## Magnetic Droplet

When a solitary wave travels atop the surface of a fluid, its shape generally changes with time, with some of its components traveling at velocities slightly different than others. In nonlinear media, this spreading effect may be countered by a slimming effect stemming from the non-linearity, which generates an object with perfectly preserved shape, called a soliton. Solitons have been observed in fluids, granular media, and other systems. **Mohseni et al.** (p. 1295) detected a dissipative soliton (one that also balances gain and dissipation) in a magnetic system, in the form of a magnetic droplet consisting of a core of spins pointing opposite to the external magnetic field. The droplet exhibited peculiar dynamics and could be controlled by electric current.

## Abstract

Dissipative solitons have been reported in a wide range of nonlinear systems, but the observation of their magnetic analog has been experimentally challenging. Using spin transfer torque underneath a nanocontact on a magnetic thin film with perpendicular magnetic anisotropy (PMA), we have observed the generation of dissipative magnetic droplet solitons and report on their rich dynamical properties. Micromagnetic simulations identify a wide range of automodulation frequencies, including droplet oscillatory motion, droplet "spinning," and droplet "breather" states. The droplet can be controlled by using both current and magnetic fields and is expected to have applications in spintronics, magnonics, and PMA-based domain-wall devices.

Dissipative solitons are localized excitations realized by a balance between nonlinearity, dispersion, gain, and loss (*1*, *2*). They can be experimentally observed in optical (*3*, *4*), chemical (*5*, *6*), granular (*7*), and liquid (*8*) dissipative systems. Large amplitude nanoscale dynamics in magnetic thin films with perpendicular magnetic anisotropy (PMA) inherently possess all mechanisms supporting dissipative solitons except for gain. Spin-transfer torque (STT) (*9*–*12*) provides for the injection of angular momentum from spin-polarized electrons into a magnet. Using STT as the gain mechanism in nanocontact (NC)–based spin-torque oscillators (STOs), a magnetic dissipative soliton—the so-called "magnetic droplet"—was recently proposed (*13*–*15*). Using NC-STOs, we created and investigated magnetic droplet dynamics experimentally.

Classical conservative solitons, such as light pulses in a virtually lossless optical fiber, preserve their shape by balancing the opposing effects of dispersion (spreading) and nonlinearity (focusing). Similarly, if damping is ignored the Landau-Lifshitz equation for an extended two-dimensional magnetic thin film with PMA can sustain a family of conservative magnetic solitons, known as "magnon drops" (*16*, *17*). All spins in a magnon drop precess in phase around the film normal, with a precession angle 0 < Θ(0) < π at the center of the drop and 0 < Θ(r) < Θ(0) decreasing exponentially fast, with radius to 0 in the far field. The family of stationary magnon drops can be parameterized by the precessional frequency *f*_{0}, satisfying *f*_{Zeeman} < *f*_{0} < *f*_{FMR}, where *f*_{FMR} is the ferromagnetic resonance (FMR) frequency, and *f*_{Zeeman} is the Zeeman frequency. Magnon drops can be strongly nonlinear, exhibiting almost fully reversed cores [Θ(0) → π] for *f*_{0} close to *f*_{Zeeman}. Whereas conservative magnon drops balance exchange (dispersion) with anisotropy (nonlinearity) for each *f*_{0}, the dissipative magnetic droplet must also balance energy gain (STT) with dissipation (damping), singling out a particular droplet precession frequency for a given drive current and applied field (Fig. 1C) (*13*). More generally, dissipative soliton systems, such as the NC-STOs studied here, are natural environments for studying pattern formation. Dissipative solitons are often robust attractors and can exhibit exotic dynamics, such as time-periodic breathing (*1*). It has been claimed that NC-STOs with in-plane anisotropy and applied field exhibit nonlinear localization in the form of a weakly nonlinear spin wave bullet with precession angles much less than 90° (*18*, *19*). In contrast, the fully nonlinear dissipative droplet studied here necessarily involves precession angles greater than 90° (*13*), exhibiting a clear experimental signature and rich nonlinear behavior.

