Systematic investigation of surface acoustic wave-driven ferromagnetic resonance: Model and simulation

Abstract

A theoretical model is presented to describe the surface acoustic wave-driven ferromagnetic resonance (SAW-FMR), systematically and effectively explaining the results of the previous experiment. In our model, the precessional cone angle \delta \theta and the power attenuation \mathrm{\Delta }P are employed to represent the intensity of FMR. The expression of \delta \theta can be divided into three independent parts: a constant term, a frequency-dependent term, and a surface acoustic wave (SAW) equivalent field term; each part is expressed explicitly with various parameters. To deeply understand the special field-sweeping SAW power attenuation spectrum patterns observed in previous experiments, we creatively interpret \delta \theta as a combination of the resonant frequency f_0 spectrum and SAW equivalent field \left|\sin ({2\phi }_M)\right| spectrum. It is the distinctive SAW equivalent term \left|\sin ({2\phi }_M)\right| that makes SAW attenuation spectrum an incomplete pattern compared to traditional electromagnetic wave (EMW). Furthermore, we discuss the influence of multiple factors on the SAW-FMR, including the external magnetic field magnitude and direction, SAW frequency and amplitude, as well as the magnetocrystalline anisotropy direction and distribution. To validate our theoretical model, micromagnetic simulations are also carried out in the corresponding situations.

Graphical abstract

Keywords

ferromagnetic resonance / surface acoustic wave

1 Introduction

Surface acoustic wave (SAW) is a kind of elastic wave propagating along the surface of an object, which can be generated by applying an alternating voltage signal to the interdigital transducer (IDT) [1−5]. The propagation velocity of SAW is much slower than that of the electromagnetic wave (EMW), i.e., SAW has a shorter wavelength than EMW at the same frequency. Therefore, the microwave devices (such as resonator, circulator, and isolator) based on SAW have smaller sizes than those based on EMW. Furthermore, the technology associated with SAW is highly mature in the manufacturing industry. Thanks to more advanced etching techniques, nanoscale IDTs can be integrated onto the substrate, ensuring the quality of SAW signals [6, 7]. These reasons make SAW a suitable candidate for the miniaturization of microwave devices. Essentially, the SAW can bring a space- and time-varying strain field to the sample interface, which has an alternating influence on the magnetization through the inverse magnetostriction effect. This effect provides a possibility to regulate the magnetization through SAW. Recently, the SAW has been reported to induce/assist magnetization reversal [8−11], motivate magnetic resonance [12−20] or spin wave resonance [21, 22], generate an acoustic voltage signal [23−26], and drive a spin current into an adjacent metal (which is named “spin pumping”) [27−30].

Surface acoustic wave-driven ferromagnetic resonance (SAW-FMR) refers to the phenomenon where the magnetic moments precess around the effective magnetic field due to SAW, with the significantly absorption of the SAW energy and conversion to other types of energy [12, 31, 32]. The research on the SAW-FMR plays a fundamental role in the development of acoustic spintronics devices [33, 34], such as acoustic-resonator, acoustic-filter, and acoustic-circulator. The SAW-FMR is an magneto-acoustic coupling effect, which can be influenced by various factors related to both the SAW and the magnet. An external constant field, with a changeable magnitude and direction, is a classic and convenient way to influence the SAW-FMR. The resulting field-sweeping spectrum typically shows a “butterfly” pattern in the experiment [35, 36]. The angle between magnetocrystalline anisotropy K and the propagation direction of the SAW also shows an impact on the “butterfly” pattern [37, 38]. Besides these factors, the SAW frequency is also an important factor that affects the SAW-FMR [14, 39]. In addition, the volume of the magnetic film [40] and the SAW propagation distance [12] have an exponential relationship with the energy attenuation due to the SAW propagation character, which may also affect the SAW-FMR spectra. The SAW-FMR in different materials has also been compared and shows different types of spectra [18]. Although previous experiments show various discrete results, a more systematic and understandable theory is required, which may pave a more practical strategy to better regulate the SAW-FMR.

In this work, we model a magnetic film with an in-plane uniaxial anisotropy. Based on the Landau−Lifshitz−Gilbert (LLG) equation, we present a concise theory and derive the expression of the precessional cone angle and the power attenuation. This allows a better qualitative understanding of the various influencing factors, thereby providing a more accessible way to comprehend the characteristics of SAW-FMR. Using our theory, we systematically calculate and explain the influence of the external magnetic field orientation and magnitude, the SAW frequency and amplitude, as well as the magneto crystalline anisotropy orientation and distribution. Additionally, the “reversal peak” is also demonstrated and discussed by our theory, which has not been thoroughly explained in previous work. Furthermore, micromagnetic simulation is employed in SAW-FMR and perfectly verifies the calculation results. These findings can clear the obstacles for the comprehension of SAW-FMR and further promote the research on SAW-driven magnetization precessional switching, acoustic spin pumping, and high-frequency acoustic wave devices.

