Theory for a dissipative droplet soliton excited by a spin torque nanocontact
Abstract
A novel type of solitary wave is predicted to form in spin torque oscillators when the free layer has a sufficiently large perpendicular anisotropy. In this structure, which is a dissipative version of the conservative droplet soliton originally studied in 1977 by Ivanov and Kosevich, spin torque counteracts the damping that would otherwise destroy the mode. Asymptotic methods are used to derive conditions on perpendicular anisotropy strength and applied current under which a dissipative droplet can be nucleated and sustained. Numerical methods are used to confirm the stability of the droplet against various perturbations that are likely in experiments, including tilting of the applied field, nonzero spin torque asymmetry, and nontrivial Oersted fields. Under certain conditions, the droplet experiences a drift instability in which it propagates away from the nanocontact and is then destroyed by damping.
pacs:
75.40.Gb 85.75.d 75.40.Mg 76.50.+g 75.30.Ds 75.75.+aI Introduction
The concept of a soliton as a localized particlelike excitation that preserves its shape can be extended to systems that are far from thermodynamic equilibrium through the concept of a “dissipative soliton” Akhmediev and Ankiewicz (2008a). This allows us to analyze a broad range of physical, chemical, and biological nonlinear systems in which localized excitations are observed. Driven magnetic systems, especially those of technological interest, exhibit strongly nonlinear dynamics and are an ideal experimental domain for exploring the dissipative soliton model.
In this paper, we provide an analytical theory for a novel, localized oscillation mode in a spin torque oscillator with a free layer having perpendicular magnetic anisotropy. The salient features of this mode include a frequency well below that of uniform ferromagnetic resonance, a weak dependence of frequency on bias current, and a precession angle at the maximal value of . Combining numerical micromagnetic simulations with an asymptotic analysis of the equations of motion, we identify this mode as a dynamic, dissipative magnetic soliton that is closely related to the “magnon droplet” predicted by Ivanov and Kosevich in 1977 Ivanov and Kosevich (1977); Kosevich et al. (1990). The mode central region exhibits magnetization pointing nearly opposite to its equilibrium direction with a perimeter manifesting precession. From our asymptotic analysis, we derive conditions on perpendicular anisotropy and bias current for the nucleation and existence of the dissipative droplet. Using our numerical simulations, we analyze the stability of the dissipative droplet soliton as a function of applied magnetic field, bias current, and spin torque asymmetry.
Solitons in conservative systems occur when nonlinear terms in the equation of motion balance the effects of dispersion Ablowitz and Segur (1981). A classic example is a light pulse moving in a lossless optical fiber: the change of refractive index with frequency (dispersion) tends to make the pulse spread out, but for a certain pulse shape, the change of refractive index with light intensity due to the optical Kerr effect (nonlinearity) exactly balances the dispersion. Pulses having this shape can propagate without spreading and are called solitary waves or solitons. The balance between nonlinearity and dispersion typically allows for the existence of a continuous family of solitons that can be excited in the system, rather than a single solution. In the optical fiber example, the family can be parametrized, for example, by pulse amplitude, and there is a continuous range of amplitudes that satisfy the soliton balancing condition.
Dissipative solitons Akhmediev and Ankiewicz (2008a) are characterized by an additional balancing condition between gain and loss that typically allows only a single solution for a given set of external parameters. Although conservative soliton models can explain weakly nonlinear behavior seen in magnetic systems of exceptionally low damping Zvezdin and Popkov (1983), damping is not a small effect for many magnetic systems of both fundamental and technological interest. By combining classical soliton theory with bifurcation theory of nonlinear dynamics and concepts of selforganization Akhmediev and Ankiewicz (2008b), the dissipative soliton concept provides a framework for describing a broad range of solitonlike behaviors. Here, we apply this concept to a nanoscale ferromagnetic system in which both damping and a driving force (spin torque) are important.
