In a vortex-state magnetic nanodisc1,2,3, the static magnetization curls in the plane, except in the core region, where it points out of plane4,5, either up or down, leading to two possible stable states of opposite core polarity p. Dynamical reversal of p by large-amplitude motion of the vortex core6,7,8,9 has recently been demonstrated experimentally10,11,12,13,14, raising the prospect of practical applications, in particular in magnetic-storage devices15. Here we demonstrate coherent control of p by single- and double-microwave-pulse sequences, taking advantage of the resonant vortex dynamics in a perpendicular-bias magnetic field16. Experimental optimization of the microwave-pulse duration required for switching p also yields information about the characteristic decay time of the vortex core in the large-oscillation regime. This time is found to be less than half the length seen in the small-oscillation regime, suggesting a nonlinear behaviour of magnetic dissipation.
Magnetic vortices are topological solitons with rich dynamical properties. The lowest-energy excitation of the vortex ground state is the so-called gyrotropic mode3, corresponding to the gyration of the vortex core around its equilibrium position with a frequency in the sub-gigahertz range17,18. It is now established experimentally14 that the excitation of this gyrotropic motion leads to a dynamical distortion of the vortex-core profile, as predicted by micromagnetic simulations and theoretical analysis8. This distortion increases with the linear velocity of the vortex core and opposes the core polarity, until the critical velocity (γ is the gyromagnetic ratio of the magnetic material and Aex its exchange constant) is reached and the vortex-core polarity is reversed9.
In zero magnetic field, dynamical control of the polarity is difficult owing to the degeneracy of the gyrotropic frequencies associated with opposite polarities p=±1, which can lead to multiple core switching7,11. Still, selective core-polarity reversal is possible using a circularly polarized microwave magnetic field because the sense of the core rotation is linked by a right-hand rule to its polarity12. Control of polarity switching can also be achieved by precise timing of non-resonant magnetic-field pulses13,19, in a similar fashion as domain-wall propagation in magnetic nanowires20.
Resonant amplification21 of the vortex gyrotropic motion enables us to reverse the core polarity with minimum excitation power12,14,15, as it enables us to concentrate the energy in a narrow frequency band. In this scheme, the damping ratio is an important parameter because it controls the minimum amplitude of the resonant excitation required to switch the core9. Here, it is shown that the damping ratio close to the reversal threshold is significantly larger than that measured in the small-oscillation regime. We associate this with the nonlinear nature of the reversal process6,8.
Investigation of vortex-core reversal using time-resolved imaging techniques based on X-ray magnetic circular dichroism is very powerful, as it enables direct determination of the vortex-core trajectory14. However, it requires averaging of many events and the interpretation can become more complicated owing to multiple reversal processes. In order to investigate the coupling between resonant microwave pulses and vortex dynamics near the core-reversal threshold, we use a magnetic resonance force microscope (MRFM) in combination with a bias magnetic field applied perpendicularly to the disc plane, that induces two distinct resonant gyrotropic frequencies associated with opposite core polarities16,22. This frequency splitting is used for a simple, single-shot reading15 of the polarity state using an MRFM, before and after microwave pulses have been applied to the vortex-state nanodisc.
The MRFM detection set-up23, illustrated in Fig. 1a, is described in the Methods section. It enables us to probe the magnetization dynamics in individual nanostructures24, and in particular to spectroscopically determine the resonance frequency of the gyrotropic mode in a vortex-state nanodisc16. The studied nanodisc with thickness 44 nm and diameter 1 μm is made of NiMnSb alloy (see the magnetic-force-microscope image in Fig. 1b), an ultralow-damping epitaxial material25 (the typical Gilbert constant αLLG is between 0.002 and 0.003). A gold antenna patterned on top enables us to apply pulses of linearly polarized microwave magnetic field h in the plane of the disc.
In our experiments, we choose the bias perpendicular field μ0H=65 mT to unambiguously discriminate the two different gyrotropic frequencies, or polarity states: resonant frequencies f−=217 MHz and f+=254 MHz respectively correspond to p=−1 (core antiparallel to H, see Fig. 1d) and p=+1 (core parallel to H, see Fig. 1c). The microwave power employed to read the polarity state with an MRFM is weak enough (P=−19 dBm) that p is not reversed during the reading sequence15. We also emphasize that the bias perpendicular field μ0H=65 mT is almost five times smaller than the static field required to switch the core polarity16,26, so no significant asymmetry between the two possible reversal processes (p=−1 to p=+1 and vice versa) is induced, as will be clear from the data presented below.
