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Emergent behaviours in complex systems arise from simple interactions between individual components that lead to global complexity [1]. Emergence is found in nature as a mechanism behind intelligent biological processes [2], but thus far, little has been done to exploit the computational power that emergence could offer for machine learning hardware. Here we demonstrate how the emergent properties of a magnetic system can be exploited to perform meaningful computation in materio. Using a combination of X-ray photoemission electron (X-PEEM) microscopy and polarised neuron reflectivity (PNR), we show that arrays of interconnected magnetic nanowire rings demonstrate emergence as a result of complex domain wall (DW) pinning events. The stochasticity associated with DW pinning produces emergent, dynamic equilibria in DW population and array magnetisation [3]. We then present a phenomenological model of these stochastic pinning events and show that the model provides excellent agreement to our experimental data (Fig. 2). The system’s nonlinear response and its inherent memory allowed us to implement a machine learning paradigm called ‘reservoir computing’ (RC). In RC, artificial neural networks are replaced by time-dependent ‘reservoirs’ which capture the advantages of recurrent neural networks (RNNs), without the associated difficulty of training RNNs [4]. Using the model, we demonstrated the feasibility of the ring arrays performing machine learning tasks, including speech processing and signal classification. We believe the sub-millimetre size and ability to solve complex tasks makes the arrays exciting candidates for microprocessors, with parallel computation and memory capability, enabling more energy efficient data processing tasks due to the reduced computational expense.References: [1] - P. Curseu, J. Inf. Technol., 21, (2006), [2] - B. Lindsey et al, Comprehensive Physiology, 2, (2012). [3] - R. Dawidek et al. [Under Review], [4] - M. Lukoševičius and H. Jaeger, Comput. Sci. Rev., 3, (2009).

MMM 2020

Neuromorphic Computation Using Emergent Magnetisation Dynamics

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The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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While chiral spin structures stabilized by Dzyaloshinskii-Moriya interaction (DMI) are candidates as novel information carriers, their dynamics on the fs-ps timescale is little known [1-4]. Since with the bulk Heisenberg exchange and the interfacial DMI two distinct exchange mechanisms are at play, the ultra-fast dynamics of the chiral order needs to be ascertained and compared to the dynamics of the conventional collinear order. Using an XUV free-electron laser we determine the fs-ps temporal evolution of the chiral order in domain walls in a magnetic thin film sample by an IR pump - X-ray magnetic scattering probe experiment. Upon demagnetisation we observe that the dichroic (CL-CR) signal connected with the chiral order correlator mzmx in the domain walls recovers significantly faster than the (CL+CR) sum signal representing the average collinear domain magnetisation mz2+mx2 [5]. We explore possible explanations based on spin structure dynamics and reduced transversal magnetisation fluctuations inside the domain walls and find that the latter can explain the experimental data leading to distinctly different dynamics for collinear magnetic order and chiral magnetic order.References: [1] Beaurepaire, E. et al., Phys. Rev. Lett. 76, 4250–4253 (1996). [2] Koopmans, B. et al., Nat. Mater. 9, 259–265 (2010). [3] Battiato, M. et al., Phys. Rev. Lett. 105, 027203 (2010). [4] Pfau, B. et al., Nat. Commun. 3, 1100 (2012). [5] Kerber N. et al., arXiv:2002.03971 (2020).

Chiral versus collinear magnetic order dynamics: faster chiral recovery after optical excitation revealed by femtosecond XUV scattering

Thermal coupling parameters between different baths (electron, phonon, and magnon) in a nanoscale ferromagnetic Ni layer is investigated. To observe the local thermal interactions, we use time-resolved pump-probe spectroscopy. We collect both thermoreflectance and magneto-optical Kerr rotation, which represent the phonon temperature and magnon temperature, respectively, in a picosecond time scale. To analyze the experimental data, we employ the three-temperature thermal model including a spatial diffusion and local coupling mechanisms. We determine the thermal coupling parameters of Ni and find that the coupling parameter between phonon and magnon baths has a comparable magnitude to that between electron and magnon. The phonon-magnon coupling is responsible for the fast restoration of magnetization right after the ultrafast demagnetization (Fig. 1). In addition, we compare the coupling parameters with direct heating of Ni and in-direct heating of Ni via Au (Fig. 1). The in-direct heating suppresses the non-thermal electrons that reach to Ni, so that we can investigate the effect of thermal electrons on the ultrafast demagnetization. Both direct and in-direct heating situations lead to similar values of the coupling parameters, and the discrepancy is less than 30 %. We conclude that the thermalized electrons play the major role in the ultrafast demagnetization of Ni.References: Kyuhwe Kang and Gyung-Min Choi, J. Magn. Magn. Mater., Vol. 514, p.167156 (2020)