Spin torque Slonczewski (1996); Berger (1996); Slonczewski (1999) occurs when a current is driven through a structure with alternating magnetic and nonmagnetic layers in which spindependent conductance at the interfaces results in a spinpolarized electron flow. When the polarized electrons enter a ferromagnetic layer whose magnetization is not collinear with the electron spins, the transmitted spins are rotated toward and the angular momentum absorbed by the ferromagnetic layer is known as the spin torque. Typical devices have two ferromagnetic layers through which current is driven: a thick “fixed” layer that determines the direction of electron polarization, and a thin “free” layer whose orientation can be readily changed by spin torque. For current of the appropriate polarity, spin torque opposes the intrinsic damping torque in the system, and currents above a threshold produce dynamic states in which of the free layer can be manipulated without applying a magnetic field. This effect has been used to control switching of nanoscale magnetic elements Myers et al. (1999), with potential applications in computer memory and data storage. The effect has also been used to produce coherent, frequencytunable microwave oscillations Kiselev et al. (2003) in a nanoscale device known as a spin torque oscillator (STO), with potential applications in integrated microwave circuits for mixing and active phase control. Recent reviews cover the physics of spin torque Ralph and Stiles (2008) and its possible applications Silva and Rippard (2008).
The equations of motion for in the presence of spin torque (presented below) are inherently nonlinear, and their full solution for a general case is often studied by use of numerical methods. Analytical methods can sometimes be applied by invoking restrictions such as high symmetry, spatially uniform (the “macrospin” model), and small precession amplitude (small angle between and its equilibrium direction). These restricted cases have been used to explain experimental results with mixed success. The local nature of spin torque allows it to drive large amplitude excitations in which varies on the scale of the magnetic exchange length (typically a few nanometers), something that applied magnetic fields cannot do. As we show here, this regime of strongly nonlinear, strongly nonuniform, sustained magnetodynamics is amenable to theoretical examination using numerical and analytical approaches. This regime is also experimentally accessible in STOs.
We note that a different type of magnetic soliton generated by spin torque, called a spin wave “bullet”, was predicted by Slavin and Tiberkevich in 2005 to occur in the pointcontact geometry with magnetic films exhibiting inplane oriented anisotropy and inplane applied magnetic field Slavin and Tiberkevich (2005). For this case, the precession frequency decreases with increasing current, which can result in localization if the frequency falls below the bottom of the spin wave band at the ferromagnetic resonance (FMR) frequency. The weakly nonlinear bullet soliton is a solution to a Nonlinear Schrödinger type equation with thirdorder nonlinearity in the excitation amplitude. As such, its predicted experimental signature consists of subtle shifts in microwave output frequency and threshold current relative to that expected for a nonlocalized mode. In contrast, the droplet soliton studied here exhibits dramatic differences in behavior from that of a nonlocalized mode. This is due to the fact that it is a strongly nonlinear solution of the full equations of motion, rather than simply a third order expansion.
Domain walls Schryer and Walker (1974), magnetic bubbles Nielsen (1976), and vortices Guslienko et al. (2005) are examples of well studied, strongly nonlinear, localized structures that occur in magnetic materials. The droplet differs from these static structures in that it is inherently dynamic; the frequency of spin precession within the droplet is always greater than zero. In this work, we focus on the twodimensional (2D), nontopological droplet, but we note that droplets in two and threedimensions come in topological flavors as well Kosevich et al. (1990).
We begin in the next section by presenting an asymptotic analysis of the model equations for the dissipative droplet in a highsymmetry case. We will also derive the droplet’s frequency vs. current relation in this section. Section III is devoted to the study of droplets in physically realistic situations incorporating the currentinduced Oersted field as well as canting of the applied field and fixed layer. Section IV details experimentally accessible nucleation conditions for a droplet that take advantage of a small amplitude instability. In Sec. V, we discuss possible extensions of the theory and we relate the droplet to other excitations in thin magnetic films. We conclude in Sec. VI with a summary of the droplet’s unique properties. Appendices A and B provide details of our stability calculation and numerical method, respectively.
Ii Droplet in a Nanocontact
We consider the LandauLifshitzSlonczewski equation in nondimensional formSlonczewski (1996); Ralph and Stiles (2008) describing the free layer magnetization in polar coordinates (; bold symbols represent vectors in or , contextually dependent; e.g., )
(1) 
Figure 1 is a schematic of the nanocontact geometry and coordinate systems considered in this work. The magnetization and fields (e.g., ) are normalized by the saturation magnetization , time is normalized by ( is the gyromagnetic ratio and is the permeability of free space), and space is normalized by the exchange length ( is the exchange parameter and is the reduced Planck’s constant). We consider a free layer whose thickness is much smaller than the magnetic excitation wavelengths, so that a 2D model with local dipolar fields is justified Gioia and James (1997). The precessional term is driven by the effective field incorporating exchange , a uniform applied field (), the currentinduced Oersted field (the definition of is given in Ref. Hoefer et al., 2008 and summarized in Sec. III.1), the perpendicular field due to crystal anisotropy (), and the demagnetizing field for a thin film . The LandauLifshitz damping coefficient is . The spin torque term involves , where is the current, ( and are the polarization and spin torque asymmetry parameters, respectively Slonczewski (1996), is the electron charge, is the nanocontact radius), (), , where is the Heaviside step function, is the nondimensional contact radius, and is the uniform fixed layer magnetization.