To test the theoretical predictions for a magnetic droplet (*13*), we fabricated NC-STOs based on orthogonal pseudospin valve stacks (Fig. 1C), in which the magnetization of the Co fixed layer lies in the plane for zero applied field, whereas that of the Co/Ni multilayer free layer lies along the film normal because PMA is sufficiently strong to overcome the demagnetization field (*20*–*22*).

The field dependence of the microwave signal from a NC-STO with 63-nm NC diameter in low to moderate perpendicular fields (Fig. 1A) shows the expected linear FMR-like field dependence (*20*, *21*). However, at a critical field of μ_{0}*H*_{droplet} = 0.65 T, the precession frequency exhibits a dramatic drop to a frequency that lies between the Zeeman and FMR frequencies, with a simultaneous jump in the integrated power (*P*). A similarly dramatic transition can be observed (Fig. 1B) as a function of current in a constant field of 0.8 T with similar changes in frequency and power. To gain further insight into the magnetic state as a function of field and current, the magnetoresistance {MR = [*R*(*H*) – *R*(*H* = 0)]/*R*(*H* = 0), where *R* is the device resistance} was measured both at –6 mA and at a lower current of –1 mA (Fig. 1A, inset). Below 0.65 T, the MR exhibits an identical linear decrease for both currents, which is consistent with a linearly increasing out-of-plane component of the fixed layer magnetization and an increasingly parallel state of the NC-STO. At exactly μ_{0}*H*_{droplet} = 0.65 T, MR [current (*I*) = –6 mA] exhibits a jump of 0.1%, and its field dependence changes sign; the NC-STO state thus becomes increasingly antiparallel with increasing field. Contrarily, MR (*I* = –1 mA) does not show any sign of transition and continues to decrease linearly, eventually saturating in a field of 1.6 to 1.8 T (fig. S1) (*22*), which is consistent with the expected saturation field for the Co layer.

Both the dynamic and static observations are consistent with the formation of a magnetic droplet in the free layer. The large drop in frequency and the sign change of the field-dependent resistance further indicate a substantially reversed central region. This is corroborated by the large increase in microwave power because a reversed droplet will have a large area of spins precessing around the equator, whereas the precession angle of the FMR-like mode is very limited close to the threshold for STO dynamics (*10*, *13*). Last, according to the theory of the magnetic droplet, its frequency (*f*_{droplet}) should increase linearly with field at a slope of γ/2π (γ is the gyromagnetic ratio, so γ/2π = 28.7 GHz/T) and decrease very weakly with current, which is in agreement with Fig. 1, A and B, respectively.

From the MR value at the transition, we can conclude that the Co layer tilt angle required to nucleate a droplet at –6 mA is ~50°. Assuming that the nucleation is primarily driven by the perpendicular component of the spin-polarized current density, we expect the required nucleation current to be inversely proportional to the perpendicular component *M*_{z} of the fixed layer magnetization. Because *M*_{z} is linearly proportional to the perpendicular field for the easy-plane Co (fig. S1) (*22*), we can directly test this assumption by plotting the nucleation current (*I*_{droplet}) versus inverse applied field, 1/*H* (Fig. 1D). We indeed observed that *I*_{droplet} is inversely proportional to the applied field and that the slope of this dependence scales with NC area, confirming that the droplet nucleation is governed by the perpendicular component of the spin-polarized current density, regardless of the applied field and NC size.

