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In the last decade, two revolutionary concepts emerged in nano-magnetism from research for advanced information processing and storage technologies. The first suggests the use of magnetic domain walls (DWs) in ferromagnetic nanowires to permanently store information in DW racetrack memories. The second proposes a hardware realisation of neuromorphic computing in nanomagnets using nonlinear magnetic oscillations in the GHz range. Both ideas originate from the transfer of angular momentum from conduction electrons to localised spins in ferromagnets, either to push data encoded in DWs along nanowires or to sustain magnetic oscillations in artificial neurones. Even though both concepts share a common ground, they live on very different time scales which rendered them incompatible so far. Here, we bridge both ideas by demonstrating the excitation of magnetic auto-oscillations inside nano-scale DWs [1] using pure spin currents originating from the spin-Hall effect. While the auto-oscillations are in a domain wall are shown for a bilayer from a heavy metal (HM) and a ferromagnet (FM) we also demonstrated the realization of a bipolar spin-Hall Nano-oscillator using a FM/HM/FM trilayer system [2].References: [1] N. Sato, K. Schultheiss, L. Körber, N. Puwenberg, T. Mühl, A. A. Awad, S. S. P. K. Arekapudi, O. Hellwig, J. Fassbender, and H. Schultheiss, Phys. Rev. Lett. 123, 057204 (2019). [2] T. Hache, Y. Li, T. Weinhold, B. Scheumann, F. J. T. Gonçalves, O. Hellwig, J. Fassbender, and H. Schultheiss, Appl. Phys. Lett. 116, 192405 (2020).

MMM 2020

Domain wall based spin-Hall nano-oscillators INVITED

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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technical paper

The aim of ab-initio spin dynamics is to go beyond the limitations of standard atomistic spin dynamics, which is based on a mapping of the original electronic Hamiltonian to an effective spin Hamiltonian. This spin Hamiltonian assumes constant magnetic moment lengths and constant interaction parameters and is therefore only valid in the magnetic ground-state configuration. Recent studies [1] have shown that non-collinearity in the magnetic state reduces the spin-spin interaction and gives improved accuracy compared to experiment. However, the proposed method in Ref. [1] is not applicable for the general case and reveals limitations of the above mentioned mapping [2]. In ab-initio spin dynamics the magnetic moment lengths and the quasi-classical torques acting on the atomic magnetic moments are calculated from a quasi-equilibrium electronic ground state within the adiabatic approximation [3], where constraining fields are required to stabilize non-collinear magnetic moment configurations [4]. The constraining field has therefore to cancel the internal effective magnetic field, which is expected to be minus the gradient of the energy with respect to the magnetization. In practical calculations the constraints on the moment directions are often only implemented approximately without constraining fields, which can introduce serious errors [5]. We present surprising numerical results from constrained density functional (cDFT) calculations which demonstrate that the constraining field and the effective field obtained from the energy gradient are not exactly opposite, as would be expected [4]. Furthermore, we provide a theoretical explanation of this discrepancy based on a theorem that relates the constraining field and the energy gradient: the constraining field theorem. We argue that in the case of cDFT the correct effective magnetic field is actually given by the energy gradient and not by the constraining field.References: [1] Szilva et al., Phys. Rev. Lett. 111, 127204 (2013). [2] Szilva et al., Phys. Rev. B 96, 144413 (2017). [3] Antropov et al., Phys. Rev. Lett. 75, 729 (1995). [4] Stocks et al., Philos. Mag. B 78, 665 (1998). [5] Singer et al., Phys. Rev. B 71, 214435 (2005).