Numerical computations of Eq. (1) will be presented in Sec. III by means of a method discussed in Appendix B. For the rest of this section, we will focus on a highsymmetry case where
(2) 
For the analysis of this case, it is helpful to consider Eq. (1) in spherical coordinates (see Fig. 1)
(3)  
(4) 
where
The polar angle satisfies , while the azimuthal angle is interpreted modulo .
In the symmetric case of Eq. (2), we can remove the explicit dependence on from Eqs. (3) and (4) by introducing the following rescaling:
(5) 
Recall that we are assuming . Then, with the scalings in Eq. (5) and dividing Eqs. (3) and (4) by , we can, without loss of generality, take . For the rest of this Section II, we will use the scalings in Eq. (5) so that .
In this Section, we consider localized magnetization configurations that satisfy
with sufficiently rapid decay. As such, we define the magnetic energy in terms of exchange and anisotropy energy via
Note that the damping and spin torque terms break energy conservation
(6) 
ii.1 Conservative Droplet Soliton
In the absence of damping and spin torque (, ), Eqs. (3) and (4) admit a continuous family of exponentially localized, nontopological solutions known as magnon droplet solitons Ivanov and Kosevich (1977); Kosevich et al. (1990). These solutions can be parametrized by the frequency and written as
(7) 
They satisfy a balance between exchange (dispersion) and anisotropy (nonlinearity) through the nonlinear eigenvalue problem , or
(8) 
with the boundary conditions
(9) 
The conservative droplet solutions in Eqs. (7), (8), and (9) have the following properties. The polar angle varies with radial distance and is independent of time; thus the spatial distribution of is static and azimuthally symmetric. The azimuthal angle is independent of position and varies linearly in time; thus all points precess at a common frequency and in phase. It has been shown that Ivanov and Kosevich (1977); Kosevich et al. (1990) satisfies
(10) 
Therefore, the total precessional frequency varies between the Zeeman frequency and the frequency of spatially uniform precession about , (the FMR frequency).
While the conservative droplet does not have a closedform analytical expression, we calculate it by numerically solving Eq. (8) subject to the boundary conditions (9) with the function bvp4c in Matlab. A plot of several conservative droplet profiles is shown in Fig. 2. We see that the amplitude at the origin decreases as the frequency is increased. While it may appear in Fig. 2 that , in fact all conservative droplets with satisfy owing to their nontopological structure.
The energy for the droplet satisfies Ivanov and Kosevich (1977); Kosevich et al. (1990)
(11) 
The fact that the energy is a decreasing function of frequency has been used to argue that the conservative droplet is stable in a 2D material Kosevich et al. (1990). The 2D conservative droplet embedded in an infinite, 3D magnet is known to be unstable. However, preliminary studies suggest that, for sufficiently thin films, the 2D conservative droplet is stabilized. This work is beyond the scope of this paper and will appear elsewhere.
As we now show, the conservative droplet soliton can be generalized to damped/driven systems such as a nanocontact. Whenever we refer to “droplet”, we mean the dissipative droplet studied in the future sections. We will always use “conservative droplet” to describe the solution of Eq. (8) that is monotonically decaying to zero as .
ii.2 Dissipative Droplet Soliton
We now extend the analysis of Kosevich, Ivanov, and Kovalev Ivanov and Kosevich (1977); Kosevich et al. (1990) to the case of the dissipative droplet solution, where damping is no longer assumed to be negligible, and spin torque is included in the analysis. In addition to the balance between exchange and anisotropy that was required in the conservative droplet case, a further balance between uniform damping (loss) and spatially localized spin torque (gain) will be derived for the droplet to be sustained. We will assume that the spin torque and damping, while small, are not zero and are of the same magnitude so that
We look for an asymptotic solution in the following form
(12) 
We have set to distinguish the frequency of the droplet from that of the conservative droplet . We will conclude that there is no frequency shift due to damping and spin torque so that here. However, other perturbations beyond those considered here could lead to a frequency shift. Inserting the expansions (12) into Eqs. 3, 4, and equating like orders in gives the following equations for the perturbations and :
(13)  
(14) 
where the selfadjoint, Schrödinger operators and are
Note the following important property:
(15) 
The rest of this section is concerned with the solution of the perturbative equations (13) and (14) in two separate cases. We use standard solvability arguments for forced, linear differential equations to remove secular terms (see e.g. Ref. Kevorkian and Cole, 1996) in order to derive an expression for the current at which the balancing condition for the droplet is achieved.
ii.2.1 Case
We first consider the case where the fixed layer is oriented perfectly normal to the film plane so that . In this regime, we can study the effect of variable spin torque asymmetry .
