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Stochastic ejection of nanocontact droplet solitons via drift instability

The magnetic droplet soliton is a large amplitude, coherently precessing wave state that exists in ferromagnetic thin films with perpendicular magnetic anisotropy. To effectively sustain a droplet, magnetic damping can be locally compensated in a nanocontact region that imparts spin-transfer torque; this has been successfully deployed in experiment to directly image the droplet and probe its dynamics electrically. However, theory predicts and experiments indicate the existence of a drift instability whereby the droplet is ejected from the spin-transfer-torque-active region and subsequently decays, an effect that may be enhanced or possibly induced by thermal fluctuations. Using soliton perturbation theory and large deviation theory, this work determines the soliton ejection rate and the most likely path an ejected soliton tracks in the presence of thermal fields. These results lead to an effective lower bound on the stability of magnetic droplet solitons in spin-transfer torque nanocontact devices operating at finite temperature and point to ways in which droplets can be made more robust.


INTRODUCTION
The pursuit of fast, scalable nonvolatile storage and new ways to process digital information have motivated much recent study of the formation and control of localized structures in ferromagnetic materials. These structures are supported by an energetic balance among contributions from material anisotropy, exchange energy, dipole or self-induced field energy, and forcing due to external fields, current-induced spin torque, and magnetic damping. Some localized structures, such as skyrmions, are created with a topology that cannot continuously deform to the trivial state, lending promise for additional structural stability in the absence of a driving force. Other structures, such as dissipative solitons, are maintained by a balance between driving and damping terms. In both cases, structures that are stable under deterministic dynamics are generally only quasi-stable under the inevitable influence of thermal fluctuations or other stochastic phenomena.
In this paper, we study the effects of weak thermal fluctuations on the magnetic droplet soliton [1], a large-amplitude, precessing mode that manifests as a balance between nonlinearity due to uniaxial anisotropy and dispersion due to the exchange interaction. Because of the droplet's precessional dynamics, its practical realization was predicted [2] and then observed both indirectly via the giant magnetoresistance (GMR) effect [3,4] and directly with x-ray transmission microscopy [5,6] in a spin-transfer torque driven nanocontact device that imparts a current-induced local torque to compensate uniform magnetic damping.
Additional experiments [7,8] used the GMR effect to measure the nanocontact device's frequency characteristics, highlighting a low-frequency signal that was attributed to a drift instability whereby the soliton is expelled from the nanocontact and then decays, after which a new droplet forms under the nanocontact and the process repeats [2]. This drift instability presents a significant challenge to the robust control of droplet solitons and the reliable operation of future spintronic devices. Due to growing interest in magnetic droplet solitons, experiments continue in spin torque nanocontacts [6,[9][10][11] and in spin Hall driven nanoconstrictions [12].
The existence and dynamics of droplets in nanocontacts have been successfully described using soliton perturbation theory [13,14], that yields a system of ordinary differential equations (ODEs) for the droplet's parameters that we call the modulation equations. The ODEs' fixed points correspond to sustained droplets, which, for a certain parameter regime, are unstable and manifest an increase in the droplet's speed, i.e., the drift instability [15].
Their stochastic dynamics under a thermal random field were explored theoretically [15].
The objective of that study was to compute the approximate variance in droplet parameters, including its frequency, linewidth and position relative to the spot of the nanocontact supporting it. The Ornstein-Uhlenbeck process derived from linearizing the stochastic model equations about the dissipative droplet state was shown to provide statistics that agree well with simulations of the nonlinear stochastic model and with micromagnetic simulations performed using open source code MuMax3. That study also contained the description of the aforementioned drift instability occurring at critical bias current.
An equally important figure of merit for magnetic droplets in the presence of a thermal field is the probability (alternatively, the rate) of ejection, where the droplet escapes the attractive potential provided by the finite spatial extent of the nanocontact and then decays due to magnetic damping. In the physically relevant limit of weak thermal effects, exits occur rarely, with a probability or rate determined by large deviation theory. In this limit, the asymptotic scaling of exit probabilities and rates with vanishing noise strength is provided by a rate function related to the exit path or paths that minimize an action functional. This asymptotic scaling provides an order-of-magnitude estimate for the ejection rate, which would otherwise be estimated through sampling using, for example, a software package such as MuMax3 [16]. These simulations must resolve dynamics on gyromagnetic time scales of nanoseconds, while our estimates show that exits occur on time scales of micro-to milliseconds, depending on the operating regime. Monte Carlo methods to estimate ejection rates or ejection probabilities over finite times are therefore unreasonably expensive.
