Dynamically controlled energy dissipation for fast magnetic vortex switching

Moving beyond traditional magnetic recording technology, the manipulation of more complex ferromagnetic (FM) microstructures such as domain walls, vortices, or skyrmions is likely to lead to novel applications in spin-based data storage, logic, sensing, and quantum information processing.^1–5 The vortex state in micromagnets has attracted attention with potential applications in magnetic information transfer,^6,7 magnetic logic,⁸ microwave amplification,⁹ and spin qubit addressability.^10,11

The relatively long relaxation time (∼100 ns) of FM vortex dynamics presents a limitation for fast operation of proposed devices. Precise and repeatable control of the vortex would require long wait times between sequential manipulations as the system dissipates energy to reach a new equilibrium. Therefore, there is a need to understand how a vortex core can be controllably translated while suppressing subsequent long-lived dynamics. Previous work has shown that vortex core oscillations may be resonantly amplified or dampened using a series of magnetic field pulses timed to push the core towards or away from its equilibrium position.¹² Here, we instead exploit the damping term in the vortex equation of motion to rapidly translate the vortex core to a new equilibrium. We monitor the vortex dynamics excited by short magnetic field pulses using time-resolved differential Kerr effect microscopy. We find that an additional short magnetic field pulse can provide the necessary energy dissipation to significantly damp the long-lived dynamics following a field step.

The ground state magnetization configuration of a thin ferromagnetic disk with negligible magnetocrystalline anisotropy is a vortex state [see Fig. 1(a)].¹⁴ This state is characterized by a curl of magnetization in the x – y plane circulating about a central core with a diameter of ∼10 nm. The magnetization in the vortex core is oriented out-of-plane parallel to $p z ̂$⁠, where p = ±1.^15–17 The vortex core can be translated across the disk by application of an in-plane magnetic field B. For small displacements, the rigid vortex model¹³ may be applied to describe the energy of the vortex state

where the stiffness k and displacement susceptibility χ₀ are material and geometry dependent parameters, $c = ± 1$ corresponds to the circulation direction of the in-plane magnetization, and $x = ( x , y )$ is the displacement of the vortex core from the center of the disk. Following the minimum of U₀, the equilibrium position $x 0 = c χ 0 ( − B y , B x )$ of the vortex core is displaced linearly with $| B |$⁠.^13,18 Defects in the magnetic material generate local pinning of the vortex core, which lowers the energy of the vortex state.^19–21 Pinning sites can be introduced into this model by adding local potential minima

where $w n ( x − x n )$ are peaked functions describing the pinning potential produced by a defect centered at a position $x n$⁠.^22,23

The non-equilibrium dynamics of the vortex core can be described using Thiele's equation of motion for two-dimensional magnetic vortices²⁵ 

where α is a damping factor, $G = − 2 π L M s γ − 1 p z ̂$ is the gyrotropic vector of a disk with thickness L, saturation magnetization M_s, and gyromagnetic ratio γ. In the micron-sized disks we study here, the inertial mass is negligible and will be ignored.²⁴ The three terms of Eq. (3) can be interpreted as a gyroforce F_G, a damping force F_α, and a restoring force F_U, respectively, with the dynamics governed by the balance of these forces $F G + F α + F U = 0$⁠. $F G$ and $F U$ are conservative forces, with only $F α$ capable of dissipating energy. In the absence of damping and pinning, F_G and F_U are in balance, leading to a perpetual circulation of the vortex core about its equilibrium position at the gyrotropic frequency $f g = k / 2 π | G |$⁠.^24,25 A nonzero F_α dissipates energy, resulting in relaxation of the vortex core towards the equilibrium position, reducing the amplitude of the gyrotropic motion over time. In this work, we aim to control F_α to enable fast damping. Since the damping parameter α cannot be easily tuned in real time, we tailor the trajectory of the vortex to yield $x ̇ ( t )$ that produces the desired F_α.

The vortex state was prepared in a Permalloy Ni_0.81Fe_0.19 disk with a diameter of D = 2 μm and thickness of L = 40 nm. The disks were fabricated via electron-beam lithography, electron-beam evaporation, and lift-off atop a gold co-planar wave guide (CPW). The defects in the material resulting from the fabrication process act as pinning sites for the vortex core.