Discrepancy Between Constraining and Effective Fields in Ab-Initio Spin Dynamics

Ferromagnetic materials are used in a wide range of electromagnetic applications (energy converters, sensors, inductances …). In this domain, we observed a growing interest in the development of simulation tools reducing the experimental campaigns and improving the knowledge and the performances. Accurate simulation results can only be obtained by coupling precise electromagnetic equations to the exact material laws (hysteresis, saturation, frequency dependence) [1]-[3]. Under the influence of an external magnetic excitation, the local magnetic state through the ferromagnetic specimen is ruled by the combination of both the magnetic domain kinetics and the external magnetic field diffusion. The usual methods for the simulation of the magnetic behavior are all based on the separation of the magnetic contributions, where the microscopic Eddy currents due to the domain wall motions and the macroscopic ones due to the extern magnetic field variations are considered separately [4]. This separation remains artificial, since practically both losses mechanisms occurs simultaneously and interact on each other [5]. In this study, an alternative solution is proposed through the resolution of an anomalous fractional magnetic field diffusion (1, 2 or 3D depending on the experimental situation). The fractional order constitutes an additional degree of freedom in the simulation scheme [6]-[8]. It is identified through comparisons to experimental results. By adjusting precisely this order, very accurate local and global simulation results can be obtained on a very broad frequency bandwidth. It allows to predict precisely the dynamic magnetic behavior of classic ferromagnetic components.References: [1] B. Zhang et al., IEEE Transaction on Magnetics, vol. 54, iss. 11 (2018) [2] B. Zhang et al., IEEE Transaction on Magnetics, Vol. 54, iss. 3, (2017). [3] B. Gupta et al., IEEE Transaction on Magnetics, Vol. 54, Iss. 3, (2018). [4] G. Bertotti, IEEE Transaction on Magnetics, vol. 24, iss. 1, pp. 621-630, (1980). [5] M. A. Raulet et al., IEEE Transactions on Magnetics, Vol. 40, iss 2, pp. 872-875, (2004). [6] B. Ducharne et al., International journal of dynamics and control, vol. 6, pp. 89-96, (2018). [7] B. Ducharne et al., Journal of Applied Physics, vol. 117, iss. 24, (2015). [8] B. Ducharne et al., The European Physical Journal Plus, 135:325, (2020).

Fractional equation for the anomalous magnetic diffusion in ferromagnetic materials

Recently we studied magnetic inhomogeneity effects in magnetocaloric properties of a 7.6 nm La0.7Sr0.3MnO3 (LSMO) thin film to understand the nature of magnetic phases transitions1. We proved this sample has two main magnetic phase transitions: superparamagnetic ferromagnetic and ferromagnetic-paramagnetic phases transitions, at T = 240 K (= TB) and T = 270 K (= TC), respectively. Here we want to understand the spin dynamics and relaxation in this sample. Analysis of the results from our investigations of the temperature dependence of ac susceptibilities (χ/ and χ//) and magnetic viscosity in a 7.6 nm thin film of La0.7Sr0.3MnO3 grown on SrTiO3 (001) substrate are presented. The Curie temperature (TC) of this film is magnetic field (H) dependent varying as: [TC(H)-TC(0)] ~ Hn, n ~ 0.15. Temperature dependence of χ/ and χ// measured at ac frequencies fm in the range of 0.1 to 10 kHz shows a broad peak near 230 K associated with the blocking temperature of spin clusters present in the 1.4 nm surface layer and a frequency dependent peak near 270 K associated with TC (Fig. 1). The mean relaxation time τ determined from the Cole-Cole analysis of and is shown to fit the Vogel-Fulcher law: τ = τ0 exp[ΔE / kB(T-T0)] , with TB = 245 K, ΔE / kB = 270 K = TC, and τ = 1.2 10-9 (s) (Fig. 2). Magnetic viscosity S (measured in H = 0 for a sample cooled in H = 50 Oe) determined from the fit to the magnetization: M (t) = M (0) – S Ln t, with time t up to 2 hours, shows a peak at 230 K above which the M (2 hrs) switches to negative values for temperatures up to TC(0). It is argued that this negative magnetization results from interaction between the superparamagnetic like spin clusters and the ferromagnetic phase.References: 1 N. Mottaghi, R.B. Trappen, S.Y. Sarraf, M.S. Seehra, and M.B. Holcomb, J. Alloys Compd. 154200 (2020).

Spin dynamics and relaxation in thin film La0.7Sr0.3MnO3/SrTiO3: ac magnetic susceptibility and magnetic viscosity investigations.

Iron oxide nanoparticles (NP) are under investigation for many biomedical applications [1], including magnetic hyperthermia. The heating efficiency of magnetic hyperthermia depends on how well the single excitation frequency matches the Néel and Brownian relaxation frequencies of the NPs. We directly measured the magnetostatic interaction induced frequency distribution with a custom-built AC susceptometer [2]. Our results show that both the peak and amplitude of this distribution can be tuned by addition of a small DC field, thus enabling a better match to the AC excitation field frequency. We measured the AC magnetic susceptibility (ACS) of 10.7±1.2 nm Fe3O4 NPs dispersed in eicosane (0.15 vol%). The ZFC-FC curve shows a blocking temperature (TB) of ~ 105 K, and the greatest response of the real (χ’) and imaginary (χ”) parts of the ACS was found between 110 and 150 K. Fig. 1 shows the imaginary component of the ACS as a function of frequency, for HDC = 0, at different temperatures. The imaginary part shows a very broad peak at 110K. At higher temperatures the dispersive peak sharpens and the peak shifts to higher frequencies. Both the breadth of the peak and its shift with temperature show that magnetostatic interactions affect high frequency response even for dilute samples. We extended our ACS measurement at 140 K with additional DC fields, ranging from 0 to 4mT. Fig.2 shows the non-monotonical trend of the imaginary part of the ACS at various DC magnetic fields. Below 1.25 mT the peak frequency decreases and the amplitude increases with increasing DC fields. At and above 1.25 mT these trends reverse. The dissipation at 100 kHz increases by a factor of 3.8 between 0 and 1.25 mT. The peak DC field is comparable to the average magnetostatic interaction field. Applying the DC field overcomes part of the spin freezing and enables a larger fraction of the spins to contribute to the dissipation. This methodology can be extended to polydisperse NPs at higher concentrations at body temperature.References: [1] Q.A. Pankhurst, et al., J. Phys. D Appl. Phys. 36, R167 (2003). [2] E. Riordan, et al., Review of Scientific Instruments 90, 073908 (2019).

