Generation and stability of dynamical skyrmions and droplet solitons

Nahuel Statuto; Joan Manel Hernàndez; Andrew D Kent; Ferran Macià

doi:10.1088/1361-6528/aac411

The control of magnetic states in nanostructures without using magnetic fields is now possible with the discovery of the spin-transfer torque (STT) effect [1–3]. A spin-polarized current can transfer angular momentum to a magnetic material [4] and modify its magnetization. The STT effect is used in the control of both static and dynamic magnetic states; one can switch the magnetization direction of a magnetic layer within a nanopillar or create coherent spin waves in an extended thin film [5]. In particular the STT effect can be used to nucleate and control solitonic modes—magnetization states that behave as particles. These self-localized magnetic objects include magnetic domains, vortices, bubbles, or skyrmions [6–13] and they are receiving a growing interest since they can be topological and can present non trivial topology and, thus, more stable against perturbations such as thermal fluctuations or fabrication defects [14–16]. Besides the possibility of nucleation and control of static solitonic modes, including skyrmion creation [17, 18], the STT effect is also used to excite their dynamical counterparts consisting in oscillating modes that are unstable in dissipative materials—damping is present in all magnetic materials and suppresses these excitations. However, damping can be now compensated locally by the STT effect, for example, with an electrical point contact providing a spin-polarized current [19–22].

Dissipative magnetic droplet solitons (droplets hereafter) are nonlinear localized wave excitations consisting of partially reversed precessing spins that can be created in films with perpendicular magnetic anisotropy (PMA) [23]. Droplets have been experimentally created using the STT effect in electric nanocontacts to PMA films [24–30]. Droplets are magnetic nano-oscillators and have a growing interest as key elements in neuromorphic computation [31, 32] and in communication devices [33]. Droplets are topologically trivial objects—they can be created continuously from a uniform ferromagnetic state where all spins are aligned in the same direction. A similar magnetic object having topologically non-trivial spin texture could be created in a similar experimental geometry: a dynamical skyrmion (DS). Zhou et al [34] have shown with micromagnetic simulations that DS can be nucleated and sustained with a large spin-polarized current in a nanocontact and are, indeed, fundamental solutions for the magnetization excitations in a film with PMA [35]. Liu et al [36] presented an experimental observation of a solitonic mode modulation that could indicate the existence of a DS. So far, the topology modification of droplets has been associated to the Dzyaloshinskii–Moriya interaction (DMI) present in some magnetic films [37].

A schematic plot of both solitonic modes, droplet and DS, is shown in figure 1 where the blue region represents magnetization pointing out-of-plane (θ = 0) and the brown, in the opposite direction (θ = 180°). The magnetization of a droplet or a DS is precessing, with a small amplitude near its center and with a larger amplitude at the boundaries. The lower panels of figure 1 show the magnetization orientation in a transversal cut of both droplet and DS. The main difference is in the region separating the center of soliton from the rest of film's magnetization; droplets have no topology (the magnetization shown in figure 1(a) can be transformed continuously into a ferromagnetic state with all spins aligned in any arbitrary direction) whereas the DSs have topology (such a transformation is not possible). The topology can be described by the skyrmion number (S), which is calculated mathematically as $S=-\tfrac{1}{4\pi }\int {\bf{m}}\cdot ({\partial }_{x}{\bf{m}}\times {\partial }_{y}{\bf{m}})$ . A droplet has S = 0, and a DS has S = 1.

Here, we investigate the conditions that lead to either droplet or DS formation and we study their stability in nanocontacts to ferromagnetic thin films with PMA and without interfacial DMI. Our micromagnetic simulations show that the Oersted fields associated with the localized electrical current, the initial magnetization state, and the rise time of the injected current, play a key role on determining whether droplet or DS form. DS are more stable to perturbations and can be sustained with much lower currents than droplets. We also provide characteristic features of droplets and DS that could distinguish the two magnetic objects experimentally.

Results

Simulations details

We consider a circular nanocontact to a ferromagnetic thin film with PMA. The parameters for the material are taken from experiments using 4 nm thick Co and Ni multilayers [27, 38]. Magnetization saturation, M_s = 5 × 10⁵ A m⁻¹, damping constant, α = 0.03, uniaxial anisotropy constant, K_u = 2 × 10⁵ J m⁻³, exchange stiffness constant, A = 10⁻¹² J m⁻¹, and a nanocontact diameter of 150 nm for most of the presented results. We modeled the magnetization dynamics in the nanocontact by solving the Landau–Lifshitz equation adding the STT term [1] with a constant spin-polarization. We performed micromagnetic simulations using the open-source MuMax code [39] using a graphics card with 2048 processing cores. We considered the effects of Oersted fields but we did not include interfacial DMI. The applied magnetic field is perpendicular to the film plane. Temperature effects are not considered unless otherwise stated (full codes are available in the supplementary materials online at stacks.iop.org/NANO/29/325302/mmedia).