To monitor the vortex motion, we use a differential magneto-optical Kerr effect (MOKE) technique. We first use a continuous-wave laser focused onto the disk using a 100× oil-immersion objective (NA = 1.4) to measure how the equilibrium position of the vortex evolves in response to an applied magnetic field. This measurement of equilibrium vortex displacement $Δ y 0$ vs. field allows us to map out the pinning potential through which the vortex travels (see Ref. 26 for details of this setup). We then use a pulsed laser to measure the non-equilibrium dynamics of the vortex in response to magnetic field pulses. The experimental setup for time-resolved measurements is shown in Fig. 1(b). A supercontinuum fiber laser produces an optical pulse train with a repetition frequency of 3.9 MHz to act as the probe beam. The spectrally broad output of the probe beam is filtered by linearly graded high and low-pass filters to a wavelength of λ = 660 nm and a linewidth of σ = 10 nm. A digital delay generator (DDG) is synchronized with the optical pulse train to drive current pulses in the CPW with controllable relative timing between the current pulses and optical pulses. The current pulses in the CPW generate a time-dependent magnetic field $B = B ( t ) x ̂$ experienced by the vortex. The optical pulse train is modulated at 100 Hz, and the electrical pulses are inhibited with a frequency of 15 kHz to enable lock-in detection. We measure the longitudinal Kerr rotation of the probe beam as a function of the relative delay t_d between the optical and electrical pulses, yielding relative vortex displacement $Δ y ( t d ) = y ( t d ) − y 0$⁠, where y₀ is the signal measured with the electrical pulses inhibited.

The equilibrium vortex displacement $Δ y 0$ is shown in Fig. 2(a) in response to static magnetic field B_x. $Δ y 0$ is roughly linear with ascending (blue) and descending (red) B_x, as described by U₀. The small jumps observed in $Δ y 0$ are a result of the vortex core transitioning through the various local minima of the pinning potential U_p. From the data shown in Fig. 2(a), we can construct an effective 1D pinning potential u_p(y), shown in Fig. 2(b), which describes the pinning landscape seen by the vortex core as it translates in response to B_x (see Ref. 23). For example, three plateaus in $Δ y 0$ are labeled A–C in Fig. 2(a), which arise when the vortex core is pinned in the correspondingly labeled sites in Fig. 2(b).

We will now look at the vortex dynamics as we switch between an initial pinning site to another target pinning site and back again. First, we accomplish this transition with a sharp step in the magnetic field (the “vortex step”), with amplitude $B 0 = 2.30$ mT as shown in Fig. 2(d) for a transition from sites A to C. The measured displacement $Δ y$ of the vortex core in response to the vortex step is shown in Fig. 2(c). Before the application of the pulse, the vortex core resides in its initial equilibrium position $Δ y = Δ y 0 = − 50$ nm. The field step shifts the minimum of the potential to the target site at $Δ y 0 = 130$ nm. The vortex core, now out of equilibrium, undergoes damped gyrotropic motion as described by Eq. (3), where the amplitude of the oscillations decreases with increasing time as energy is dissipated and the vortex core relaxes into its new equilibrium position. After the step down to the initial field value at t_d = 50 ns, the vortex again undergoes gyrotropic motion as it settles back into its initial state. To quantify the vortex dynamics, we fit the data in the range $t d =$ 2–50 ns to a decaying sinusoidal function $Δ y = A sin ( 2 π f g t ) e − t / τ$⁠, where f_g, τ, and A correspond to the gyrotropic frequency, relaxation time, and oscillation amplitude, respectively. The resulting value $τ = 20 ± 5$ ns represents the timescale required for the system to dissipate the additional energy introduced by the step in field.

To shorten the equilibration time into a target pinning site, we add a short pulse (the “damping pulse”) with amplitude B_p to the vortex step to initially drive gyrotropic motion with larger amplitude, and therefore larger $x ̇$⁠, and hence larger damping force F_α. The motion driven by the additional damping pulse dissipates energy at a faster rate than the vortex step alone, causing the vortex to approach the target site more rapidly. If the damping pulse ends when the vortex position is near the target site, the vortex core is already near the target equilibrium, and any subsequent oscillations are substantially suppressed. We demonstrate the effect of the damping pulse by adding a pulse with B_p = 2.2 mT and duration t_p = 7.7 ns [red line in Fig. 3(b)] to the vortex step. The resulting vortex displacement $Δ y$ is plotted in Fig. 3(a) (red), with the dynamics from the vortex step alone shown in blue. Although we observe a large initial increase in $Δ y$ in response to the damping pulse, once the damping pulse ends, the gyrotropic precession of the vortex core is significantly damped in comparison to the response from the vortex step only.

