RC circuit based on magnetic skyrmions

TL;DR

The paper shows that a magnetic skyrmion in a racetrack with a quadratic PMA landscape can replicate the dynamics of a classical RC circuit, mapping capacitor voltage to skyrmion position via $V\leftrightarrow x$. By deriving and validating a Thiele-equation-based RC form, it demonstrates DC charging/discharging with a nanosecond time constant $\tau_s$ and AC low-pass filtering with cutoff $\omega_s=1/\tau_s$, confirmed by micromagnetic simulations. The work also highlights potential neuromorphic applications, noting the RC-LIF equivalence and the ability to transform waveforms (e.g., square to triangle) via the skyrmion device. This approach provides a pathway to skyrmion-based, high-speed, energy-efficient analog components and artificial neurons, operable at hundreds of gigahertz scales in principle. Overall, the study establishes a direct, quantitative link between skyrmion dynamics and fundamental electronic circuit behavior, broadening the functional repertoire of skyrmion-based devices.

Abstract

Skyrmions are nano-sized magnetic whirls attractive for spintronic applications due to their innate stability. They can emulate the characteristic behavior of various spintronic and electronic devices such as spin-torque nano-oscillators, artificial neurons and synapses, logic devices, diodes, and ratchets. Here, we show that skyrmions can emulate the physics of an RC circuit, the fundamental electric circuit composed of a resistor and a capacitor, on the nanosecond time scale. The equation of motion of a current-driven skyrmion in a quadratic energy landscape is mathematically equivalent to the differential equation characterizing an RC circuit: the applied current resembles the applied input voltage, and the skyrmion position resembles the output voltage at the capacitor. These predictions are confirmed via micromagnetic simulations. We show that such a skyrmion system reproduces the characteristic exponential voltage decay upon charging and discharging the capacitor under constant input. Furthermore, it mimics the low-pass filter behavior of RC circuits by filtering high-frequencies in periodic input signals. Since RC circuits are mathematically equivalent to the Leaky-Integrate-Fire (LIF) model widely used to describe biological neurons, our device concept can also be regarded as a perfect artificial LIF neuron.

Figures (5)