In this work, we demonstrate that, even in regimes where the droplet fixed point is nominally stable under deterministic dynamics of the modulation ODEs, the proximity of a saddle point associated with the drift instability provides an important mechanism for droplet ejection. We use a combination of analytical and numerical techniques to compute the action associated with exits through all saddles with a single unstable manifold, including the saddles with nonzero velocity bifurcating through the drift instability [15] and those associated with zero velocity originating in a saddle-node bifurcation in the precessional frequency [13]. Our most important physical result is the determination of the mean time it takes for a deterministically stable droplet to be ejected from the nanocontact due to thermal noise. Based on physical parameters from recent experiments [7,17], we estimate this time to be approximately 50 nanoseconds or a rate of droplet ejection of 20 MHz, which is within an order of magnitude of low-frequency observations from recent experiments. We estimate that this rate can be manipulated and, importantly, significantly decreased through appropriate choice of operating parameters.
where 0 < α 1 is the damping parameter, H(x) is the Heaviside function, and i = I/I 0 is the STT current nondimensionalized by with H k the anisotropy field, δ the free layer thickness, η the spin torque efficiency, and ρ * = R * /L. In Eqn. 1, time has been nondimensionalized using the precessional time scale where γ is the gyromagnetic ratio and µ 0 is vacuum permeability. In Finally, thermal effects are modeled using the fluctuation-dissipation principle [18] and are described by the final term in Eqn. 1, which is taken to be space-time white noise with where I is the identity matrix, T is temperature, and β 2 = T /T 0 , with It is important to note that the stochastic differential equation (SDE) Eqn. 1 must be interpreted in the sense of Stratonovich to preserve the unit magnitude ofm (|m(x, t)| = 1 for all x ∈ R 2 , t ≥ 0).
The droplet is thus characterized by six free parameters: azimuthal phase φ 0 , position ξ, precessional frequency ω > 0, and velocity v. It was demonstrated in Ref. [14] using soliton perturbation theory in the limit of large nanocontact radius ρ * 1, weak damping and STT current i = O(α) 1, and low temperature β 0 1, that the droplet parameters where the process where It should be pointed out that in deriving Eqn. 10 from Eqn. 1 (i.e., from the general modulation equation (4.4) in Ref. [14]), the stochastic driving term is not strictly normalizable (i.e., does not have bounded variance) due to the 1/ρ contribution in the integrand near ρ = 0 (the deterministic term does not suffer this difficulty since 1/ρ is integrable but not square-integrable). Using the argument that the primary contribution from thermal fluctuations must come from values at the perimeter of the droplet ρ ≈ 1/ω 1, however, we disregard the contribution near ρ = 0 -where the droplet is exponentially close to pointing down -to provide a well-posed diffusion process.
Since no other parameters depend on φ and since spatial isotropy allows the coordinate frame to be aligned with ξ with nothing lost but an irrelevant angle, the full dynamics of Eqns. 7 through 10 are captured by the four-dimensional system with where and The nonlocal terms in Eqns. 13 through 16 are given by The terms in (7) through (10), and consequently (13) through (16), proportional to α or i correspond to contributions from magnetic damping or STT, respectively.

LARGE DEVIATION THEORY
Stochastic dynamical systems in the form of the Ito diffusion have long been the subject of study in physical systems where rare events are important phenomena to understand. It is now well understood [19] that, in the limit as → 0, the probability of paths connecting any two states u 1 and u 2 approaches zero with a scaling that is exponential in 2 . The rate of this asymptotic decay is dictated by minimizers of the Wentzell-Freidlin action functional, where the admissible set includes all absolutely continuous paths u(t) connecting u 1 and u 2 .
In the case of an exit from the basin of attraction of a deterministically stable fixed point, the time T over which the exit occurs clearly plays an important role in determining the probability of exit. If Ω is the basin of attraction of a fixed point u 0 , then where t E = min(t : u(t) / ∈ Ω) is defined as the first time of exit and ∂Ω is the boundary of Ω.
Over arbitrary long time scales, exits occur with probability one. The question is therefore not whether an exit will occur, but rather how long one should expect to wait before it does.
Under broad conditions, minima of the quasi-potential occur at 1-saddles, which represent the most likely points of exit from the basin of attraction of the stable fixed point.
Gradient flows (23) where f (u) = −∇V (u) for some potential function V (u) represent a special case in the theory of large deviations. In this case, it can be shown [19] that the quasi-potential between a minimum of V and the nearest saddle point is simply given by twice the difference in potential between the two points. One-dimensional SDEs are automatically gradient flows since one can simply define While the above consideration provides the exponential scaling law for transition probabilities and mean times to exit, a more detailed analysis is required [20][21][22] to improve on this estimate by providing the prefactor of the exponential, for example. In this work we discuss only the rate function, leaving a more accurate approximation for further study.

