# Manipulation of Time- and Frequency-Domain Dynamics by Magnon-Magnon Coupling in Synthetic Antiferromagnets

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{3}[9,10], compensated ferrimagnet gadolinium iron garnet [11], or magnetic metal-insulator hybrid structure [12,13]. Interestingly, the tunable magnon-magnon coupling in these systems provides the opportunity to use interactions between the magnon branches as a means to control/manipulate magnons in the device-based antiferromagnetic spintronics.

## 2. Simulation Model

_{1}nm)/Ru/CoFeB (d

_{2}nm) trilayer patterned in a circular shape of 100 nm × 100 nm. Here we consider two samples: One is symmetric SAF structure with d

_{1}= d

_{2}= 2.0 nm (Sample-I) and the other is an asymmetric SAF with d

_{1}= 2.0 nm and d

_{2}= 4.0 nm (Sample-II). In this study, the dynamics of the trilayer samples were simulated by using the open-source simulation software OOMMF (National Institute of Standards and Technology, Gaithersburg, MD, USA) [26], which is based on the Landau–Lifshitz–Gilbert equation:

_{s}is the saturation magnetization of CoFeB, α is the Gilbert damping factor, and γ is the gyromagnetic ratio.

**H**

_{eff}is the effective magnetic field that includes the intralayer exchange field, demagnetizing field

**H**

_{d}, interlayer exchange field

**H**

_{IEC}between the upper and lower CoFeB, and external magnetic field

**H**

_{0}. The effective magnetic field is:

_{ex}is the exchange stiffness, μ

_{0}is the vacuum permeability. ${H}_{\mathrm{IEC}}={J}_{\mathrm{IEC}}/({d}_{j}{M}_{s})$, here J

_{IEC}is the interlayer exchange coupling constant, with J

_{IEC}> 0 for ferromagnetic coupling while J

_{IEC}< 0 for antiferromagnetic coupling. d

_{j}is the thickness of CoFeB layer (j = 1 or 2 refers to the upper or lower layer). In this study, we suppose the thickness of non-magnetic layer (d

_{Ru}) is 1.1 nm (the second peak of antiferromagnetic coupling) [21]. Additionally, the typical material parameters of CoFeB (in CGS units) are used [27,28]: ${M}_{\mathrm{s}}=1000\text{}\mathrm{emu}/{\mathrm{cm}}^{3}$, ${A}_{ex}=2.0\times {10}^{-6}\text{}\mathrm{erg}/\mathrm{cm}$, $\alpha =0.01$ and ${J}_{\mathrm{IEC}}=-0.2\text{}\mathrm{erg}/{\mathrm{cm}}^{2}$. Here, the magnetic anisotropy is ignored because its energy is almost unaffected by the in-plane orientation of the sublayer magnetizations [23,24]. All the simulations are performed without taking temperature into account.

## 3. Results and Discussion

**H**

_{0}is along the x-direction. For the symmetrical SAF (Sample-I), as shown in Figure 1a,c, there are only two equilibrium states: spin-canted state and parallel saturation state. Without the external magnetic field, the initial state of ${\mathit{m}}_{1}$ and ${\mathit{m}}_{2}$ are antiparallelly aligned due to the antiferromagnetic coupling. Under the action of

**H**

_{0}with the strength of $0<{H}_{0}<{H}_{\mathrm{s}}$, the ${\mathit{m}}_{1}$ and ${\mathit{m}}_{2}$ are rotated into a spin-canted state within the film plane, where H

_{s}= 2H

_{IEC}represents the saturation magnetic field. The angles between

**m**

_{i}and H

_{0}satisfy: φ

_{2}= −φ

_{1}= cos

^{−1}(H

_{0}/H

_{s}). When ${H}_{0}>{H}_{\mathrm{s}}$, the ${\mathit{m}}_{1}$ and ${\mathit{m}}_{2}$ orient parallel to the direction of bias magnetic field (i.e., x-axis). Taking the parameter values of ${M}_{\mathrm{s}}$, d, and J

_{IEC}, we get the strength of H

_{IEC}= 1100 Oe, which is in good agreement with the simulation results (the saturation field H

_{s}= 2300 Oe and thereby H

_{IEC}= H

_{s}/2 = 1150 Oe).

_{0}is larger than ${H}_{\mathrm{cri},2}={H}_{\mathrm{IEC}1}+{H}_{\mathrm{IEC}2}$, both ${\mathit{m}}_{1}$ and ${\mathit{m}}_{2}$ are forced to align in the x-direction and the SAF reaches a saturation state.