Thermal coupling parameters between electron, phonon, and magnon of Nickel

All-optical spin switching and demagnetization are two sides of the same physics. We find a way to unify them through the same model that takes into account the Heisenberg exchange between spins and spin-orbit coupling. To have efficient spin switching, the electron initial momentum direction must closely follow the spin's orientation, so that the orbital angular momentum is transverse to the spin, and consequently the spin–orbit torque lies in the same direction as the spin. The demagnetization can be understood from the spin-orbit coupling-mediated spin-wave excitation. The efficiency of laser-induced demagnetization is very high. We also observe the collapse of the spin-spin correlation length within 20 fs, consistent with the experimental findings [2]. Upon laser excitation there is a massive spin wave generated. The wave starts at the center of the laser beam and propagates outwards. Figure 1(b) shows the effect of the linearly polarized light at 123 fs after laser excitation. The spins close to the center bend strongly to the –z-axis. The situation is quite different with a right-circularly polarized laser (see Fig. 1(c)), where the spins are much more strongly affected. For left-circularly polarized light, the effect is much weaker. The demagnetization results from spin misalignment across the sample. This is similar to the traditional spin wave theory, with the low-energy spin wave excitation across several hundred lattice sites, which is far beyond the capability of the time-dependent density functional theory. Our simulation shows that the spin-spin correlation is reduced within 20 fs [3], consistent with the experimental results.References: [1] G. P. Zhang and M. Murakami, Journal of Physics: Condensed Matter 31, 345802 (2019). [2] P. Tengdin et al., Science Advances 4, eaap9744 (2018). [3] G. P. Zhang , M. Murakami , Y. H. Bai, Thomas F. George, and X. S. Wu, J. Appl. Phys. 126, 103906 (2019).

Unification of ultrafast demagnetization and switching

Helicity-independent all-optical switching (HI-AOS) of rare-earth/transition metal ferrimagnets is being intensively studied [1-6] for the last few years due to its potential applications towards faster read-write spintronic devices. The underlying magnetization dynamics has been discussed using two different approaches; i) atomistic Landau-Lifshitz-Gilbert model [3-4] and ii) microscopic three temperature model (M3TM) [5]. Recently, the ultrafast switching (< 10 ps) of ferromagnetic Co/Pt stack (exchange coupled with GdFeCo ferrimagnet) has been demonstrated showing a greater potential for technological applications [6]. We have extended the M3TM approach, to investigate similar systems by including a RKKY exchange (JRKKY) coupled ferromagnetic (Co/Pt) layer along with a ferrimagnet (GdCo) alloy. JRKKY, estimated as a fraction of CoPt exchange (JCoPt = 1.26X10-21 J), depends on the thickness of the Pt spacer layer separating Co/Pt from GdCo. Positive and negative sign of JRKKY means ferromagnetic and anti-ferromagnetic type coupling respectively. The dynamics for each of the magnetic sub-lattices have been plotted in Fig. 1. MCo crosses zero in < 1 ps while MGd is still in its demagnetized state and thereby forming a transient ferromagnetic state which is a necessary condition for HI-AOS [4-5]. Co/Pt remains demagnetized and its switching depends on the strength and sign of JRKKY. The white and dark region in Fig. 2 depicts relaxed and randomly demagnetized state of the material. The light green denotes switching of both the Co/Pt and Gd23Co77, while the dark green region denotes switching of only Gd23Co77. We found that the switching time of Co/Pt significantly depends on the composition of the GdCo alloy, absorbed power and the strength of JRKKY. Our simulation qualitatively explains the switching of Co/Pt in < 10 ps as experimentally observed by Gorchon et al. [6] and establishes the role of indirect exchange coupling in such switching.References: 1. I. Radu, K. Vahaplar, C. Stamm et al., Nature 472, 205 (2011) 2. A. Ceballos, A. Pattabi, A. El-Ghazaly et al., arXiv:1911.09803 3. U. Atxitia, P. Nieves and O. Chubykalo-Fesenko, Phys. Rev. B 86, 104414 (2012) 4. F. Jakobs, T. Ostler, C. H. Lambert et al., arXiv:2004.14844 5. M. Beens, M. L. M. Lalieu, A. J. M. Deenen et al., Phys. Rev. B. 100, 220409(R) (2019) 6. J. Gorchon, C. H. Lambert, Y. Yang, et al., Appl. Phys. Lett. 111, 042401 (2017)

