Recently, reservoir computing, originally developed from a Recurrent Neural Network (RNN) based framework [1,2], has gained significant attention due to its faster and simpler processing of sequential or temporal data. In RC, the inputs are mapped into a high-dimensional space in the reservoir as shown in figure 1(a). Then, a one-dimensional readout is trained to extract the features of the high-dimensional pattern for classification. The primary advantage of RC is that, unlike an RNN, RC does not require training of the large complex reservoir; only training the output weights are sufficient for its successful operation. This straightforward and comparatively faster training approach has significantly reduced learning-cost compared to standard RNNs [3] which makes RC amenable to implement in power and hardware resource restrained edge devices that can learn in-situ [4].
Since the basic principle of a reservoir is to transform temporal/sequential inputs into a high-dimensional space and extract the features of the input by simple algorithms such as linear regression, any nonlinear dynamical system that exhibits a complex dynamic response to inputs can be utilized as a potential reservoir. Researchers have proposed various physical systems for implementing physical RC, including optical system [5,6], memristors [7], and spintronic oscillators [8].
One such system is based on skyrmions. Skyrmions are topologically protected, localized, particle-like spin structures [9,10,11,12,13,14]. The requirement of very small depinning current compared to domain walls in racetrack memory devices [11,15,16,17] and their scalability have made skyrmions a potential candidate for high density and low power spintronic devices [18,19,20]. For example, our group has shown the feasibility of voltage control of skyrmions [21] and that skyrmion mediated voltage-controlled memory devices [22] can be scaled aggressively [23,24]. Similarly, skyrmion can be a viable option for low-power, small-scale reservoir devices [1,25]. In a skyrmion based PRC, the reservoir layer of RC is replaced by a skyrmion hosting thin film as shown in figures 1(a)-(c). Recently, D. Pinna et al. showed with simulations that random magnetic texture can be used as an ideal candidate for reservoir computing [26]. This was able to achieve simple temporal pattern recognition tasks by exploiting complex time varying resistance due to dynamics of skyrmions in response to an input AC voltage pulse. However, many aspects of skyrmion based PRC are unknown and yet to be explored for its successful implementation. In this paper, we utilize the nonlinear magnetization dynamics of skyrmions in response to the variation of perpendicular magnetic anisotropy (PMA) [27,28] for physical RC. We applied voltage induced strain to manipulate the PMA of a thin film which in effect induces breathing of skyrmions and showed that an energy-efficient straintronic skyrmion based PRC can perform simple temporal pattern recognition task with 100% accuracy in the presence of room-temperature thermal perturbation. In order to determine whether the skyrmion-based PRC can be applied to long-term prediction, we also tested it using Mackey-Glass time series. Additionally, in the proposed configuration, the state of the skyrmion based reservoir computer block is amenable to reading out with MTJs with enhanced sensitivity and reproducible indicating that the reservoir block can perform recognition task for multiple cycles. Furthermore, in this work, we use thin film hosting single/multiple skyrmions to study the effect of coupling through dipole interaction and spin-wave on computing metrics (STM and PC) and performing simple temporal pattern recognition tasks.
In section 2, the reservoir model and method for providing an input to the reservoir block in the form of voltage generated strain as well as reading the output using MTJs is described. In section 3, the STM and PC capacity of the reservoir is investigated to assess the impact of spin-wave and dipole interaction on these metrics, thus studying the connection between the physics of coupling between these skyrmions and their computational performance. Section 4 discusses the performance of reservoir computer in the presence of room-temperature thermal noise, reproducibility of the output states from the reservoir, method of enhancing sensitivity, and prediction of chaotic time-series with skyrmion based PRC. Section 5 highlights the amount of energy dissipated for each input pulse while performing a simple pattern recognition task. Section 6 concludes with a discussion on the future direction for this work.