First, we explore in Fig. 2a the efficiency of single microwave pulses to switch the vortex core, depending on their duration, frequency and power. Π− pulses are defined as single pulses that reverse the polarity from p=−1 to p=+1 and Π+ pulses from p=+1 to p=−1 (see Fig. 2b). The experimental data are acquired as follows: for Π− (Π+) pulses, the p=−1 (p=+1) state is first reset using an initialization pulse whose result is known to be fully deterministic15. Then, a single microwave pulse of given duration w, frequency f and power P is applied, and the final polarity state is read using an MRFM. An opaque coloured pixel marks pulse settings for which reversals are recorded with a 100% success rate, whereas a blank pixel means that no reversal is recorded. The contour plots presented in Fig. 2a with different shades of red (blue) show the superposition of the results for Π− (Π+) pulses with three different durations w:100, 50 and 20 ns.
The frequency splitting introduced by the perpendicular bias field clearly appears in Fig. 2a. Due to the resonant character of the investigated switching process, a pronounced minimum in the power level required to reverse the vortex core is observed at a frequency f−* (f+*) close to the gyrotropic frequency associated with the initial core polarity, for each duration of the Π− (Π+) pulses. This frequency discrimination introduced by H enables most of the recorded results to be fully deterministic.
The minimum power P* required to reverse the core polarity increases as the pulse duration w decreases. We have plotted the dependencies on w of the optimal frequency f+* (Fig. 2d) and of the optimal pulse energy E*=P*w (Fig. 2e). In these plots, w ranges from 1 μs down to 3 ns, with experimental points obtained from the analysis of data sets similar to those presented Fig. 2a. There is a clear minimum in the optimal pulse energy that occurs around w=50 ns. The position of this minimum yields an estimation of the characteristic decay time. If the pulse duration exceeds this characteristic time, the vortex-core dynamics reaches steady state before the end of the pulse; that is, the pulse duration is uselessly too long. In contrast, if the pulse duration is shorter than this characteristic time, some energy is wasted outside the resonance line. Due to the transient response of the vortex core to the excitation pulse, the optimal frequency f+* measured for pulses shorter than 50 ns shifts to higher frequency9 (see Fig. 2d), a general behaviour of any harmonic oscillator forced on timescales shorter than its decay time.
To be more quantitative, we have carried out a numerical calculation based on the analytical approach developed for zero applied magnetic field9, with the new ingredient that, owing to the bias perpendicular field, f− and f+ are distinct frequencies. The trajectory of the vortex core subjected to the microwave field pulse is calculated using Thiele’s equation27, and Guslienko’s criterion8 for core velocity is used to determine whether the polarity is switched by the end of the pulse. The behaviour of the optimal pulse energy as a function of w can be reproduced with an analytical formula where the damping ratio dforced* is an adjustable parameter (blue solid line in Fig. 2e). The best fit is obtained for dforced*=0.018, corresponding to a characteristic decay time τforced=1/(dforced*2πf+)≃35 ns. The predicted optimal pulse duration is wmin=1.26τforced≃44 ns (see Methods for details). Fig. 2c illustrates the good agreement of the model with the experimental data of Fig. 2a on the whole range of pulse parameters.
The significant result here is that the damping ratio dforced*=0.018 inferred in the forced regime close to the core-reversal threshold is more than twice the value measured in the small-oscillation regime of the gyrotropic mode, found to be dvortex=0.0075 (corresponding to a decay time τvortex=85 ns). The damping ratio dvortex is itself about three times larger than the Gilbert constant, measured from the linewidth in the perpendicularly saturated state of the NiMnSb disc and found to be αLLG=0.0025 (see Supplementary Fig. S1). The difference between dvortex and αLLG corresponds to the expected increase due to topological renormalization27,28. This enhancement is produced by the strongly inhomogeneous spatial distribution of magnetization in the undistorted vortex state compared with the perpendicularly saturated state. Our experimental results of Fig. 2 give a quantitative estimation of the damping ratio dforced* for the vortex distorted by large oscillations8,14. In a further step, it would be interesting to evaluate how much of the observed increase of dforced* against dvortex is due to topological renormalization between the distorted and undistorted vortex states. As can be seen in Fig. 2e, a practical consequence of this increase is to decrease the optimal pulse duration wmin and to increase the corresponding minimum pulse energy.