To better understand the experimental results, we numerically simulate the vortex dynamics using Eq. (3). We use the literature value of $M s = 8 × 10 5$ A/m, with $α = 0.04 | G |$ and f_g = 156 MHz set to match the measured values. The one-dimensional pinning potential $u p ( y )$ shown in Fig. 2(b) was extended into the two-dimensional $U p ( x )$ by approximating the individual pinning sites as symmetric Gaussian wells [color plot in Fig. 3(c)]. With the initial vortex position set in pinning site A, Thiele's equation was then integrated numerically for 50 ns using the MATLAB ODE solver. In reality, other pinning sites exist away from those shown along the measured line, but these measured sites are the most relevant as they overlap with the possible equilibrium vortex positions x₀ in the fields applied here. The equation of motion depends on the gradient of total potential, so the relative contributions of the pinning potential U_p as compared to the free potential U₀ can be understood by comparing the gradients of each. A typical pinning site such as C produces a maximum gradient $∇ U p ≈ 200$ meV/nm, while the gradient of the free potential $∇ U 0 = k ( x − x 0 )$⁠, with $k ≈ 7$ meV/nm². For a pinning site to trap the vortex, it is required that $| ∇ U p | > | ∇ U 0 |$ and therefore $| x − x 0 | ≲ 30$ nm. For x farther from equilibrium, the pinning site cannot trap the vortex and instead just perturbs the vortex motion. The strength of this perturbation scales as the ratio of $k | x − x 0 |$ to the maximum of $∇ U p$⁠. This discussion justifies the inclusion of pinning sites only along x = 0 in Fig. 3(c). To confirm the insensitivity to other pinning sites, we have repeated the simulations with pinning sites scattered across the entire 2D range, with the width, depth, and spacing randomly chosen from a distribution matching the observed sites (not shown). No significant difference was observed with these additional pinning sites added to the model.

The curves plotted in Fig. 3(c) show the simulated vortex core trajectories $x ( t )$ from the vortex step alone (blue) and with the additional damping pulse (red) with markers placed at $1$ ns intervals. The vortex step alone produces decaying gyrotropic motion at frequency f_g about the target site. The addition of the damping pulse initially drives much larger amplitude gyrotropic motion with the same frequency for a single period. Because the oscillation period is the same in both cases, the larger amplitude motion has greater velocity $x ̇$ and therefore dissipates energy at a faster rate $d U / d t = F α · x ̇ = − α x ̇ 2$⁠. With this enhanced damping, the position of the vortex approaches the target site after a single period of oscillation. Due to the negligible inertial mass of the vortex core, after the damping pulse ends, the system is found near equilibrium conditions and the forces acting on the vortex are nearly eliminated, resulting in a significant reduction of the subsequent gyrotropic oscillation amplitude about the target site. Figure 3(d) shows the vortex speed $| x ̇ ( t ) | ∝ | F α |$ corresponding to the two trajectories plotted in Fig. 3(c). With the only vortex step (blue), the velocity and hence damping force show a continuous smooth decay, with occasional bumps as the vortex passes through pinning sites. In contrast, when the damping pulse is added (red), the speed is initially much larger, giving rise to enhanced energy dissipation during the damping pulse. This energy dissipation is tuned to bring the vortex near to the target position at the end of the damping pulse. After the damping pulse ends, the velocity drops below that of the vortex step alone and subsequently finishes relaxing in the same way as the single step case but significantly advanced in time.

We can estimate B_p required to relax into the target site after a single gyrotropic period. The vortex step produces an equilibrium displacement $Δ y = χ 0 B 0$⁠, and hence, energy $Δ U = 1 2 k χ 0 2 B 0 2$ must be dissipated (ignoring the small correction due to pinning). During the damping pulse, we approximate the speed $| x ̇ | ≈ 2 π f g χ 0 ( B p + B 0 / 2 )$ as a constant equal to the average of the speeds at the beginning and end of the pulse. At time $1 / f g$⁠, the dissipated energy $Δ U ≈ α x ̇ 2 / f g ≈ ( α / | G | ) 2 π k χ 0 2 ( B p + B 0 / 2 ) 2$⁠. Solving yields

For the value $α / | G | = 0.04$ observed here, $B p ≈ 0.9 B 0$⁠, in reasonable agreement with the pulse amplitudes used in Figs. 3 and 4.

We now map out the vortex dynamics vs. damping pulse duration t_p, as the vortex switches from initial site A to target site B. Figure 4(a) shows a collection of measurements of $Δ y ( t d )$ in response to the magnetic field profile shown in Fig. 4(d), with varying t_p. The amplitude of the damping pulse is chosen so that the trajectory approaches the target site after one or two periods of the large-amplitude gyrotropic motion. For comparison, we simulate the vortex trajectory resulting from these ideal pulse shapes [Fig. 4(d), dashed line], with y(t) shown in Fig. 4(b). In the region t_d < t_p, large oscillations are visible in both the experiment and the simulation while the damping pulse is being applied. After the damping pulse (t_d > t_p), we observe the dynamics as the vortex relaxes into site B, with amplitude depending on the vortex position $x ( t d = t p )$ when the damping pulse ends.