In addition to the scalings in Eq. (5), we can also scale out the dependence on the applied field when with the substitution
(16) 
Then, and satisfy Eqs. (3) and (4), as before, but with . This transformation shows that the vertical applied field simply shifts the precessional frequency by .
We seek a solution to Eqs. (13) and (14) in the form , , , and (i.e., linear phase evolution with time) which results in the following nonhomogeneous equation for :
(17) 
This equation is solvable if and only if the nonhomogeneous terms are orthogonal to the kernel of . By use of Eq. (15), multiplying the righthand side of Eq. (17) by and integrating from to infinity we obtain the existence condition for a dissipative droplet, , where
(18) 
The choice singles out a specific value for the current as a function of the droplet frequency . This value of the current provides the necessary balance between damping and spin torque, in addition to the balance between exchange and anisotropy, in order to sustain the droplet. Therefore, we call the sustaining current.
We can also understand the choice for the sustaining current (18) by computing the rate of change in the energy for the droplet from Eq. (6):
(19) 
Therefore, the energy of the droplet is approximately conserved for the choice . Figure 3 shows a snapshot in time of a dissipative droplet.
Figure 4 represents the numerical evaluation of Eq. (18) and shows the dependence of on . The droplet frequency has two branches as the sustaining current is varied, but only one branch is stable. For a given droplet frequency , consider increasing the current slightly above the sustaining value, . From Eq. (19), the droplet energy will increase, and from Eq. (11), this increase in energy corresponds to a decrease in droplet frequency for a stable droplet Ivanov and Kosevich (1977); Kosevich et al. (1990). Thus, the upper branch in Fig. 4, for which increasing current causes an increase in frequency, is unstable.
Figure 4 shows that there is a minimum sustaining current , maximum frequency , and minimum in the center of the droplet where
While the specific choice of spin torque cutoff function (e.g. , we could have used a smooth cutoff, as opposed to a sharp, Heaviside cutoff) may be important in numerical applications Berkov and Gorn (2007), it has only a slight effect on the droplet sustaining current. For the Heaviside cutoff considered here, Eq. (18) simplifies to
(20) 
Equation (20) reveals the explicit dependence of the sustaining current on two key spin torque parameters: the contact radius and the spin torque asymmetry . We now investigate properties of the dissipative droplet as a function of these two parameters. Figures 5(ad) show the dependence of , , , and the droplet radius on the contact radius for various choices of the spin torque asymmetry. The droplet radius is defined to be the radius at half maximum:
(21) 
Figure 5(a) shows that droplets excited by smaller contacts require larger sustaining currents. The dependence of on and for small contact radii can be made explicit by an asymptotic evaluation of the denominator in eq. (20) giving
(22) 
Therefore, grows like for small contact radii in agreement with Fig. 5(a). Interestingly, we find that the droplet is constrained to have a frequency (Fig. 5(b)), significantly below the FMR frequency of 1. Also, since (Fig. 5(c)), the droplet always has some region near its center that is partially inverted with respect to the surrounding magnetization. As shown in Fig. 5(d), the droplet is well localized within the nanocontact; e.g. , when . There is a minimum droplet radius of about in dimensional units. Finally, spin torque asymmetry has only a small, perturbative effect and does not substantially alter the droplet solution.
ii.2.2 Case
We consider now the case without spin torque asymmetry (i.e. ) in which case we can study the effect of varying the angle of the fixed layer on the droplet dynamics. In this section, we use the rescalings in Eq. (5) but do not use the transformation (16).
We solve Eqs. (13) and (14) with , , and , leading to the following system of nonhomogeneous equations:
(23)  
(24)  
(25) 
Note that, in contrast to the case , the overall phase no longer evolves linearly in time. Applying to Eq. (23) and to Eq. (24) gives the decoupled system:
These equations are always solvable if and are strictly positive operators. One can show that , so that . One can also show that , where is the smallest eigenvalue of and is strictly negative. Then, . We have verified by numerical computation of that so that as required, and Eqs. (23) and (24) are solvable.
