Self-organization in the one-dimensional Landau-Lifshitz-Gilbert-Slonczewski equation with non-uniform anisotropy fields
Mónica A. García-?ustes, Fernando R. Humire, Alejandro O. Leon
SSelf-organization in the one-dimensionalLandau-Lifshitz-Gilbert-Slonczewski equation with non-uniformanisotropy fields
Mónica A. García-Ñustes ∗ Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Chile
Fernando R. Humire † Departamento de Física, Facultad de Ciencias,Universidad de Tarapacá, Casilla 7-D Arica, Chile
Alejandro O. Leon ‡ Center for the Development of Nanoscience andNanotechnology, CEDENNA, Santiago, Chile
Abstract
In magnetic films driven by spin-polarized currents, the perpendicular-to-plane anisotropy isequivalent to breaking the time translation symmetry, i.e., to a parametric pumping. In this work,we numerically study those current-driven magnets via the Landau-Lifshitz-Gilbert-Slonczewskiequation in one spatial dimension. We consider a space-dependent anisotropy field in theparametric-like regime. The anisotropy profile is antisymmetric to the middle point of the sys-tem. We find several dissipative states and dynamical behavior and focus on localized patternsthat undergo oscillatory and phase instabilities. Using numerical simulations, we characterize thelocalized states’ bifurcations and present the corresponding diagram of phases. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ n li n . AO ] J a n . INTRODUCTION Non-equilibrium systems present complex dynamics [1, 2], including pattern formation [2,3], localized states [2], and chaotic behaviors [1]. Nanometric magnetic systems exhibit quasi-Hamiltonian dynamics, perturbed by a relatively small injection and dissipation of energy [4].Domain walls [5, 6], self-sustained oscillations [7, 8], textures [9–12], topological [13, 14] andnon-topological [12, 15–17] localized states are examples of non-equilibrium states of drivennanomagnets. These are solutions of a nonlinear partial differential equation that governsthe magnetization dynamics in the continuum limit, namely the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) model [4]. Even though Landau and Lifshitz published the first form ofthis equation in 1935 [18], it is still widely investigated and revised to incorporate recentlydiscovered effects. For example, electric currents with a polarized spin [19, 20] are modeled asa non-variational term that injects energy and favors limit cycles [7, 8]. Also, the dispersion ofmagnetization waves can be generalized to include anisotropic terms that create topologicaltextures [13]. A third relevant mechanism is the modulation of the magnetic anisotropyfields by applied voltages in insulating structures. Since these fields are responsible for thesaturation of the magnetization near equilibria (i.e., they are the most relevant nonlinearitiesof the LLGS equation), their tuning by electric fields promises a space or time-dependentcontrol of both the linear and nonlinear parts of the LLGS system.The voltage-controlled magnetic anisotropy (VCMA) effect [21–24] promises memory de-vices with low power consumption due to the absence of Joule dissipation. Furthermore,VCMA can induce several magnetization responses. For example, a voltage pulse can as-sist [22] or produce [23] the switching of the magnetization between two equilibria. Voltagesoscillating near the natural frequency generate resonances [24]. On the other hand, if thevoltage oscillates at the twice natural frequency, parametric instabilities [25, 26], Faraday-type waves, and localized structures emerge [17]. These phenomena also appear in other parametrically driven systems that are excited by a force that simultaneously depends ontime and the state variable. Magnetic anisotropies can be manipulated by applying a strainto the magnetic medium, via the modulation of its thickness and doping with heavy atoms.While temporally modulated magnetic anisotropies receive considerable attention, the self-organization arising from its static non-uniform counterpart has not been fully explored.In this work, we study the Landau-Lifshitz-Gilbert-Slonczewski equation in one spatialimension, representing a magnet subject to a magnetic field, a spin-polarized charge cur-rent [7, 8, 19, 20], and a non-uniform perpendicular magnetic anisotropy. We focus on theparameter values where the current acts as additional damping and stabilizes an equilibriumthat would otherwise be unstable. In this regime, the magnet behaves as a parametrically-driven system, even if its parameters are time-independent. The parametric nature of themagnet manifests as a breaking of symmetry induced by the magnetic anisotropy field, inthe same way as a time-varying force breaks the time-translation invariance in parametri-cally driven systems. Furthermore, the LLGS equation can be mapped to the parametricallydriven