#### 3.1. Dynamic Resonance Properties of Symmetrical SAF

_{0}= 600 Oe for a symmetrical SAF with two identical FM layers (d

_{1}= d

_{2}= 2.0 nm). In the case of transverse pumping, only AM resonance is excited at a low frequency of 4 GHz while for the longitudinal pumping case only OM resonance is excited at a high frequency of 15 GHz, as shown in Figure 2a,b, respectively. This result can be well explained as follows: by considering that the resonance response signal is characterized by the rf components of the net magnetization $\mathit{m}=\left({\mathit{m}}_{1}+{\mathit{m}}_{2}\right)/2$ along the pumping field direction. For the low frequency resonance state, as shown in Figure 2c, d, both y- and z-components of ${\mathit{m}}_{1}$ and ${\mathit{m}}_{2}$ precess in phase while the x-component oscillates out-of-phase. In contrast, for the high frequency resonance state, as shown in Figure 2e,f, the x-component of ${\mathit{m}}_{1}$ and ${\mathit{m}}_{2}$ precess in-phase while both y- and z-components precess out-of-phase. Therefore, for the AFMR measurement with a transverse pumping microwave field (along the y-axis) to the bias magnetic field (x-axis), the in-phase AM resonance state (taken from the m

_{y}or m

_{z}) is only observed while the OM resonance state is hidden. In contrast, for the longitudinal pumping microwave field (along the x-axis), the observed resonance signal comes from the net magnetization m

_{x}(because the net m

_{y}= 0 or m

_{z}= 0) but we classify this resonance state as the OM state due to the out-of-phase in m

_{y}(or m

_{z}) component.

_{0}applied in the x-direction. In the spin-canted region, the frequency of the in-phase AM f

_{AM}increases with the increasing field while the out-of-phase OM frequency f

_{OM}decreases gradually until it reaches zero at the critical filed ${H}_{\mathrm{s}}=2300$Oe. Theoretically, we could assume the whole FM layer is a single-domain and possesses a uniform magnetization precession within each layer. Thus, within the macrospin approximation, Equation (1) can be expanded as:

_{1}= d

_{2}= 2.0 nm), the angular frequencies of the two modes are expressed as [22]:

#### 3.2. Dynamic Resonance Properties of Asymmetrical SAF

_{1}= d

_{2}, and the magnon-magnon coupling does not occur due to symmetry-protection [9]. It has been theoretically predicted that the symmetry breaking will lead to a magnon-magnon coupling between the pure AM and OM, accompanied by an anti-crossing gap opening in frequency spectra [33]. This can be done in SAF structures by changing the two FM layers either with different materials or different thicknesses. To verify whether the intrinsic asymmetry can induce the coupling between the AM and OM, we simulated an asymmetrical SAF with different thicknesses, d

_{1}= 2 nm for the bottom layer and d

_{2}= 4 nm for the top layer. All other conditions are the same as the symmetrical SAF. The external bias field is applied along the x-direction.

_{up}and f

_{down}refer to the minimum frequency of up branch and the maximum frequency of down branch. The simulation shows that the magnon-magnon coupling strength is$g$ = 1.5 GHz in our sample. Theoretically, the resonance frequency can also be derived from the eigenvalue equation of Equation (4) [33]:

_{0}increases, the down mode changes from pure AM to OM, while the up mode changes from pure OM to AM. Remarkably, the phase difference of the new hybrid mode is not 0$\xb0$or 180$\xb0$but almost 90$\xb0$ at the strongest coupling field (H

_{0}= 1500 Oe). Actually, this process of change can also establish the relationship between the phase difference and the magnon coupling strength.

_{1}= d

_{2}and asymmetrical SAF with ${d}_{1}\ne {d}_{2}$. For symmetrical SAF, only pure in-phase AM and out-of-phase OM are observed and the phase difference $\delta \phi ={\phi}_{1}-{\phi}_{2}$ between ${\mathit{m}}_{1}$ and ${\mathit{m}}_{2}$ is constant ($\delta \phi =0\xb0$ for the AM and $180\xb0$ for the OM magnons), as shown in Figure 5a. For the asymmetrical SAF, as shown in Figure 5b, the phase difference $\delta \phi $ varies with the external magnetic field H