Our proposed device is shown in figure 1(d), which is made up of two main components: a skyrmion based MTJ stack [29,30,31] and a piezoelectric substrate. The MTJ stack consists of a fixed layer, a tunnel barrier and a thin film hosting single/multiple skyrmions, which acts as a free layer. We consider heavy metal (HM)/CoFeB/MgO/CoFeB as the MTJ stack materials where the interface of HM/CoFeB and CoFeB/MgO gives rise to Dzyaloshinskii-Moriya interaction (DMI) and PMA respectively, which are two essential parameters for the formation and stabilization of magnetic skyrmions [32]. Four patterned electrodes on the piezoelectric substrate are positioned on four sides of the MTJ stack as shown in figure 1(d) to provide a voltage-generated localized strain large enough to control the thin film's magnetic anisotropy. When PMA is modulated by applying a voltage pulse between the electrodes on top and bottom of the piezoelectric material, skyrmion breathing is induced. The skyrmion core size varies due to the breathing of skyrmions which changes the magnetoresistance across the MTJ stack. This change in magnetoresistance is read out at regular interval by applying a read voltage pulse across the MTJ as shown in figures 1(d)-(e), which are then used for reservoir computing.
In this study, we considered ferromagnetic thin film of square geometry with 1000 nm, 1500 nm, and 2000 nm lateral dimensions for the implementation of the reservoir block. To demonstrate the effect of spinwave interaction and dipole coupling on the performance of skyrmion based PRC, we formed two types of thin films: continuous (unconfined) and discontinuous (confined), which is shown in figures 2(a)-(c). In continuous type of thin film, the breathing skyrmions can interact with each other through spin-wave and dipolar interaction whereas in discontinuous thin films, skyrmions can interact through dipolar interaction only since each of them is confined in a block of ~500nm×500nm dimension and separated by 30 nm gap (cells removed). Particularly, we placed single skyrmion in continuous/discontinuous thin film of all three lateral dimensions and four, nine, and sixteen skyrmions in thin film of 1000 nm, 1500 nm, and 2000 nm lateral dimension respectively. We note that continuous thin films are read as a single MTJ, while discontinuous thin films can also be read as a single MTJ with several blocks where the gap between the blocks in the free layer is filled with SiO2 as shown in figure 1(e). [33 , 34 ]. We observe that for each of the skyrmions the magnetization profile looks like a 360 0 Neel domain which verifies that the stabilized states are skyrmion states].
The magnetization dynamics of the breathing skyrmions induced by PMA modulation in the thin film is simulated by solving the Landau-Lifshitz-Gilbert (LLG) equation using the MUMAX3 simulation package [35 ]:
Where 𝛾 and 𝛼 represent the gyromagnetic ratio and the Gilbert damping coefficient respectively. 𝑚 ⃗⃗ stands for the normalized magnetization vector, which is found by normalizing the magnetization vector (𝑀 ⃗⃗ ) with respect to saturation magnetization (Ms). The thin films are discretized into cells with dimensions of 2nm×2nm×1nm, which are much shorter than the exchange length ( √ 2𝐴 𝑒𝑥 𝜇 0 𝑀 𝑆 2 ). In equation ( 1), 𝐵 ⃗ 𝑒𝑓𝑓 is the effective magnetic field, which includes the following components:
In equation ( 2), 𝐵 ⃗ 𝑑𝑒𝑚𝑎𝑔 and 𝐵 ⃗ 𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒 represent the effective field due to demagnetization energy and the Heisenberg exchange interaction respectively. 𝐵 ⃗ 𝐷𝑀 is the effective field due to Dzyaloshinskii-Moriya interaction which is defined as:
Where D is the DMI constant and mx, my, and mz are the components of unit magnetization vector 𝑚 ⃗⃗ along x, y, and z direction, respectively.
𝐵 ⃗ 𝑎𝑛𝑖𝑠 is the effective field due to the perpendicular anisotropy and expressed by the following equation:
Where 𝐾 𝑢1 is the first order uniaxial anisotropy constant with 𝑢 ⃗ a unit vector in the anisotropy direction.
