Skyrmion Gas Manipulation for Probabilistic Computing

The topologically protected magnetic spin configurations known as Skyrmions offer promising applications due to their stability, mobility, and localization. We emphasize how to leverage the thermally driven dynamics of an ensemble of such particles to perform computing tasks. We propose a device employing a Skyrmion gas to reshuffle a random signal into an uncorrelated copy of itself. This device is demonstrated by modeling the ensemble dynamics in a collective coordinate approach where Skyrmion-Skyrmion and Skyrmion-boundary interactions are accounted for phenomenologically. Our numerical results are used to develop a proof of concept for an energy efficient (approximately in the range of microwatts) device with a low area imprint (roughly on the order of $\mu {\mathrm{m}}^2$). Whereas its immediate application to stochastic computing circuit designs will be made apparent, we argue that its basic functionality, reminiscent of an integrate-and-fire neuron, qualifies it as an alternative bioinspired building block.

Figure 1

Bare-bones schematic of a generic Skyrmionic device constructed with a thin-film ferromagnet–heavy-metal bilayer. One or more Skymions are injected into the geometry where manipulation occurs before readout.

Figure 2

(a) and-gate implementation of the multiplication operation under the stochastic computing paradigm. The $p$ value of the output signal is equal to the product of the input signals’ $p$ values. (b) Whenever correlations exist between the input signals, the and gate does not perform the expected multiplication. (c) Forcing the input signals through a Skyrmionic reshuffler allows for a correct multiplication operation even in the presence of strong correlations between the input signals.

Figure 3

The proposed device consists of two magnetic chambers into which Skyrmions are injected depending on the state of an input telegraph noise signal. The net drift of the Skyrmion particles due to a constant current flow along with the thermal diffusion in the chambers leads to an exit order that can be significantly different from that of entry. This behavior is employed to reconstruct a new outgoing signal with the same statistical properties as the first, as well as being uncorrelated from it.

Figure 4

Two Skyrmions, initially separated by 50 nm, are allowed to evolve, via micromagnetic simulation, for 65 ns in the absence of thermal noise and applied currents. The topological constants are shown as a function of time. After a very brief relaxation phase, both the gyrovector $G$ (left panels) and the dissipative dyadic $D$ (right panels) stabilize around fixed values. Numerical deviations from constancy are due to the limited magnetic texture that is considered when performing integrals (3) and (4). To better qualify the relaxation process, the average Skyrmion radius and Skyrmion asymmetry (a value of 1.0 representing perfectly circular particles) are plotted as a function of time.

Figure 5

Two Skyrmions, initially separated by 50 nm, are allowed to evolve for 65 ns in the absence of thermal noise and applied currents. Their extracted trajectories (left panel) are employed to compute the inter-Skyrmion repulsion-force intensity (right panel), plotted as a function of distance in units of an average Skyrmion diameter (approximately 39 nm), which is then fit phenomenologically according to Eq. (10). Over the course of the simulation, the two Skyrmions are seen to spiral away from each other as a direct consequence of the gyrotropic terms appearing in the Thiele-dynamical model (an initial frame of the simulation showing the Skyrmion initial position is given for reference).

Figure 6

Net force experienced by a solitary Skyrmion initially nucleated at a distance of $d=50\mathrm{nm}$ from the boundary of a 1024-nm-diameter circular geometry and allowed to evolve for $1\mu \mathrm{s}$. Over the course of the simulation, the Skyrmion’s trajectory (left panel) is seen to move—clockwise—both along and away from the magnetic boundary (the thick black curve) as a result of the gyrotropic effects captured by the Thiele-dynamical model (an initial frame of the simulation showing the Skyrmion initial position is given as reference). The red line (right panel) shows a fit to the numerically derived force as a function of the distance from the boundary in units of the average Skyrmion diameter.

Figure 7

47 Skyrmions, initially placed randomly inside a 1024-nm-diameter circular geometry, are allowed to evolve for 150 ns in the absence of thermal noise and applied currents. The topological constants are shown as a function of time. After a very brief relaxation phase, both the gyrovector $G$ (left panels) and the dissipative dyadic $D$ (right panels) stabilize around fixed values. The numerical deviations from constancy are due to the limited magnetic texture that is considered when performing integrals (3) and (4).