_{0}, showing the $\delta \phi $ changes from 0$\xb0$ to 180$\xb0$ for the down magnon mode while changes from 180$\xb0$ to 0$\xb0$ for the up mode. An obvious hybrid characteristic is shown.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Chumak, A.V.; Vasyuchka, V.I.; Serga, A.A.; Hillebrands, B. Magnon spintronics. Nat. Phys.
**2015**, 11, 453. [Google Scholar] [CrossRef] - Yu, H.; Xiao, J.; Schultheiss, H. Magnetic texture based magnonics. Phys. Rep.
**2021**, 905, 1. [Google Scholar] [CrossRef] - Cheng, R.; Xiao, J.; Niu, Q.; Brataas, A. Spin Pumping and Spin-Transfer Torques in Antiferromagnets. Phys. Rev. Lett.
**2014**, 113, 057601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Baltz, V.; Manchon, A.; Tsoi, M.; Moriyama, T.; Ono, T.; Tserkovnyak, Y. Antiferromagnetic spintronics. Rev. Mod. Phys.
**2018**, 90, 015005. [Google Scholar] [CrossRef] [Green Version] - Rezende, S.M.; Azevedo, A.; Rodríguez-Suárez, R.L. Introduction to antiferromagnetic magnons. J. Appl. Phys.
**2019**, 126, 151101. [Google Scholar] [CrossRef] [Green Version] - Cheng, R.; Xiao, D.; Brataas, A. Terahertz Antiferromagnetic Spin Hall Nano-Oscillator. Phys. Rev. Lett.
**2016**, 116, 207603. [Google Scholar] [CrossRef] - Vaidya, P.; Morley, S.A.; van Tol, J.; Liu, Y.; Cheng, R.; Brataas, A.; Lederman, D.; del Barco, E. Subterahertz spin pumping from an insulating antiferromagnet. Science
**2020**, 368, 160. [Google Scholar] [CrossRef] [Green Version] - Li, J.; Wilson, C.B.; Cheng, R.; Lohmann, M.; Kavand, M.; Yuan, W.; Aldosary, M.; Agladze, N.; Wei, P.; Sherwin, M.S.; et al. Spin current from sub-terahertz-generated antiferromagnetic magnons. Nature
**2020**, 578, 70. [Google Scholar] [CrossRef] - MacNeill, D.; Hou, J.T.; Klein, D.R.; Zhang, P.; Jarillo-Herrero, P.; Liu, L. Gigahertz Frequency Antiferromagnetic Resonance and Strong Magnon-Magnon Coupling in the Layered Crystal CrCl
_{3}. Phys. Rev. Lett.**2019**, 123, 047204. [Google Scholar] [CrossRef] [Green Version] - Sklenar, J.; Zhang, W. Self-Hybridization and Tunable Magnon-Magnon Coupling in van der Waals Synthetic Magnets. Phys. Rev. Appl.
**2021**, 15, 044008. [Google Scholar] [CrossRef] - Liensberger, L.; Kamra, A.; Maier-Flaig, H.; Geprags, S.; Erb, A.; Goennenwein, S.T.B.; Gross, R.; Belzig, W.; Huebl, H.; Weiler, M. Exchange-Enhanced Ultrastrong Magnon-Magnon Coupling in a Compensated Ferrimagnet. Phys. Rev. Lett.
**2019**, 123, 117204. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Cao, W.; Amin, V.P.; Zhang, Z.; Gibbons, J.; Sklenar, J.; Pearson, J.; Haney, P.M.; Stiles, M.D.; Bailey, W.E.; et al. Coherent Spin Pumping in a Strongly Coupled Magnon-Magnon Hybrid System. Phys. Rev. Lett.
**2020**, 124, 117202. [Google Scholar] [CrossRef] [Green Version] - Chen, J.; Liu, C.; Liu, T.; Xiao, Y.; Xia, K.; Bauer, G.E.W.; Wu, M.; Yu, H. Strong Interlayer Magnon-Magnon Coupling in Magnetic Metal-Insulator Hybrid Nanostructures. Phys. Rev. Lett.
**2018**, 120, 217202. [Google Scholar] [CrossRef] [Green Version] - Parkin, S.S.P.; More, N.; Roche, K.P. Oscillations in exchange coupling and magnetoresistance in metallic superlattice structures: Co/Ru, Co/Cr, and Fe/Cr. Phys. Rev. Lett.
**1990**, 64, 2304. [Google Scholar] [CrossRef] - Grunberg, P.; Schreiber, R.; Pang, Y.; Brodsky, M.B.; Sowers, H. Layered magnetic structures: Evidence for antiferromagnetic coupling of Fe layers across Cr interlayers. Phys. Rev. Lett.
**1986**, 57, 2442. [Google Scholar] [CrossRef] - Ruderman, M.A.; Kittel, C. Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons. Phys. Rev.
**1954**, 96, 99. [Google Scholar] [CrossRef] - Kasuya, T. A Theory of Metallic Ferro- and Antiferromagnetism on Zener’s Model. Prog. Theor. Phys.