A simulation study of RKKY exchange mediated ultrafast all-optical switching of GdCo/[Co/Pt] heterostructures

The relaxation of the magnetisation in magnetic materials is of great importance from both an applied and fundamental point of view. Conventional modelling of magnetisation dynamics employs the Landau-Lifshitz-Gilbert (LLG) equation[1] where the damping is included phenomenologically by a term that describes the coupling of the magnetic modes (given primarily by the atomic spin) to the non-magnetic modes (lattice vibrations and electron orbits) which are assumed to be in equilibrium. Recent studies of the dynamics induced by THz laser pulses has highlighted the necessity of understanding magnetisation relaxation beyond this assumption. In reality the spin and lattice dynamics mutually influence one another, hence it is necessary to employ a unified model of molecular and spin dynamics [2], Spin-Lattice dynamics (SLD). The transfer of energy and angular momentum between the lattice and the spin system is realised by the pseudo-dipolar coupling [3], which arises from the spin-orbit interaction and can be parameterised by magneto-elastic experiments. Our results show that the equilibrium magnetisation is independent of the thermostat used and by coupling the spin system only to the lattice vibrations the magnetisation temperature dependence can be reproduced without the need of a phenomenological spin damping. The autocorrelation of the coupling field (which acts as an induced noise on the spin systems) shows well defined excitation peaks at nonzero frequencies corresponding to both phonon and magnon modes, proving a successful coupling between the two subsystems (Fig.1). The magnon-phonon damping is evaluated within this framework (Fig. 2) and agrees well with the damping measured in magnetic insulators where it depends more on magnon-phonon coupling than electronic effects. Our model opens the possibility to describe the distinct dynamics of spins and phonons, necessary for the understanding of ultrafast magnetisation dynamics experiments.References: [1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [2] P.-W. Ma, C. Woo, and S. Dudarev, “Large-scale simulation of the spin-lattice dynamics in ferromagnetic iron,” Phys. Rev. B, vol. 78, no. 2, p. 024434, Jul. 2008. [3] Aßmann, Matthias, and Ulrich Nowak. "Spin-lattice relaxation beyond Gilbert damping." Journal of Magnetism and Magnetic Materials 469 (2019): 217-223.

Advanced Atomistic Modelling of Magnetisation Dynamics with Spin and Lattice Degrees of Freedom

The recently proposed probabilistic spin logic (PSL) offers promising solutions to novel computing applications[1], including some that have previously been covered by quantum computing. The PSL’s basic element is probabilistic bit (p-bit). A p-bit outputs random signals continuously while the mean of the output signal is input-dependent. When multiple p-bits are coupled through a network, a PSL is built. So far, several task implementations of PSL, including invertible logic gate, have been simulated numerically. Here, we report an experimental demonstration of a magnetic tunnel junction (MTJ) based hardware implementation of PSL[2]. Differently from the original p-bit proposals, we propose using two biasing methods, magnetic field and voltage, on the MTJ to excite and fine tune the random fluctuation of MTJ[3]. Therefore, our proposed hardware implementation of p-bit carries the benefit of extra tunability of random signal properties, which are eventually used to compensate the variations of mean and rate of random signals due to the intrinsic device-to-device variations of MTJs. As a result, multiple p-bits are tuned to exhibit similar properties even they are built with highly varied MTJs. Another benefit of the proposed p-bit design is that the similar principle could work for not only thermally-unstable MTJs but also thermally-stable MTJs, which are matured for STT-MRAM technologies and are more ready for integration. Then three p-bits are connected by a coupling network so that a PSL of an invertible logic gate is experimentally demonstrated (Fig. 1). As shown in Fig. 2, the invertible AND gate hops between all possible combinations under the AND logic constraint, C=AB. Along with more results from the demonstration, the PSL of an invertible AND logic gate is shown to function as intended. And thanks to the proposed p-bit hardware design that offers high tunability, the entire PSL system does not suffer from the slowest of the system, and shows much greater performance and potential for further scaling.References: [1] K. Y. Camsari, B. M. Sutton, and S. Datta, “P-bits for probabilistic spin logic,” Applied Physics Reviews, vol. 6, no. 1. AIP Publishing LLC, p. 011305, 11-Mar-2019. [2] Y. Lv, R. P. Bloom, and J. P. Wang, “Experimental Demonstration of Probabilistic Spin Logic by Magnetic Tunnel Junctions,” IEEE Magn. Lett., vol. 10, 2019. [3] B. R. Zink, Y. Lv, and J.-P. Wang, “Independent Control of Antiparallel- and Parallel-State Thermal Stability Factors in Magnetic Tunnel Junctions for Telegraphic Signals With Two Degrees of Tunability,” IEEE Trans. Electron Devices, vol. 66, no. 12, pp. 5353–5359, Dec. 2019.