damped Nonlinear Schrödinger equation (pdNLS), which is the paradigmatic modelof parametrically forced systems. In this transformation, the current acts as damping, theapplied magnetic field is a frequency shift or detuning, and the anisotropy field is equivalentto a parametric injection of energy. Regarding the anisotropy field, the considered field pro-file is the sum of two Gaussian functions with opposite signs. We find that localized patternsemerge when the anisotropy field is large enough, as occurs in parametrically driven systemswith a heterogeneous excitation [27].Those localized patterns can be dynamic or stationary, depending on the parametervalues. For example, when the absolute value of the anisotropy at the left and the right arenot the same, localized patterns drift. Via the calculation of the eigenvalues of the fixedpattern near the drifting transition, we find that a stationary instability is responsible forthis bifurcation. This instability is of a subcritical type, which creates bistability betweendrifting and pinned patterns. If the modulus of the left and right anisotropy fields aresimilar, an oscillatory (Andronov-Hopf) instability occurs when the modulus of the chargecurrent is below a threshold. Similar results replicate as the separation distance betweenthe left and the right of the anisotropy varies.The anisotropy profile considered here resembles some properties of the so-called parity-time ( PT )- symmetric systems, which are characterized by internal gain and loss but conservethe total energy. Recently, the generalization of PT - symmetric systems to include injectionand dissipation of energy as a small perturbation, i.e., quasi- PT -symmetric systems , havegathered some attention, see [28] and references therein. Phase transitions in out-equilibriummagnetic systems, via the breaking of a PT -symmetry, have been reported [29–31]. Forexample, in Refs. [29, 30], with the use of a spin Hamiltonian with an imaginary magneticfield, one arrives at the LLGS equation [29]. A phase transition between conservative andon-conservative spin dynamics is discussed in terms of the PT -symmetry and extended tospin chains [30]. Stable solitons in nearly PT -symmetric ferromagnet with the spin-torqueoscillator with small dissipation [31].The article is organized as follows. In the next section, we introduce the LLGS equationand transform it into an amplitude equation with broken phase invariance (i.e., the pdNLSequation). In Sec. III, we show our numerical results, while our conclusions and remarks arein Sec. IV. II. LANDAU-LIFSHITZ-GILBERT-SLONCZEWSKI EQUATION
Let us consider a ferromagnetic medium with two transverse lengths small enough toguarantee that the magnetization remains uniform in those directions (i.e., the magnetiza-tion varies along one axis only). Its magnetization is M ( x, t ) = M s m ( x, t ) , where M s is thenorm, and m is the unit vector along the orientation of M . The dimensionless space andtime coordinates are x ∈ [0 , L ] and t ∈ [ t , t f ] , respectively, where L is the length of themagnet, and t and t f are the initial and end time of the simulation. The magnet is part ofa so-called nano-pillar structure, see Fig 1(a). There is an applied magnetic field h = h e x ,where { e x , e y , e z } are the unit vectors along the corresponding Cartesian axis. The spa-tiotemporal dynamics of the magnetization obeys the Landau-Lifshitz-Gilbert-Slonczewskiequation, which in its dimensionless form reads [4] ∂ t m = − m × h eff + g m × ( m × e x ) + α m × ∂ t m , (1) h eff = h e x + ∂ xx m − h d ( x ) m z e z , (2)where the first term of Eq. (1) induces energy-conservative precessions around the effectivemagnetic field h eff . It has contributions from the external field h e x , the exchange (or dis-persion) field in one spatial dimension ∂ xx m , and the anisotropy field − h d ( x ) m z e z . Thefunction h d is the sum of the magnetocrystalline anisotropy and the demagnetizing fieldin the local approximation. The effective field is the functional derivative of the magneticenergy E M , h eff ≡ − δE M /δ m , where E M = (cid:90) L dx (cid:20) − hm x + h d ( x )2 m z + | ∂ x m | (cid:21) . (3)The second term of the LLGS equation is the spin-transfer torque [19, 20] induced by thecurrent and parametrized by g . This is a non-variational effect that can inject into (for > ) and dissipate (for g < ) the magnetic energy. Finally, the third term of Eq. (1)is a phenomenological Rayleigh-like dissipation, ruled by the dimensionless parameter α .Typical values [4] of g and α are g ∼ − and α ∼ − − − , respectively.Two equilibria of Eq. (1) exist for all parameter values. In the first equilibrium, the mag-netization is parallel to the spin polarization of the charge current ( m = e x ) and the otherwhere they are antiparallel ( m = − e x ). Positive (negative) values