We are interested in the solvability condition for in Eq. (25), which is
(26) 
Note that the applied field appears only as a shift in the droplet frequency, as in the case studied in the previous section. The expression (26) for the sustaining current agrees with the previously derived sustaining current in Eq. (18) when and , as required. Thus, the dominant effect of rotating the fixed layer is to increase the sustaining current in proportion to .
Iii Physical Perturbations of a Dissipative Droplet
So far, we have considered the dissipative droplet solution only for a simplified geometry where asymptotic methods can be applied. In these cases, the external field is both uniform and oriented perfectly perpendicular to the film plane. By making this geometrical simplification, we were able to factor out the contribution of the external field from the droplet solution. However, in a real pointcontact system, the external field is neither uniform nor perfectly perpendicular. In particular, the current flowing through the contact is an additional source of spatially inhomogeneous magnetic field, the Oersted field, and the applied uniform magnetic field in actual experiments is usually tilted away from the perpendicular axis. We employ micromagnetic simulations to investigate how these physically important perturbations to the external field alter the ideal droplet solution. The numerical details used for our simulations are presented in Appendix B.
We find that the combination of external field tilt and the Oersted field breaks the symmetry of the solution such that the droplet is no longer centered in the middle of the contact. As a result, the solution takes on a nontrivial inhomogeneous phase structure where the phase of the spin precession closer to the center of the contact precedes the phase further from the center of the contact. In addition, the spatial structure of the droplet is no longer perfectly circular. For some particular combinations of simulation parameters, the droplet breaks away from the contact altogether and dissipates, a behavior we call a drift instability. When this occurs, the droplet may maintain its form for many precession cycles, but it eventually decays, since outside the contact there is no spin torque excitation to balance damping.
iii.1 Oersted Field
First, we consider the effect of the current induced Oersted field while keeping the applied field and fixed layer oriented almost normal to the film plane (canted by and , resp.). The reason for this slightly asymmetric configuration is to test whether highsymmetry solutions are structurally stable to small changes in the system parameters. Such a configuration is experimentally possible, in principle. Our model for the Oersted field was presented in Ref. Hoefer et al., 2008 and takes the form
where
(27) 
The function given in Ref. Hoefer et al., 2008 involves integrals of Bessel functions and depends on the geometry of the currentdensity distribution. The parameters defining in Ref. Hoefer et al., 2008 are , , and . The other, more dominant, term in Eq. (27) is the field generated by an infinitely long conducting wire. As an example, for the simulation in Fig. 6, we have , while , an order of magnitude difference.
Figure 6 illustrates how the Oersted field changes the structure of the droplet. In contrast with the symmetric case in Fig. 3, the azimuthal angle shows significant spatial variations. The droplet is also slightly shifted offcenter. The strong phase variations are indicative of a tendency for the droplet to propagate Piette and Zakrzewski (1998). In some cases, although not for the simulation in Fig. 6, the droplet breaks free from the nanocontact. This drift instability is discussed further in Sec. III.2.
In Fig. 7 we compare the numerically computed perturbed droplet frequency as a function of current with the sustaining current from Eq. (20). We find that the droplet frequency is approximately shifted down by an overall amount of , but otherwise follows the same trend as the symmetric result. This behavior demonstrates that the analysis of the previous section, despite the necessary high symmetry restrictions, yields relevant, qualitative information about localized structures excited in a physically realizable nanocontact.
As shown in Sec. IV, when the current exceeds a threshold value; e.g., the Slonczewski critical currentSlonczewski (1999), a droplet can nucleate for sufficiently large anisotropy. The vertical line in Fig. 7 is the numerically computed threshold current. It differs from Slonczewski’s result because , , and . This threshold for droplet nucleation suggests a hysteretic effect that will be discussed in Sec. IV.3.
Our micromagnetic simulations also show that the perturbed droplet, for certain parameter choices (e.g., the crosses in Fig. 7), undergoes a drift instability. This behavior will now be investigated further.
iii.2 Canted Applied Field, Fixed Layer, and Oersted Field
In this section, we investigate the combined effects of the Oersted field as well as canting of the applied field and fixed layer. Figure 8 is a time sequence showing the evolution of a strongly perturbed droplet over one precessional period. In contrast to the nearly symmetric configuration of Fig. 6, where the droplet was slightly shifted to the left, the droplet is slightly shifted down, toward the region of lower inplane field.