Experimental Demonstration of Probabilistic Spin Logic by Magnetic Tunnel Junctions

In the development of dedicated hardware for neuromorphic applications, the realization of suitable memristive devices as synaptic elements is a key factor. The main drawback of the existing devices is the reliability of the intermediate states and their scalability. In this study we propose a memristive device based on magnetic tunnel junctions (MTJ) and whose functionalities do not depend directly on the size of the device. The state of the memristor relates to the tunnel magnetoresistance (TMR) variation with the relative angle between the two in-plane magnetizations. The specificity of our concept consists in the integration of an isotropically coercive medium as magnetic free layer able to stabilize its magnetization along whatever in-plane direction. Here the current driven rotation of the magnetization is investigated on devices of 100nm of diameter. The spin transfer torque (STT) of the in-plane MTJ is considered together with the contribution of the additional perpendicularly magnetized polarizer. For a proper ratio of the two STT contributions, the magnetization switches from the parallel to the anti-parallel state following a circular out-of-plane trajectory as for the precessional switching [4]. We prove experimentally that by properly designing the voltage pulses we can discretely rotate and stabilize the magnetization along different in-plane directions, reaching an almost continuous range of intermediate resistances. The resistance variation turns out to be monotonous with the voltage polarity confirming the memristive behavior of the device under a train of pulses. A study on the thermal stability of the intermediate states shows a resistance variation below 2% for a 3min timescale. Moreover, the Joule effect, occurring during the application of pulses, adds a random contribution to the resistance jumps. The observed resistance noise is associated to the thermally activated switching of small magnetic areas of the samples, probably due to the granular nature of the sputtered samples. The larger the diameter, the lower the noise due to the increased number of grains.References: 1. Mansueto, M. et al. Phys. Rev. Appl. 12, 044029 (2019). 2. Mansueto, M. et al. 2020 IEEE International Memory Workshop (IMW) 1–4 (2020). 3. Mansueto, M. et al. in preparation 4. Vaysset, A. et al. Appl. Phys. Lett. 98, 242511 (2011)

Spintronic memristor based on the angular variation of TMR

Reservoir computing (RC) [1] is a subset type of recurrent neural network, where only the weights of the output layer are altered during training. This training scheme does not require multi-layer backpropagation and gradient descent, making RC simple to train. This technique is therefore well suited for resource constrained hardware environments, since expensive circuitry for updating internal reservoir weights is unnecessary. Some spintronic devices can exhibit highly nonlinear behavior that is extremely well-suited for implementing RC in hardware. We have proposed a novel reservoir comprising a planar arrangement of nanomagnets each having perpendicular magnetic anisotropy (PMA) [2]. Our solution differs from [3], which uses voltage pulses to move information within the reservoir, whereas ours uses the natural relaxation of the nanomagnets. Fig. 1(a) shows an example layout of nanomagnets with two input nanomagnets (A and B). The effect of nanomagnet magnetic fields upon adjacent nanomagnets exhibits two features: non-linear interaction and variable interaction strength, making the proposed implementation well-suited for RC. Information is input by stimulating individual nanomagnets or portions of the nanomagnet reservoir with external magnetic fields. The magnetizations of various nanomagnets are read electrically via magnetic tunnel junctions. A complementary single layer circuit is used to perform vector-matrix multiplication on the magnetization values and the trained output weights to obtain the output vector. Micromagnetic simulations are performed with MuMax3 for the nanomagnet network pictured in Fig. 1(a). The nanomagnet reservoir was simulated with an input stream comprising triangle or square waves. Fig. 1(b-d) shows various snapshots of the network during the simulation; colors indicate magnetization directions. Fig. 1(e) shows the reservoir inputs and outputs, and Fig. 1(f) tracks the magnetization over time for three nanomagnets. The reservoir successfully identified the waveforms with 100% accuracy for both the training and testing data.References: [1] H. Jaeger, Bonn, Germany: German National Research Center for Information Technology GMD Technical Report., Vol. 148, p.13 (2001) [2] P. Zhou, N. McDonald, A. Edwards, L. Loomis, C. Thiem, and J. Friedman. arXiv:cs.NE/2003.10948 (2020) [3] H. Nomura, T. Furuta, K. Tsujimoto, Y. Kuwabiraki, F. Peper, E. Tamura, S. Miwa, M. Goto, R. Nakatani, and Y. Suzuki. 2019, Japanese Journal of Applied Physics., Vol. 58, p.070901 (2019)

