# Micromagnetics of Microwave-Assisted Switching in Co-Pt-Based Nanostructures: Switching Time Minimization

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## Abstract

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_{3}Pt nanosrtuctures as a function of applied DC field and microwave frequency. In all the cases, the existence of microwave excitation can lower the switching field by more than 50%. However, this coercivity reduction comes at a cost in the required switching time. The optimal frequencies follow the trends of the ferromagnetic resonances predicted by the Kittel relations. This implies that: (a) when the DC field is applied along the easy axis, the coercivity reduction is proportional to the microwave frequency, whereas (b) when the coercivity is lowered by applying the DC field at an angle of 45° to the easy axis, extra MAS reduction requires the use of high frequencies.

## 1. Introduction

_{2}granular medium [8], it was observed that the coercivity decreases linearly with an increase in microwave frequency, and the coercivity reduction ratio can be as high as 80%.

_{3}Pt nanodisks. Co-Pt alloys are typical materials proposed for high-density magnetic recording media [34,35,36]. In particular, the equiatomic chemically ordered CoPt (as well as FePt) alloys consist of alternative Co and Pt layers along the c-axis of the tetragonal L1

_{0}structure. This atomic arrangement gives high anisotropy [37]. These high anisotropy materials can be combined with semihard phases such as the, also chemically ordered, Co

_{3}Pt to reduce the coercive field in favor of writability [38]. The key advantage of such composite structures is that the reversal process which is induced by thermal activation, is more homogeneous than the reversal induced by applying an external field. Thus, due to the different reversal modes of the field-induced switching process and the temperature-induced switching process, the ratio of the energy barrier over the coercive field can be optimized to achieve high thermal stability without the loss of writability [14]. An extra interesting fact about these phases is that they can be produced with intermediate degrees of chemical ordering, achieving tailor-made properties [39]. The micromagnetic simulations have been performed using the mumax3 package [40,41]. This is an open-source graphics-processing-unit accelerated micromagnetic simulation package that gives possibility to perform faster, larger, and more complex simulations. The results of the quasistatic properties can be compared with the predictions of the Stoner–Wohlfarth model: this is an exactly solvable model for coercivity based on the simplifying assumption of homogeneous reversal in single-domain particles with uniaxial anisotropy [42]. Assuming homogeneous magnetization, the multiparameter problem of a spatially inhomogeneous magnetic state is reduced to finding the minimum of one-parameter free energy function. This parameter is the angle of the magnetization with the easy axis, which also defines the angle to the applied field. The Stoner–Wohlfarth model has large applicability despite the fact that, even in single-domain particles, the reversal might not be uniform. However, it turns out that non-homogeneous modes have nucleation fields with a similar angular dependence [43]. In order to study cases with clear deviations from homogeneous reversal, we have also modeled 60 nm diameter disks. As MAS is based on the resonant excitation of the precessional motion, we compare the results with the predicted homogeneous resonances of the system given by the Kittel equations [42]. For a uniaxial particle with effective anisotropy H

_{K}along its symmetry axis

_{,}under an inversed field H applied along its axis, the Kittel equation is simply $f=\gamma \left({H}_{\mathrm{K}}-H\right)$, where γ is the gyromagnetic ratio, typically γ = 28.025 GHz/T. Note, that in this case, the anisotropy field also defines the Stoner–Wohlfarth coercivity. The corresponding equation for the field applied at an angle of 45° to the easy axis is derived in Appendix A.

## 2. Micromagnetic Simulation Details

_{S}= 800 kA/m and uniaxial anisotropy K

_{mc}= 4.9 MJ/m

^{3}[42]) or two phases, where a thin semihard layer (Co

_{3}Pt type with saturation magnetization M

_{S}= 1114 kA/m and uniaxial anisotropy K

_{mc}= 0.6 MJ/m

^{3}[42]) is on top of the hard layer. The magnetocrystalline easy axis was set either along the disk normal (z-axis) or at 45° to the disk normal for the tilted media. A small misalignment of 1 deg was introduced to avoid numerical errors that would arise in the cases where the axes of the magnetocrystalline, shape anisotropy, and applied field all coincide. A perfect alignment, apart from not being relevant to real conditions, leads to zero torque: in this case, the reversal is mainly governed by some incubation time related to the random thermal motion of the magnetization.