As the system parameters are changed, the shifting of the droplet center can be large enough to actually dislodge the droplet from the nanocontact. An example of this drift instability is shown in the panels of Fig. 9. The current was taken to be less than the current for Fig. 8, which did not experience a drift instability. A number of precessional periods pass before the droplet breaks free. Once free, it propagates, but because it no longer satisfies the required balancing condition between damping and spin torque, it loses amplitude and decays. Once the droplet has drifted outside of the nanocontact, a new one is formed if the nucleation conditions are satisfied (see Sec. IV). We have also observed droplets that drift several nanocontact diameters before decaying, i.e. the central magnetization lifts up so that . The manifold in parameter space in which the droplet manifests a drift instability appears to be complicated. Nevertheless, we readily find parameter regimes where the droplet apparently does not experience the drift instability, as in Figs. 6, 7, and 8.
Notice that the droplet propagates down, in the direction. Recall that the canting direction of the applied field is along . Due to the symmetry of the Oersted field, the direction of propagation of the drifting droplet appears to track the azimuthal angle of the applied field minus , if is sufficiently large. For example, if the applied field is canted along , then the droplet will drift along if unstable. This can be understood as a consequence of magnetostatic interactions between the effective dipole moment of the droplet and the field gradient associated with the Oersted field. Given the canting of the applied field, the effective dipole moment of the droplet acquires an inplane component that is drawn to the edge of the contact where the Oersted field is also in the direction such that the Oersted field gradient acts to trap the droplet. In the case of Fig. 6, where the applied field is canted only , the droplet is observed to drift to the left rather than down. Therefore, the strength of the Oersted field and the inplane component of the applied field have a strong effect on the existence and dynamics of a drift instability.
Iv Nucleation of a Dissipative Droplet
Figure 10 shows the birth of the droplet pictured in Fig. 8 starting from a state pointing uniformly in the direction. For sufficiently large perpendicular anisotropy and current, the magnetization inside the nanocontact reverses and nucleates a droplet. In this section we show that the reversal mechanism is caused by an instability of small amplitude waves. We will study Eq. (1) in the weakly nonlinear regime and find that the small amplitude Slonczewski mode that exists near the threshold for the onset of dynamics is stable/unstable depending on whether is less than/greater than a critical value . We find that due to exchange effects and converges to 1 as the contact size is increased. The value of is important for the possible experimental observation of a dissipative droplet.
iv.1 Stability Analysis of a Macrospin
Before studying the PDE (1), we consider the macrospin model where spatial variation is neglected. The stability analysis for this model is suggestive and mathematically simpler. However, we find that the critical anisotropy field in the macrospin case satisfies and depends strongly on the applied field, which differs from the result obtained by analyzing the full PDE model, where we find with weak applied field dependence.
We consider Eqs. (3) and (4) neglecting all spatial variation and interlayer dipole coupling in the symmetric regime and ; i.e., we are assuming that the initial condition for the system is in the parallel state. In this case, the equation for is decoupled from , so we can just study the scalar, firstorder ODE
(28) 
By linearizing Eq. (28) about the equilibrium , we find that it becomes unstable when
in agreement with previous numerical and mathematical analyses of this system contained in Ref. Bazaliy et al., 2004. We seek a periodic equilibrium solution just above threshold by taking
Then, we have
This solution exists (is real valued) as long as
(29) 
Furthermore, one can show that this equilibrium is stable. Therefore, when , the equilibrium undergoes a supercritical Hopf bifurcation as the current is increased beyond . When , there is no periodic solution, and the system switches from to the fully reversed state when exceeds .
iv.2 Stability Analysis of the Micromagnetic System
We consider Eq. (1) in the symmetric regime with , , , and the substitutions
where . Then approximately satisfies a complex Nonlinear Schrödinger type equation
(30) 
Similar to the macrospin case, when is increased past a threshold value, the zero solution becomes unstable. The threshold, critical current and onset frequency were found by Slonczewski as a solution to a linear eigenvalue problem Slonczewski (1999) (see Eq. (35)). Incorporating weak nonlinear effects, Ref. Hoefer et al., 2005 showed that a small amplitude periodic solution exists as a modulation of the Slonczewski mode for and . This timeperiodic, weakly nonlinear mode was found to be numerically stable. In this section and Appendix A, we extend these results to and , where is defined through and is given by Eq. (38). This behavior is analogous to the supercritical Hopf bifurcation for the macrospin, studied in the previous section.
