SUPPLEMENTARY INFORMATION

Transcription

1 Supplementary Information for Commensurability and chaos in magnetic vortex oscillations Sebastien Petit-Watelot 1,2, Joo-Von Kim 1,2,*, Antonio Ruotolo 3,4, Ruben M. Otxoa 1,2, Karim Bouzehouane 3, Julie Grollier 3, Arne Vansteenkiste 5, Ben Van de Wiele 6, Vincent Cros 3, Thibaut Devolder 1,2 1 Institut d Electronique Fondamentale, Univ. Paris-Sud, Orsay, France 2 UMR 8622, CNRS, Orsay, France 3 Unité Mixte de Physique CNRS/Thales and Univ. Paris-Sud, 1 avenue A. Fresnel, Palaiseau, France 4 Department of Physics and Materials Science, City University of Hong Kong, Kowloon, Hong Kong 5 Department of Solid State Sciences, Ghent University, Krijgslaan 281-S1, B-9000 Ghent, Belgium 6 Department of Electrical Energy, Systems and Automation, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium *Correspondence to: joo-von.kim@u-psud.fr Abstract This document contains supplementary information pertaining to the study presented in the paper cited above. In particular, we elaborate on the experimental and simulation methodology, and provide additional experimental and simulation results that lend further support to the claims made in the paper. NATURE PHYSICS 1

2 1. High-frequency electrical measurements The frequency domain setup, shown in Fig. S1, is always used first to identify the regions of microwave oscillations. An rf switch allows to route the signal to the time domain analysis configuration (Fig. S1): a resistive power splitter separates the signal into two parts. The first part is low-pass filtered below 1 GHz and sent to a single shot oscilloscope. The other part of the signal is bandpass filtered (between f1 to f2) and sent to the trigger input of the oscilloscope. Depending on the type of spectrum measured, the passband (f1 to f2) of the analogue filter is configured to isolate the fundamental frequency f1 < F < f2 to trigger on it and then to perform averaging to improve the signal to noise ratio. By comparing the averaged and the single shot curves, we have checked that this procedure was free of artefacts. This procedure could not be implemented under all experimental conditions. First, we only have access to a limited batch of analog filters. Second, our analogue filters have a roll-off of typically 6 db per octave, such that they can isolate the fundamental frequency only provided that none of the k = {2,..., N} harmonics carries more that 6k db more power than the fundamental. Some representative frequency and time domain spectra are given in Fig. S2 for an applied field of µ0h = 13 mt, which corresponds to the data presented in Fig. 2. In the low-current regime (Fig. S2a), we observe a power spectrum with a large harmonic content. The origin of these harmonics can be seen in the corresponding averaged time trace of the voltage oscillations, given in the inset, which shows strong non-sinusoidal behaviour. This is consistent with a noncircular vortex trajectory around the nanocontact. For higher currents in which modulation due to relaxation oscillations is present, the locking ratio can be directly deduced from the sidebands in the power spectral density (PSD) and the time traces. In the latter, one can clearly see the amplitude modulation as a result of periodic core reversal, where a reduction in amplitude appears at every five (Fig. S2b), three (Fig. S2c) and two (Fig. S2d) oscillation periods, corresponding to the locking ratios of fmod / f0 = 1/5, 1/3 and 1/2, respectively. 2. Simulation methods The initial state used for each simulation run was obtained by mimicking the experimental procedure used to nucleate the vortex oscillations. We started the simulation with the free layer in a uniform magnetised state, along the x direction, and applied a magnetic field in the film plane of µ0h = 1 mt along the +x direction with an applied current of 10 ma. This simulated the field ramp used 2 NATURE PHYSICS

3 to switch the free layer magnetisation in the experiment. The ensuing dynamics was computed over 100 ns. We observed that the reversal involves domain nucleation at the centre of the simulation grid, around the nanocontact, which then grows as domain walls sweep across the film (Fig. S3). During this process, a number of vortex-antivortex pairs are created and annihilated, leading to the final steady state which consists of one vortex orbiting around the nanocontact and an antivortex pinned at the bottom edge of the simulation grid (Fig. S3h). This steady state configuration then served as the starting point for each of the simulation runs presented in the paper (Figs. 3 and 4). For the applied perpendicular magnetic field considered, we first ran the simulations over 50 ns, under the starting current of 16 ma, so that the new steady state configuration was reached after some initial transient dynamics. (Note that no in-plane fields are considered in these calculations.) We then continued the simulations for 200 ns to record the magnetisation dynamics, which allowed the power spectrum to be obtained for that current value. This calculation was then repeated for decreasing currents, in steps of 0.1 ma, where the final micromagnetic state at one current step served as the initial state for the next. For the incommensurate states in Fig. 4, the simulation times for certain current steps were extended to a value between 500 ns and 5 µs to ensure that no repeating motifs were observed. 3. Simulations: PSD without antivortex under experimental conditions The presence of the antivortex in the simulations, as discussed in the main text, appears from the initialisation procedure in which the free layer magnetisation is reversed under an applied current of 16 ma flowing into the nanocontact. To verify that the antivortex is important for describing the experimental spectra, we performed simulations in which the gyration involves only a single vortex state but under the same experimental conditions (µ0h = 13 mt). This was achieved by using a generated micromagnetic configuration with a single vortex state as an initial state and running the simulations for 50 ns at I = 16 ma. This ensured that the transient dynamics are unimportant and that steady state gyration was reached. The micromagnetic state at the end of this initial run then served as the initial state for the calculation of the power spectra as a function of current. Starting at I = 16 ma, the simulations were run for 200 ns for each current step of 0.1 ma, where the final micromagnetic state at the end of a simulation run for one current step serves as the initial state for the following current step. NATURE PHYSICS 3