_{ex}= 10 pJ/m. The lateral (along the disk diameters) cell size was set to 1.0 nm, whereas along the disk axis, it was set to 0.5 nm. These values are much smaller than the characteristic exchange length scale, ${L}_{\mathrm{ex}}=\sqrt{2{A}_{\mathrm{ex}}/{\mu}_{0}{M}_{\mathrm{S}}^{2}}$, which is close to 5 nm for both phases. This choice ensures the minimization of discretization errors. One way to check this issue is by ensuring that further reduction does not change the results (errors are smaller than the used data point symbols). Furthermore, the maximum angle between two simulation cells can be checked at all times. We have found that this value (which is maximized close to the coercive field) at all cases is less than 0.009 rad for single-phase CoPt materials and less than 0.2 rad at the interface of bilayer CoPt/Co

_{3}Pt nanodisks.

## 3. Results

_{dc}is the coercivity without MAS. The color code represents the time of the reversal. For this disk geometry, the demagnetization factor along the z direction is N = 0.7341. For a hard phase with anisotropy K

_{mc}and saturation magnetization M

_{S}, the coherent rotation coercivity is given by the relation:

_{dc}= 11.7 T, which agrees with the data of Figure 1.

_{dc}) can be achieved if high frequencies (180 GHz) are available, but we must also note that in this case, the reversal time increases to 0.5 ns, compared to just 0.1 ns at higher fields. The reversal can remain fast (0.13 ns) and still have a reduction by 40% of H

_{dc}with 100 GHz.

_{3}Pt layer of 2 nm, the DC coercivity is reduced to 4.2 T, but the reversal time is doubled (Figure 2). Using frequencies close to 100 GHz, the reversal field can be substantially lowered down to 0.6 T (86% of H

_{dc}), but the reversal time approaches 1 ns.

_{K}= 12.3 T. However, the switching occurs at lower fields than the H

_{K}/2 (predicted by the Stoner–Wohlfarth for θ = 45°) and the optimal frequencies are lower than the Kittel resonances of the system.

_{dc}) using frequencies close to 90 GHz. At 2.95 T, the switching time is 0.05 nsec, but increases to 1 nsec at 1.8 T. One common feature of the tilted cases is that the resonance frequency does not depend sensitively on the applied DC field.

## 4. Discussion and Conclusions

_{eff}that governs the dynamics is the applied field minus the effective anisotropy field. The larger their difference, the higher the frequency of the precession and the faster the dynamics. Thus, the presession frequency is initially $\gamma \left(H-{H}_{\mathrm{K}}\right)$, but as the magnetization is reversed towards the field direction, it increases to $\gamma \left(H+{H}_{\mathrm{K}}\right)$. Although H

_{eff}changes during the reversal, the sign of H

_{eff}and the chirality of the precession are always the same. On the other hand, using MAS, the reversal starts from fields below the anisotropy field. In this case, not only the precession frequency, but also the chirality required for the reversal, change [26] due to the sign change of the effective field $\gamma \left(H-{H}_{\mathrm{K}}\right)$. Our simulations show that, using a fixed frequency, 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. It is reasonable then to ask if there is any point to extend the duration of the microwave pulse beyond the first stages of the reversal and specially beyond the point where the precession changes sign. However, we found no case in which the switching time was reduced by stopping the microwave pulse before the reversal is completed. The pulse rise time is also an important parameter which depends on the STO design. We examined the effect of the rise time by varying its value from 5 ps to 400 ps. The reversal time, in general, is not a monotonous function of the rise time, but presents many peaks. However, at all cases examined, the reversal time is increased with respect to the value obtained for the rise time approaching zero, but by an amount less than the rise time. The results of the simulations presented here cover diverse situations of MAS application and can provide general guidelines for the optimization of practical MAMR systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**m**. We, therefore, can write for each component, ${\tilde{\mathit{m}}}_{i}={\mathit{m}}_{i}+\delta {\mathit{m}}_{i}\xb7{e}^{i\omega t},i=\mathrm{x},\mathrm{y},\mathrm{z}$, where the ${m}_{i}$ are the static magnetization components and $\delta {m}_{i}$ are the small amplitudes of variation due to the precession.