In this section, we show that weakly nonlinear modulations of the Slonczewski mode are no longer stable when . This behavior is analogous to the switching exhibited by the macrospin for and above threshold. While we are unable to analytically follow the dynamics of the instability, numerical simulations such as the droplet birth sequence shown in Fig. 10 demonstrate that the magnetization reverses inside the nanocontact and develops into a dissipative droplet.
For the stability analysis, we seek a multiple scales solution representing a modulation of the Slonczewski mode in the form
(31) 
where is the amplitude of the mode at the origin, which is modulated by with the “slow” time, is the frequency of the Slonczewski mode with threshold current , and represents a deviation from . The explicit form for and the implicit equations for and are given in Appendix A.
Invoking a solvability condition at , Eq. (37) gives the nonlinear amplitude equation
(32) 
with complex linear and nonlinear coefficients and . There is a timeperiodic solution of Eq. (32) for a specific choice of
(33) 
This solution represents the nonlinear frequency shift to the Slonczewski frequency . The stability analysis in appendix A shows that the solution (33) is unstable when . An explicit formula for is given in Eq. (38). We evaluate the integrals numerically and plot as a function of for specific parameter values in Fig. 11.
There is a critical value of the anisotropy field satisfying
(34) 
above which the weakly nonlinear Slonczewski mode is modulationally unstable. In other words, weak modulations of the Slonczewski mode will grow exponentially in time when . Figure 12 shows the dependence of , computed numerically by solving Eq. (34) with Eq. (38), as the contact radius and spin torque asymmetry are varied and . We see that is strictly greater than one and that larger perpendicular anisotropy is required to enable the nucleation of a droplet in smaller nanocontact devices. For a given contact size, larger spin torque asymmetry permits nucleation of a droplet at smaller values of .
We have also investigated the dependence of on the applied field magnitude . Since appears in only multiplied by , we have , which is small. This is confirmed by numerical calculations of the dependence on for the values of and plotted in Fig. 12. We find that varies by at most 3 % for with . Importantly, the lower bound for is preserved: when . This behavior stands in stark contrast to the macrospin result for in Eq. (29) that strongly depends on the applied field.
From Eq. (33), we see that is negative when . Numerically, we find that , so that when , which corresponds precisely to the instability criterion. Furthermore, when , any finiteamplitude excitation will tend to grow, thus the timeperiodic solution is unstable. This behavior is similar to the macrospin model discussed in the previous section where, for , a small amplitude periodic solution did not exist.
Numerical simulations of Eq. (1) confirm the foregoing analysis, even in the nonsymmetric cases with , , and . There is a critical value of the current above which large amplitude dynamics ensue. When is above , we find that the magnetization inside the nanocontact reverses to form a localized, coherently precessing, fully nonlinear magnetic solitary wave. We identify this solitary wave with the dissipative droplet soliton found in the asymptotic analysis of Sec. II.
Due to the numerically robust formation of the droplet for a variety of initial data and across a large parameter regime, we view it as a global attractor. As long as the current is above threshold, a droplet is observed to nucleate.
iv.3 Hysteresis
We have shown in this section that a dissipative droplet will form when the current exceeds the threshold for instability of small amplitude waves inside the contact. In the high symmetry case of Eq. (2) and , the threshold is the Slonczewski critical current , which is plotted in Fig. 7 (triangle) along with the numerically calculated threshold current from micromagnetic simulations that incorporate the Oersted field and a canted fixed layer (, vertical line). We observe that the minimum droplet sustaining current is below the threshold for droplet nucleation. This suggests a hysteretic effect, whereby a droplet can be nucleated at a current above threshold and remains stable when followed by an adiabatic decrease of the current below threshold. We have performed this experiment numerically for the perturbed droplets of Fig. 7. We use an already nucleated droplet (at ) as the initial condition for a new simulation with , below threshold. We find that sufficiently close to, but below, threshold, the droplet’s frequency slightly increases but remains stable. For further decrease of the current to , however, the droplet undergoes a drift instability similar to behavior shown in Fig. 9. Since the current is below threshold a new droplet does not form.
V Discussion
The Slonczewski form for the spin torque term we are considering here, Eq. (1), was derived strictly for the case of spatially uniform magnetization distributions Slonczewski (1996). Nonlocal generalizations of the spin torque term to nonuniform magnetization distributions have been derived in the small amplitude regime Stiles et al. (2004) and applied to single layer nanocontact simulations Hoefer et al. (2008). Because the spin torque and damping terms are treated as perturbations giving rise to a dissipative droplet solution, we expect that the specific form for the spin torque term, and the damping term for that matter, will not yield qualitative changes in the structure of the droplet. The asymptotic analysis presented here is applicable to an arbitrary spin torque term as long as its net effect is to oppose the inherent damping in the system.