Creation process

To excite a droplet in a ferromagnetic layer with PMA using a spin-polarized current in a nanocontact, the STT must compensate the damping. There is a threshold current that depends on the NC size, the saturation magnetization, the film thickness, the spin-polarization of the current, and the external field [23, 28, 30, 40]. Such a threshold does not depend on the initial magnetization state or on whether the spin-polarized current is applied adiabatically or abruptly. For currents above the threshold, the magnetization in the NC forms a droplet state in a process that can take less than a nanosecond [41]. Once the droplet is created, the current in the nanocontact is still required to sustain the magnetic excitation—although smaller current values than the threshold current are needed [24, 27]. Droplet states can be inferred by measuring the dc resistance of the nanocontact—a reversal of the magnetization produces a change in the nanocontact resistance [24–30]. Further, the magnetization dynamics of droplets can be detected experimentally through the ac electrical resistance oscillations in the nanocontact [24–27, 29, 30] caused by the precessing magnetization in the droplet.

DS may also form in a NC to a ferromagnetic layer with PMA [34, 36] when a sufficiently large spin-polarized current is applied. The difference between droplet and DS is in the topology of spins at the boundary, which provides additional stability [34, 36]. For this reason, we are interested in determining the differences in stability between droplet and DS as well as the experimental conditions under which DS form.

In simulations we choose an initial magnetization state close to equilibrium—all spins nearly aligned with the applied field—and then we apply a spin-polarized current and record the evolution of magnetization in an area that is 5 times larger than the contact diameter. The initial magnetization state has to be non-collinear with the spin-polarized current in order to allow the STT to have an effect. We, thus, need to provide an initial condition for the mangetization. One option is to provide an arbitrary angle θ_I and ϕ_I for each spin followed by a magnetic relaxation for a given time—this goes asymptotically to a configuration close to equilibrium (θ = 0 for all spins). Another option is to use an initial magnetization state close to the equilibrium given by a small θ_I and a constant value of ϕ_I for all spins. In our simulations we study the effect of initial magnetization states on the formation of droplet and DS and thus we use an initial magnetization state given by an angle θ_I as a parameter in simulations.

Figure 2(a) shows the magnetization precession frequency and amplitude within the NC as a function of the applied spin-polarized current under an applied field of 0.5 T. Each point corresponds to a different current value applied to an initial magnetization condition θ_I = 1.5°. For values of current below 10 mA the magnetization in the NC (the average value) has a small oscillation with a frequency close to the ferromagnetic resonance frequency. Above 10 mA there is an abrupt decrease of the frequency together with an increase of the precession amplitude, which corresponds to the creation of a droplet. If we continue applying current of larger amplitude (always starting from a same initial state), we reach a second threshold at 28 mA where a DS forms, having an almost identical magnetization precession frequency (blue dots) but a much smaller precession amplitude (red dots). The precession amplitude of spins is much larger at the boundary of the soliton than in the central part. Thus, the average nanocontact precession amplitude is mostly driven by the edge precession. In the DS the spins at the boundary precess at a similar amplitude than in droplets but the fact that they are not in phase causes a reduction in the amplitude of the contact's overall magnetization oscillation—which is a feature that can be used to identify DS experimentally. The same argument applies to describe the smooth decrease in the precession amplitude of the NC magnetization in the droplet state as the current increases from 10 mA to 28 mA; the phase of droplet becomes less and less uniform along the overall droplet edge when increasing the applied polarized current [41].

We next study the time evolution of magnetization during the process of droplet and DS formation. Figure 2(b) shows the magnetization evolution in the NC region, m_z, as a response of an applied current for a droplet (yellow line) at 26 mA and for the DS (red line) at 36 mA; both time traces correspond to points in figure 2(a) having an initial magnetization state with θ_I = 1.5°. We see that the higher applied current values produce a faster magnetization reversal, which is something that occurs no matter whether the final state is a droplet or a DS and is caused by the larger STT effect—being proportional to the applied current [41]. We can also observe that the DS (red line) presents a larger oscillation of the magnetization indicating there is a breathing of the localized object at the precession frequency [34, 36]. We note here that the magnetization, m_z, averaged over the NC (plotted in figures 2(b) and (c)) is a relevant quantity for experiments as it can be directly associated to the NC resistance.

