Generation and annihilation time of magnetic droplet solitons
Jinting Hang, Christian Hahn, Nahuel Statuto, Ferran Macià, Andrew D. Kent
GGeneration and annihilation time of magnetic droplet soli-tons
Jinting Hang +1 , Christian Hahn +1 , Nahuel Statuto , , Ferran Maci`a , and Andrew D. Kent ∗ Center for Quantum Phenomena, Department of Physics, New York University, New York, NewYork 10003 USA Department of Condensed Matter Physics, University of Barcelona, 08028 Barcelona, Spain Institut de Ci`encia de Materials de Barcelona (ICMAB-CSIC), Campus UAB, 08193 Bellaterra,Spain + These authors contributed equally to this work. ∗ Corresponding author: [email protected]
Magnetic droplet solitons were first predicted to occur in materials with uniaxial magneticanisotropy due to a long-range attractive interaction between elementary magnetic excita-tions, magnons. A non-equilibrium magnon population provided by a spin-polarized currentin nanocontacts enables their creation and there is now clear experimental evidence for theirformation, including direct images obtained with scanning x-ray transmission microscopy.Interest in magnetic droplets is associated with their unique magnetic dynamics that canlead to new types of high frequency nanometer scale oscillators of interest for informationprocessing, including in neuromorphic computing. However, there are no direct measure-ments of the time required to nucleate droplet solitons or their lifetime—experiments to dateonly probe their steady-state characteristics, their response to dc spin currents. Here wedetermine the timescales for droplet annihilation and generation using current pulses. An-nihilation occurs in a few nanoseconds while generation can take several nanoseconds to amicrosecond depending on the pulse amplitude. Micromagnetic simulations show that thereis an incubation time for droplet generation that depends sensitively on the initial magneticstate of the nanocontact. An understanding of these processes is essential to utilizing theunique characteristics of magnetic droplet solitons oscillators, including their high frequency,tunable and hysteretic response. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r agnetic droplet solitons form in spin transfer torque oscillators that consist of nanome-ter scale contacts to thin films with perpendicular magnetic anisotropy
3, 4, 8–10 , as predicted by the-ory . They have been studied in thin films with two ferromagnetic layers, a free layer and a fixedspin-polarizing layer, separated by a thin metallic layer. Droplets consist of a partially reversedcoherently precessing magnetization in and near the nanocontact region. Through the giant mag-netoresistance effect, droplet formation leads to a step increase in nanocontact resistance as wellas a sharp peak in the high-frequency spectra at the spin-precession frequency when overcominga threshold current
3, 4 . There have been a number of proposed applications of magnetic dropletsolitons that rely on their unique magnetic dynamics . First, their output signal can be largerbecause of their large angle spin excitations giving a larger magnetoresistive response
12, 13 . [Whilemagnetic droplet solitons have been studied thus far in metallic multilayers, forming droplets in amagnetic tunnel structure would provide a larger output signal because of their larger magnetoe-sistive response.] Second, their response is highly tunable with current and hysteretic . Third,their oscillation frequencies can be one to two orders of magnitude larger than that of magneticvortex oscillators , recently used in neuromorphic computing demonstrations
15, 16 , thus enablinghigher speed device operation. Applications require knowledge of the time required to nucleatedroplet solitons and their lifetime as well as an understanding of the nucleation and annihilationprocesses. Thus far experiments only probe the steady-state characteristics of droplet solitons, i.e.their properties when the current has been on or off for long periods compared to the time scale oftheir intrinsic dynamics.In this article we report measurement of the current pulse times for droplet generation andannihilation. In these studies we use the fact that droplet solitons exhibit magnetic bistability; thatthere are large ranges of applied current and field at which both droplet and non-droplet states arepossible . The nanocontact state thus depends on its prior history. By biasing a nanocontact ina bistable condition and applying current pulses we determine the pulse amplitude and durationneeded to generate and annihilate droplet solitons.The experiments were performed on nanocontacts to metallic multilayers that consist of aperpendicularly magnetized free layer and easy-plane fixed layer (Fig. 1a). These layers are sep-2rated by a non-magnetic spacer layer that decouples them magnetically while enabling a flow ofspin-polarized current. (See the Methods Section for the layer compositions.) All measurementsreported here were done at room temperature.A step increase/decrease in dc-resistance is associated with the creation/annihilation of thedroplet state, which can be used for determining the critical current needed to drive the dropletdynamics. The step in resistance corresponds to an abrupt change in the high-frequency spectraof the nanocontact, consistent with droplet creation (a step decrease in the noise frequency) andannihilation (a step increase in the noise frequency), as reported in Refs.
