point-sources due to highly localised excitons bound to the defect-site [9]. Alternatively, nitrogen vacancies in diamond exhibit spin-dependent fluorescent rates required for high resolution magnetometry [10]. The magnetic defect in the ASI system acts as a source of emergent monopoles at lower energy-barriers than a pristine lattice. Recent results of the delicate interplay between defect and lattice states are studied in the context of signal propagation applications by magnetic imaging techniques, including magnetic force microscopy (MFM); Lorentz TEM; and electron holography. We reveal how the magnetic defect affects the surrounding structure compared to those formed independently from the defect, e.g., from discontinuous lattice-edges. The second presented system is a quasi-hexagonal ASI lattice, which is arranged to encompass competing interactions from both ferromagnetic/antiferromagnetic coupling between parallel islands and geometric frustration (Fig. 1a) [11]. By including parallel islands along just one axis, the lattice exhibits an additional shape anisotropy under field directions (Fig 1b-c). We demonstrate that the inclusion of these coupled islands promotes the non-uniform break-down of Ising "macro-spins" in the islands under a specified field protocol (Fig. 1d). Through MFM and micromagnetic modelling, we investigate the disrupted periodicity of the energy states across the lattice and the predictability of non-Ising state formation. Control over the local coupling effects have application in programmable logic, e.g. non-Boolean or probabilistic computing [12].

**References:**

[1] R. F. Wang et al., Nature, vol. 439, no. 7074, pp. 303–306, Jan. 2006. [2] M. Tanaka et al., Phys. Rev. B - Condens. Matter Mater. Phys., vol. 73, no. 5, pp. 3–6, 2006. [3] M. B. Jungfleisch et al., Phys. Rev. Appl., vol. 8, no. 6, pp. 1–7, 2017. [4] A. Farhan et al., Phys. Rev. B, vol. 96, no. 6, p. 064409, Aug. 2017. [5] S. Ladak et al., Nat. Phys., vol. 6, no. 5, pp. 359–363, May 2010. [6] S. H. Skjærvø et al., Nat. Rev. Phys., vol. 2, no. 1, pp. 13–28, Jan. 2020. [7] S. Gliga, E. Iacocca, and O. G. Heinonen, APL Mater., vol. 8, no. 4, p. 040911, Apr. 2020. [8] H. Arava et al., Nanotechnology, vol. 29, no. 26, 2018. [9] A. Srivastava et al., Nat. Nanotechnol., vol. 10, no. 6, pp. 491–496, Jun. 2015. [10] G. Balasubramanian et al., Nature, vol. 455, no. 7213, pp. 648–651, 2008. [11] R. Puttock et al., Small, vol. 2003141, pp. 1–8, Sep. 2020. [12] Project supported by the UK government department for Business, Energy and Industrial Strategy through NMS funding (Low Loss Electronics). LTEM and EH was available through funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 823717 – ESTEEM3.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/80e75899528a7a4590fb6f72ac0d7385.jpg)
Figure 1. (a) AFM image of four primitive cells of the quasi-hexagonal ASI lattice where three vertex-shapes are outlined. MFM images of unimodal (b) and bimodal (c) configurations favourable when the field is aligned along y- and x-axes, respectively. (d) Four possible magnetic configurations in parallel islands upon application of a critical field.