The initial state determines whether the response to an applied current is a droplet or a DS. In figure 2(c) we plot time traces for the magnetization, m_z, in the NC for a same applied current value, 30 mA, but different initial magnetization states, θ_I = 2.5° (yellow line) and θ_I = 0.8° (red line). We note that the initial state with θ_I = 2.5° evolves to a droplet state whereas the initial state with θ_I = 0.8° evolves to a DS state. The current threshold for DS formation thus has a dependence on the magnetization initial state. Additionally, we measured the precession frequency of droplet and DS for the case presented in figure 2(c). For the same current both the droplet and DC have nearly the same precession frequency: f = 14.60 GHz for droplet and f = 14.58 GHz for DS, which is not seen in the transition at 27 mA of figure 2(a) due to the small difference. We attribute such a small variation in frequency to the small changes in the size of the magnetic object and therefore in the value of internal magnetic fields—mainly dipolar fields. We note here that theory predicts a variation in frequency with current for droplet states [23]. However, our current densities and material's parameters do not produce a considerable change in the size of the solitonic modes and, thus, we do not observe any variation of frequency with applied current, similar to experimental studies [24–30].

In order to understand how the threshold current for DS formation depends on the initial magnetization state, we repeat the process used in figure 2(a) with different initial magnetization states (different θ_I) and we identify the current values that result in a droplet or a DS. Figure 3 shows the phase diagram of droplet and DS formation as a function of applied current and initial magnetization angle. We see that the threshold for droplet formation is always the same independent of the initial magnetization state; different initial states cause the process of droplet formation to become faster or slower (see traces for time evolution in the insets of figure 3) [41]. On the other hand, the threshold for DS formation has a strong dependence on the initial magnetization angle, θ_I, increasing with larger angles. An additional map is provided in the supplementary materials showing the phase diagram of droplet and DS formation as a function the polarization of the applied current and the initial magnetization angle (θ_I) for a fixed current of 30 mA. In that case the Oersted-field effects are fixed and only STT effects vary with spin-polarization. At a small polarization, there is a small STT effect and no excitations are present independent of the initial magnetization. As the current polarization increases we found first the onset of droplet states and with a further increase the onset of DS. Again the droplet threshold does not depend on the initial state whereas the DS threshold has a strong dependence requiring larger values of polarization at larger angles of the initial magnetization angle, θ_I.

**Figure 3.** Phase diagram of the droplet and DS formation creation of both solitonic modes as a function of the applied current, I (with polarization p = 0.45), and the initial magnetization angle, θ_I. For currents below 10 mA neither droplet nor DS can be excited, orange region. When the current is higher than 10 mA a droplet is excited and droplet's threshold current does not depend on θ_I, yellow region. If the current is further increased, a DS is created, green region. The current threshold for DS (red line) is higher than the droplet and depends on the initial magnetization state, θ_I. Insets correspond to time evolution curves of nanocontact magnetization at different conditions.
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Next we study the influence of other factors such as the size of the nanocontact and the spin-polarization. An increase of polarization from p = 0.45 to p = 0.6 reduces the droplet threshold by 2 mA and reduces the DS threshold by 5 mA. The contact size determines the net current required to excited solitonic modes. We computed the thresholds for contact diameters of 50 and 100 nm and obtained values of 4 and 6 mA for the droplet threshold—which represents a decrease of 3 and 5 mA with respect to the diameter of 150 nm presented in figure 3. Here we note that the threshold does not scale exactly with the current density because there are always Oersted fields associated with the currents that depend also on the contact size. We observed a larger reduction of 5 and 9 mA for the DS formation. Both diagrams are presented in the supplementary materials.

The applied magnetic field increases, the required current for soliton formation [24–30]. We have studied the effect on the droplet and DS thresholds in a range of applied fields between 0 and 2 T and found that both current thresholds increase equally (and linearly) with the applied field (see supplementary materials).

The temperature modifies the initial magnetization state—a temperature of 300 K provides a fluctuation of spins larger than θ = 10° for our studied configuration. However, the effect of temperature cannot be compared directly to the case studied in figure 3 because temperature also provides fluctuations of spins during the creation process. We have simulated a new phase diagram of the droplet and DS formation as a function of temperature in figure 4. We applied a spin-polarized current after an initial relaxation. We can see in figure 4 that droplet thresholds are the same we obtained with no temperature whereas DS thresholds increase with increasing temperature.

