Introduction

Past achievements in spintronics have largely relied on an endowment of electron-charge-based electronics with novel magnetization excitation phenomena such as single-domain switching1,2, vortex-core switching in nanomagnets3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18, domain-wall motions in nanowires19,20,21,22,23 and spin-wave excitations in magnonic crystals24,25. Ultrafast (~ps) magnetization excitations and their collective modes in nanomagnets have attracted interest owing to their suitability as information carriers with the additional benefit of much reduced energy dissipation compared to other electronics-based circuit devices.

A prototypic example is the study of the fundamental dynamics of vortex gyrations in single nano-size disks and their coupling in one- or two-dimensional arrays of vortex-state disks26,27,28,29. Since the gyration mode, representing the orbiting motions of an individual vortex core around its equilibrium position at characteristic frequencies from several hundred MHz up to ~1 GHz30,31,32,33,34, is well understood and lends itself to accurate analytical predictions35, the vortex gyration has attracted much attention it can be used as carrier for information26,27,28,29 or as a source for microwave emission36,37. For implementation in future potential devices, however, the gyration amplitude must be large enough for sufficient signal gains and signal transfer fidelity in integrated functional blocks along with significant reductions in the operating power consumption26,27,38.

Here, we experimentally investigate the resonant amplification of vortex oscillations in nanodisks by optimally tuned field pulses, by time-resolved soft X-ray microscopy measurements. The application of a series of coherent magnetic field pulses of ~1 Oe triggers a resonant enhancement of the gyration and subsequently an increased vortex-core oscillation amplitude and, therefore, signal gain. This mechanism provides a novel and efficient way to operate reliably future devices at significantly reduced power consumption. We show that this behavior can be explained by individual force terms and their balance, as obtained from combined results of micromagnetic numerical and analytical calculations.

Results

Vortex oscillation

Figure 1a shows the spin configuration, i.e., the magnetic vortex structure, in a single circular magnetic disk. It is characterized by an out-of-plane magnetization (referred to as polarization) in the core region which can point either upward or downward and an in-plane curling magnetization (referred to as chirality) around core which can point either counter-clockwise or clockwise39,40,41. The vortex is a stable ground state and results from the competition between the short-range exchange and long-range magnetostatic interaction. For small deviations of a vortex core from its equilibrium position, the motion of a vortex core with its velocity v is known to be well described by Thiele's equation42,43: , where is the gyroforce vector of a given core polarization p and is the damping tensor with the identity matrix and its damping constant D < 0 [Ref. 35]. The is the potential energy for a displaced core position X and can be expressed as , where κ is the stiffness coefficient, the first term W(0) the potential energy for a vortex at its initial position X = 0 and the second term the potential energy for the shifted vortex-core position35. The last term, WH, is the Zeeman energy due to a driving force if any field is externally applied. With this approach, the force-balance equation is essential to understand and predict vortex motions, which is given as FG + FD + FW + FH = 0, where each term corresponds to the gyroforce , the damping force , the restoring force and the Zeeman force , respectively35,42,43. If there is no external driving force and damping, the restoring force is in balance with the gyroforce , resulting in a persistent motion of the vortex core. If, however, there is intrinsic damping of the constituent magnetic materials (nonzero damping parameter ) a damping of core oscillations will result. In order to allow vortex-core motions to compensate or overcome such damping, external forces such as the static, alternating, or pulse form of magnetic fields or spin-polarized currents need to be applied.

Figure 1
figure 1

Concept of resonant amplification of vortex oscillations by coherent serial pulses.

(a) Top: Perspective-view image and cross-section of magnetic vortex state. The color and height of the surface indicate the in-plane and out-of-plane components of local magnetizations, respectively. Bottom: Potential energy surface and analytically calcualted vortex-core trajectories. The vortex-core trajectories under the Gaussian-pulse train (pulse number N = 2) are plotted on the potential energy surface. For the cases I and II, the time intervals employed were and , respectively. The blue and red lines indicate the trajectories of vortex-core positions driven by the first and second pulses, respectively (see the pulse profiles in (b)). (b) Profiles of train of Gaussian-pulse currents along single-strip electrode with pulse width (σ), time interval (ΔT) and pulse number (N = 4).