To quantify the effect of the additional damping, the experimental and simulated dynamics in Fig. 4 were fit to a decaying sinusoid at t_d > t_p. The resulting oscillation amplitude A and relaxation rate τ^–1 are plotted in Fig. 4(c) vs. t_p, with experimental error bars, indicating the confidence interval from the fit. (The error bars on the fit to the simulation are much smaller and thus not shown. For data points in which the oscillation amplitude becomes very small, fitted values of τ^–1 become dominated by error, so these data points have been excluded from the plot of τ^–1.) The experiment (red) and simulation (blue dashed line) both display minima with a period of 1∕f_g ≈ 6 ns, corresponding to the completion of the first two gyrotropic orbits. At these minima, the damping pulse places the vortex in proximity to the target site, and the subsequent dynamics are most strongly suppressed. At the maxima of A vs. t_p, the damping pulse drives the vortex to the apex of its oscillation, quite far from the target site, resulting in large oscillations after the pulse. At these maxima, the data show a somewhat reduced τ that is not captured by the simulation. This may be caused by stronger dephasing of the vortex dynamics when it is undergoing particularly large-amplitude motion. The discrepancies in amplitude and time delay between the simulation and experiment can be corrected by taking into consideration the finite rise times of the field step and vortex dynamics. Repeating the simulations shown in Fig. 4(b) with a more realistic field profile [solid blue line in Fig. 4(d)] can yield values of A vs. t_p [solid blue line in Fig. 4(c)] that closely match the experiment.

The key to a fast switching protocol is to bring the vortex to the target position in as short a time as possible. Because the vortex inertia is negligible here, once the vortex is at the target position, the magnetic field can be switched to the final value, and the vortex will stop at the target site. With this view, the vortex step alone is the slowest possible switching, with the vortex position approaching the target site as $t → ∞$⁠. By adding an additional pulse (as we do here), the vortex position can approach the target site at finite t. To illustrate this general idea, we simulate the vortex path in the presence of two pinning sites, an initial site and a target site, following a single magnetic field step with amplitude B_x. We then find the minimum distance d_m between the target site and the vortex path and the time t_m at which this minimum occurs. Efficient switching is possible when $d m → 0$ at the smallest possible t_m. Figure 5(a) shows the resulting d_m vs. B_x for relatively small pinning sites with a depth of w_p = 1 eV and a width of 20 nm, separated by 50 nm. The color of the data points represents t_m. The direct vortex step occurs at B_x ≈ 0.6 mT, where we see that $d m → 0$⁠, but at very long t_m (the color scale is saturated at $t m > 20$ ns). Near B_x = 2.5 mT, we observe the behavior that is the focus of this work, in which the vortex is driven far beyond the desired equilibrium position and approaches the target site near $t m = 1 / f g$⁠. We note that there also exists an even faster switching protocol at $B x ≈ 0.3$ mT. This corresponds to shifting $x 0$ approximately halfway between the initial and target sites so that the vortex approaches the target at t_m = 1∕2f_g, without the need for the enhanced energy dissipation mechanism. The problem with this protocol, however, is that it is easily disrupted by the presence of pinning, as illustrated in Fig. 5(b), in which we perform the same simulation with stronger yet still typical pinning sites (w_p = 3 eV). We observe that the vortex now remains trapped in the initial site beyond the B_x required for the faster switching protocol. The method that we demonstrate here, on the other hand, is largely unaffected by the stronger pinning sites. This can also be understood by the relative gradients of the free and pinning potentials U₀ and U_p as described above. For the small pulse, $| x − x 0 | ≈ 25$ nm, so that $| ∇ U 0 | ≈ 175$ eV/nm, which is comparable to the maximum gradient from a typical pinning site. Thus, the trajectory will be highly susceptible to pinning for such a small pulse. In the method we focus on here, the vortex is quite far from equilibrium during the damping pulse (⁠ $| x − x 0 | ≈ 300$ nm), and the pinning sites only add a small perturbation to the vortex trajectory, with $| ∇ U 0 | ≈ 2000$ eV/nm, larger than the typical maximum pinning site gradient by a factor of ten.

For proposed devices that rely on translating FM vortices, the slow relaxation of the vortex's gyrotropic oscillations presents an obstacle to fast, reliable sequential operations. Here, we have shown how the vortex dynamics can be controllably used to shorten the relaxation time down to a single gyrotropic period; in this case, 1∕f_g ≈ 6 ns. This time could be reduced further for more strongly confined vortices which would have higher f_g. More generally, these results indicate that the fast dynamics of the vortex state, rather than being a limitation for future technology, can be well-characterized, controlled, and ultimately exploited.

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC008148.