 Magnetic defect-driven dynamics in artificial spin ice

There is a growing pursuit to create neuromorphic devices that are able to perform machine learning tasks, both in an energy efficient manner and compatible with CMOS architecture. These approaches are based on taking the complex response of materials and mimicking aspects of a given machine learning model. Typically, what cannot be mimicked directly in a material, such as learning, is supplemented by other devices, circuitry, or external computers. These various approaches often face a number of bottlenecks, related to scaling, functionality independent of computers, and energy efficient operation. Ultimately constructing autonomous materials, namely individual materials that independently adapt their physical properties based on input data, requires deeper fundamental research linking physical processes to machine learning concepts. In this way, it is necessary to link quantum phenomena in materials to concepts like plasticity, memory, and learning. In this way, new types of quantum materials can be designed for autonomous recognition, as well as new functionalities can be explored, such as quantum machine learning. Energy-based models, like the Boltzmann machine, provide a natural link between physics and machine learning functionality via the Ising model and the physics of multi-well potentials and glasses (1, 2). I will discuss a new model system, based on the recently developed concept of orbital memory (3), which mimics the behavior of a Boltzmann machine directly in its stochastic dynamics and in one elemental material, down to the atomic-scale (4). Using scanning tunneling microscopy (STM) to fabricate nano-scale arrays of cobalt atoms on semiconducting black phosphorus, we observe stochastic noise described by the tunable multi-well landscapes seen in stochastic energy-based models. Utilizing a concept based on separation of time scales and the anisotropic behavior of black phosphorus (5), I will show that these multi-well landscapes are adaptive and exhibit inherently coupled neural and synaptic dynamics, at the atomic-scale. I will discuss perspectives of learning in these systems, and how such model systems provide fundamental insight into the future design of quantum-based brain-like devices. This work was done in collaboration with H.J. Kappen, Brian Kiraly, Elze J. Knol, and Werner M.J. van Weerdenburg. This work was funded by NWO-Vidi Manipulating the interplay between superconductivity and chiral magnetism at the single-atom level” with project number 680-47-534. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 818399).

**References:**

 1. A. Kolmus, M. I. Katsnelson, A. A. Khajetoorians and H. J. Kappen, New J. Phys. 22 (2), 023038 (2020). 2. U. Kamber, A. Bergman, A. Eich, D. Iusan, M. Steinbrecher, N. Hauptmann, L. Nordström, M. I. Katsnelson, D. Wegner, O. Eriksson and A. A. Khajetoorians, Science 368 (6494), eaay6757 (2020). 3. B. Kiraly, A. N. Rudenko, W. M. J. van Weerdenburg, D. Wegner, M. I. Katsnelson and A. A. Khajetoorians, Nat. Commun. 9 (1), 3904 (2018). 4. B. Kiraly, E. J. Knol, H. J. Kappen and A. A. Khajetoorians, arXiv, 2005.01547 (2020). 5. B. Kiraly, E. J. Knol, K. Volckaert, D. Biswas, A. N. Rudenko, D. A. Prishchenko, V. G. Mazurenko, M. I. Katsnelson, P. Hofmann, D. Wegner and A. A. Khajetoorians, Phys. Rev. Lett. 123 (21), 216403 (2019).