**Figure 4.** Phase diagram of the droplet and DS formation. The initial condition is here obtained after magnetic relaxation at a given temperature in the presence of an applied field of 0.5 T. Top panels show the evolution of magnetization (m_z in colorscale and m_x and m_y with vector arrows) for a T = 300 K and a current of I = 27.5 mA. A dynamical skryrmion formed. For currents below 10 mA neither droplet nor DS can be excited at any temperature, orange squares. When the current is higher than 10 mA a droplet is excited and the droplet's threshold current does not depend on temperature, blue circles. If the current is further increased, DS are created, red circles, at a different currents for each temperature.
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Stability

Both droplet and DS exhibit magnetic bistability over considerable ranges of applied current and magnetic field [24–30, 34, 36]. We investigate here the conditions that produce the annihilation of the solitonic modes when a lower degree of spin-transfer torque—a lower current—is applied. In figure 5(a) we show two curves corresponding to the average magnetization within the NC, m_z, as the applied current decreases from an initial value of I = 30 mA. A droplet and a DS are created at 30 mA (using θ_I = 3° and θ_I = 0.1° respectively). The droplet collapses at about 9 mA whereas the DS requires a much lower current value of 4 mA to vanish revealing that the DS remains stable over a larger range of applied currents or in other words, the DS requires smaller current values to be sustained.

**Figure 5.** Stability of droplet and DS (a) curves of annihilation of droplet (red line) and DS (blue line) with decreasing the applied current. The curves correspond to the averaged normalized magnetization within the nanocontact, m_z, as the applied current decreases from an initial value of I = 30 mA. Both droplet and a DS are created at 30 mA (using θ_I = 3° and θ_I = 0.1° respectively). The droplet collapses at about 9 mA whereas the DS collapses at a lower current of about 4 mA. (b) Time evolution of m_z for a droplet (red line) and a DS (blue line) in the presence of an in-plane magnetic field. Both droplet and DS states are created at 30 mA using different initial states (same as in (a)) and after stabilization the applied current is reduced to 10 mA, black squares in (a), and a small in-plane field of 50 mT is applied. The magnetization of droplet and DS behaves completely differently; the droplet's magnetization oscillates caused by a drift resonance (∼40 MHz) while the DS's magnetization, although it initially oscillates, it stabilizes after ∼80 ns and remains with its initial (S = 1) topology.
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Next, we investigate the effect of a magnetic field gradient in the NC. A small constant in-plane field, a small change in anisotropy, or a variation in the film's thickness combined with the Oersted fields from the charge current could result in a gradient of effective magnetic field in the NC that dephases the precession of magnetization in different locations of the NC and eventually may annihilate the magnetic excitation. Experiments revealed that the low frequency noise in droplets [28, 30, 40] is associated with a periodic process of shifting, annihilation, and creation. Simulations showed that an asymmetry of the effective field causes a drift instability resulting in an oscillatory signal of hundreds of MHz—a drift resonance. We excite droplet and DS states at 30 mA using different initial states (same as in figure 5(a)) and after a stabilization period we reduce the applied current until 10 mA, black squares in figure 5(a). We then apply a small in-plane field of 50 mT in order to destabilize the solitonic modes. The combination of a fixed in-plane field with the Oersted fields creates an in-plane field gradient in the nanocontact. Figure 5(b) shows the time evolution of the magnetization for a droplet (red line) and a DS (blue line). The small in-plane applied field causes a shift of the droplet away from the NC followed by a re-nucleation of a droplet state. The process of creation and annihilation is repeated at a frequency in the MHz range (∼40 MHz) [28]. The effect of an in-plane field to the DS is different; DS has an initial change as a result of the abrupt change of the magnetic field but later on the DS stabilizes again. Full videos of the evolution of droplet and DS in figure 5(b) are available in the supplementary materials.

Discussion

Two main effects are involved in the magnetization dynamics when a spin-polarized current flows through a nanocontact to a magnetic film. On the one hand, a spin-polarized current of the appropriate polarity interacts with the magnetization via the STT effect trying to align the magnetization in the opposite direction of the applied field. The STT effect is proportional to the non-collinear component of the magnetization with respect to the polarization of the current (i.e., if the magnetization is precisely aligned in the direction of the polarized current, say z for the studied case, there is no effect). On the other hand, the electrical current flowing through the nanocontact causes Oersted fields that curl the magnetization.