• Figure 1: Schematics of the device concept. (a) Top: Artist view of an RC circuit consisting of a resistor with resistance $R$ and a capacitor with capacitance $C$. The voltage at the capacitor follows the behavior discussed in Sec. \ref{['sec:RC_circuits']}. Bottom: Schematic representation of the device concept introduced in Sec. \ref{['sec:skyrmions_as_RC']} featuring a Néel skyrmion stabilized at the center of a racetrack. The racetrack is composed of a ferromagnetic (FM) layer with an engineered spatial variation of the perpendicular magnetic anisotropy (PMA), $K_{\mathrm{u}}(x)$. The skyrmion can be moved along the length of the racetrack (indicated by the green arrow) via spin-transfer torques (STT) induced by an applied current (indicated by the blue arrow). The position of the skyrmion is analogous to the voltage across a capacitor in the corresponding RC circuit. The colors in the FM layer represent variations in the PMA. (b) This panel illustrates the variation of $K_{\mathrm{u}}$ across the FM layer, as shown in panel (a). The anisotropy $K_{\mathrm{u}}$ varies quadratically. The skyrmion can move in both directions by changing the sign of the applied current, with the linear force $F_\mathrm{PMA}$ (white arrows) resulting from the PMA variation pointing towards the center of the track.
• Figure 2: Characterization of RC circuits. (a) Schematic of an RC circuit with resistance $R$ and capacitance $C$ in series. (b) The charging and discharging processes under DC voltage are depicted by the normalized voltage response $\Gamma = V_{\mathrm{out}}(t)/V_{\mathrm{in}}(t)$, plotted as a function of $m = t/\tau$, where $t$ is the time, and $\tau$ is the circuit's time constant. The input voltage is turned on at $m = 0$, starting the charging process of the capacitor. At $m = 7$, the current is turned off, and the capacitor discharges. (c) RC circuits are low-pass filters, attenuating high-frequency components from input signals such as AC currents. The normalized transfer function $\mathcal{H} = V_{\mathrm{out}}^A / V_{\mathrm{in}}^A$ [Eq. \ref{['eq:ratio_RC']}], is plotted versus $n = \omega/\omega_c$, where $\omega$ is the input signal frequency and $\omega_c$ is the cutoff frequency. (d) The phase shift $\phi$ [Eq. \ref{['eq:phase_RC']}] of the output signal in response to AC currents is plotted versus $n$. In (e), the gain function [Eq. \ref{['eq:gain_RC']}] is plotted against $\log_{\mathrm{10}}{n}$. The black dashed line in panels (c-e) marks the cutoff frequency, where $n=1$.
• Figure 3: Comparison of micromagnetic simulations with result based on the Thiele equation. (a) Skyrmion moving in the racetrack based on micromagnetic simulations (colormap corresponds to the magnetization component $m_z$). The dark blue line indicates the skyrmion's trajectory when applying a direct current of $-0.5 \times 10^{12}$ A/m$^2$ for $t = 160$ ns and back to the origin once the current is switched off. The saturation position $x_{\mathrm{sat}} = x_\mathrm{in}^0 \approx 166.5$ nm is indicated by the dashed line. (b) Time evolution of the skyrmion's position $x_\mathrm{out}(t)$ based on micromagnetic simulations, corresponding to the trajectory shown in panel (a). (c) The response $\Gamma_\mathrm{s}=x_\mathrm{out}/x_\mathrm{in}$ is plotted against $m = t/\tau$. The black curve shows the numerical data based on the micromagnetic simulation, which is proportional to the data shown in (b). The red curve resembles the exponential function derived analytically, for which the time constant $\tau = 14.84$ ns has been determined numerically, as explained in the main text. This exponential function is the same for the skyrmion system and for the RC circuit, $\Gamma_\mathrm{s}=\Gamma$, due to the established equivalence of Eq. \ref{['eq:ratio_skyrmion']} and \ref{['eq:ratio_RC']}.
• Figure 4: Alternating current (AC) behavior and low-pass filter dynamics of the skyrmion device concept. (a) Trajectory of the skyrmion $x_\mathrm{out}(t)$ driven by currents with different frequencies (different colors, normalized as $n = \omega / \omega_s$) based on micromagnetic simulations. The current amplitude is $j^\mathrm{A}=0.5 \times 10^{12}$ A/m$^2$. The dashed line marks the saturation point $x_\mathrm{sat}=x_\mathrm{in}^A$ to which the skyrmion would move for a corresponding direct current (DC) with the same magnitude. (b-d) Comparison with the analytical results established in Sec. \ref{['sec:analytical']}. Dark blue dots represent the micromagnetic simulations of the skyrmion system as simulated in (a). The red dots are added to represent the corresponding analytical values obtained from Eqs. (3–5) for direct comparison with the RC circuit. The solid dark blue line represents the analytically derived behavior from the Thiele equation [Eqs. 14-16]. These lines are equivalent to the behavior of the RC circuit. (b) Normalized transfer function, $\mathcal{H}=x_\mathrm{out}^\mathrm{A}/x_\mathrm{in}^\mathrm{A}$, plotted against the frequency quantifying the attenuation of the oscillation amplitude. (c) Gain function plotted versus $\log_{\mathrm{10}}{(n)}$. (d) Phase shift between input current and observed oscillation of the skyrmion, respectively. The turquoise lines in (b-d) are analytical results representing a system without a PMA gradient, where the corresponding functions are obtained by setting $F_{\mathrm{PMA}} = 0$ in Eq. \ref{['eq:vx_edges']} [or $K_2 = 0$ in Eq. \ref{['eq:dxdt_edges']}].
• Figure 5: Response of the skyrmion device concept to square-wave current inputs. (a) Skyrmion trajectory $x_\mathrm{out}(t)$, normalized by the oscillation amplitude $x_\mathrm{out}^\mathrm{A}$, for various driving frequencies (blue pallet colors, normalized as $n = \omega / \omega_s$). (b) Normalized input $x_{\mathrm{in}}(t)$ (proportional to the square-wave current) as a function of time normalized by the period. (c-j) Fourier coefficients $b_k$ of $x_\mathrm{out}(t)$ (blue pallet scatter dots) and $x_{\mathrm{in}}(t)$ (orange scatter dots) for the different current frequencies, as shown in (a) and (b). The orange and green dashed lines are added for comparison. They represent the coefficients' peak decay rates $\propto1/k$ and $\propto1/k^2$, corresponding to a perfect square and triangular wave, respectively.