3, 4 (see SupplementarySection II for nanocontact spectral data.) For a certain field range (see Supplementary SectionI and Ref. ), one finds a current zone where the droplet is bistable, i.e. both droplet and non-droplet state are possible for the same current. For a fixed applied magnetic field in the bistablezone, two critical currents are observed: I c below which the droplet state does not form and I c ( > I c ) above which there is only a droplet state. Between the two critical currents a hystereticresponse is observed. In Fig. 1b we show the nanocontact resistance at a fixed applied field of 0.7 Twhile ramping the current up (red curve) and down (blue curve). Here we find I c = 12 . ± . mA and I c = 14 . ± . mA. The uncertainly is the standard deviation in the currents I c and I c determined by repeated I-V measurements. The distributions of I c and I c are shown ashistograms that are overlaid on Fig. 1b. We note that the standard deviation of the generationcurrent is twice that of the annihilation current, showing there is a greater stochasticity in thegeneration process.To study droplet generation and annihilation we bias the nanocontact with a current in thehysteretic region and apply pulses to momentarily change the current. For example, to studydroplet annihilation we initialize the nanocontact in the droplet state by ramping the current up to I > I c and then reduce the current to a range I c < I < I c . At a current within the hystereticregion (indicated by the dashed line in panels b-d of Fig. 1), we apply a negative polarity pulse tomomentarily reduce the current below I c . There is a step down in nanocontact resistance if thedroplet has been annihilated, as seen in Fig. 1c. To study droplet generation, we start in a non-droplet state, again in the hysteretic region I c < I < I c (indicated by the dashed line in Fig. 1d),3 pacerpolarizernano-contactfree-layer Figure 1: a , Schematic of a multilayer with a magnetic droplet. Electrons are injected through a nanocontact to thefree layer and move from the free layer (red) to the polarizer (blue) for positive current polarity. An external fieldis applied perpendicularly to the film plane, partly canting the polarizer magnetization. b , Resistance vs. current ata fixed field of 0.7 T after subtracting a background caused by Joule heating. The overlaid histogram indicates thedistribution of generation and annihilation currents. c , The nanocontact is biased at 13.5 mA in the higher resistancestate (dashed line). A negative current pulse annihilates the droplet, as seen by the step decrease in nanocontactresistance. d , Starting in the non-droplet state a positive current pulse can generate the droplet, as seen by the stepincrease in nanocontact resistance. and apply a positive polarity current pulse to increase the current above I c . Droplet generationcauses a step increase in the nanocontact resistance as seen in Fig. 1d. We repeat these pulsesmultiple times to determine the generation and annihilation probabilities as a function of pulseamplitude and duration.The droplet annihilation probability as a function of pulse duration for different pulse am-plitudes is shown in Fig. 2. We indicate the standard error of the mean as error bars, SEM = √ ( (cid:80) ni =0 ( x i − µ ) / (( n − n )) , with x i as the individual trial result, µ being the mean value and n the number of trials, n = 50 for the results in Figs. 2 & 3. In Fig. 2, we observe a monotonicincrease of the annihilation probability as a function of pulse duration. Varying the applied pulseamplitude from 2.24 mA to 3.15 mA we also see an increase in the probability with increasingpulse amplitude, reaching 100 % annihilation probability at 2 ns for 3.15 mA.We then proceed to study the times needed to generate the droplet by initializing the nanocon-tact in the non-droplet state and applying positive current pulses. Again, we start with a bias cur-rent of 13.5 mA and apply varying pulse durations and amplitudes that briefly increases the currentabove the threshold current I c . The probability for droplet generation is shown as a function ofpulse duration for different pulse amplitudes in Fig. 3. Similarly to the annihilation probability we4 .5 1.0 1.5 2.0 2.5 3.0 t Pulse (ns) P r o b a b ili t y I Pulse [mA]2.242.522.823.15
Figure 2: The annihilation probability versus pulse duration at different pulse amplitudes. Pulses were added to a dccurrent of 13.5mA and a 0.7 T field was applied. The lines are guides to the eye. observe a monotonic increase with the pulse duration. The pulse duration is plotted on a logarith-mic scale to encompass the range of generation times observed. For the highest pulse amplitudeapplied in the annihilation experiment (3.15 mA), we obtain 50% generation probability at a pulseduration of 70 ns. We achieve 50% creation probability for a 3.93 mA amplitude and 10 ns du-ration pulse. We note that we have also examined samples with 150 nm diameter nanocontacts.These nanocontacts show similar behavior, exhibiting a monotonic increase of generation and an-nihilation probability with pulse duration; they display an even stronger asymmetry between thegeneration and annihilation pulse durations, necessitating yet one order of magnitude longer pulsesfor droplet generation than shown in Fig. 3 yet still short (ns) pulses for annihilation. (Data on a150 nm diameter nanocontact is shown in Supplementary Section III.)These results show that the time needed to enter the droplet state can be longer than forexiting it; for the same current pulse amplitude it can take orders of magnitude longer to createthe droplet than to annihilate it. This indicates a different mechanism involved in annihilation andgeneration processes. To better understand this asymmetry we modeled the generation and theannihilation processes of magnetic droplets. We consider a circular nanocontact to a ferromag-netic thin film with perpendicular magnetic anisotropy with parameters taken from experiment.(Parameters are given in the Methods Section.) The inset in Fig. 4b, left panel, shows the averagez-component of magnetization in the nanocontact m z as a function of the applied current. Clear5 t Pulse (ns) P r o b a b ili t y I Pulse [mA]2.823.153.513.93
