Simple nanomagnets execute limit cycle trajectories under ferromagnetic resonance conditions

Nanomagnetic particles respond sensitively and nonlinearly to electromagnetic radiation. Many excitation schemes are now well known. However, nonlinear dynamics determinations have not been examined in detail under FMR conditions. The nonlinear dynamics of a simple nanomagnet is studied and the excitation field, H1, is varied. We solve numerically the Landau–Lifshitz nonlinear dynamics of M ( t ) . There is a special focus on the spherical degrees of freedom: θ ( t ) , ϕ ( t ) . We find that the θ ( t ) trajectories converge asymptotically to θ asym = constant, while ϕ ( t ) is a linear function of time. The combined dynamics of θ ′ ( t ) and ϕ ′ ( t ) produce a limit cycle for each value of H1. The systematic numerical calculations and analysis show that the limit cycle θ asym follows a fourth-degree polynomial on H1 and an inverse law on frequency ν1. It is also found that the limit cycles are established after 12–20 nanoseconds. They cause M ( t ) to sweep a constant precession cone that lasts for more than 200 ns independent of initial conditions. These results bring significant novel knowledge for fast information technology.


Abstract
Nanomagnetic particles respond sensitively and nonlinearly to electromagnetic radiation. Many excitation schemes are now well known. However, nonlinear dynamics determinations have not been examined in detail under FMR conditions. The nonlinear dynamics of a simple nanomagnet is studied and the excitation field, H 1 , is varied. We solve numerically the Landau-Lifshitz nonlinear dynamics of M t .
( ) There is a special focus on the spherical degrees of freedom: t , q ( ) t . f ( ) We find that the t q ( ) trajectories converge asymptotically to asym q =constant, while t f ( ) is a linear function of time. The combined dynamics of t q¢( ) and t f¢( ) produce a limit cycle for each value of H 1 . The systematic numerical calculations and analysis show that the limit cycle asym q follows a fourth-degree polynomial on H 1 and an inverse law on frequency ν 1 . It is also found that the limit cycles are established after 12-20 nanoseconds. They cause M t ( ) to sweep a constant precession cone that lasts for more than 200 ns independent of initial conditions. These results bring significant novel knowledge for fast information technology.

Introduction
Fundamental studies of the spin dynamics and the development of nanomagnetic materials and arrays are of great relevance in the present time due to the general trend towards miniaturization of all sort of electronic devices [1,2], GHz wireless communication, the development of spintronic devices [3,4], and the potential development of ever faster magnetic memories with increased storage capacities [5,6]. A better understanding of the full dynamics and the time development of the magnetization, M t , ( ) of individual and structured nanomagnets under different excitation schemes [7][8][9], and of magnetic stripes and thin films [10][11][12], is desired in order to develop new nanomagnetic functions and devices. For example there are complete studies of nonlinear response and synchronized magnetization dynamics of spin-transfer nano-oscillators under spin current and weak microwave fields [13][14][15]. Simpler nanomagnets without demagnetizing fields, anisotropies or thermal fields are also good potential candidates for applications. Here, we focus on the nonlinear dynamics of the magnetization as given by the Landau-Lifshitz equation of motion [9,16,17] of an isolated, isotropic nanomagnet, with the magnetic structure of monodomain under excitation conditions of ferromagnetic resonance-FMR. This is to say, a Zeeman magnetic field, H 0 and a periodically varying microwave field, H , 1 are applied orthogonally to each other. The Landau-Lifshitz, nonlinear equation of motion-LL for M t ( ) has not been analytically solved for any general conditions of excitation, and/or anisotropies [9,18]. It has been analytically solved in a few simple instances under excitation conditions far from ours. [19,20]. In this work, the Landau-Lifshitz equation of motion is solved numerically, while Kittel's resonance condition holds [17,21]. The thermal field effects are not considered in this model.  [17,21]. In the linearized-approximation FMR-treatment, M t ( ) precession is supposed to be uniform and stable as long as the system absorbs the microwave energy from the excitation field, H 1 (t). Since H 1 is very small (perturbative) compared to the Zeeman field, H 0 , it is usually stated that the angle of aperture, , FMR q of the precession cone is small, ∼3°. But it is not known how the magnetization reaches such state; and within itself, what the effect of H 1 (2πν 1 t) in the motion of M t ( ) is. The time scales of these processes are of the order of nanoseconds, but more precise knowledge is lacking. So, the picture of the dynamics of M t ( ) that emerges from these treatments is broad and approximated. Our detailed numerical calculations allow finding dynamic-asymptotic states of M t , ( ) which are limit cycles denoted by ( , asym q t f ( )). t q ( ) trajectories converge asymptotically to asym q =constant, while t f ( ) is a linear function of time. H 1 , the amplitude of the excitation, is the cause of the existence of the limit cycles. asym q depends non-linearly on H 1 and varies as a power law with the excitation frequency. The limit cycles are stable, circular, with periods in the picosecond regime, and are established non-instantaneously. M t ( ) in the limit cycle state of motion sweeps a constant precession cone with the tip of M t ( ) contained always on the M | |-sphere, resembling the simplified linear FMR-precession. All these results inform us of a very rich non-linear dynamics of the magnetization while executing ferromagnetic resonance, being the non-instantaneous development of the limit cycle of M t ( ) (after many nanoseconds) perhaps the most significant feature found. Some of these complex and rich nonlinear dynamics could be harvested in order to design new functions and operations in all areas of modern technology. Msin t cos t ,