**Images:**

 An atomic-scale Boltzmann machine capable of self-adaption

Ferromagnetic devices, where current pulses flowing through the heavy metal layer below the ferromagnetic layer move the ferromagnetic domain wall through spin-orbit torque, have been proposed and experimentally demonstrated as synapses for neuromorphic computing [1–3]. But since ferrimagnets are known to exhibit faster domain wall motion compared to ferromagnets, the potential for ferrimagnetic synapses also needs to be explored [4]. Here, we first carry out micromagnetic simulations of current-induced domain wall motion in a ferromagnetic device as well as a ferrimagnetic Co/Gd-bilayer-based device. The working principle of the ferrimagnetic synapse is just the same as the ferromagnetic synapse: current flowing through the underlying heavy metal layer moves the domain wall in the ferrimagnetic bi-layer just like it moves it in the ferromagnetic layer (Fig. 1a) [4]. Motion of the domain wall leads to a change in conductance and hence an update in the weight stored in the synapse [1, 2]. In our simulation, the spin Hall angle of the heavy metal (Pt), current flow through which generates the necessary spin-orbit toque for domain wall motion, is considered to be 0.13 [4]. For the ferrimagnetic system, dynamics of the magnetic moments of both the Co and Gd layers, exchange-coupled with each other (coupling strength = -0.9 mJ/m2), has been modeled in ‘mumax3’ [4]. The temperature-dependent variation of the Co and Gd magnetic moments is also considered. From our simulations (Fig. 1b), we observe that in the ferrimagnetic system, the domain wall velocity is highest at the angular-momentum-compensation temperature, T= 165 K, at which moment in Gd layer (mGd) = 0.9 times moment in Co layer (mCo). This is consistent with the experimental observation and corresponding one-dimensional-domain-wall calculations reported by Blasing et al [4]. Fig. 2a shows that across a wide range of current density through the heavy metal layer, domain wall velocity in the ferrimagnetic system at the angular-momentum-compensation temperature (165 K) is 2X–2.5X higher than that in the ferromagnetic system at room temperature. Next, we carry out system-level simulations with the device characteristic in Fig. 2a to compare the performance of the ferromagnetic and the ferrimagnetic devices as synapses in a crossbar-array-based neuromorphic system that mimics a Fully Connected Neural Network (FCNN) with a hidden layer. Synaptic weights are updated in both the ferromagnet-synapse and ferrimagnet-synapse crossbars using the thresholding algorithm, proposed by Kaushik et al. [5], to achieve on-chip learning in the crossbars on Iris data set of flowers and MNIST data set of handwritten digits. Using this thresholding algorithm, for Iris data set, we obtain train accuracy of 100% and test accuracy of 96%, and for MNIST, we obtain train accuracy of 98% and test accuracy of 92%. We obtain these same accuracy numbers for both ferromagnetic and ferrimagnetic synapse crossbars because we allow the same number of conductance states for both these types of synapses (50). According to this thresholding algorithm, for each synapse in the FCNN, the synaptic weight/ conductance is either increased or decreased by a fixed amount or kept the same at any particular iteration [5]. Thus training can be achieved only by applying current pulses of fixed magnitude and duration to the ferrimagnetic or ferromagnetic synapses — positive polarity of current for weight increase and negative polarity for weight decrease (Fig. 1a). A shorter duration pulse needs the domain wall to move with a higher velocity compared to a longer duration pulse to cause the same weight update. Hence, a shorter pulse needs to be of higher current magnitude and will consume more energy. Fig. 2b shows how energy consumed by a pulse increases with a decrease in duration of the pulse, both in the case of the ferrimagnetic system and the ferromagnetic system. We do observe that for the same duration of current pulse, energy consumed is 4X lower for the ferrimagnetic system than for the ferromagnetic system. Similarly, if the energy consumption per pulse is kept constant, pulses of time duration 4X lower can be used for the ferrimagnet compared to the ferromagnet. As a result, when we compare the energy consumption and time taken for on-chip learning in the two aforementioned crossbar systems, we observe that for the same net energy consumption in the system, the ferrimagnetic crossbar can be trained 4X faster than the ferromagnetic crossbar. Similarly, for the same speed across the two crossbars, the ferrimagnetic crossbar can be trained at 4X lower energy than the ferromagnetic crossbar. Thus, our study shows that ferrimagnetic-bilayer-based systems can pave the way for fast and energy-efficient neuromorphic computing in the future.

**References:**

[1] A. Sengupta and K. Roy, IEEE Trans. Biomed. Circuits Syst., vol. 10, no. 6 (2016) [2] D. Kaushik, U. Singh, U. Sahu, I. Sreedevi, and D. Bhowmik. AIP Adv. vol. 10, no. 025111 (2020) [3] R. Blasing, T. Ma, S-H. Yang et al. Nature Communications, vol. 9, no. 4984 (2018) [4] T. Jin, W. Gan, F. Tan et al, J. Phys. D: Appl. Phys., vol. 52, no. 445001 (2019) [5] D. Kaushik, J. Sharda, and D. Bhowmik. Nanotechnology vol. 31, no. 364004 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a5a1e18c735248823a90d465d55cd209.jpg)
Fig. 1 (a) Schematics of ferrimagnetic (Co/Gd-bilayer-based) and ferromagnetic (Co-based) synaptic devices. The domain wall is shown through rotating green arrows, which correspond to magnetic moments. (b) Domain-wall velocity vs current density flowing through the heavy metal layer (Pt), for different values of temperature, which correspond to different ratios of Gd’s magnetic moment (mGd) to Co’s magnetic moment (mCo).

![](https://assets.underline.io/uploads/markdown_image/1/image/eab3ad1532ecb07e4520b5a23f9f3af2.jpg)
Fig. 2. Comparison of (a) domain-wall velocity vs current density characteristic and (b) energy per synaptic-weight-update pulse vs time duration of such a pulse for ferrimagnetic synapses (at angular-momentum compensation temperature, when domain-wall velocity is highest) and ferromagnetic synapses (at room temperature).

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