of g tend to destabilize(stabilize) the equilibrium m = e x . On the other hand, positive (negative) fields h favor(disfavor) the state m = e x . Then, in this type of nanometric magnets, there can be twocompeting effects, the current and the field, and non-trivial non-linear dynamics emerge.The magnetization dynamics induced by the combination of currents and fields haveseveral similarities [12] with systems subject to a forcing that oscillates at twice their naturalfrequency, namely, parametrically driven systems . This association becomes evident whenusing the so-called Stereographic mapping ψ = ( m y + im z ) / (1 + m x ) , which projects themagnetization field on the unit sphere m = 1 to the complex plane (see Ref. [12] andreferences therein). The resulting amplitude equation reads ( i + α ) ∂ t ψ = ( ig − h ) ψ − h d ψ − ψ ∗ ) 1 + ψ | ψ | + ∂ xx ψ − ψ ∗ ( ∂ x ψ ) | ψ | , (4)with ψ ∗ being the complex conjugate of ψ . The equilibrium m = e x is mapped to ψ = 0 .Note that the perpendicular anisotropy coefficient controls both the parametric injectionof energy [i.e., terms proportional to ψ ∗ in Eq. (4)] and the cubic saturation terms. Then,we expect that the modulations of h d will produce appealing dynamical effects. See alsothat the role of the nonlinear gradients ψ ∗ ( ∇ ψ ) / (1 + | ψ | ) in the dynamics can becomemore critical compared to the one of the relatively week anisotropy-induced saturations. m Spin filterspacer h e x e y e z e x Figure 1. Schematic representation of the setup understudy. A magnet with unit magnetization m is affectedby a magnetic field h and charge current. The current isspin-polarized after traversing a thicker magnet with mag-netization along the unit vector e x . We focus on the layerwith a variable perpendicular-anisotropy coefficient, whichis represented as a thin layer with variable thickness. n the next section, we numerically solve equation (1). However, for pedagogical reasons,let us analyze here its similarities to the pdNLS model.Let us consider the scaling α ∼ | g | ∼ | h | ∼ | h d | (cid:28) , which ensures that each effect(dissipation, detuning, dispersion, and phase-invariance-breaking term) appears once in theequation. This scaling is not difficult to obtain since the current ( g ) and magnetic field ( h )are control parameters that can be tuned to be of the order of the damping ( α ). Beyond thevoltage-controlled magnetic anisotropy [21–24] where h d is a control parameter but needs aninsulating layer, the anisotropy field can be engineered by bulk [32, 33] and interfacial [34]spin-orbit interactions and magnet thickness, magnetostriction [35], periodic modulation ofsurfaces [36], among other mechanisms. With those considerations, Equation (4) at linearorder, reads ∂ t ψ = − µψ − i (cid:0) ν + ∂ xx (cid:1) ψ − iγψ ∗ , (5)where µ = − g , ν = − h − h d / , and γ = h d / . The previous equation is the linear versionof the well-known parametrically driven damped NonLinear Schrödinger (PDNLS) equation,which governs the dynamics of classical systems subject to a force that simultaneouslydepend on time and the state variable. In those systems, the parameter µ accounts for theenergy dissipation, ν is the detuning between half the forcing frequency and the naturalfrequency of the system, and γ is the strength of the energy-injection.Note that in the studied system, the parameters related to the energy injection ( γ ) anddetuning ( ν ) depend on the anisotropy field h d . Also, from Eq. (5), we know that patternsor Faraday-type waves [37] emerge in the magnetic system for v ≥ and γ ≥ µ [12].The nonlinear dynamics of nanomagnets with uniform anisotropy coefficients have beenextensively studied in the literature. In the next section, we focus our attention on sys-tems with a non-uniform anisotropy. Some mechanisms that can produce a non-uniformanisotropy are films with interfacial anisotropy and variable thickness. III. NON-UNIFORM ANISOTROPY FIELD
Let us consider the anisotropy profile h d ( x ) = β zL e − ( x − xL ) σ − β zR e − ( x − xR ) σ , (6)here x L = L/ − a and x R = L/ a are the centers of the left and right Gaussians, respec-tively. The parameter a = | x L − x R | controls the separation between the Gaussian maxima, σ is the characteristic width and L is the device length. Thus, the injection of energy iscontrolled by four parameters, namely { β zL , β zR , a, σ } . When the device is large enough, thespatial average of the Eq. (6) is (cid:82) L dxh d ( x ) ≈ (cid:82) ∞−∞ dxh d ( x ) = √ πσ ( β zL − β zR ) . Then, nonet parametric-like injection of energy occurs for β zL = β zR . However, rich self-organizationis expected from this driving mechanism [28]. It is worth noting that in other parametricallydriven systems with heterogeneous forcing, the qualitative dynamics are insensitive to thespecific form of the injection