Ferromagnetic resonance
The ferromagnetic resonance condition for any magnetic specimen, simple nanomagnets included, subjected to the following external and internal magnetic fields: a static field H 0 , an excitation microwave magnetic field, H 1 (ω 1 t), of small amplitude and usually orthogonal to H 0 , and an assortment of anisotropic fields (including demagnetizing fields), is given in general by Kittel's expression [17,21]: = ( ) H T contains the contributions of the external (H 0 , H 1 ) and internal fields that the magnetic specimen, in the general shape of an ellipsoid of revolution, experiences. H D represents the demagnetizing fields, H shp and H K the anisotropy fields experienced by the magnetic specimen. ω r is the angular frequency of the resonant absorption of microwave power and γ is again the gyromagnetic ratio.
The nonlinear dynamics of M t ( ) is studied when both equations (1) and (2) are simultaneously satisfied.
For the nanomagnetic specimen, isotropic, isolated and monodomain, all the anisotropy and demagnetizing fields in H T vanish, leaving only

Just in resonant condition
To say that our nanomagnet is under ferromagnetic resonance condition is to say that (3) and (4) hold simultaneously.
In the FMR experiment H t H cos t x ( )ê is linearly polarized and under easy control at the laboratory with H 1 1mT and 2 1 1 w p n = / usually taking any value from 2 to 33 GHz and sometimes even 64 up to ∼150 GHz [22,23]. This means that we fix ω 1 , the frequency of the small, oscillating, excitation field, H , 1 and H H , = ê the external magnetic field (also known as the Zemann field), has to be varied to meet the equality in equation (4), so the resonance frequency is set to , 1 w then , and H 0 is varied until it meets the value (ω 1 /γ). It is under these resonance conditions that the solutions for the Landau-Lifshitz equation of motion are sought. Now, the Landau-Lifshitz equation of motion (1) is simply rewritten by using ,

Defining the unitary vector magnetization m M M,
= / the LL equation of motion becomes: In terms of applied torques we have m d dt t .
Where the 'free' motion torques are t ag =´( ) is dependent of time and oscillating at ω 1 . We observe that 0 dampH0 The damping torques are smaller than the 'free' motion torques by a factor α, the strength of the damping. For nanomagnets with very small damping constant α, the dynamics lose energy and damp very slowly. Notice that the external oscillating torque τ 1 , in addition to contribute to the nonlinear dynamics, is injecting energy ) because it is the 'source' that provides 2 , 1 1 w p = n / and from quantum mechanics, it provides the photons hν 1 , for the FMR transition to occur. It should be noticed that the Landau-Lifshitz equation in the form (6) contains a first term that produces motion of M(t) in more than one dimension, on the surface of a sphere, and also contains a term that produces damping (which implies dissipation of energy) in the motion of M(t). To get advantage of the fact that M is a constant of motion, we transform the LL-equation of motion to spherical coordinates. Figure  H sin H cos t cos cos sin 7

Numerical calculations and experimental parameters
The system of equations (7) and (8) has no analytic solutions, for general cases, so, it is solved numerically and the time evolution of t ,  (7) and (8) the Wolfram Mathematics 8 is utilized [24] and its function NDSolve is used extensively. This command is amply used to solve different equations in other areas of research [24,25]. The experimental parameters of damping α=0.003 and frequency of ν 1 =13.2 GHz for Fe film are used in all the calculations, as given in [26]. The gyromagnetic ratio γ=1.79×10 11 rad s −1 T −1 (10.256°/ns·mT) with g=2.10 for Fe is used [21]. The amplitude of H 1 =1mT and the frequency 2 13.2GHz 1 1 are fixed, while the initial conditions are varied.
With the numerical solutions of t q ( ) and t f ( ) in (7)  For a better appreciation of the evolution of the dynamics of M t , ( ) the plotted time interval of 70 ns, is divided into four quarters (70/4=17.5 ns): black trace for the first quarter, blue trace for the second quarter, green for the third quarter and red for the fourth quarter. Note that the magnetization vector M sweeps-precesses over the whole of the sphere in just the first quarter, ∼17.5 ns, then it moves tracing the blue spirals, then the green spirals to end up in the red circle when it reaches the fourth quarter.

