Microwave assisted switching in two-phase nanowires

Micromagnetic simulations are used to study the resonances and the microwave assisted magnetic switching (MAS) in two-phase Co/CoPt nanowires. This is chosen as an interesting model system in which the processes of nucleation, interfacial domain wall pinning, and propagation are well distinguished and can all come into play. The coupling strength is varied to cover both smooth (strong coupling) and stepped (weak coupling) hysteresis curves. At intermediate coupling strengths interfacial resonances play an important role in MAS and enable to find optimal conditions combining both low MAS frequencies and dc applied fields. There is always a tradeoff between using low fields and achieving short reversal time.


Introduction
Microwave assisted switching of the magnetization (MAS) has attracted a lot of attention due to its recent application in magnetic recording media to alleviate the conflicting constraints of increased density, high signal-to-noise-ratio, writability and thermal stability [1,2]. Using MAS, the switching field is lowered by resonantly exciting the precessional motion of the magnetic moment by a radio frequency field. This mechanism presents scientific interest as the resonant response is fundamentally different from a thermal one [3][4][5][6]. Magnetic nanowires, on the other hand, are of interest for both permanent magnet, spintronic and microwave electronics applications as well as applications that go beyond magnetism including medicine, catalysis and energy storage [7][8][9][10]. Micromagnetic simulations have shown that, in 16 nm composite nanopillars, consisting of ferromagnetically coupled soft and hard magnetic sections, MAS can be realized even if the damping constant of the hard magnetic section is large [11]. Several works have focused on the MAS in nanowires [12][13][14] but mainly in the form of permalloy thin film strip waveguides. The wall structure and dynamic response to an applied field (wall velocity and Walker fields) in nanostrips depend sensitively on the geometrical characteristics, due to the demagnetizing field arising from the precessional magnetization component perpendicular to the strip plane [15]. In contrast, wires with cylindrical symmetry may exhibit interesting reversal mechanisms which involve topologically non-trivial states [16]. During field-driven domain wall (DW) propagation in ferromagnetic nanotubes, the Walker breakdown field and velocity are found to be asymmetric for opposite directions [17].
Here, microwave assisted magnetic switching (MAS) is simulated in two phase Co/CoPt nanowires using the mumax3 finite difference micromagnetic simulation program [18]. By considering 80 nm long wires we obtain an interesting model system where the processes of nucleation, interfacial domain wall pinning, and propagation can come into play depending on the interfacial coupling strength between the two phases [19][20][21]. The coupling strength is varied to cover both smooth (strong coupling) and stepped (weak coupling) hysteresis loops.

Methods
The sample geometry is that of a cylindrical nano-element consisting of a 60 nm hard phase coupled to a 20 nm soft phase, with its axis along the z-direction. The diameter is set to 10 nm which gives radially coherent reversal for both phases (as opposed to the possible existence of curling modes [22]). The chosen hard phase length permits the determination of the velocity of domain wall propagation. For a soft phase length above 20 nm the nucleation and reversal fields do not vary considerably: as the soft phase exceeds a critical length a reversed nucleus is formed fully in the soft part and the nucleation field is mainly defined by the soft phase anisotropy. These lengths are such that can yield both smooth and stepped hysteresis loops, by variation of the interfacial coupling strength.
For the hard phase the following parameters of the equiatomic chemically ordered CoPt (with the tetragonal L1 0 structure) are used: Saturation magnetization M S = 0.8 MA/m, uniaxial anisotropy K mc = 4.9 MJ/m 3 and exchange stiffness A ex = 10 pJ/m [22]. For the soft phase the following hexagonal Co-parameters were supposed: [23]. For the dynamic properties, the damping constant of the Landau-Lifshitz-Gilbert equation was set to 0.02. The magnetocrystalline easy axis was set along the cylinder length for both phases: in this case the magnetocrystalline anisotropy is additive to the shape anisotropy of the needle which is beneficial for both magnetic recording and permanent magnet applications. Note that, Co nanowires when prepared by the polyol method tend to grow naturally with their length along the crystallographic c-axis, which is the easy axis of magnetization [7]. A small misalignment of 1.0 deg was introduced to avoid numerical errors that could arise in the case where the axes of the magnetocrystalline, shape anisotropy and applied field coincide. The cell size was set to 1.0 nm. This value is smaller than the characteristic exchange length scale which is close to 5 nm for both phases. In order to reduce staircase effects in describing the circular diameter in 1 nm cells, the edges were smoothed by using an average of 8 3 samples per cell with respect to the ideal cylindrical shape. The interfacial coupling strength was varied between zero and the theoretical value for full coupling. For simplicity we give the values of coupling strength normalized the full coupling strength denoted by κ (   0 1 k ). In mumax3 the coupling at the interface is scaled with respect to the harmonic mean of the two phases which corresponds to 29 mJ m −2 for a cell of 1 nm. It must be noted though, that even with zero exchange interfacial coupling ( there is some dipolar interaction which displaces the minor loop of the soft phase by an effective interaction field of 0.1 Tesla. A purely interfacial exchange coupling that would give an equivalent loop shift would correspond to 2.8 mJ m −2 . For weak interphase coupling the reversal is characterized by the typical stepped hysteresis starting with a reversed nucleus in the soft phase at a field value H n , followed by the full reversal at a higher negative field H C as the domain wall propagates in the hard phase through their interface. For stronger coupling H n increases, H C decreases, and they finally converge to give a single-phase-like hysteresis. For 20 nm long soft phase this occurs for 0.6. k  The results are summarized in figure 1. The interfacial coupling defines both the nucleation field (H n ) of a reversed region and the propagation field of the reversed domain formed in the soft phase to the hard one, which determines the coercivity (H C ). As the coupling becomes weaker it becomes easier for a reversed nucleus to form, but harder to propagate. The fields H n and H C , also depend sensitively on the soft phase length as this defines whether a reversed nucleus can be formed fully within the soft phase or not. For soft phase longer than 10 nm the nucleation field is not influenced by the existence of the hard one, as there seems to be enough length for the nucleus to be formed fully in the soft part. Consequently, the nucleation field is close to the anisotropy field of the soft phase. For shorter lengths the nucleus extends to the hard phase and the nucleation field is increased.
To simulate MAS, each sample was initially relaxed in zero field from a state magnetized along z. + Then a reversed dc field (along z-) was applied simultaneously with a CCW circularly polarized alternating field (B t B t cos , sin x y w w~) with frequency up to 420 GHz and amplitude 0.1 Tesla. The magnetization was monitored for time duration up to14 ns in steps of 0.5 ps. The microwave field rise time and duration were systematically varied for selected cases.