Figure 3: The generation probability versus pulse duration for different pulse amplitudes. The dc current was fixed at13.5mA and a 0.7 T field was applied. The lines are guides to the eye. hysteresis is seen, like in the experiment; higher current is required to generate the droplet withincreasing current, while droplet annihilation occurs at lower current when decreasing the current.To study droplet annihilation and generation we start at a dc current in the hysteretic zone,where both droplet and non-droplet states are possible. As in the experiment, the current is mo-mentarily reduced to study droplet annihilation and increased to study droplet creation. Fig. 4bshows the time evolution of the magnetization of the nanocontact in the annihilation process, leftpanel, and for the generation process, right panel. Fig. 4c shows the applied current as a functionof time. In the annihilation process a negative polarity pulse of − . mA is applied to a dc currentof . mA. The pulse duration needed to annihilate is . ns. On the other hand, in the generationprocess, starting in a non-droplet state, a (positive polarity) pulse of . mA is applied. Under theseconditions it takes ns to create a droplet. After ns the current is reduced to the initial current( . mA) and this current sustains the droplet. We thus see that for the same pulse amplitudethe generation and annihilation processes occur on different time scales. The generation processhas a waiting time and after that time the droplet forms rapidly, in a transition time (20 % to 80%of the initial to final m z ) of just nanoseconds . The images in Fig. 4a show the magnetization atdifferent times in the generation and the annihilation processes. We note that the final state of thegeneration process and the initial state in annihilation process are not the same, as after some timethe droplet moves in the nanocontact and the spins at the boundary of the droplet dephase (see6upplementary Section IV).It is important to note that a long droplet generation time is not fundamental; the droplet gen-eration time can be reduced and made comparable to the annihilation time by a number of means.For example, larger current pulse amplitudes lead to 10 ns generation times, as seen in Fig. 3 for I pulse = 3 . mA. Further the incubation time is a very sensitive function of the initial magnetiza-tion state of free layer. An initial state with the magnetization tilted with respect to anisotropy axisby just a few degrees leads to nanosecond generation times, as discussed in the SupplementarySections V and VI. This can be achieved by applying an in-plane field pulse (see SupplementarySection VI). Finally, while in our experiments the nanocontact is initially biased with a dc current,droplets can also be generated by applying pulses starting with a no bias current (see Supplemen-tary Section VI). A bias current is applied in our experiments for a practical reason—it enables avoltage readout of the nanocontact state after a pulse.The duration of current pulses needed for generation and annihilation are thus different. It isinteresting to compare the droplet annihilation time to the magnon relaxation time τ m = 1 / (2 παf ) ,where α is the damping constant and f is the spin-precession frequency. For α = 0 . and aspin-precession frequency of 20 GHz, the magnon relaxation time is 0.2 ns, about an order ofmagnitude smaller than the observed droplet annihilation time. In micromagnetic simulations thedroplet drifts across the boundary of the nanocontact, as seen in Fig. 4a, and dissipates withinin a few nanoseconds after having left the region experiencing a spin-transfer torque. In contrastto droplet annihilation, where the magnetization relaxes to equilibrium, the generation processcan be viewed as a transition between magnetic states. In this case, similar to the switching of amacrospin between two stable magnetic states by a spin-transfer torque, there is an incubation timeassociated with building up spin precession angle starting from a small initial angle
17, 18 . This canalso be viewed as a time required to generate a sufficient number of magnons to form a dropletsolition. Once initiated (i.e. after the waiting time) the formation process is fast, comparable tothe times needed for droplet annihilation (several ns and less). We find that the incubation timebecomes smaller with increasing initial magnetization angle and generation pulse amplitude (seeSupplementary Section IV). 7 -101-10 a m z I ( m A ) Time (ns)
No Droplet cb − − Time (ns) t Pulse =1.3 ns I Pulse =1.7 mA
Droplet I c1 Pulse m z − − I (mA)
12 13 14 15 I c2 Pulse
No Droplet t=1 ns m z m x − − No Droplet DropletDroplet t Pulse =21 ns I Pulse =1.7 mA
Figure 4: Creation and annihilation process. a , Images of the magnetization at times in the simulation. Imagescorrespond to a × nm field of view. The blue circle shows the boundary of the nanocontact. b , Timeevolution of the nanocontact magnetization for the annihilation and the creation processes, left and right images,respectively. The black squares correspond to the times shown in the images. c , Current applied as a function oftime for the annihilation and generation processes. The vertical dashed black lines shows the time where the pulsewas applied. The inset of b , left hand panel, shows the average magnetization of the nanocontact as a function of thecurrent and the resulting hysteresis: I c = 13 . mA and I c = 14 . mA and between these currents both droplet andnon-droplet states are stable.