By designing an electrode pair with a lateral dimension comparable to that of the thin film and a piezoelectric layer thickness equal to the lateral dimension of the thin film, a localized strain of sufficient magnitude to vary the magnetic anisotropy of the thin film can be created [36]. For an example, by applying 1 V between the bottom electrode and top electrodes of dimension 1200nm×1200nm each and taking 1000 nm thick piezoelectric Pb[Zr0.52Ti0.48]O3, sufficient strain can be produced to modulate the required amount of PMA energy density in a 1000nm×1000nm thin film. In the micromagnetic simulation, the stress effect is modeled by the modulating 𝐾 𝑢1 .
Fluctuating thermal field is introduced with the following equation:
where T is the temperature (K), ΔV is the cell volume, kB is the Boltzmann constant, Δt is time step and 𝜂 (step) is a random vector from a standard normal distribution which is independent (uncorrelated) for each of the three cartesian co-ordinates generated at every time step.
The following parameters for CoFeB are used for the simulation of magnetization dynamics of the skyrmions:
TABLE I. CoFeB material properties Saturation magnetization (Ms) 1.3×10 6 A/m [37] Exchange stiffness (Aex) 15×10 -12 J/m [37] DMI 0.65×10 -3 J/m 2 [32] Thickness 1 nm Damping coefficient 0.01 [38] PMA energy 1092.5 μJ/m 2 [39] PMA modulation (△PMA) 7.5×10 3 J/m 3
We analyzed the average magnetization in the z-direction of the thin films hosting single/multiple skyrmions generated in response to randomly distributed sine and square voltage pulses applied on the patterned electrodes and calculated the training and testing accuracy. Figure 3 shows an example of PMA energy density variation with input wave forms to be classified and the corresponding average magnetization response of the reservoir in the z-direction from a discontinuous thin film of 1000 nm lateral dimension hosting a single skyrmion. The frequency of the input voltage pulse is considered to be 50 MHz. We observed that for all of the cases presented in this study, the training and testing accuracy are 100%. However, performance of different configurations as a reservoir block is estimated and compared on the basis of their STM and PC capacity. Classification and STM/PC evaluation method is discussed in supplementary section 1.
From figure 4 we can see that a single skyrmion located at the center of discontinuous thin film shows the highest STM and PC capacity (CSTM=4.39 and CPC=4.62 for single skyrmion in discontinuous thin film of 1000 nm and 1500 nm lateral dimension respectively). To validate that a single skyrmion in a discontinuous thin film is the most effective and this is not the case for a set of optimized parameters, we varied the DMI and exchange stiffness and observed that the same trend is followed which has been discussed in supplementary section 2. Before explaining the reason for a single skyrmion confined in a discontinuous film performing better than multiple skyrmions in continuous/discontinuous films, we first visualize the influence of the output of a reservoir block to an input on the STM and PC capacity with an example case of single skyrmion confined in a 1000nm×1000nm thin film. Memory capacity implies that the current magnetization response should be influenced by past pulse and due to the nonlinearity, the magnetization response should be different for different combination of past and present pulses. For this purpose, we observe the average magnetization response of thin film in the z-direction for four different past-present pulse combinations [26]. Specifically, we investigate if the output of the present pulse is dependent on the past waveform by considering (a) Square-square, (b) Sine-sine, (c) Sine-square and (d) Square-sine combinations. peak whereas for sine pulse the peaks are smaller. However, depending on the past input, the magnetization response takes different shapes while retaining the dominant feature arising from the present input. The difference in magnetization response for different combinations and similarity for the same combination suggests that the single skyrmion confined in a thin film has memory and nonlinearity. We also estimated the STM and PC capacity for a single skyrmion hosted by a 1000nm×1000nm continuous thin film (CSTM=2.49, CPC=1.27) and observed that these capacities are significantly smaller compared to single skyrmion in discontinuous thin film (CSTM=4.39, CPC=2.98) of the same dimension. We note that though a single skyrmion can be stabilized in a 1000 nm×1000 nm continuous thin film but it cannot be stabilized in 1500nm×1500nm and 2000nm×2000nm thin film and therefore "single skyrmion in continuous thin film" case is not listed in figure 4. The difference in performance between the single skyrmion in a continuous and discontinuous thin film can be explained with figure 6. We see that for a single confined skyrmion, the magnetization response to sine and square wave inputs has visibly distinguishable features: sine and square wave inputs produce small and large peaks in average magnetization in the z-direction respectively (figure 6(a)). Particularly, the magnetization responses correspond to any of the four types of responses shown in figure 5. On the other hand, for a single skyrmion in a continuous thin film, these features are not visibly as identifiable as for the confined skyrmion (see figure 6(b) which is juxtaposed with figure 6(a)). We take two instances of responses for square input and observe that the confined skyrmion is forced to breathe as a nearly circular skyrmion (figure 6(c)). Also, magnetization states visited while responding to the same kind of pulse are identical. On the contrary, the initially stabilized skyrmion in the unconfined geometry has neither circular nor identical shape for the same kind of input wave as shown in figure 6(d) and the shape of the skyrmion changes arbitrarily, which led to lower capacity of the reservoir to distinguish different input waves. Thus, we have correlated the magnetization dynamics (geometric shape of breathing) with the average magnetization along z-direction (affects the magnetoresistance read out) to the RC metrics (STM and PC) ultimately.