We now turn to the modulation sidebands that appear simultaneously with the nucleation of the magnetic droplet. The droplet and its field dependence are very robust and reproducible from device to device; however, we found greater variation in the modulation, with some devices showing single (Fig. 2A) and multiple (Fig. 2B) well-defined sideband pairs, some showing strong peaks at *f*_{droplet}/2 (with some power at 3*f*_{droplet}/2) (Fig. 2C) and others not showing any modulation at any current or field. In some cases, pure single-tone operation may be preceded by modulated behavior at lower currents (Fig. 2B); the onset current is dependent on the direction of the current sweep. The low-frequency (~1 GHz) modulating signal can be measured directly (Fig. 2A). Because all of our measurements involved dc drive alone, the observed modulation is unrelated to ordinary STO modulation (*23*, *24*), in which the drive current contains an intentionally superimposed modulating current. The observed automodulation must instead be intrinsic to the droplet.

With the aid of dissipative droplet theory and micromagnetic simulations (*22*), we can identify these complex dynamics with different dynamical wave states, including quasiperiodic and periodic structures. At nucleation, the dramatic drop in frequency is associated with droplet formation (*13*), in which all the spins in the droplet precess uniformly at a single fixed frequency (Fig. 3A). In (*13*), a droplet drift instability was identified when the droplet was ejected from the NC area, eventually succumbing to damping, leaving room for the nucleation of a new droplet in a periodic fashion. This process occurs on a nanosecond time scale, which is consistent with the experimentally observed modulation sidebands. However, this explanation is not consistent with the observed hysteresis (Fig. 2B), predicted for stable, nondrifting droplets (*13*), because it precludes the periodic death and rebirth of the droplet. Upon decrease of the current below the nucleation threshold, a new droplet cannot form once the first droplet has disappeared. Micromagnetic simulations reveal two possible explanations for this discrepancy relying on the presence of a sufficiently canted polarizer for moderate fields. The drifting droplet may experience a restoring force leading to gyrotropic-like motion of the droplet within the NC area with a characteristic ~1 GHz frequency (Fig. 3B); depending on other parameters, micromagnetic simulations also reveal asymmetric droplets that "spin" on the edge of the NC area while emitting spin waves (Fig. 3C), exhibiting sidebands with frequency spacings of several GHz. For strong canting of the polarizer away from the film normal (weak fields), we observed periodic dynamics with signals at 1/2 and 3/2 of the fundamental frequency (Fig. 3D). We identify these characteristic signals with a breathing mode whose observed breathing frequency is half the precessional frequency (Fig. 2C). Generally, micromagnetic simulations at larger fields tend to exhibit stable precessional dynamics, whereas lower fields can lead to unstable behavior or modulation sidebands by way of droplet oscillations, breathing, and spinning.

Besides the fundamental interest of the creation and control of dissipative solitons in magnetic systems—including, for example, surface magnetic drops expected in thick PMA films (*25*)—the observed magnetic droplet may have an impact on applications, in particular its influence on the emerging fields of STOs, domain-wall electronics, and magnonics. For STOs, the dramatic frequency drop enables ultra-broadband frequency-shift keying (*26*), in which the carrier frequency can be switched by ~10 GHz by varying the drive current a fraction of its absolute value. Micromagnetic simulations indicate nucleation on the sub-nanosecond time scale, but the definitive modulation rate should be determined experimentally. In the emerging field of domain-wall electronics with PMA materials (*27*), the magnetic droplet may be used as a current-controlled nanoscopic domain-wall injector and hence facilitate the implementation of compact domain-wall devices. By applying local field gradients, the magnetic droplet can be transported away from the nanocontact region (*14*, *15*), carrying information on its own or spatially modifying local spin wave propagation in magnonic devices (*28*–*30*). These magnetic droplets hence may join domain walls and magnetic vortices as distinct and useful nanomagnetic objects.

## Supplementary Materials

www.sciencemag.org/cgi/content/full/339/6125/1295/DC1

Materials and Methods

Supplementary Text

Fig. S1

Reference (*31*)

## References and Notes

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**Acknowledgments:**This work was supported by the European Commission FP7 contract ICT-257159 "MACALO," the Swedish Foundation for Strategic Research, the Swedish Research Council, and the Knut and Alice Wallenberg Foundation. J.Å. is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.