Results and discusion
Computer runs for more than 200 ns do not change the red circle. The motion is damped with a damping constant α, yet, it does not end up at θasymp=0°, completely aligned with Ho. Moreover, the magnetization vector reaches a stationary precession motion, which tip is the red circle, with a large aperture cone-angle. How come, the magnetization vector does not end up at θ asymp=0°? The microwave excitation field H 1 is injecting energy in the form of quanta, hν 1 =gβH 0 , with ν 1 being the frequency of the microwave field. After one or two decades of nanosecond dynamics, energy reaches a balance, input compensates exactly the losses.
The trajectory-precession of M t ( ) over the sphere, as shown in figure 2(a), is accompanied by changes of its polar angle θ, as shown in figure 2(b). And linear progression of the precession angle t , f ( ) as shown in figure 2(c); t q ( ) changes continuously during the first two quarters of time (black and blue) from its initial polar angle θ I =162°, t q ( ) decreases rapidly until it reaches a θ MIN , still in the first quarter or interval (black), then it increases slowly and continues to increase during the second quarter (blue), until t q ( ) reaches asymptotically a final value, θ asym =21°during the third (green) and fourth (red) quarters.  FMR conditions with a precession cone much wider than expected is found for ferromagnetic resonance conditions. This dynamic is seldom described in FMR literature [21,[27][28][29][30][31][32]. asym q By the beginning of the fourth quarter (red), M t ( ) has already stationed into asym q =21°. This means that the same limit cycle is reached independently of the initial conditions, from below 21 , i q <  or, from above i q >21°. And the time needed to reach the limit cycle is shorter while the initial polar angle, , i q is smaller. Moreover, for small initial polar angles such as: 3 , i q =  below 21°, t q ( ) gradually increases, meanwhile M t ( ) spirals monotonically, yet non-linearly, opening its precession cone (not closing it as a simplified picture tells) towards θ asym and the time to reach the limit cycle, t LC is ≈18.3 ns. In contrast, for i q =162°, t LC is ≈26.7 ns. For any initial conditions, t LC is in the range 18.3-26.7 ns. And the whole of changes of t q ( ) is in the (0−t LC ) time interval and corresponds to the first two quarters of the total time (black, blue and green traces). The fact that t q ( )-trajectories, on the sphere, starting with either initial q values larger or smaller than asym q =21°, in the long run, spiral to the same ( , asym q t f ( )) state is clear evidence that this state is a stable limit cycle [9,33] ) which is the radius of a circular limit cycle, the red circle in figure 1(a). All the above dynamics develop in the slow time scale, nanoseconds, that belongs to the relaxation (dissipative) dynamics commanded by the damping constant α, i.e., the torques τ dampH0 and τ dampH1 in figure 1. As shown in figures 2(a) and (b), the evolution of t , q ( ) by itself, is capturing these many details of the rich slowtime-scale M t ( ) dynamics from the very beginning of the motion, passing through a minimum, , MIN q then opening the precession cone, slowly approaching the limit cycle, and reaching the limit cycle itself. Figure 2(c) gives the evolution, t , f ( ) of the magnetization for the different initial conditions, i q and i f =0, as in figure 2(b). All the t f ( ) curves superpose, and all have the t slope 4752 ns t , / which is the same as 13.2turns ns.
/ Hence, a f-turn is completed in t=75.75 ps, and a limit cycle is established in approximately 18 ns (18000 ps), doing about 237.6 f-turns. The time scale to complete a f-turn, tens of picoseconds, is quite fast compared with the time scale to reach the limit cycles. Two-time scales, one fast and the other one slow, are common in the dynamics of nonlinear systems [9,18,33]. Note that 13.2 turns/ns equates one f-turn to one cycle of the excitation frequency, ν 1 =13.2 GHz, which is another way to say that H H .
The frequency of the excitation and the resonance condition determine the fast-timescale dynamics of M(t). In this sense, and in this case, the slow and fast magnetization dynamics are mathematically decoupled ( t q ( )-Slow and t f ( )-Fast). And this is a welcome result, since, in general, the fasttime-scale M(t)-dynamics is concealed and obscured [9] due to the entangled effects of the acting torques. In order to decouple them, special methods must be developed. Here, the decoupling arises naturally. Angular velocity behaviours in the plane , q f ¢ ¢ ( ) are shown in figure 3, the phase plots that contain the portrait of a developing limit cycle. A horn-like spiral, , , q f ¢ ¢ ( ) forms, starting at the initial values 4752 ns, f¢ = / and i q¢ ≈0, as shown in figure 3(a). We denote the center of the horn-like spiral as , , c c f q ¢ ¢ ( ) and it traces, in time, a curve (yellow) that is named the Horn Centre trajectory (HC trajectory). At any time, the , f¢ q¢ velocity coordinates spiral around the HC, and the horn spiral ends up in a closed trajectory which is not a point, but it is centred at , c c f q ¢ ¢ ( ) of the limit cycle, as it is shown in the successive zooms that capture progressive time windows of the horn spiral motion. The trajectory can start from above, 21 , The general shape of the trajectories t q ( ) in figure 4(a) is as it is shown in figure 2(b); they all start a fast decrease in the first 10 ns, so that they also develop a minimum during the first quarter of motion, except for the H 1 =0, and the lowest microtesla excitations. Now the asymptotic horizontal tail levels off at different asym q values; the lower the magnitude of H 1 , the smaller the limit cycles asym q value, reaching just 2.1°for H 1 =100 μT; 0.0021°for 1mT, and finally no limit cycle, asym q =0°for exactly zero excitation field, as expected. So, under FMR conditions of excitation, H 1 is the field responsible to produce two torques M H which, in the long run, equilibrate the damping torques , dampH0 t and     (a) Behaviour of t q ( ) registered for 70 ns, for different values of ν 1 . There appear limit cycles again for all these microwave frequencies from 33 to 5 GHz. The trajectories are again divided into four quarters of time as before and are indicated with different colours. Note that the limit cycles are now reached at longer times, even taking more than 70 ns for ν 1 =5 GHz excitation. (b) Nonlinear power law dependence of the limit cyclesasym q on the microwave frequency, ν 1 , of the H 1 , microwave-excitation.
the lower the energy that it carries in its, h , n photons, the larger the limit cycle asym q value, reaching just 8.25°f or ν 1 =33 GHz, but 71.74°for 5 GHz; all the other parameters remain the same. Moreover, these limit cycles are reached more slowly and longer t LC times are required for lower frequencies, ν 1 . So, under FMR conditions of excitation, ν 1 of the excitation modifies the limit cycles-, asym q and therefore the precession-cone dynamics of M follows a power law relation that is given by the equation: asym q =(265. n = the slope is 10 800°/ns, which means a period of 33.33 ps for one f-turn, and the f-period remains one-to-one with one cycle of the excitation, as expected. The selected range of values for ν 1 falls within experimental standard FMR parameters. And 5 GHz, which is the lowest frequency limit studied, covers the upper limit that wireless communications have recently reached. These limit cycles and precession-cone dynamics appear very slowly for the low end of the frequencies studied. For frequencies smaller than 5 GHz, the establishment of a very wide precession-cone can take 70 ns, or even more than 200 ns with an aperture larger than 71.72°. The precession cones, just described, seem qualitatively quite distant from the FMR-like narrow-precession cones.