Next, we investigate in Fig. 3 the vortex-core dynamics in the free-decay regime by monitoring the core reversal induced by two consecutive microwave pulses. In these experiments, the pulse duration is set to w=9 ns and the power to P=−1.8 dBm, that is, slightly below the minimum power P*=−1 dBm required to reverse the core polarity p for such a short pulse. In contrast to a single pulse, two pulses separated by a delay τ (Fig. 3a, Π−–τ–Π−; Fig. 3b, Π+–τ–Π+) can reverse p. The striking oscillatory dependence on the pulse carrier frequency and on the delay τ observed in Fig. 3a,b elucidates the phase-coherent coupling between the vortex gyrotropic motion and the microwave excitation. During the pulse duration the gyrotropic motion is forced at the pulse frequency, whereas in the free-decay regime the core oscillates towards its equilibrium position at its natural frequency, thereby acquiring a phase shift with respect to the excitation carrier. As a result, the efficiency of the second pulse to drive the vortex core to the reversal threshold, which is estimated out of ten attempts in Fig. 3, depends on the microwave frequency and on the delay between pulses in an oscillatory manner (the oscillation period scales as the inverse frequency detuning).
To illustrate further this effect of coherence, similar experiments with a π phase shift introduced between the two pulses have been carried out. It is clear that the regions where successful reversal are observed in Fig. 3c and d are complementary to those in Fig. 3a and b, respectively. Thus, phase control of the microwave excitation can trigger vortex-core switching.
Using the same approach as before, it is possible to calculate numerically such experimental phase diagrams. An excellent agreement with experiments is achieved, as can be seen in Fig. 4a. In Fig. 4b, we have plotted the calculated vortex-core trajectory and velocity as a function of time corresponding to two consecutive Π+ pulses at f=253 MHz separated by a 60 ns delay. In the top graphs, there is no phase shift between pulses and the time delay is such that the second pulse is efficient to amplify the gyrotropic motion from the beginning of the pulse, in contrast to the bottom graphs, where a π phase shift is set between pulses. As a result, the vortex core is not reversed by the end of the second pulse in the latter case, whereas it is in the former. The obtained data sets also enable us to fit the free-decay time, and the best agreement with the experiment is obtained for τfree≃53 ns. Analysis of the difference of this value from τforced should further shed some light on the precise nonlinear nature of magnetic dissipation close to the reversal threshold and on its dependence on the amplitude of the vortex-core motion.
The magnetic nanodisc (thickness 44 nm, diameter 1 μm) was patterned by standard e-beam lithography and ion-milling techniques from an extended film of NiMnSb grown by molecular-beam epitaxy on an InP(001) substrate29. A 50-nm-thick Si3N4 cap layer was deposited on top of the disc for protection and a broadband coplanar microwave antenna (300-nm-thick Au) was subsequently evaporated on top of the patterned disc. The width of the antenna constriction above the disc is 5 μm (see Supplementary Fig. S2).
Injecting a microwave current from a synthesizer inside the antenna produces an in-plane linearly polarized microwave magnetic field h, oriented perpendicular to the stripe direction. To apply two consecutive pulses separated by a variable delay τ, the carrier of the cw excitation is split into two branches, each of them being gated by independent mixers before being recombined. A phase shift can be introduced between the two branches using a delay line. The typical rise and fall times of pulses are 0.8 ns and the control on the delay τ is better than 0.1 ns. The calibration of the amplitude of h yields the value μ0h=1.05 mT with an error bar of 25% for a 0 dBm input power in the antenna. Therefore, the microwave power range in Fig. 2a corresponds to microwave field amplitude ranging from 0.2 to 0.7 mT, the power in double-pulse experiments of Fig. 3a to μ0h=0.85 mT and the power employed to read the polarity to μ0h=0.1 mT.