From figure 4 we can see that multiple skyrmions in discontinuous thin film of 1000 nm, 1500 nm, and 2000 nm lateral dimension have higher STM and PC capacity than continuous thin film. Figure 7 (a) shows the magnetization state visited at 1800 ns as an example for breathing dynamics of multiple skyrmions in continuous and discontinuous thin film. For continuous thin film, the skyrmions are slightly displaced and distorted whereas in discontinuous thin film skyrmion breathes retaining their nearly circular shape without being dislocated. The displacement and distortion in shape of skyrmions causes the continuous film reservoir to generate irregular responses (similar to "single skyrmion in continuous thin film" case) when excited with input waveforms of same sequence which in effect degrades the STM and PC capacity of thin films hosting multiple skyrmions. Figure 7 (b), (c) show an example case of magnetization response in the z-direction from a discontinuous and continuous thin film respectively of 2000 nm lateral dimension hosting sixteen skyrmions for the present pulse of "sine-square" wave sequences. We can see that the responses from continuous thin film vary more irregularly (figure 7(c)) due to the distortion and displacement of skyrmions compared to discontinuous thin film which indicates that multiple skyrmion based discontinuous thin film performs better than continuous thin film. Therefore, we find that skyrmion-based straintronic PRC can classify simple sine and square pulses with 100% accuracy, regardless of whether it is based on a single skyrmion or multiple skyrmions, continuous thin films, or discontinuous thin films. However, discontinuous thin films are more efficient as a reservoir block compared to continuous thin film because skyrmions breathe in response to input pulses while maintaining a nearly circular shape due to the edge effect. The most efficient reservoir block among all the configurations shown in Fig. 2 is a single skyrmion in a discontinuous thin film having the highest STM and PC capacity.
From the non-thermal study, we observe that single skyrmion in a discontinuous thin film shows the highest STM and PC capacity and we further evaluate the performance of this RC in the presence of roomtemperature thermal perturbation. We increased the PMA energy from 1092.5 μJ/m 2 to 1150 μJ/m 2 to stabilize the skyrmion with room-temperature thermal noise as shown in figure 8(a). Also, to obtain responses from thermally stable skyrmion, the modulation of PMA energy was increased from 7.5×10 3 J/m 3 to 10×10 3 J/m 3 . Other than these two values, all other parameters were kept as same as non-thermal case and we observed that single skyrmion in a discontinuous thin film can perform recognition task with 100% accuracy. However, the STM and PC capacity decreases (CSTM=1.6240, CPC=0.7322) in the presence of thermal perturbation.