function if its maximum value and characteristic width aresimilar [27]. Furthermore, there is agreement between experiments in fluids vibrated with asquare-like spatial profile and theory with Gaussian and parabolic modelings [27].In addition to the spatial instability mentioned in the previous section, another bifurcationtakes place when µ changes its sign. It occurs when the spin-polarized current injects energyinto the system and induces an Andronov-Hopf (oscillatory) instability. At the bifurcationpoint, µ ≡ , the oscillator has injection and dissipation of energy only in the nonlinearterms of its equation of motion, and the system has some similarities with the ones with PT -symmetry.We integrate Eq. (1) using a fifth-order Runge-Kutta algorithm with parameter valuessimilar to the following set: α = 0 . , g = − . , h = − . , β zL = 1 , β zR = 1 , a = 6 , and σ = 3 . . The simulation box is discretized as L = dx ( N − , with step size dx = 0 . and N = 512 simulation points. We use specular boundary conditions, i.e., the magnetizationgradient vanishes at the borders ( ∂ x m ) (0 , t ) = ( ∂ x m ) ( L, t ) = 0 . The magnetization norm, [ (cid:80) i m i /N ] / is monitored and never deviates from 1 more than − , which is enoughprecision for this work. A. Limit of strong interaction
Let us start with the limit where the two Gaussians are close to the center but a dif-ferent amplitude ( β zL (cid:54) = β zR ), see Fig. 2(a). As it occurs in parametrically driven systemswith homogeneous [37] and heterogeneous [27] energy injections, the parametrically-inducedspatial instability creates a texture, as shown in Fig. 2(b) and (c) for the spatiotemporaldiagrams and snapshots of the m y variable. We fix β zR = 1 and change the values of β zL .or β zL < β c , the patterns exhibit a drifting-like dynamics where they continuously travelto the center of the simulation. The left-traveling waves have a larger amplitude becausethey are excited by a stronger forcing β zR > β zL . The left panel of Fig. 2(b) and (c) illus-trate the drifting solutions. When the β zL value is increased above a threshold, stationarypatterns emerge subcritically, creating a hysteresis zone in β c ≤ β zL ≤ β c . In the region β c ≤ β zL ≤ β c , only the stationary patterns are observed, while for β zL > β c the pat-tern undergoes an oscillatory instability that produces breathing-like motions of the patternphase. The maximum value of the m y component, max [ m y ( x, t )] , is a single number thatmeasures the amplitude of the pattern. On the other hand, the standard deviation ∆ m y ofthe temporal series m y ( x , t ) , where x is the position where the max [ m y ( x, t )] is reached,provides useful information of the oscillatory character of the solutions. Figures 2(d) and (e)reveal the diagram of phases of the system in the strong interaction limit using max [ m y ( x, t )] and ∆ m y , respectively.The analytic study of the instabilities of non-uniform states in systems with non-uniform parameters is a hard task. However, one can conduct a numerical study asfollows. Let us start writing the magnetization vector in the Spherical repsentation, m = sin θ (cos φ e x + sin φ e y ) + cos θ e z . The dynamical variables θ ( x, t ) and φ ( x, t ) arethe polar and azimuthal angle, respectively. Integrating Eq. (1), one obtains the station-ary solution of the localized pattern, { θ ( x ) , φ ( x ) } , and the small deviations around thisstate satisfy { δθ , φ } = (cid:80) j e λ j t (cid:126)u j . The eigenvalues λ j can be obtained numerically afterdiagonalizing the Jacobian matrix of Eq. (1) for the state { θ ( x ) , φ ( x ) } . Starting froma numerically solved stationary pattern state, Fig. 3 shows the stability spectrum nearthe subcritical transition to a drifting pattern in (a), in the zone of stationary patterns in(b), and close to the oscillatory instability in (c). The spectrum allows us to categorizethe transitions at β zL = β c and β zL = β c as stationary and Andronov-Hopf bifurcations,respectively.Similar results replicate as the separation distance between the left and the right of theanisotropy varies. For β zL = 0 . and β zR = 1 fixed, we varied the separation parameter a in the interval ≤ a ≤ . . Figure 4(a) displays the different dynamical regimes forinteracting localized patterns as a function separation parameter a . For values smallerthan a min = 2 , the pattern disappears due to the partial cancellation the Gaussians. Adrifting-like behaviour is observed for a min < a < a c = 5 [shown in Fig. 4(b)]. Given both zL = β zL = β zL = h d ( x ) x x L x R β zL h d ( x ) β zL h d ( x ) β zL x x L x R x x L x R (a) x x L x R x x L x R x x L x R t t t t i2 t i1 m y ( x,t ) m y ( x,t ) m y ( x,t ) x x L x R x x L x R x x L x R m y ( x , t i ) m y ( x , ) m y ( x , t i ) (b)(c) m a x [ m y ( x , t )] β zL β zL Δ m y Stationarypattern Oscillatorypattern D r i f t Bistability