Due to the symmetries in the problem when , one might expect the Oersted field to prefer the excitation of a localized vortex structure; e.g., a topological soliton Kosevich et al. (1990). We observe no such excitations in our simulations. Twodimensional, conservative topological solitons are stable but they have higher energy than the conservative nontopological soliton Kosevich et al. (1990), providing one possible explanation for this behavior. Another possible explanation lies in the form of the spin torque term. In contrast to the local form for the spin torque considered here, singlelayer nanocontact simulations that incorporate a nonlocal spin torque have demonstrated the formation and numerical stability of a precessional vortex in highsymmetry configurations without perpendicular anisotropy Hoefer et al. (2008). A more realistic model of the spin torque may favor a topological soliton in certain cases.
Unfortunately, we as yet have no detailed physical understanding of either the dissipative droplet asymmetry or the drift instability in the presence of the combined effects of an Oersted field and a tilted applied field. In part, our lack of understanding stems from the limited theory for the spatial propagation of the droplet Piette and Zakrzewski (1998); Cooper (1998). Extension of the theory presented here to cases where the droplet is accelerated by field gradients and other such forces may provide more physical insight to aid in understanding why the droplet is subjected to displacing forces.
The physical appearance of the droplet is reminiscent of the magnetic bubble that was once the subject of intense investigation as a possible alternative to ferritecore computer memory Nielsen (1976). Indeed, we can identify the zerofrequency droplet as a topologically trivial magnetic bubble Komineas and Papanicolaou (1996) with a winding number of zero, though in the theory presented here, the droplet frequency approaches zero only in the limit of infinitely large diameter. However, we expect that the droplet will converge to the bubble structure at a finite diameter, contingent on the inclusion of nonlocal magnetostatic energy in the calculation; even when considering free layer films as thin as 3 nm, such a term will eventually overcome the positive contribution of exchange energy to the oscillator frequency, stabilizing the droplet as a static structure. In this sense, then, we can think of the conservative droplet soliton, a delicate balance between exchange and perpendicular anisotropy, as a dynamically collapsing bubble, and the dissipative droplet soliton as an imminently collapsing bubble that is critically stabilized by the localized injection of spin torque.
Vi Conclusion
We have derived equations for a dissipative droplet soliton through an asymptotic analysis of the LandauLifshitzSlonczewski equation for a pointcontact spin torque oscillator with perpendicular anisotropy in the free layer. The droplet soliton is a localized, dynamic, solitary wave solution consisting of partially reversed magnetization directly under the contact and a zone of large amplitude precession in a region bounding the reversed magnetization. The diameter of the precessing boundary is approximately the contact diameter. The droplet frequency is always strictly less than the ferromagnetic resonance frequency for the film, and is also a monotonically decreasing function of droplet diameter. The balance between spin torque and damping required to sustain the droplet determines the relation between bias current and frequency. The instability of small amplitude solutions that leads to formation of the reversed domain in the droplet requires a minimum perpendicular anisotropy that is a function of the contact radius and spin torque asymmetry.
Some of the unique, identifying properties of the dissipative droplet that could be observed experimentally include

The droplet’s frequency is well below the ferromagnetic resonance frequency.

Sufficiently far from the minimum sustaining current, the droplet’s frequency has a weak dependence on current.

The droplet may manifest a drift instability, which would reveal itself as a transitory cessation in ac oscillations until the droplet again forms.

Hysteresis in current is expected, unless a drift instability results in a finite droplet lifetime.
We have investigated the nucleation and stability of the dissipative droplet soliton through numerical simulations. We find that droplet formation begins once the current in the pointcontact is sufficient to instigate the small amplitude Slonczewski mode, characterized by spin waves that radiate away from the pointcontact. For sufficiently strong perpendicular anisotropy, this mode is subject to a modulational instability and rapidly evolves into the reversed magnetization profile of the droplet soliton. We find that the droplet is stable in certain parameter regimes with regard to the inhomogeneous Oersted field and to variations in spin torque asymmetry and applied field angle. Finally, the droplet is subject to a drift instability that is a complicated function of the parameters employed in this theory.
Appendix A Modulational Instability of Slonczewski Mode
Here we provide the details of our stability analysis for small amplitude, modulated waves excited in a nanocontact.