Before applying the input voltages, we first stabilize the skyrmion hosted in the thin film. We set single/multiple skyrmions in thin films of 1000 nm, 1500 nm, and 2000 nm lateral dimension and relax those for 500 ns to reach a stable state as shown in figure 2. Randomly distributed sine and square pulses are applied globally on these stable nanomagnets. Classification, prediction, memory capacity, and nonlinearity are evaluated based on the output (magnetoresistance) from the reservoir block. Once the randomly distributed waves are applied on the thin film, the skyrmion may dislocate or distort, which reduces the reproducibility of the output. Therefore, the skyrmion-hosting thin film is further stabilized for 500 ns after the application of pulse train and the reproducibility is checked. For single skyrmion in a 1000nm×1000nm thin film we can see in figure 8(b) that the magnetization in response to two sets of same voltage pulses are superposed (nearly superposed) for non-thermal (thermal) case, indicating the reproducibility of the capacities (see supplementary section 3 for the repeatability of reservoir computing with multiple skyrmions). Therefore, STM and PC capacities for two pulse trains are identical for nonthermal case but the capacities vary slightly in the presence of thermal noise. We note that the STM capacities are 1.624 and 1.4884 and PC capacities are 0.7322 and 0.4404 for first and second pulse train respectively in the presence of room-temperature thermal noise. However, we believe that with an appropriate choice of material parameters, the STM and PC capacities, as well as reproducibility, can be improved in the presence of room-temperature thermal perturbation.
We have considered the average magnetization of the whole thin film for determining the readout and single skyrmion confined in a discontinuous thin film is shown to have the highest STM and PC capacity. In commercial applications where readability can be an issue, the output signal of the single skyrmion confined at the center of a discontinuous thin film can be enhanced by taking the average magnetization from the central region as shown in the figure 9 (magnetoresistance change is proportional to the change in average magnetization in the z-direction of the soft layer of a perpendicular MTJ).
Sine and square pattern recognition have relatively good tolerance to prediction errors since it requires to determine if the predicted value is bigger or smaller than a threshold value. However, time-series prediction is much more challenging since the quantitative difference between the actual and predicted values count and might accrue in consecutive predictions. Therefore, in order to better illustrate the potential of skyrmion based PRC we tested a single skyrmion in a discontinuous thin film based PRC using Mackey-Glass time series [40 ]. Despite the system's deterministic form, forecasting is challenging, which has led to the frequent use of this chaotic system as a benchmark for forecasting task tests [41 , 42 , 43 ]. The chaotic Mackey-Glass time series is generated by the nonlinear time delay differential equation [40 , 42 , 43 ]:
Where x(t) is the value of the time-series at time t, τ=17, n=10, β=0.2, x(0)=1.0 and γ=0.1. Runge-Kutta method is used to solve the Mackey-Glass equation, which is then normalized into the [-1,1] range. The pulse width of the input signal is kept 20 ns, the same as for the recognition task, and the PMA is modulated linearly. From figure 10 we can see that a single skyrmion in a discontinuous thin film-based reservoir can predict the Mackey-Glass time series for various virtual nodes.
Energy is dissipated in two ways while using the proposed device to perform reservoir computing: reading the MTJ states and heating provided by charging current (I) running into the capacitor while generating strain to modulate PMA. The required maximum stress (σ=
) for the mentioned PMA modulation is 161.29 MPa considering CoFeB as the free layer material with saturation magnetostriction, 𝜆 𝑠 =31 ppm [44] and Young's modulus, E=160 GPa [45]. This can be generated by a strain of 10 -3 , which can be developed by applying an electric field, E=1.0 MVm -1 by designing four electrodes with a lateral dimension comparable to that of the thin film and a piezoelectric layer thickness equal to the lateral dimension of the thin film [36,46]. Thus, the voltage applied to the top electrodes for a thin film of 1500 nm lateral dimension is, V= 1.5V. The piezoelectric layer between four top electrodes and a bottom electrode as shown in figure 1(d) create four capacitors in parallel where alternating voltage pulses are applied from a 1.5 V input. The frequency of the applied voltage pulses is 50 MHz and considering an input resistance of 50 Ω, the energy cost (VItcosθ, where θ is the phase angle) for each input voltage pulse of 20 ns period (t) is 0.1fJ. For reading an MTJ state around 1fJ energy is required [47] and therefore for reading 50 states per input pulse, ~50fJ energy is dissipated. Thus, the total energy dissipated in a skyrmion based straintronic PRC is ~50fJ per input pulse. We compare to the energy dissipated in skyrmion based PRC with an equivalent CMOSbased echo-state network (ESN). The ESN can be simulated to replicate the performance of a skyrmion based PRC [48 ]. The PC capacity of a single skyrmion hosted by a discontinuous thin film of 1500 nm lateral dimension is ~5, which is similar to the PC capacity of the spintronic nano-oscillator based RC and it has been shown that to achieve this PC capacity with CMOS-based ESN, ~200 pJ energy is required for each input pulse [48 ]. Thus, skyrmion based PRC requires three-four orders of magnitude less energy compared to CMOS based ESN.