Stationarypattern Oscillatorypattern D r i f t Bistability (d) (e) β c β c β c β c β c β c Figure 2. Dynamics of closely interacting localized patterns. The dynamical regimes of drifting,stationary and phase-oscillatory patterns are exemplified in the left, central, and right panels. (a)shows the anisotropy (or, equivalently, the parametric injection) profile as a function of the posi-tion x for β zL = 0 . (left), β zL = 0 . (center), and β zL = 0 . (right). The Gaussian functionsare very close, which produces a strong coupling of the localized patterns. Panel (b) is the spa-tiotemporal diagram of m y for the same β zL values of (a). As these diagrams illustrate, for small β zL , the patterns have a drifting-like phase dynamics. For moderate values, the localized patternis stationary. For quasi-symmetric injection parameters, β zL (cid:46) β zR = 1 , a breathing-type phasedynamics appears. (c) shows the snapshots of m y at the times when the maximum value is reached,max [ m y ( x, t )] . (d) Diagram of phases using the maximum value of the magnetization y − th com-ponent, max [ m y ( x, t )] , for a given set of parameters. (e) The oscillation amplitude of the pattern ∆ m y for a set of parameters. The number ∆ m y is defined as the standard deviation of m y ( x , t ) ,where is the position where the maximum max [ m y ( x, t )] is achieved. Gaussians are interacting strongly in this limit, the drift is more complex that one observedin the previous case, Fig. 2(b). The right-traveling waves struggle to enter into the leftregion. In contrast, stationary and oscillatory regimes, shown in Fig. 4(c) and 4(d), are .5 (a) -1.503-3 I m [ λ j ] -2 -1 0 Re[ λ j ] I m [ λ j ] -2 -1 0 Re[ λ j ] (c) I m [ λ j ] -2 -1 0 Re[ λ j ] (b) β zL = β zL = β zL = Figure 3. Stability spectrum in the closely interacting limit. The eigenvalues λ j are plotted in thecomplex plane, with Re( λ j ) and Im( λ j ) being the real and imaginary parts of λ j , respectively. (a)A stationary instability takes place when reducing β zL . Indeed, when β zL → β c , the eigenvalueswith the largest real part reduce their imaginary component and cross the vertical axis at the origin.For β zL < β c , only drifting-like motions are observed, see Fig 2. On the other hand, for β c <β zL < β c , both the stationary and the localized drifting patterns are stable. (b) Spectrum far fromany bifurcation bifurcations, all eigenvalues have a negative real part, which characterizes stablestates. (c) Onset of the Andronov-Hopf instability. The leading modes have a finite frequency. For β c < β zL , the localized patterns exhibit a phase-breathing-like oscillatory dynamics, as illustratedon the spatiotemporal diagram of Fig 2(b). similar to the ones of Fig. 2. The stationary-drifting bifurcation is subcritical-type, showinga bistability region in the interval a c < a < a c = 7 . , while a supercritical one describesthe emergence of oscillatory patterns at the critical value a c . Small differences observed inpaths for a < a c and a > a c are numerical. B. Diluted limit
In this subsection, we present our results in the limit where the Gaussians are far away,and consequently, the dissipative structures at the right and left zones are only slightlyinteracting. When the injection peaks are well separated, the amplitude mismatch β zL − β zR is not important for the dynamics. Thus, we use the values β zL = β zR = 1 and vary thecurrent parameter g . When g → , the dominant dissipation mechanism vanishes, and theequilibrium m = e x loses its stability. Then, when reducing | g | , we expect the emergence ofdynamical states. Figure 4 summarizes our results. For very negative values of the current g , that is for significant dissipation µ , only one pattern emerges, see Fig. 4(a) and (b). ] x t -0.500.5 L m y (x,t) (b) [ ] x t -0.500.5 m y (x,t) (c) L [ ] x t -0.500.5 m y (x,t) (d) L0 Figure 4. Dynamics of interacting localized patterns as a function of the separation parameter a . (a)Diagram of phases using the maximum value of the magnetization y -th componente, max[ m y ( x, t ) ]as a function of the separation parameter, for a given set of parameters. For fixed value β zL = 0 . and β zR = 1 , the patterns have a drifting-like phase dynamics for a < a c = 5 . For a c < a < a c astationary pattern is observed. A breathing-type phase dynamics appears for a > a c = 7 , . Thedynamical regimes of drifting, stationary and phase-oscillatory patterns are exemplified in (b), (c)y (d), respectively. When the dissipation goes to zero, the stationary pattern loses stability via a supercritical[see Figs. 5(c) and (d)] Andronov-Hopf [see Figs. 5(e)] bifurcation. The new state emitsevanescent waves from its core. For even larger values of the current, the right Gaussianstabilizes a static texture.