**1956**, 16, 45. [Google Scholar] [CrossRef] - Zhang, Z.; Zhou, L.; Wigen, P.E.; Ounadjela, K. Angular dependence of ferromagnetic resonance in exchange-coupled Co/Ru/Co trilayer structures. Phys. Rev. B
**1994**, 50, 6094. [Google Scholar] [CrossRef] [PubMed] - Rezende, S.M.; Chesman, C.; Lucena, M.A.; Azevedo, A.; de Aguiar, F.M.; Parkin, S.S.P. Studies of coupled metallic magnetic thin-film trilayers. J. Appl. Phys.
**1998**, 84, 958. [Google Scholar] [CrossRef] - Belmeguenai, M.; Martin, T.; Woltersdorf, G.; Maier, M.; Bayreuther, G. Frequency- and time-domain investigation of the dynamic properties of interlayer-exchange-coupled Ni
_{81}Fe_{19}/Ru/Ni_{81}Fe_{19}thin films. Phys. Rev. B**2007**, 76, 104414. [Google Scholar] [CrossRef] [Green Version] - Waring, H.J.; Johansson, N.A.B.; Vera-Marun, I.J.; Thomson, T. Zero-field Optic Mode Beyond 20 GHz in a Synthetic Antiferromagnet. Phys. Rev. Appl.
**2020**, 13, 034035. [Google Scholar] [CrossRef] [Green Version] - Sud, A.; Zollitsch, C.W.; Kamimaki, A.; Dion, T.; Khan, S.; Iihama, S.; Mizukami, S.; Kurebayashi, H. Tunable magnon-magnon coupling in synthetic antiferromagnets. Phys. Rev. B
**2020**, 102, 100403. [Google Scholar] [CrossRef] - Dai, C.; Ma, F. Strong magnon–magnon coupling in synthetic antiferromagnets. Appl. Phys. Lett.
**2021**, 118, 112405. [Google Scholar] [CrossRef] - Shiota, Y.; Taniguchi, T.; Ishibashi, M.; Moriyama, T.; Ono, T. Tunable Magnon-Magnon Coupling Mediated by Dynamic Dipolar Interaction in Synthetic Antiferromagnets. Phys. Rev. Lett.
**2020**, 125, 017203. [Google Scholar] [CrossRef] [PubMed] - He, W.; Xie, Z.K.; Sun, R.; Yang, M.; Li, Y.; Zhao, X.-T.; Liu, W.; Zhang, Z.D.; Cai, J.-W.; Cheng, Z.-H.; et al. Anisotropic Magnon–Magnon Coupling in Synthetic Antiferromagnets. Chin. Phys. Lett.
**2021**, 38, 057502. [Google Scholar] [CrossRef] - Donahue, M.J.; Porter, D.G. OOMMF User’s Guide; Interagency Report NISTIR 6376; NIST: Gaithersburg, MD, USA, 1999. Available online: http://math.nist.gov/oommf (accessed on 10 October 2021).
- Kanai, S.; Yamanouchi, M.; Ikeda, S.; Nakatani, Y.; Matsukura, F.; Ohno, H. Electric field-induced magnetization reversal in a perpendicular-anisotropy CoFeB-MgO magnetic tunnel junction. Appl. Phys. Lett.
**2012**, 101, 122403. [Google Scholar] [CrossRef] - Devolder, T.; Bianchini, L.; Miura, K.; Ito, K.; Kim, J.-V.; Crozat, P.; Morin, V.; Helmer, A.; Chappert, C.; Ikeda, S.; et al. Spin-torque switching window, thermal stability, and material parameters of MgO tunnel junctions. Appl. Phys. Lett.
**2011**, 98, 162502. [Google Scholar] [CrossRef] [Green Version] - Sorokin, S.; Gallardo, R.A.; Fowley, C.; Lenz, K.; Titova, A.; Atcheson, G.Y.P.; Dennehy, G.; Rode, K.; Fassbender, J.; Lindner, J.; et al. Magnetization dynamics in synthetic antiferromagnets: Role of dynamical energy and mutual spin pumping. Phys. Rev. B
**2020**, 101, 144410. [Google Scholar] [CrossRef] - Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev.
**1948**, 73, 155. [Google Scholar] [CrossRef] - Chen, X.; Zheng, C.; Zhang, Y.; Zhou, S.; Liu, Y.; Zhang, Z. Identification and manipulation of spin wave polarizations in perpendicularly magnetized synthetic antiferromagnets. New J. Phys.
**2021**, 23, 113029. [Google Scholar] [CrossRef] - Chen, X.; Zheng, C.; Zhou, S.; Liu, Y.; Zhang, Z. Ferromagnetic resonance modes of a synthetic antiferromagnet at low magnetic fields. J. Phys. Condens. Matt.
**2021**, 34, 015802. [Google Scholar] [CrossRef] - Li, M.; Lu, J.; He, W. Symmetry breaking induced magnon-magnon coupling in synthetic antiferromagnets. Phys. Rev. B
**2021**, 103, 064429. [Google Scholar] [CrossRef]