8n summary, we report measurement of the timescales for droplet generation and annihila-tion, which has led to an improved understanding of these processes. This understanding allowsnew means to control and modify these processes. For example, the droplet incubation time canbe greatly reduced by increasing the pulse amplitude or creating an initial tilt of the magnetization.We also note that the measured annihilation time underestimates the actual droplet lifetime, as it ispossible for the droplet to move away from the nanocontact and still exist after application of thepulse. Finite temperature and noise processes must also play a role in generation and annihilationprocesses. These require additional analysis and micromagnetic simulations that we have not con-sidered in this work. It will be interesting to study dynamical skyrmions
13, 19 , which are expectedto be longer lived excitations, as well as to examine the effect of temperature on the stability andformation of droplet solitons.
Methods
The multilayers are composed of 10 Ni Fe |
10 Cu | [0.2 Co | × dielectric layer. Permalloy(Ni Fe ) is the fixed layer and CoNi is the free layer and has perpendicular magnetic anisotropy.The Cu layer between the fixed and free layer is thinner than its spin-diffusion length enablingeffective spin-transport between the layers. The effective perpendicular anisotropy field M eff ofthe CoNi layer was determined using ferromagnetic resonance spectroscopy with the applied fieldperpendicular to the film plane, µ M eff = µ ( H k − M s ) = 0 . T, where µ is the permeability offree space, H k perpendicular anisotropy field, and M s is the saturation magnetization .E-beam lithography was used to define the nanocontact, a − nm diameter aperture inresist. The resist pattern was transferred into the SiO dielectric capping layer using reactive ionetching. The resulting device structure is show in Fig. 1(a), where the point-contact is indicatedas a non-shaded area on top of the free layer. Electrons flow from the free layer to the polarizinglayer for positive current polarity ( I > ). The field is applied perpendicular to the film planeto cant the magnetization of the polarizer partly out of the plane. The free layer magnetization9ynamics is illustrated by a grid of arrows. In the area of current flow, the out of plane magnetizedfree layer has nearly reversed its magnetization, as it is the case when I > I c . We use the highfrequency port of a bias-T to couple short pulses into a nanocontact and contact the sample usinga ground-signal-ground probe.The parameters for the micromagnetic simulations were: saturation magnetization, M s =5 · A/m, damping constant α = 0 . , uniaxial anisotropy constant K u = 2 · J/m , exchangestiffness A = 10 − J/m and a nanocontact diameter of nm. We assumed an electrical currentwith a spin polarization p = 0 . passing through the nanocontact and an applied field of . Tout of the film plane. We performed micromagnetic simulations using the open-source MuMax code with a graphics card with 2048 processing cores. We considered the effects of Oersted fieldsbut not interfacial Dzyaloshinskii-Moriya interactions (DMI) or finite temperature. The simulationof m z as a function of the applied current in the inset of Fig. 4b, left panel, was performed with ns of simulation time for each applied current. Pulse current simulation start with a dc current of I = 13 . mA, which is in the hysteretic zone. The full code is available in Supplemental SectionV. Acknowledgements
F.M. acknowledges financial support from the Ram´on y Cajal program through Grant No. RYC-2014-16515 and from MINECO through the Severo Ochoa Program for Centers of Excellencein R&D (Grant No. SEV-2015-0496). N.S. acknowledges funding from SURDEC through theresearch training grant FI-DGR. Research at UB is partially supported through project MAT2015-69144-P (MINECO/FEDER, UE). Research at NYU was supported by Grant No. NSF-DMR-1610416.
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F.M. acknowledges financial support from the Ram´on y Cajal program through GrantNo. RYC-2014-16515 and from MINECO through the Severo Ochoa Program for Centers of Excellence inR&D (Grant No. SEV-2015-0496). N.S. acknowledges funding from SURDEC through the research train-ing grant FI-DGR. Research at UB is partially supported through project MAT2015-69144-P (MINECO/FEDER,UE). Research at NYU was supported by Grant No. NSF-DMR-1610416.
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