Reservoir Computing with Planar Nanomagnet Arrays

We experimentally demonstrate the application of superparamagnetic magnetic tunnel junctions (MTJs) in stochastic logic operations. The two resistance states of an MTJ are used as logical 1 and 0. When the energy barrier to magnetization reversal is low compared to the thermal energy, the MTJ switches between the 1 and 0 states stochastically. The MTJ resistance is instantaneously digital yet analog in time average. Spin transfer torque (STT) enables electronic control of the time averaged resistance [1]. Three 60 x 90 nm CoFeB MTJs were connected to the logical circuit, and their resistances were measured simultaneously (Figure 1(a)). The probabilities P of the eight possible microstates (x1,x2,y) were analyzed pairwise. We denote by (0,0) the state 000 that is consistent with the truth table of an AND gate together with 001 that is not. The preference for 001 relative to 000, C(x1=0,x2=0), was defined as the normalized difference between the probabilities P(001) and P(000). After connecting feedback, when, P(000) will increase and P(001) will become smaller, effectively decreasing C(0,0). The condition will be the same for C(0,1) and C(1,0) but not C(1,1), since P(111) increases but P(110) decreases. Figure 1(b) shows experimental results with the feedback connected, showing a statistical preference for valid AND gate states. Invertibility is based on what happens when the output is pinned to a particular value. Pinning y to 1 increases the probabilities for all the states with y = 1 (Figure 1(c)), while pinning y to 0 increases the probability for all the states with y = 0 (Figure 1(d)). This is qualitatively consistent with the energies of the different states.References: Bapna, M., & Majetich, S. A. (2017). Current control of time-averaged magnetization in superparamagnetic tunnel junctions. Applied Physics Letters, 111(24), 243107.

Superparamagnetic Tunnel Junctions for Invertible Stochastic Logic Gates

Interacting nanomagnets with low thermal stability have been proposed for probabilistic computing, which has potential uses in low power sensing and logic, as well as in encryption and decryption [1]. Superparamagnetic tunnel junctions (SP-MTJs) are an ideal type of nanomagnet because their time-averaged magnetization of the free layer can be programmed with a voltage or current (Figure 1). We have previously used SP-MTJs for random number generation (RNG), and analog multiplication [2]. While voltage-controlled magnetic anisotropy (VCMA) has advantages for RNG due to ultra-low power consumption, spin transfer torque (STT) or spin orbit torque (SOT) is superior for logic devices due to the wider tuning range. For small sizes, magnetic tunnel junctions can be superparamagnetic near zero bias voltage, and the average retention time can be as short as a few hundred nanoseconds, so GHz speeds are feasible [3]. Pairs of superparamagnetic tunnel junctions fluctuate independently when isolated (Figure 2, top), but when coupled, the output of one is the NOT of the other (Figure 2, bottom). For AND and NAND gates the coupling between three SP-MTJs is optimized by Boltzmann machine simulations to statistically favor the set of microstates consistent with the truth table. Experimental results show that fast operation can be achieved with a hybrid SP-MTJ – CMOS circuit and without the need for a separate microprocessor. Prospects for optimizing the speed and performance of these devices will be discussed, and benchmarked with non-magnetic stochastic computing technologies.References: 1. K. Y. Camsari, et al., Phys. Rev. X 7, 031014 (2017). 2. M. Bapna and Sara A. Majetich, Appl. Phys. Lett. 111, 243107 (2017). 3. B. Parks, A. Abdelgawad, T. Wong, R.F. L. Evans, and S. A. Majetich, Phys. Rev. Applied 13, 014163 (2020).

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