In the creation process of solitonic modes there is a competition between the two mentioned effects. The STT effect increases rapidly as the magnetization tilts from the state perpendicular to the film plane, θ_I = 0°, and thus if the initial magnetization state is sufficiently far from such a state, the solitonic mode forms without topology resulting in a droplet state. On the other hand if the initial state is closer to θ_I = 0° the effect of STT produces a much slower variation of the magnetization and there is a time lapse where the effect of the Oersted fields provides topology to the magnetization in the NC, which eventually results in the formation of a DS. In summary, the farther from equilibrium the initial magnetization state is, the larger the current density required to create a DS is.

Preparing experimentally an initial state close to θ_I = 0° requires low temperatures. These states are usually avoided in other situations such as in a magnetic tunnel junction devices because they reduce the switching probability at a given time [42]. It is possible, on the other hand, to prepare initial states with ${\theta }_{I}\ne 0$ by increasing the temperature, by applying a short in-plane field pulse or by using a polarized current not collinear with the magnetization state (e.g., using canted polarizers). We provide simulations in supplementary materials showing that the formation of a DS is avoided with a short in-plane magnetic field pulse or by using a canted polarizer.

There is another ingredient that plays a role in defining whether a droplet or a DS forms: the speed of ramping the polarized current from zero, or from a small value, to a high value that nucleates solitonic modes. Simulations in the diagram shown in figure 3 are done with a sharp step of current. However, we have seen that using ramping currents with a rise time (10%–90%) larger than 700 ps suppresses the formation of DS in favor of droplets. We include in supplementary materials a simulation comparing current rise times of 100 ps and 1 ns with identical initial conditions showing the formation of DS and droplet states respectively.

We, thus, speculate about the possibility of observing DS experimentally. Typically, an experimental setup used for the study of droplets contains a free layer with PMA where the solitonic modes may form, which corresponds to our simulated CoNi layer, and a fixed layer that is used as a spin polarizer for the current, [24–30, 43]. To create a DS instead of a droplet we need to either depart from an initial magnetization state close to all perpendicular or produce torques associated with the Oersted fields larger than those associated with the polarized currents. In the first case we can try to apply large out-of-plane fields in order to set and appropriate initial magnetization state or lower the temperature to reduce the thermal noise that might produce fluctuations of the magnetization. It could be that experiments performed at low temperatures and large fields [27, 40] have already created DS. The second case consists in providing a large current that is not polarized, producing large Oersted fields but no STT effect. With the same configuration, the current polarization has to increase so that the STT becomes predominant and promotes the creation of a solitonic mode. If the magnetization was already curled due to the Oersted fields it could result in the creation of a solitonic mode with topological protection: a DS.

This realization is feasible by using a perpendicular polarizer [43] with appropriate coercivity so that it switches at a field that can create the DS. One could then apply a constant current to the nanocontact while the magnetic field sweeps, the perpendicular polarizer would then switch and create an abrupt change in the spin-polarization of the applied current while maintaining the Oe fields. We added in the supplementary materials simulations where the polarization is switched on in presence of an applied current (with no spin-polarization) and found that DS can be create at certain values of the polarization with no need of an artificial canted initial magnetization state—the Oe fields associated to the electrical current with no spin-polarization provide an initial magnetization state with ${\theta }_{I}\ne 0$ .

It is necessary however to distinguish experimentally the two solitonic modes once they are created. The differences in precession frequency are too small to serve as a signature of droplet or DS. Instead, studying the stability of the solitonic modes is the best option. One could study the hysteretic response or the response to small in-plane fields and the appearance of low frequency noise as seen in figure 5.

In conclusion we have shown that both droplet and DS can be created with a same configuration of applied field and spin-polarized current by controlling the initial magnetization state, the degree of spin-polarized current, or the speed at which the current—or the polarization—is changed. We also studied the difference in stability between droplet states and DS and found that DS is not only more stable against effective field variations but DS also requires much lower currents to be sustained. Our results provide a pathway for experimental studies of DS and their stability.

Acknowledgments

FM acknowledges financial support from the Ramón y Cajal program through RYC-2014-16515 and from MINECO through the Severo Ochoa Program for Centers of Excellence in R&D (SEV-2015-0496) and (MAT2017-85232-R). JMH and NS acknowledge support from MINECO through MAT2015-69144-P. Research at NYU was supported by Grant No. NSF-DMR-1610416 and DARPA award No. D18AP00009.

Generation and stability of dynamical skyrmions and droplet solitons

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