Conclusions
The limit cycle states of the magnetization nonlinear dynamics are found for isolated and isotropic nanomagnets through numerical solutions of the Landau-Lifshitz equation of motion under FMR conditions. These limit cycles are reached when t q ( ) trajectories, in the long run, converge asymptotically to asym q =constant, while If H 1 is turned off, the limit cycle disappears, and the magnetization ends its dynamic alignment along the applied field H 0 . The limit cycleasym q follows a 4th-degree polynomial function on H 1 and a power law on frequency . 1 n asym q and t f ( ) produce the precession cones that M t ( ) draws on the surface of the M=constant-sphere once the time t LC is reached. For the parameters used here, the limit cycles are circular, stable with periods in the picoseconds range, and its radius is a nonlinear function of H 1 , typically ∼0.35 M , | | and asym q independent of initial conditions. The frequency, , 1 n of the excitation field H 1 modifies the precession-cone dynamics of M following an inverse relation. For the 5 GHZ low end of the frequencies, the establishment of a very wide precession-cone can take 188 ns with periods of ∼200 ps each f-turn. The precession cones described here seem qualitatively quite distant from the FMR-like narrow-precession cones. Then, H 2 t 1 1 pn ( )plays a crucial role in the non-linear dynamics of M t ( ) under FMR conditions beyond that of being a simple perturbative excitation. The results are significantly more detailed and go beyond what has been reported previously.
All these results inform a very rich nonlinear dynamics of the magnetization of an isotropic nanomagnet while executing ferromagnetic resonance, being the non-instantaneous development of the limit cycle of M t ( ) (after many nanoseconds) the most outstanding feature of all. Controllability of limit cycles by an external microwave field, H 1 , on a simple nanomagnet could prove very valuable in technological applications.