Tuning the Dynamics in Fe3O4 Nanoparticles for Hyperthermia Optimization

Antiferromagnetic (AFM) spin Hall oscillators driven by a sub-threshold spin current produce ultra-short (~ 5 ps) spikes in response to a weak external stimulus and can be used as ultra-fast artificial neurons [1, 2]. One of specific features of AFM oscillators is the effective inertia that originates from exchange coupling between two AFM magnetic sublattices and manifests itself, for instance, in hysteretic behavior of the threshold generation current [3]. Here, we theoretically investigate influence of the exchange inertia on the dynamics of AFM-based artificial neural networks. The exchange inertia leads to delay of the spike generated by an AFM neuron relative to the input stimulus signal. The delay time depends on the coupling strength between AFM neurons and can vary from ~ 10 ps to several 100s ps. This effect can be used to create a tunable periodic spike generator by coupling several AFM neurons in a ring structure, in which output of one neuron serves as an input for the next neuron. The exchange inertia can also be used to create an “inhibitor” neuromorphic circuit, in which propagation of a spike is suppressed in the presence of another (“inhibiting”) signal. The inhibitor works by splitting the signal path in two branches and controlling the delay time in one branch by the inhibiting signal. In the presence of the inhibiting signal, the spikes in two branches arrive at the output neuron at different times and do not induce an output spike. It should be noted, that such “inhibitor” circuit is impossible to realize with inertia-less integrate-and-fire neurons. Combining the inhibitor circuit with a ring structure allows one to realize a controllable neuromorphic memory loop (see example simulations in Fig. 1). These and other examples demonstrate that exchange inertia effects can be utilised to achieve new functionalities of AFM-based artificial neural networks.References: [1] R. Khymyn, I. Lisenkov, J. Voorheis, et al., Sci. Rep. 8, 15727 (2018). [2] O. Sulymenko, O. Prokopenko, I. Lisenkov, et al., J. Appl. Phys. 124, 152115 (2018). [3] R. Khymyn, I. Lisenkov, V. Tiberkevich, et al., Sci. Rep. 7, 43705 (2017).

Dynamic effects of exchange inertia in artificial neural networks based on antiferromagnetic oscillators

Memristors – two-terminal circuit elements whose resistance is ideally proportional to the integral of input over time – enable the implementation of synapses in artificial neural networks. However, “ideal” memristive functionality has not been yet achieved. We experimentally demonstrate the possibility to implement almost ideal memristors using thin-film ferromagnet/antiferromagnet (F/AF) bilayers, where frustrated exchange interaction at the F/AF interface results in the correlated spin liquid state of the AF characterized by viscous magnetization dynamics [1]. We utilize a bilayer formed by a 10 nm-thick F Permalloy (Py) and an 8 nm-thick AF NiO film. External magnetic field H serves as the driving input in our demonstration. The angle of magnetization rotation is proportional to the integral of the field over time, which is converted into resistance variations by anisotropic magnetoresistance (AMR). The resulting magnetoelectronic hysteresis loop is consistent with that expected for an ideal memristor. Analysis of the time-domain response shows deviations from ideal behaviors, which are associated with the inhomogeneity of viscous dynamical properties. We also discuss how the demonstrated memristive functionality can be achieved with current-driven nanostructures by utilizing spin torque [2]. The demonstrated memristors are amenable to downscaling and should exhibit high endurance, making them particularly attractive for neuromorphic applications.References: 1. S. Urazhdin, W. Li and L. Novozhilova, JMMM, Vol. 476, pp.75-85 (2019) 2. G.X. Chen, S. Ivanov, and S. Urazhdin, arXiv:2006.07996 (2020)