Full size image

Tuning resonant amplification of vortex gyrations

Here, we focus on a mathematically simple but technologically very useful pulse form as a driving force. Figure 1b shows a serial-pulse train composed of Gaussian-shaped unipolar pulses of identical height I0 and width σ and time interval ΔT. In the case of a damped oscillator, the motion of the vortex core can be resonantly amplified by applying a series of driving forces synchronized with the oscillatory motion of the vortex. Arbitrary driving forces would increase the oscillation amplitude whenever the driving force is in the same direction as the motion of the oscillator, but would result in a forced damping when in the opposite direction. Thus, to resonantly build up vortex-core oscillations, the driving force needs to form an in-phase repeating pattern of the same frequency as the eigenfrequency of the oscillator. Similarly, if a train of Gaussian pulses of width (σ) and time interval (ΔT) (as shown in Fig. 1b), tuned to the intrinsic resonant frequency (ωD) of the given vortex, were applied either in-phase or out-of-phase relative to the vortex gyration, the serial successive forces would result in acceleration or damping of the vortex gyration, according to the time interval. The trajectories of the vortex core driven by two Gaussian pulses are illustrated in Fig. 1a. When the time interval is tuned to an integer of the vortex' precessional period, the amplitude of the vortex gyration can be resonantly increased, as seen in case I. By contrast, the vortex gyrations are rapidly decreased if the time interval between the first and second pulses is a half integer of the vortex precessional period, as seen in case II. On the basis of this approach, we used unipolar field pulses with an optimal width σ = 1/ωD and coherent time interval ΔT = 2π/ωD (see Supplementary text online).

Soft X-ray imaging of vortex oscillations

In order to experimentally demonstrate the above described concept of resonant vortex oscillations, we fabricated a sample, as shown in Fig. 2. The sample contains several single dots, each of which consists of Permalloy (Py: Ni80Fe20) of diameter 2R ranging from 2.0 to 6.0 μm and a given thickness L = 40 nm (for experimental details see Methods). Trains of Gaussian current pulses applied along the y axis are simply expressed mathematically by , where N is the number of pulses in a given train. Core motions in the Py disk of diameter 2R = 3.0 μm were obtained by spatiotemporal-resolved full-field magnetic transmission soft X-ray microscopy (MTXM) which features a 70 ps temporal resolution and a 25 nm lateral resolution utilizing a stroboscopic pump-and-probe microscopy technique44. X-ray magnetic circular dichroism (XMCD) contrast near the Fe L3 absorption edge (around 707 eV) provides magnetization orientation contrast. In our experimental setup, the in-plane curling magnetizations of the vortex were directly imaged as seen in Fig. 2b. The polarization and chirality of the Py dot were downward (p = −1) and counter-clockwise (C = +1), respectively, as determined by the rotation sense of the vortex gyration and the in-plane MTXM contrast. The optimal pulse-width (σ) value, as obtained from an experiment with single Gaussian-pulse excitation (N = 1), was 1.45 ns. From this value, we could estimate a possible eigenfrequency of the real sample, here ωD = 2π × 110 MHz, according to the explicit analytical form of σ = 1/ωD [Refs. 17, 18].

Figure 2
figure 2

Sample geometry for time-resolved soft X-ray microscopy measurement.

(a) Schematic drawing of sample geometry consisting of Py disks with indicated diameter 2R = 3.0 μm and thickness L = 40 nm and electrode of Ti (5 nm)/Cu (70 nm)/Au (5 nm). The electrode is positioned on the y-axis, thus yielding magnetic fields along the x-axis. The left-upper inset shows an optical microscopy image of the sample wherein Py disks of different diameters 2R = 2.0~6.0 μm are placed. The sample surface was mounted at an angle of 60° with respect to the direction of the incident X-rays. (b) MTXM image of ground vortex state of Py disk of diameter 2R = 3.0 μm and thickness L = 40 nm, obtained at the Fe L3 edge. The vortex state is characterized by the CCW in-plane and downward core orientations.

Full size image

As an illustration of the nontrivial motion of the vortex core in the experiment, the time-varying oscillatory x and y components of the vortex-core position and their trajectories on the disk plane as excited by two (N = 2) Gaussian pulses of field strength H0 = 4.6 Oe are plotted in Figs. 3a and 3b, respectively, for ΔT = 2π/ωD = 9.09 ns and ΔT = 1.5 × (2π/ωD) = 13.64 ns. Representative serial snapshot images of the XMCD contrast are shown in each second panel. In the case of , where ΔT is the same as one cycle of vortex gyration, orbiting core displacement was increased with time upon application of the second pulse. By contrast, for , the core gyration amplitude rapidly decreased after application of the second pulse.

Figure 3
figure 3

(a) Amplification and (b) damping of vortex oscillations.

The first row shows the profiles of two serial field pulses of (a) ΔT = and (b) ΔT = . In the second row of (a) and (b), there are serial snapshot XMCD images of the temporal evolution of vortex-core motions. The red and yellow symbols in the first row indicate the snapshot time for each image. The third row indicates the x- and y-components and displacement of the vortex-core positions (left) from the center position (x, y) = (0, 0) and the constructed vortex-core trajectories (right), as obtained from the time-resolved MTXM images; the fourth row shows the corresponding simulation results.