Figure 8

47 Skyrmions, initially placed randomly inside a 1024-nm-diameter circular geometry, are allowed to evolve for 150 ns in the absence of thermal noise and applied currents. (a) Their extracted trajectories are then used (an initial frame of the simulation showing the Skyrmion initial positions is given as reference), in conjunction with their extracted gyrotropic constants [see Fig. 7 and Eq. (1)], (b)–(d) to compute the effective forces acting on each particle (${\mathbf{F}}_{\mathrm{ext}}={\widehat{\mathbf{G}}}^{\alpha }·\dot{\mathbf{x}}$). The net forces acting on each sampled Skyrmion is shown (the colored crosses) and compared to the additive two-Skyrmion prediction (9)—both with and without the boundary interaction term (the green and red fits, respectively)—to exemplify the importance of all of the effects discussed in the text.

Figure 9

Simulation of the Skyrmion reshuffler operating at a temperature of 300 K employing two identical chambers (with 1024-nm diameters) to scramble 0.5-ms-long, randomly generated telegraph noise input streams for different current intensities as measured at the 100-nm-wide conduits. (Right panel) Temporal evolution of correlation as the output stream is generated (data correspond to the $p=0.62$ simulation). (Left panel) Final product-moment correlation coefficient at each current intensity for five different input $p$ values. The blue curve fits data to the scaling $\rho =c_A/(c_B+J^{-3/2})$ argued for in the discussion following Eq. (14).

Figure 10

Output signal correlation as a function of different chamber sizes driven at a current intensity of $1.7\times {10}^{11}\mathrm{A}/{\mathrm{m}}^2$. Increasing the chamber size results in a longer diffusion time of the particles inside the chamber, leading to a better decorrelation of the output signal. The red curve fits data to the scaling $\rho =c_A+c_B/\sqrt{L}$ argued for in the discussion following Eq. (14).

Figure 11

A neuron collects neuronal spiking activity from nearby neurons. The incoming spikes lead to an increasing voltage potential between the inside and outside of the receiving neuron. Once the potential difference reaches a certain threshold, the receiving neuron discharges by releasing a voltage spike to its axon terminals, where the signal is transmitted through synapses to other neurons. Because of voltage leakage across the cell membrane, the neuron can be thought of as a leaky capacitor with low breakdown voltage.

Figure 12

The Skyrmion neuron consists of a magnetic chamber into which Skyrmions are injected depending on the state of one or multiple input telegraph noise signals. A voltage gate, initially in an on state, impedes the Skyrmions from drifting to the output conduit, thus leading to particle accumulation. A reading element placed inside the chamber can be used to estimate the Skyrmion density inside the chamber and switch off the voltage gate when a critical density is reached.

Figure 13

Maximum number of approximately 39-nm Skyrmions that can be stabilized in a circular chamber as a function of chamber diameter at 300 K.

Figure 14

Topological charge, damping, average radius, and asymmetry observed through micromagnetic simulations of two repelling skyrmions via the material parameter set $S1$.

Figure 15

Topological charge, damping, average radius, and asymmetry observed through micromagnetic simulations of two repelling Skyrmions via the material parameter set $S2$.

Figure 16

Topological charge, damping, average radius, and asymmetry observed through micromagnetic simulations of two repelling Skyrmions via the material parameter set $S3$.

Figure 17

Topological charge, damping, average radius, and asymmetry observed through micromagnetic simulations of two repelling Skyrmions via the material parameter set $S4$.

Figure 18

Topological charge, damping, average radius, and asymmetry observed through micromagnetic simulations of two repelling Skyrmions via the material parameter set $S5$.

Figure 19

Topological charge, damping, average radius, and asymmetry observed through micromagnetic simulations of two repelling Skyrmions via the material parameter set $S6$.

Figure 20

Topological charge, damping, average radius, and asymmetry observed through micromagnetic simulations of two repelling Skyrmions via the material parameter set $S7$.