DETERMINISTIC DYNAMICS
In the four-dimensional system described by Eqns. 13 through 16, the first three parameters only depend on v ⊥ through the diffusion tensor. A complete description of the deterministic dynamics is therefore afforded by the reduced systeṁ The case v = 0 has been thoroughly studied in Ref. [13], where it was shown that multiple fixed points are possible with ξ = v = 0 depending on the number of roots of where we note that Γ(0 + ) = αh 0 > 0. Figure  The full stability of the centered fixed points is dictated by the Jacobian evaluated at (0, ω * , 0), where Bifurcation to the drift instability described in Ref. [15] occurs when i/α is sufficiently large to force λ 3 > 0, which occurs when The tangent direction of this unstable manifold is providing an indication of the ejection mechanism associated with the drift instability.
Analysis of non-centered fixed points with v = 0 (ξ = 0) is greatly simplified by the fact that they satisfy a two-dimensional algebraic system with the velocity then provided by For larger values of h 0 , however, the bifurcation is subcritical in i/α, with a sheet of saddles having only one unstable direction in close proximity to the stable centered equilibria.
These present the primary ejection mechanism when thermal effects are included, and will be discussed in the next section.
An alternative to the strategy above for solving Eqns. 37 and 38 that provides additional insight is to consider them asymptotically for large ρ * and small ω (i.e., the regime of validity for Eqns. (27) through (29)). We observe in that case that the support of each integral lies in the intersection between the circular disk of radius ρ * and a "washer" with O(1) width and central radius 1/ω, with the disk and washer centers separated by ξ, as depicted in Fig. 7. The two points of intersection lie at angles ±ϕ * satisfying so that by moving into droplet-centered coordinates, we have x≤ρ * where 0 ≤ ϕ * < π by construction. The implicit assumption in this construction is that the disk and annulus have nontrivial intersection, which approximately equates to the condition Treating the remaining integrals similarly, we have that Equating expressions for ω thus gives From Eqn. 43, we require that π − ϕ * > παh 0 /i, so that which confines the droplet center to within the nanocontact (see Fig. 7). Thus it is only within this region that the drift bifurcation is subcritical, such that the region where a stable fixed point coexists with a saddle point associated with the drift mode is the intersection of (47) and the half-plane bounded by (35).
The unique solution for ϕ * produces non-centered fixed points with approximate values Both of these quantities are positive. The associated parallel velocity is provided by which is well-approximated near the critical line i = αh 0 by The perpendicular velocity v ⊥ associated with non-centered fixed points is v ⊥ = 0. Hence, for each (ω v , ξ v , v v ) fixed point, there is a corresponding collection of fixed points for the original six-dimensional dynamical system (7-10) with droplet positions ξ on the circle |ξ| = ξ v .
Having identified all centered and non-centered fixed points, we now turn to the calculation of the action of paths connecting them.

ACTION AND OPTIMAL PATHS
In the limit of vanishing noise strength, exits from the basin of attraction of a stable fixed point collapse at an exponentially growing probability onto paths that lead to the nearest 1-saddle, where proximity is defined using a metric weighted by the diffusion tensor. In the four-dimensional system given by Eqns. 13 through 16, there are two candidates for the saddle that represent the most likely exit point, either a decay mode or a drift mode. We study these in turn to identify their associated limiting exit rates.
The first saddle is that identified at ξ = v = 0 for values of i/α that lie between the saddle-node bifurcation occurring at approximately i/α = h 0 and the value of i/α where the saddle either runs off to infinity (for small values of h 0 ) or is destroyed in a saddle-node bifurcation with another stable fixed point (for sufficiently large values of h 0 ). Labelling the centered stable fixed point by (0, ω * , 0) and the saddle by (0, ω s , 0), we postulate based on the form of the action (24) and numerical results that optimal paths connecting these two fixed points with optimal action S satisfy ξ = |v| = 0. Dependence of the diffusion tensor on the state variables precludes a definite statement that this is true, but we have observed it in all cases we have computed. Since this exit mechanism occurs strictly via changes in ω, thereby mimicking the impact of damping on solitons [13], we refer to it as a decay mode of exit.
If we assume this to be true, the action collapses to where, by construction, Γ(ω) ≤ 0 if ω * ≤ ω ≤ ω s and equality holds only at the endpoints.