Results: dynamic properties
Since MAS is based on the resonant excitation of the precessional motion, we first examine the resonances of the system. The resonance frequencies are calculated following the methods and considerations described in reference [24]: for each magnetic state (determined by the value of applied reversed dc field), an exciting external field having a time dependence described by a sinc f t 2 c ( ) p function, with f 0.98 THz, c = is applied and the resulting magnetic response is Fourier analyzed to get the fluctuation amplitude of the magnetization at each frequency. The sampling time step was set to 0.5 ps and the total time 2048 ps. These settings give frequencies up to 1000 GHz with a step of 0.488 GHz. The amplitude of the sinc function was H 10 mT. rf 0 m = Though a twomacrospin model can qualitatively describe the resonances, at least for κ 0.2, it cannot capture the physics of this system because there are separate resonances located at specific parts of the cylinder: namely, the two different edges, the two different sides of the hard/soft phase interface, the centers of the two phases. In order to elucidate the origin of different resonances, we have produced diagrams which depict where each resonance is localized along the length of the nanowire. These diagrams are obtained by taking separately the Fourier transform of the magnetic response of each specific (1 nm) part along the length of the sample. As an example, the sample with κ = 0.2 in a reversed field 3.5 T is shown in figure 2: At 95 GHz the interface is excited on the soft phase side. At 137 GHz the excitation is located at the soft phase edge of the sample. At 168 GHz the interface is excited on the hard phase side. The excitations of the center of the hard phase (260 GHz) and the one at the hard phase edge (254 GHz) are close and appear as one broadened resonance if this spatial assignment is not done.
The resonances as function of the applied reversed field, for different values of coupling strength (κ = 0, 0.1, 0.2, 0.3, 0.5 and 1.0) are shown in figure 3. The lines with a downward slope correspond to the parts of the nanowire that are not reversed, for which the external reversed field is subtracted from the effective field. Conversely, the lines with an upward slope correspond to the parts of the sample that are reversed, for which the external reversed field is added to the effective field. Thus, there is a discontinuity in the response of the soft phase which appears at low fields at the point where the soft phase reverses, followed by a discontinuity at higher fields at the point where the hard phase reverses. For full coupling (κ = 1.0) the two phases reverse simultaneously, and the two discontinuities coincide into one. At intermediate values of coupling, complex diagrams, with several resonances are observed. Based on several diagrams as that of figure 2. the resonance assignments noted in figure 3 are done. In general, the resonances resulting from the two sides of the interface, the two edges of the sample and center modes are distinguishable, especially for intermediate coupling strengths. For full coupling no resonances located at the interface are observed. For zero coupling the soft phase center mode is very close (though at somewhat higher frequency) to those of its two ends (at the edge of the sample and at the interface with the hard). The frequencies of the two modes which are located at the two ends of the soft phase are not completely equivalent, due to the presence of magnetostatic coupling at the end that is in contact with the hard phase. However, all three frequencies are close and appear in figure 3 as one line.