A Π+ pulse with settings w=50 ns, f=f+, P=−11 dBm (ref. 15) is used to set the initial polarity state to p=−1. In fact, such an initialization pulse does not affect the p=−1 state and it transforms p=+1 into p=−1 (see Fig. 2a). Similarly, a Π− pulse (w=50 ns, f=f−, P=−11 dBm) is used to set the initial polarity state to p=+1.
The MRFM set-up23 is located inside a vacuum chamber (10−6 mbar) and operates at room temperature. The cantilever is an Olympus Biolever (spring constant k≃5 mN m−1) with a 800-nm-diameter sphere of soft amorphous Fe (with 3% Si) glued to its apex. MRFM spectroscopy is achieved by placing the centre of the magnetic spherical probe above the centre of the NiMnSb nanodisc. The separation between the sample and the probe is s=1.5 μm. The probe senses the dipolar force Fz proportional to the perpendicular component Mz of the magnetization of the nanodisc (see Fig. 1a). Ferromagnetic resonance spectra are obtained as a function of the microwave excitation frequency at a fixed bias field H. The microwave modulation is a cyclic absorption sequence, where the microwave power is switched on and off at the cantilever resonance frequency, fc≃11.8 kHz. The MRFM signal originates from the cyclic diminution of Mz of the nanodisc synchronous with the absorption of the microwave field16,23. The resulting force modulated at the mechanical resonance of the cantilever force produces a cantilever vibration amplitude enhanced by its quality factor Q≃4,000, which is optically detected.
The calculations presented in Figs 2 and 4 are based on the linearized Thiele equation27, an effective equation of motion for the vortex-core position X in the disc plane: The first term is the gyroforce (G is the gyrovector), the second one is the damping (d=−D/|G|), the third one is the restoring force (κ is the stiffness coefficient) and the last one is the Zeeman energy with the spatially uniform external field h ( is the unit vector normal to the disc plane and details of μ and previously mentioned coefficients can be found in refs 3, 28). The resonant frequency of vortex-core gyration is ωG=κ/|G|. From equation (1) we can find the instantaneous position X(t) and the velocity v(t) of the vortex core in the presence or absence of the harmonic excitation field h(t), as derived in the supplementary documents of ref. 9. If Guslienko’s criterion on critical velocity8 is met at some time of the simulated pulse sequence (v(t)>Vc), the vortex core is assumed to have switched. Experimental sample size, gyrotropic frequencies f− and f+ associated with each polarity, and nonlinear redshift of the frequency (3%) are used in our calculation. The minimal excitation amplitude hc(ω,w) to reach Vc by the end of the pulse depends9 on the excitation frequency ω and on the pulse duration w. Minimization of the energy E∝whc2(ω,w) with respect to ω yields the dependence on w of the optimal frequency f+* plotted in Fig. 2d and of the optimal pulse energy E* plotted in Fig. 2e: In this expression, a is an experimental conversion factor between the input power in the antenna and h2 determined from calibration; R is the radius of the nanodisc. Therefore, two independent parameters are used to adjust the calculation with the data in Fig. 2e: the damping ratio d to fit the overall shape (position of the minimum) and the critical velocity Vc to fit the absolute value of the energy. The optimal pulse duration wmin corresponding to the minimum energy Emin* follows from equation (2): wmin=1.26/(dforced*ωG)=1.26τforced (for the blue solid line in Fig. 2e, dforced*=0.018, τforced=35 ns, hence Emin* is reached for wmin=44 ns). The fitted critical velocity in our NiMnSb disc is Vc≃190 m s−1, in good agreement with the expected value9 of 225 m s−1(the exchange constant of NiMnSb is Aex=6 pJ m−1; ref. 30). The same parameters are used in the calculations of Figs 2 and 4. In Fig. 4, the only new fitting parameter is the characteristic time τfree of the vortex-core free decay (the relaxation is assumed to be exponential). We also allow a fine adjustment of the phase shift between the two pulses to obtain the best agreement with the data (25° in the ‘zero-phase-shift’ experiment and 190° in the ‘π-phase-shift’ experiment; these small differences are ascribed to an imperfect delay line in the experimental pulse set-up).
This research was partially supported by the French Grant Voice ANR-09-NANO-006-01, EU Grants DynaMax FP6-IST-033749 and Master NMP-FP7-212257. The authors acknowledge discussions with V. S. Tiberkevich, A. N. Slavin and K. Y. Guslienko.
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Nature Physics (2012)