Memristive Functionality of a Thin-film Ferromagnet/Antiferromagnet Bilayer

Memristors are circuit elements whose resistance is ideally proportional to the total charge that passes through them, which can provide synaptic functionality in artificial neural networks. Many implementations of generalized memristors – devices whose resistance depends on their electronic history – have been explored, but ideal memristor functionality has not yet been achieved. We utilize analytical arguments and numerical simulations to show that ideal memristors can be realized using spin transfer torque (STT)-driven viscous magnetization dynamics. The latter can be achieved in the spin liquid state of thin-film F/AF heterostructures resulting from frustrated exchange interaction [1]. We show that the magnetization dynamics can be efficiently driven by STT, despite large damping associated with viscous dynamics. For a magnetic system with negligible in-plane anisotropy and large damping, magnetization response to current pulses with varied duration Δt and amplitude I0 depends only on the total pulse charge Q [Fig.1], consistent with ideal memristive behaviors. Additionally, we show that current-induced Joule heating can serve as an extra state control variable, resulting in the dependence of current-driven resistance variation on the relative timing of the current pulses [Fig.2]. This additional second-order memristive functionality [2] provides natural implementation for the spike-timing dependent plasticity in spintronic neuromorphic network with non-overlapping spikes. Our work thus opens a new route for the development of advanced spintronic neuromorphic hardware.References: [1] Urazhdin, Sergei, Weijie Li, and Lydia Novozhilova. "Magnetic freezing transition in a CoO/Permalloy bilayer revealed by transverse ac susceptibility." Journal of Magnetism and Magnetic Materials 476 (2019): 75-85. [2] Kim, Sungho, et al. "Experimental demonstration of a second-order memristor and its ability to biorealistically implement synaptic plasticity." Nano letters 15.3 (2015): 2203-2211.

Ideal memristor based on viscous magnetization dynamics driven by spin torque

Spin Torque Nano-Oscillators are magnetic devices that use tunnel magnetoresistance to emit microwaves. They hold great promise for neuromorphic computing [1], as we can use their non-linear dynamic to emulate neurons and they can be integrated by millions in CMOS chips. In previous works, it has been shown that their transient or coupled dynamics can be used to perform pattern recognition [2], [3]. In order to build deep neural networks [4] it is now crucial to find a way to interconnect different layers of neurons with artificial tunable synapses. Here we present an architecture using spintronic resonators to weight the microwave encoded signals of artificial RF spintronic nano-neurons. These synaptic resonators leverage the spin-diode effect [5,6]. The resistance of the devices oscillates at the same frequency as the signal they receive. The rectified voltages are proportional to the power of the microwaves and depend strongly on the frequency mismatch between the signal and the resonance frequency of the device. Hence to tune these resonance frequencies is analog to tune the synaptic weights. And because of the frequency selectivity of our resonators, each of them rectifies only the signal of its corresponding neuron. We use an analytical model and dynamical simulations to show that a chain of spintronic resonators wired in series, rectifying different microwave signals, is equivalent to the key operation of neural networks, called Multiply-And-Accumulate (MAC), despite the nonlinearity of these components. We also demonstrate experimentally with Magnetic Tunnel Junctions as resonators and several RF generators, the validity of this MAC operation. Finally, we simulate a full network with spintronic oscillators and resonators to test our architecture. We manage to achieve state-of-the-art recognition on a handwritten digit dataset. These are encouraging results for high density hardware neural network, as it is the first demonstration of a fully-spintronic radio-frequency neural network.References: [1] J. Grollier, D. Querlioz and M. D. Stiles, Proceedings of the IEEE, Vol. 104, p. 2024-2039 (2016) [2] J. Torrejon, M. Riou et al, Nature, Vol. 547, p. 428 (2017) [3] M. Romera, P. Talatchian, S. Tsunegi et al, Nature, Vol. 563 (7730), p. 230-234 (2018) [4] Yann LeCun et al, Nature, Vol. 521, p. 436-44 (2015) [5] C. Wang, Y.-T. Cui et al, Phys. Rev. B Vol. 79, p. 224416 (2009) [6] D. Marković, N. Leroux, A. Mizrahi, et al, Phys. Rev. Appl, Vol. 13 (4), p. 044050 (2020)

Radio-frequency synapses for neural networks made of spin-torque nano-oscillators