Full size image

Analytical calculation and numerical simulation

To interpret the experimentally observed core gyration amplitude asymmetry (amplification versus damping), we conducted micromagnetic simulations (see Methods for details). The results are shown in the fourth rows of Figs. 3a and 3b and are in excellent agreement with the experimental results, except for slight deviations in their oscillation amplitudes and frequencies. These discrepancies can be attributed to sample imperfections or several possible simulation/experiment deviations in the dimensions, material parameters and field strength. Then, based on Thiele's analytical approach42,43, for a given magnetization structure taken at a certain time and a given core velocity v, all of the forces can be calculated directly from the integrations of the magnetization configurations that were obtained from the micromagnetic numerical simulations. The individual forces are as follows: with , the damping force , where a damping tensor with the damping constant α, the gyromagnetic ratio γ and the saturation magnetization . The restoring force and Zeeman force are given as with the effective field . From these relations, the Zeeman force can be descried as , where with ( for the “side-charge-free” model35).

Accordingly, the resultant core motion can be determined from the numerically estimated individual forces, as represented by the magnitudes and directions of all of the forces. For the two cases of and , temporal variations of the force magnitude and orientation are plotted in Figs. 4a and 4b, respectively. The definition of the force orientation (θ) is illustrated in Fig. 4c. The first Gaussian pulse excites the precessional motion of the vortex core from its initial position X = 0; if the second Gaussian pulse is applied, the Zeeman force FH is generated and thus, to compensate the Zeeman force FH, the magnitude and orientation of gyroforce FG and restoring force FW change. Fig. 4a and 4b show the angles and amplitudes of the Zeeman force FH and restoring force FW in response to the vortex-core position vector X and applied magnetic field H. When the second Gaussian pulse is applied with the time interval (ΔT) of an integer of the vortex precessional period , the Zeeman force FH induced by the second pulse is in-phase relative to the vortex core orbiting around the center position. To satisfy the force balance, the angle of the gyroforce FG has the negative angle and the magnitude of gyroforce FG and restoring force FW increase as shown in Fig. 4a. As a result, the angle of the vortex velocity vector v is an acute angle due to the relation of . This means that the radius of the vortex gyration is increased according to the resultant velocity vector, as shown in the first schematic image in Fig. 4d (the damping force FD is not shown here for simplicity). In the same manner, the Gaussian-pulse train with the time interval of a half integer of the vortex precessional period can reduce the radius of the vortex gyration, as shown in the second image in Fig. 4d.

Figure 4
figure 4

Force-balance relation in vortex oscillation.

(a, b) The second row presents the strengths of the individual force terms, that is, the gyroforce FG, the Zeeman forces FH, the restoring force FW and the damping forces FD, according to the applied Gaussian-pulse fields shown in the first row. The bottom row shows the orientations of the individual force terms, that is, the angles θ with respect to the vortex-core-position vector (X), as defined in (c). (d) Schematic illustration of orientation and strength of employed forces under resonant amplification (left) and forced damping (right), where for simplification, damping force FD is not shown. v corresponds to the core motion velocity vector. The red dot indicates the core position.

Full size image

Furthermore, we experimentally observed the maximum core displacement |X|max versus ΔT/ΔT0 ( = 9.09 ns, to ) by applying a sequence of two Gaussian pulses of field strength H0 = 4.6 Oe and optimal width = 1.45 ns. The experimental results (closed circles) of |X|max are compared with the analytical calculations (solid line), as shown in Fig. 5. As predicted by the analytical form of , the integer values of m evidenced amplification of vortex gyration, whereas the half integers showed further damping of the vortex gyration. This implies that the time at which the second pulse arrives plays a role in the in-phase and out-of-phase motions of the vortex core.

Figure 5
figure 5

Maximum vortex-core displacement |X|max for different time intervals, = 0.5~2.0, where , along with identical parameters of pulse number N = 2, = 1.45 ns and H0 = 4.6 Oe.

The yellow dots and solid line correspond to the experimental results and analytical calculation, respectively.

Full size image

Impact of pulse number on the resonant amplification

Next, we experimentally examined the core displacement as a function of time for a different pulse number N ( = 1, 2, 3, 4, 5) for the optimal width σ = 1.45 ns, time interval ΔT = 9.09 ns and field strength H0 = 4.6 Oe, as plotted in Fig. 6. The inset in Fig. 6 shows the corresponding simulation results for the real Py disk and explains well how the core displacement increases with the number of coherent optimized field pulses. Figure 7a shows the maximum displacement of the core position for N = 1, 2, 3, 4 and 5, as obtained from Fig. 6 and provides evidence that the amplification of the vortex-core gyration by N = 5 is three times greater than that by N = 1. In Fig. 7b, for N = 10, the maximum displacement is plotted with H0, indicating that with application of more Gaussian pulses, a sufficient displacement of core motion, 200 nm (1/7.5 of the disk radius), can be achieved with the extremely low field strength of 2 Oe. Interpolation of this data showed that with 1 Oe, 100 nm displacement can be achieved.