4 The results of the simulations are presented in Fig. S4. The variation of the power spectra with current differs qualitatively from the behaviour seen in the experiment (Fig. 2a) and in simulations with an antivortex present (Fig. 3a). First, no harmonics are seen in the low current limit, which indicates that the vortex trajectory around the nanocontact is circular. This behaviour is expected from theory for a circular nanocontact 1. Second, relaxation oscillations, which appear at around 13.5 ma, result mainly in incoherent modulation of the gyrotropic motion, with very little evidence of phase-locking. This is in stark contrast to the dynamics presented in Fig. 3a, where the dynamics is largely dominated by phase-locked (or commensurate) states. These simulation results show that circular orbits are inconsistent with the phase-locking behaviour seen in the experiment. 4. Simulations: Sensitivity to initial conditions A key signature of a chaotic state is its sensitivity to initial conditions. To check whether the fmod / f0 = case presented in the main text (Fig. 4) exhibits this feature, we performed the following analysis using the simulated time traces. Over a simulation run lasting 5 µs, we identified 127 points within 1-2 nm of each other on the vortex core trajectory in the vicinity of the y axis crossing, for motion in the counterclockwise direction, which served as initial conditions for our analysis. The time evolution of these points were then followed over 10 ns, as shown in Fig. S5. While the points remained clustered over the first few nanoseconds, their trajectories diverged rapidly after that, leading to the scattered positions shown in Fig. S5d after 10 ns. This behaviour is consistent with chaotic dynamics. References 1. Kim, J.-V. & Devolder, T. Theory of the power spectrum of spin-torque nanocontact vortex oscillators. arxiv cond-mat.mtrl-sci, (2010). 4 NATURE PHYSICS

5 Figures I dc v ac 10 kω +68 db RF switch 50/3 Ω 50 Ω Spectrum analyser Nanocontact Low-pass 1 GHz Oscilloscope 50 Ω 50 Ω +22 db Band-pass f 1 -f 2 Oscilloscope Trigger Figure S1 Experimental setup used for the frequency and time domain electrical measurements. A current source supplies a dc current Idc to the magnetic nanocontact. The vac source with the 10 kω series resistance represents the lock-in amplifier used for dc resistance measurements of the device. An rf switch allows either frequency or time domain measurements to be taken in constant experimental conditions. NATURE PHYSICS 5

6 PSD (nv 2 /Hz) a b PSD (nv 2 /Hz) c Frequency (MHz) d Frequency (MHz) Figure S2 Representative power spectra of vortex oscillations at four different currents. (a) 9.3 ma, (b) 11.8 ma, (c) 13.8 ma and (d) 16 ma. The insets show averaged time-domain traces at each current; three oscillation periods are shown in (a), while three modulation periods are shown in (b)-(d). 6 NATURE PHYSICS

7 a b m y m x c d +1 m z 1 e f g h y V AV z x Figure S3 Snapshots of the micromagnetic state during the vortex nucleation process. Each image corresponds to the top view of the entire simulation grid, whose lateral dimensions are nm. Snapshots are given for (a) the initial state, and at instances of (b) 0.2 ns, (c) 0.4 ns, (d) 0.8 ns, (e) 1.2 ns, (f) 1.6 ns, (g) 2 ns and (h) 10 ns after the start of the simulation, respectively. The magnetisation component in the film plane is given by the colour wheel, while the component perpendicular to the plane is given by a grey level. The positions of the vortex (V) and the antivortex (AV) are shown in (g). NATURE PHYSICS 7

8 Frequency (MHz) PSD (arb. units) Current (ma) Figure S4 Current-dependence of the simulated power spectrum in the absence of an antivortex. The power spectrum at each current step was computed from simulated dynamics over 200 ns. Above the onset current of relaxation oscillations (around 13.5 ma), the behaviour is dominated by incoherent modulation (incommensurate phase), which differs qualitatively from the experimental spectra. 8 NATURE PHYSICS

9 a b c d 100 nm Figure S5 Sensitivity to initial conditions. Snapshot of the simulated dynamical system for 127 closely spaced initial points for the vortex core position, indicated by the red dots, at four different times: (a) t = 0 ns, (b) t = 0.1 ns, (c) t = 1 ns and (d) t = 10 ns. The grey lines represent the entire vortex trajectory computed over 5 µs and the yellow circle represents the nanocontact, drawn to scale. NATURE PHYSICS 9