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**Figure 1.**Conditions (exciting frequency vs. applied reversed field) under which reversal occurs within 1 ns for a CoPt disk with diameter D = 16 nm and thickness t = 4 nm. The color code represents the switching time in nanoseconds. A particular color is used for switching time, which is between the values indicated at its two edges. The solid black line represents the Kittel condition $f=\gamma \left(11.7T-H\right)$.

**Figure 2.**Conditions (exciting frequency vs. applied reversed field) under which reversal occurs within 1 ns for a CoPt disk with diameter 16 nm and thickness 4 nm, covered by a 2 nm Co

_{3}Pt layer. The color code represents the switching time in nanoseconds. A particular color is used for switching time, which is between the values indicated at its two edges. The solid black line represents the Kittel condition $f=\gamma \left(4.2T-H\right)$.

**Figure 3.**Conditions (exciting frequency vs. applied reversed field) under which reversal occurs within 1 ns for a CoPt disk with diameter 16 nm and thickness 4 nm, when the applied field is at 45° to the easy axis. The color code represents the switching time in nanoseconds. The solid black line represents the Kittel condition in this case.

**Figure 4.**Conditions (exciting frequency vs. applied reversed field) under which reversal occurs within 1 ns for a CoPt disk with diameter 16 nm and thickness 4 nm, covered by a 2 nm Co

_{3}Pt layer, when the applied field is at 45° to the easy axis. The color code represents the switching time in nanoseconds. A particular color is used for switching time, which is between the values indicated at its two edges.

**Figure 5.**Conditions (exciting frequency vs. applied reversed field) under which reversal occurs within 1 ns for a CoPt disk with diameter D = 60 nm and thickness t = 4 nm. The color code represents the switching time in nanoseconds. A particular color is used for switching time, which is between the values indicated at its two edges. The solid black line represents the Kittel condition $f=\gamma \left(11.2T-H\right)$.

**Figure 6.**Conditions (exciting frequency vs. applied reversed field) under which reversal occurs within 1 ns for a CoPt disk with diameter D = 60 nm and thickness t = 4 nm, covered by a 2 nm Co

_{3}Pt layer. The color code represents the switching time in nanoseconds. A particular color is used for switching time, which is between the values indicated at its two edges. The solid black line represents the Kittel condition $f=\gamma \left(2.9T-H\right)$.

**Figure 7.**Summary of minimum magnetization reversal time using MAS, as a function of the applied reversed field, for different CoPt (hard) and CoPt/Co

_{3}Pt (hard/soft) nanostructures.

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**MDPI and ACS Style**

Thanos, C.; Panagiotopoulos, I.
Micromagnetics of Microwave-Assisted Switching in Co-Pt-Based Nanostructures: Switching Time Minimization. *Magnetism* **2023**, *3*, 61-70.
https://doi.org/10.3390/magnetism3010006

**AMA Style**

Thanos C, Panagiotopoulos I.
Micromagnetics of Microwave-Assisted Switching in Co-Pt-Based Nanostructures: Switching Time Minimization. *Magnetism*. 2023; 3(1):61-70.
https://doi.org/10.3390/magnetism3010006

**Chicago/Turabian Style**

Thanos, Christos, and Ioannis Panagiotopoulos.
2023. "Micromagnetics of Microwave-Assisted Switching in Co-Pt-Based Nanostructures: Switching Time Minimization" *Magnetism* 3, no. 1: 61-70.
https://doi.org/10.3390/magnetism3010006