Coherent manipulation of quantum information is pivotal for real quantum computing applications. Promising approaches employ hybrid quantum systems based on coupled superconducting qubits and resonators and magnetic elements [1-6], where strong coupling rates (exceeding 100 MHz) are easily reached. An advantage of magnetic elements is that their resonance frequencies are tunable in a wide range by a bias magnetic field. However, all previous studies in this area focus on quasi-static systems where magnetic resonance frequency is constant during the signal lifetime. Here, we propose a new method of coherent information manipulation in hybrid magnetic/superconducting systems based on dynamic control of the magnetic element at timescales comparable or less than characteristic coherence times. This method opens new possibilities for quantum information transduction, processing, and entanglement generation, that are not possible to realize in static hybrid systems. Fig. 1 shows a simple example of how to coherently transfer information from one superconducting resonator P1 to another P2 using intermediate magnetic resonator Pm, where the resonance frequency is rapidly swept across the resonance frequencies of resonators P1,2 with ramp rate ρ = dωm/dt = 2π×2.5 MHz/ns. As one can see, such “passage” results in almost complete information transfer to the P2 resonator. Notably, the population of the magnetic resonator Pm stays low (< 0.2) during the whole process (see bottom of Fig. 1), which explains the weak sensitivity of this process to magnetic damping. Dynamically changing the resonant magnon frequency can be used for information transfer between disparate systems, creating quantum gates, and realizing quantum data exchange between two elements. This dynamic frequency change may perform information processing and can be used to study fundamental properties of magnons, such as if we do 50% information transfer, thereby generating magnon-mediated quantum entanglement.References: [1] H. Huebl, C. W. Zollitsch, J. Lotze, et al., Phys. Rev. Lett., vol. 111, no. 127003, 2013. [2] Y. Tabuchi, S. Ishino, T. Ishikawa, et al., Phys. Rev. Lett, vol. 113, no. 083603, 2014. [3] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett., vol. 113, no. 156401, 2014. [4] Y. Tabuchi, S. Ishino, A. Noguchi, et al., Science, vol. 349, no. 6246, pp. 405-408, 2015. [5] Y. Li, T. Polakovic, Y.-L. Wang, et al., Phys. Rev. Lett., vol. 123, no. 107701, 2019. [6] J. Hou and L. Liu, Phys. Rev. Lett., vol. 123, no. 107702, 2019.

Magnon Mediated Coherent Information Manipulation and Processing INVITED

Mutually synchronized spin-torque nano-oscillators (STNOs) are one of the promising platforms for bioinspired computing and microwave signal generation [1]. Using STNOs one can achieve 90% recognition rate in spoken vowels [2]. However, in order to do more complex tasks, larger-scale synchronized oscillators are needed, something that is not easily done with the STNOs demonstrated so far. In my talk, I will describe a different type of spin current-driven device called spin Hall nano-oscillators (SHNOs), which can generate microwave frequencies over a very wide frequency range [3]. The SHNOs are based on 50 – 120 nm wide nano-constrictions in Pt(5)/Hf(0.5)/NiFe(3) trilayers (all numbers in nm). When multiple nano-constrictions are fabricated close to each other (300 – 1200 nm separation) they can mutually synchronize and chains of up to nine nano-constrictions have been demonstrated to exhibit complete synchronization [4]. For the first time, we can now also synchronize two-dimensional SHNO arrays with as many as 8 x 8 = 64 SHNOs [5]. The mutual synchronization is observed both electrically and using scanning micro-BLS microscopy. Both the output power and linewidth of the microwave signal improves substantially with increasing number of mutually synchronized SHNOs, such that quality factors of about 170,000 can be reached. Following the approach of Romera et al, we also demonstrate neuromorphic computing using a 4 x 4 SHNO array with two injected microwave signals as inputs. Given their high operating frequency (~10 GHz), easy fabrication, and highly robust synchronization properties, nano-constriction SHNO arrays are likely the most promising candidates for neuromorphic computing based on oscillator networks.References: [1] Torrejon J., et al. Neuromorphic computing with nanoscale spintronic oscillators. Nature 547, 428 (2017) [2] Torrejon J., et al. Vowel recognition with four coupled spin-torque nano-oscillators. Nature 563, 230 (2018) [3] Chen T., et al. Spin-torque and spin-Hall nano-oscillators. Proc. IEEE 104, 1919 (2016) [4] Awad A. A. et al. Long-range mutual synchronization of spin Hall nano-oscillators. Nature Physics 13, 292 (2017) [5] Zahedinejad M., et al. Two-dimensional mutually synchronized spin Hall nano-oscillator arrays for neuromorphic computing. Nature Nanotechnology 15, 47 (2020)

Domain wall based spin-Hall nano-oscillators INVITED

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