As explained on p. 9, one-dimensional flows are trivially gradient flows and it is immediate that optimal paths satisfyω which integrates to give the Wentzell-Freidlin action where Q is the quasi-potential defined by with Li 2 (z) the dilogarithm function defined by The quasi-potential (55) has been defined such that lim ω→∞ Q(ω) = 0. As explained earlier, depending on the parameters h 0 and i/α, the saddle point (0, ω s , 0) may not exist. In these cases, we identify an exit via the decay mode with ω → ∞, such that the associated Wentzell-Freidlin action simplifies to S = −Q(ω * ). As illustrated in Fig. 8, expression (55) for the quasipotential Q blows up as ω → 0. This is expected since the noise strength goes to zero as ω → 0. Figure  The contribution from nonzero v ⊥ is expected to be negligible because v ⊥ = 0 at all fixed points, however. For simplicity, we therefore restrict our computations to the minimizers of the action restricted to three-dimensional dynamics, giving where u = (ξ, ω, v ), The diffusion tensor appearing in Eqn. 57 is The drift f (u) is not, in general, either gradient or trivially non-gradient and the action must therefore be computed numerically. The left image in Fig. 10 depicts the approximate MTE obtained by using (25) with the action computed using the generalized minimum action method (or GMAM, see Ref. [23] for details) at parameter values where the base stationary state is stable and coexists with the saddle corresponding to the drift mode. As explained on p. 14 and observed in Figs. 4 through 6, for a given nanocontact radius ρ * , this is approximately the set of current values h 0 < i/α < 2h 0 . The right image in Fig. 10 is the approximate MTE associated with leaving the basin of attraction of stable non-centered fixed points through the now-unstable centered fixed point.
In the parameter regime operable in Fig. 9 and the left image of Fig. 10, the droplet soliton is potentially destroyed via two candidate noise-related mechanisms. The first is the drift mechanism described above, where the position and precessional frequency are driven by noise to the point where the droplet loses stability to a transient traveling droplet which is then ejected from the nanocontact. The MTE associated with this mechanism is plotted on the left in Fig. 10. The second is the decay mode described earlier, whereby an improbable accumulation of fluctuations that drive its precessional frequency sufficiently high as to fall into a regime where damping is no longer counterbalanced by the current passing through the nanocontact. The stochastic dynamics of this mechanism has an associated MTE depicted in Fig. 9. It is clear from this comparison that the action associated with the drift mode is substantially smaller for all parameters investigated here, indicating that this is by far the most dominant mechanism for exit of a centered droplet soliton over the range of values of magnetic field h 0 and rescaled current i/α for which the original fixed point is stable and the saddle associated with the drift mode exists and has a one-dimensional unstable Right: MTE from stable non-centered fixed points through centered saddle points. This calculation is relevant for a different region of parameter space. Note that the exit times associated with this droplet annihilation mechanism are much lower than the exit times shown in Fig. 9, and these events are therefore much more likely to be observed on any finite time scale.
manifold. As discussed earlier, this is true for a subset of rescaled current values satisfying i/α < 2h 0 . For large current but small external field h 0 , where the non-centered fixed points are stable, it appears that the only exit mechanism for those non-centered droplets is through the centered saddle point with a MTE somewhat higher than the exit times associated with annihilation of centered droplets; see Fig. 10, right for the MTE associated with this ejection mechanism.

CONCLUSIONS
Perturbation theory for a droplet soliton that experiences uniform magnetic damping, local spin transfer torque via a nanocontact, a vertical applied field, and thermal fluctuations, results in a six-dimensional system of stochastic ordinary differential equations. In this paper, we have reduced this system to an equivalent-in the deterministic case-or approximately equivalent-in the stochastic case-three-dimensional system that is amenable to detailed analysis. We completely determined the nature of the fixed points for this dynamical system. One family of fixed points, called nanocontact-centered, exist and are deterministically stable in a restricted regime of current and applied field. These fixed points represent dissipative droplet solitons centered within the nanocontact and precessing at a fixed frequency. In addition to previously studied nanocontact-centered fixed points, we also identify a new class of non-centered fixed points. These latter fixed points correspond to droplet solitons whose centers are displaced relative to the center of the nanocontact, with the resulting force balanced by an anisotropy in the soliton's in-plane magnetization. These non-centered fixed points are stable in an appropriate parameter regime, but we have not found any parameters at which the centered and non-centered droplet solitons are simultaneously stable. By combining our fixed point analysis with large deviation theory, we identify the mostly likely path through phase space along which a droplet subject to weak thermal noise is expected to decay. A direct computation of the Wentzell-Freidlin action shows that the most likely decay process for stochastically perturbed, centered droplet fixed points is to drift out of the nanocontact. The computed action also provides an estimate for the rate of droplet expulsion from the nanocontact, which sets an effective lower bound for droplet stability in these devices at finite temperature. Based on the parameter values in