Our work shows that the intrinsic nonlinear behavior of skyrmions in response to a voltage generated strain can be exploited for implementing extremely energy-efficient reservoir computing. This skyrmion based straintronic PRC can classify simple temporal inputs with 100% accuracy in the presence of roomtemperature thermal perturbation. We have also shown that a single skyrmion in a discontinuous thin film based PRC can perform chaotic system prediction task successfully. Furthermore, we have studied the effect of the nonlinear and coupled magnetization dynamics (breathing of skyrmions) on reservoir computing metrics (STM and PC) and specifically showed that for achieving high STM and PC, skyrmion(s) are required to be confined and retain nearly circular shape while breathing in response to input. Though it is expected that spin-wave and dipolar interaction between skyrmions would enhance the mapping ability of the reservoir block to high-dimensional space by adding complexity; we observe the opposite, the spin-wave and dipolar interaction deteriorate the performance of the reservoir computer. In multiple skyrmion-based continuous thin film there are only nonlinear type of interactions, which includes nonlinear breathing dynamics of skyrmion, nonlinearly varying dipolar interaction and nonlinear spin-wave interaction. The presence of nonlinear spin-wave interaction potentially displaces and distorts the skyrmions in continuous thin film and therefore the STM and PC capacity of multiple skyrmion based continuous thin film decreases. When the multiple skyrmions are separated by a gap in a discontinuous thin film, there is no spin-wave interaction and the nonlinearity in discontinuous film results from the nonlinear breathing of skyrmion and nonlinearly varying dipolar interaction. As there is no spinwave interaction in addition to the existence of boundary effect in discontinuous thin film, the skyrmions are not displaced and distorted and provide better STM and PC capacity compared to multiple skyrmions in continuous thin film. When a single skyrmion is placed at the center of a thin film and is separated by a gap from its ferromagnetic periphery, there remains the linearly varying dipolar interaction from the periphery and nonlinear breathing of skyrmion. The coexistence of these linear and nonlinear interaction makes a discontinuous thin film hosting a single skyrmion a "mixture reservoir" and this coexistence of linearity and nonlinearity can be attributed to the highest performance provided by a single skyrmion in a discontinuous thin film in a similar way reported by M. Inubushi and K. Yoshimura [49].We note that single skyrmion performing better than multiple skyrmions is true for our case where input is applied simultaneously to all skyrmions. Applying inputs separately to different skyrmions may produce different results (while also being more challenging to fabricate) but this is beyond the scope of the current paper. We also observed that the skyrmion hosting thin film can go through multiple cycles since capacities are reproducible. Moreover, we showed that the output signal from a single skyrmion based reservoir can be enhanced. In summary, our work provides a pathway to realize highly energy-efficient and highperformance physical reservoir computing using voltage induced strain driven breathing of skyrmions. Such a physical RC paradigm can be well suited to learning relatively simple tasks in-situ on edge devices where energy and hardware resources are severely constrained. In future, for scaling the proposed skyrmion based reservoir, 2D magnetic materials can be used since it has been recently demonstrated that DMI increases when magnetic materials have a small thickness or monolayer as 2D magnetic materials [50], and high DMI is necessary for scaling skyrmions [23,24,51,52]. Therefore, 2D magnetic materials based straintronic physical reservoir computing can be a fascinating topic to further investigate in future research.