IV. CONCLUSIONS AND REMARKS
Nanoscale magnetization dynamics attract considerable attention due to their appealingnon-equilibrium behaviors and the associated technological applications. A driving mech-anism recently discovered is the modulation of the anisotropy field via the application ofvoltages, tuning the magnet thickness, and interfacial doping. The temporal modulation of m a x [ m y ( x , t )] Δ m y Stationarypattern Oscillatorypattern Stationarypattern Oscillatorypattern (c) (d) x x L x R x x L x R x x L x R t t t m y ( x,t ) m y ( x,t ) m y ( x,t ) x x L x R x x L x R x x L x R m y ( x , ) m y ( x , t i ) m y ( x , t i ) (a)(b) -0.34 -0.3 -0.26 -0.22 -0.18 g t i2 t i1 −0.6 (e) -0.34 -0.3 -0.26 -0.22 -0.18 g g = - I m [ λ j ] -0.2 -0.1 0 Re[ λ j ] -0.3 Figure 5. Dynamics in the diluted regime. The separation between the anisotropy field peaks is a = 20 , which guarantees that the excited localized states interact only slightly. (a) Spatiotemporaldiagram for the stationary localized pattern (left), oscillating texture (center), and both an oscil-latory and a static pattern (right). Panel (b) shows a snapshot of the m y variable at a given time.(c) and (d) show the bifurcation diagram of the left localized pattern in terms of the maximum andstandard deviation of the m y temporal series, equivalent to the graphs of Figs. 2(d) and (e). Thestability spectrum is shown in (e), and it reveals a pair of eigenvalues with large imaginary com-ponents. As the current parameter g increases, the real part of those eigenvalues becomes positive,leading to the self-sustained oscillations. the anisotropy function has gathered considerable attention. However, texture formation anddynamics by heterogeneous anisotropy fields remain mostly unknown. This work is a stepin this direction. We studied the self-organization of a one-dimensional magnetic mediumdriven by an applied magnetic field, a spin-polarized electric current, and a non-uniformanisotropy field. The well-known Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation de-scribes this system. We concentrated on the parameter sets where the magnet is equivalentto a parametric resonator, even if the magnet is only subject to time-dependent forcingmechanisms.The starting point of this manuscript is a simple model based on the one-dimensionalnd local approximations. However, the numerous and challenging behaviors found justifythe use of such approximations as a necessary first step before exploring more elaborateconditions.The profile of the anisotropy field is a sum of two Gaussian functions with oppositesigns. We found a family of localized pattern states. They can be stationary, drifting, oroscillatory textures. In the strong interaction regime, the localized drifting patterns originatefrom a stationary instability of a static texture, as shown via the calculation of the linearspectrum of the pattern. On the other hand, when the Gaussian functions have similarabsolute values, then the patterns undergo an oscillatory (Andronov-Hopf) bifurcation witha breathing phase mode. When the anisotropy field is composed of well-separated peaks,only one transition is observed, namely, localized patterns transit between stationary tophase-oscillatory regimes when the dissipation or injection parameters are modified.Our findings may motivate the use of localized patterns as effective individual oscillationswith the capacity to couple and interact, and could potentially be used as units of a nano-oscillators network. ACKNOWLEDGMENTS
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