Figure 6
figure 6

Vortex-core displacement as function of time according to different number of coherent pulses.

The trains of Gaussian magnetic field pulses under the optimal conditions ns, ΔT = ns and field strength H0 = 4.6 Oe were applied. The pulse profile according to the time is shown in the first row. The right inset shows the corresponding simulation results obtained under the same conditions as those prevailing in the experiment.

Full size image
Figure 7
figure 7

Experimental results of maximum vortex-core displacement versus pulse number in (a) and versus H0 in (b).

The trains of Gaussian magnetic field pulses under the optimal conditions ns, ΔT = ns were applied. For comparison, the results in (a) also are plotted in (b).

Full size image

Discussion

We experimentally demonstrated that the resonant excitation and amplification of vortex gyrations can be achievable with extremely low power consumption as much as ~1 Oe by application of a series of coherent Gaussian pulses of optimal interval and pulse width . Signal gain can be defined as the mean ratio of the final vortex-gyration amplitude to the initial vortex gyration. Therefore, a large displacement of vortex gyration indicates a sufficient signal gain detectable through Tunnel magnetoresistance45 (TMR) or Giant magnetoresistance46,47 (GMR) in possible spin value structures. Since signal gain in electronic devices is a measure of the capacity to increase the power or amplitude of a signal from input to output. The present work quantitatively clarifies the fundamental of vortex excitations and their amplifications and might provide a way to significantly amplify vortex-gyration-based signal gains in future devices at considerably reduced power consumption (~1 Oe in field strength).

Methods

Sample fabrication

For the measurement of soft X-ray transmission through the sample, all of the disks and electrodes were prepared on 100-nm-thick silicon nitride (Si3N4) membranes. The 40 nm-thick Py films were deposited by magnetron sputtering under base pressures of less than 5 × 10−9 Torr. Py disks of radius 2R = 2.0~6.0 μm were patterned by typical e-beam lithography (Jeol, JBX9300FS) and subsequent lift-off processes. Each disk was placed on a strip electrode composed of Ti (5 nm)/Cu (70 nm)/Au (5 nm) and patterned, like the Py disks, by e-beam lithography and subsequent lift-off processes.

Time-resolved soft X-ray microscopy measurement

Snapshot images of vortex-core oscillations were measured by full-field soft X-ray transmission microscopy at the Advanced Light Source (Beamline 6.1.2) in Berkeley, California, USA, using a stroboscopic pump-and-probe technique44. The microscopy boasts a temporal resolution of 70 ps and a lateral resolution of less than 25 nm. The in-plane magnetization contrast was provided by X-ray magnetic circular dichroism (XMCD) near the Fe L3 absorption edge (around 707 eV), according to the in-plane geometry whereby the sample was mounted at an angle of 60° with respect to the propagation direction of incident X-rays. In order to achieve adequate XMCD contrasts, 10 to 20 individual images were accumulated. The structural contrast was normalized to an image obtained under a saturation field. Based on a stroboscopic pump-and-probe technique44, Trains of Gaussian-shaped field pulses were synchronized to X-ray pulses of ~3 MHz frequency and the time delay between the pulse trains and the incident X-ray was varied from 0 to 60 ns in increments of 1.14 ns (eight images per cycle of vortex gyration, at a frequency of 110 MHz). In the experiment, a Gaussian current pulse was applied with an arbitrary waveform generator. The height of the output pulse voltage was measured as 350~370 mV with an oscilloscope with an impedance of Z = 50 Ohm, which corresponds to 0.7~0.74 transmittance. On the basis of Ampere's Law, we estimated the resultant Oersted field strength and spatial distribution with the measured output voltage and the impedance of the oscilloscope employed. Accordingly, the field strength was determined to be H0 = 4.6 Oe at the center of the Py disk (20 nm from the electrode surface). The local distribution of the magnetic field was calculated using the analytical equations shown in Ref. 48.

Micromagnetic simulations

Micromagnetic simulations were carried out for a model system of the same dimensions as those of the sample and under the same conditions as those in the experiment. To numerically simulate the vortex-core oscillations, we employed the OOMMF code49, which utilizes the Landau-Lifshitz-Gilbert equation of motion of magnetization:) [Ref. 50 and 51], with the gyromagnetic ratio γ (2.21 × 105 m/As) and the phenomenological damping constant α (0.01). We used a unit cell size of 4 × 4 × 40 nm3. To match the eigenfrequency of the model to that of the sample ( MHz), we used Ms = 7.6 × 105 A/m for the numerical simulations4,52,53. The other simulation parameters for Py were as follows: exchange stiffness Aex = 1.3 × 10−11 J/m and magnetocrystalline anisotropy constant K = 0.