Results: microwave assisted reversal
Starting from the diagrams of figure 3, one can check whether microwave assisted reversal (MAS) occurs for each different point on these diagrams, i.e. when a microwave field with a specific frequency is applied simultaneously with a specific reversed dc field. Two typical examples of thus obtained magnetization versus time curves (for weak and strong coupling) are presented in figure 4.
The reversal, as expected, begins from the soft phase and it is completed within the time denoted as t soft . The domain wall, that is formed as a result of the reversal within the soft part, gets pinned at the interface. In many cases the reversal stops at this stage, leaving the two phases oppositely magnetized. If the conditions are favorable, the domain starts to propagate in the hard phase at a time denoted as t propag leading to full reversal at t = t rev . For weak coupling (as the κ = 0.2 in the top panel of figure 4) the soft phase reverses fast but it takes higher fields to propagate to the hard one in comparison to the strong coupling cases (as the κ = 0.5 in the bottom panel of figure 4). The almost linear dependence in the time interval t propag < t < t rev permits to define a propagation velocity taken equal to its slope. This velocity has a dependence on frequency. We must note that the thus calculated velocity values are under the assumption that there is only one domain wall moving from the interface to the hard phase. However, several possibilities occur: if there is a simultaneous reversal from the interface and the edge of the hard phase there will be two domain walls and the effective speed is doubled. If there is a simultaneous reversal from the two edges of the wire and a nucleus at the center, there will be four domain walls and the effective speed is quadrupled. As an example, the frequency dependence of the propagation velocity for the κ = 0.2 sample at an applied reverse field of 8.5 T is given in figure 5. We can see that away from resonance the values are close to υ ∼ 50 m s −1 but close to resonance there are regions with υ ∼ 100 m s −1 and υ ∼ 200 m s −1 .
Using families of curves as the ones presented in figure 4, one can summarize the effects of MAS on the reversal time of the systems by superimposing on the frequency versus reversed field resonance plot of each system (given in figure 3) the points for which MAS at the specific frequency and field would lead to reversal (figure 6). These points are colored according to the reversal time.
For weak coupling the soft phase reverses fast, but its propagation to the hard part requires strong fields. Interfacial resonances on the two sides of the interface are observed, but do not coincide. The effect of MAS is pronounced, given that the required propagation fields with MAS can be lowered to one half of those without MAS. However, the required frequencies are quite high as they tend to coincide with the ferromagnetic resonances of the high anisotropy hard part. This disadvantage is alleviated at the intermediate coupling strength: in this case an extra branch, located at lower frequencies, develops gradually. This is due to the fact that interfacial resonances start to come into play and govern the MAS mechanism. For κ = 0.5 frequencies even lower than 10 GHz can be effective in reducing the required dc field up to 40% but the reversal times are long. When there is strong coupling between the two phases, the required dc fields are low, but interfacial resonances are suppressed, and the effect of MAS is not significant.

Discussion and conclusions
In short, MAS is optimal at intermediate coupling strengths, where interfacial resonances play an important role and an extra low frequency resonance branch emerges, enabling us to find useful conditions combining both low frequencies and fields. However, there is a tradeoff between using low fields and achieving short reversal time, which can be explained by the fact that higher fields, in general, lead to higher precession frequencies.
Though a macrospin model can qualitatively describe the resonances, at least for  0.2, k it cannot capture the physics of the MAS process because there are resonances located at the edges and at the interface which are crucial for the reversal. Thus, only micromagnetic theory can explain the complex behavior of such systems.  Similarly to our previous work [25], the simulations show that the resonance is limited to the initial stages of the reversal, but as the reversal proceeds, the precession frequency quickly deviates from the microwave frequency. However, we found no case in which the switching time was reduced by stopping the microwave before the reversal is completed. The reversal time in most of the cases is mainly limited by the domain wall propagation in the hard phase.
In conclusion, micromagnetic simulations are used to study the microwave assisted magnetic switching (MAS) in two phase Co/CoPt nanowires. This is an interesting model system where the processes of nucleation, interfacial domain wall pinning, and propagation are well distinguished and can all come into play depending on the conditions. The coupling strength is varied to cover both smooth (strong coupling) and stepped (weak coupling) hysteresis curves. Three qualitatively different regimes can be distinguished: (i) Weak coupling: In this case the reversal easily nucleates within the soft phase, but large fields are required for its propagation to the hard phase. The MAS is effective at the frequencies which correspond to the resonances of the hard phase. Thus, the disadvantage is that high frequencies are required in order to use low dc fields.  (iii) Strong coupling: The reversal propagates easily to the hard phase at low fields, which are defined by the soft phase coercivity. Therefore, the presence of the hard phase does not add to the field stability and furthermore the effect of MAS is minimal in reducing the required field. Still the switching time can be reduced by resonance of the hard phase.
In general, there is always a tradeoff between using low fields and achieving short reversal time. Producing two-phase nanowires with controlled interfacial coupling is experimentally feasible: The most standard technique in producing magnetic nanowires is by alternating electrodeposition the constituent phases in the pores of anodically oxidized alumina nanopores [9,10]. Strong coupling between different phases can also be achieved during polyol and sol-gel synthesis of nanocomposites [26].The coupling depends sensitively on the quality and matching at the interfaces. In the case of direct coupling, the coupling strength depends on the growth conditions and can be very strong leading to simultaneous reversal of the two phases. To achieve controlled intermediate coupling non-magnetic 3d element layers of appropriate thickness can be inserted. Therefore, the analysis presented here is of both theoretical and applied interest.