**Figure 1.**In-plane hysteresis loop as a function of the external field H

_{0}along x-direction for a symmetrical SAF (Sample-I) (

**a**) and asymmetrical SAF (Sample-II) (

**b**), respectively. The corresponding equilibrium angles of the magnetization vectors of two FM layers for the symmetrical SAF (

**c**) and asymmetrical SAF (

**d**) are also shown.

**Figure 2.**Top panel: The simulated AFMR response signal for the transverse pumping (

**a**) and longitudinal pumping (

**b**) at H

_{0}= 600 Oe. An acoustic resonance mode (AM) occurs at low frequency f = 4 GHz while the optic resonance mode (OM) occurs at f = 15 GHz. The color-coding refers to the resonance amplitude of magnetization response. The insert diagram shows the corresponding diagram of magnetization precession, where

**m**(

_{1}**m**) represents the magnetization unit vector for ferromagnetic layer 1(2). Bottom panel: The magnetization oscillations of

_{2}**m**(black) and

_{1}**m**(red) in the acoustic mode (

_{2}**c**) and optic mode (

**e**), where m

_{y}and m

_{z}precess in phase in the acoustic mode while out of phase in the optic mode. The corresponding net magnetization

**m**= (

**m**+

_{1}**m**)/2 oscillations are also shown in (

_{2}**d**) for AM and (

**f**) for OM.

**Figure 3.**(

**a**) Dispersion relation of frequency versus external magnetic field H

_{0}for the symmetrical SAF (Sample-I). The open circles represent the simulation results and dash lines represent the analytical calculations; (

**b**,

**c**) Comparison of the phase difference for the AM and OM resonant states at three different fields. Here ${\phi}_{i}\left(i=1,2\right)$ is the azimuth angle of ${\mathit{m}}_{i}$. It clearly shows the time-dependent in-phase precessions for the AM and antiphase precessions for the OM resonance state.

**Figure 4.**(

**a**) Dispersion relation of frequency in asymmetrical SAF (Sample-II) with the external magnetic field H

_{0}applied along the x-axis direction. The open circles represent the simulation results while dotted lines represent the corresponding theoretical calculations, showing an obvious coupling gap. Phase difference of down mode (

**b**) and up mode (

**c**) at 1300 Oe, 1400 Oe,1500 Oe, 1600 Oe, and 1700 Oe. The phase difference between

**m**

_{1}and

**m**

_{2}varies with the external magnetic field, indicating a new hybrid mode.

**Figure 5.**(

**a**) Phase difference of pure AM (black curve) and pure OM (red curve) for the symmetrical SAF (Sample-I); (

**b**) Phase difference of down branch mode (red curve) and up branch mode (black curve) for the asymmetrical SAF (Sample-II).

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**MDPI and ACS Style**

Chen, X.; Zheng, C.; Zhou, S.; Liu, Y.; Zhang, Z.
Manipulation of Time- and Frequency-Domain Dynamics by Magnon-Magnon Coupling in Synthetic Antiferromagnets. *Magnetochemistry* **2022**, *8*, 7.
https://doi.org/10.3390/magnetochemistry8010007

**AMA Style**

Chen X, Zheng C, Zhou S, Liu Y, Zhang Z.
Manipulation of Time- and Frequency-Domain Dynamics by Magnon-Magnon Coupling in Synthetic Antiferromagnets. *Magnetochemistry*. 2022; 8(1):7.
https://doi.org/10.3390/magnetochemistry8010007

**Chicago/Turabian Style**

Chen, Xing, Cuixiu Zheng, Sai Zhou, Yaowen Liu, and Zongzhi Zhang.
2022. "Manipulation of Time- and Frequency-Domain Dynamics by Magnon-Magnon Coupling in Synthetic Antiferromagnets" *Magnetochemistry* 8, no. 1: 7.
https://doi.org/10.3390/magnetochemistry8010007