2 Model and method

In the SAW-FMR experiments, a multilayer heterostructure film is usually employed. As is depicted in Fig.1(a), a ferromagnetic Ni film is deposited on the piezoelectric LiNbO₃ substrate, and two interdigital transducers (IDTs) are etched on the two sides of the piezoelectric substrate. When an alternating voltage signal is applied to one of the IDTs, an elastic wave signal is excited in the LiNbO₃ and interacts with the magnetic film to induce FMR. When the SAW propagates to the other IDT, the elastic wave signal is converted back into a voltage signal to be detected. The SAW transmission parameters (S₂₁) can indicate the attenuation of the SAW and further reflect the resonance intensity in the ferromagnetic film. In addition, the acoustic voltage signal is a newly emerging method to detect the SAW-FMR these days.

To develop a theoretical model of SAW-FMR, we should make some assumptions as follows:

(i) The thin film is in the single-domain state to ensure all magnetic moments rotates coherently. The parameters are assumed to be uniform, and we neglect any spatial distribution. The anisotropy distribution model is employed to calculate the larger sample (in Section 4.4). Note that our model neglects the energy absorption in various magnetic configurations, which is complex and needs further research in the future.

(ii) The SAW is assumed to be a Rayleigh wave and only the normal strain component is included in our model. The shear strain is neglected because it only has a slight influence on the amplitude of FMR. As shown in the previous literatures [14, 24, 41], the overall pattern of SAW-FMR remains unchanged; only the nonreciprocity is introduced due to the effect of the shear strain. From the perspective of energy, SAW is considered as a type of strain energy in our model.

(iii) In our work, the external magnetic field orientation and magnitude, the SAW frequency and amplitude, and the magneto crystalline anisotropy orientation and distribution are all adjustable to find the optimum for SAW-FMR.

The micromagnetic parameters used in the calculation and simulation are as follows: the ferromagnet is a 20 nm × 20 nm × 2 nm Ni film, whose saturation magnetization M_s = 4.14 × 10⁵ A/m, exchange stiffness constant A = 1.05 × 10⁻¹¹ J/m, magnetocrystalline anisotropy constant K_u = 5 × 10³ J/m³ in the film plane, damping constant α = 0.01. The magnetoelastic coupling coefficient b₁ is set to 1.02 × 10⁷ J/m³. The SAW frequency is in the band of GHz and the amplitude of corresponding strain is in the range of 1 × 10⁻⁵ to 1 × 10⁻⁴. The shear strain component of Rayleigh-type SAW is neglected in our work. The external magnetic field is applied in the film plane; hence the equilibrium state of magnetization is in the same plane. Mumax3, a GPU-accelerated micromagnetic simulation software [42, 43], is employed in our work. In the simulation, the whole film is meshed into several 2 nm × 2 nm × 2 nm elements by the Finite Difference Method. Then the fourth-order Runge-Kutta method is used to solve the LLG equation numerically. The temperature is set to 0 K to avoid interference from the thermal disturbance.

We adopt a result-verification strategy in our work to study the influence of various parameters on the SAW-FMR. Firstly, we analytically derive the expression of the precessional cone angle and power attenuation and use it to calculate the results in different cases. Then, micromagnetic simulation is utilized to corroborate the calculation results. Additionally, the previous experiment results also support the calculation results obtained by our model.

3 Theory

3.1 Precessional cone angle

The precessional cone angle is the angle between the magnetization and the static effective field in precession, which can represent the FMR intensity. In order to find its expression, the LLG equation in the spherical coordinate system can be obtained (the process can be found in Appendix A):

(1)\{\begin{array}{l}-\delta \dot{\theta }={\displaystyle \frac{\gamma }{{\mu }_0{M}_s\sin \theta }}{\displaystyle \frac{\mathrm{\partial }{F}_{tot}}{\mathrm{\partial }\varphi }}+\alpha \sin \theta \delta \dot{\varphi },\\ \delta \dot{\varphi }={\displaystyle \frac{\gamma }{{\mu }_0{M}_s\sin \theta }}{\displaystyle \frac{\mathrm{\partial }{F}_{tot}}{\mathrm{\partial }\theta }}+{\displaystyle \frac{\alpha }{\sin \theta }}\delta \dot{\theta },\end{array}

where \gamma is the gyromagnetic ratio (\frac{\gamma }{{\mu }_{0}2\pi }\approx 28\mathrm{G}\mathrm{H}\mathrm{z}/\mathrm{T}), \alpha is the Gilbert damping, M_s is the saturation magnetization, and F_{tot} is the total free energy density of the ferromagnetic film. The static energy density in the absence of SAW is given to find out the equilibrium state of the magnetization and further derive the resonant frequency:

(2)\begin{array}{rl}{F}_{sta}=& E_{exch}+E_{ani}+E_{Zee}+E_{dem}\\ =& -K_u{\cos }^2⟨\mathit{M},\mathit{K}⟩-{\mu }_0{M}_{S}H\cos ⟨\mathit{M},\mathit{H}⟩\\ & +\frac{1}{2}{\mu }_0{M}_S^2\left(N_x{m}_x^2+N_y{m}_y^2+N_z{m}_z^2\right)\\ =& -K_u{[\sin {\theta }_M\cos ({\phi }_M-{\phi }_K\left)\right]}^2\\ & -{\mu }_0{M}_{S}H\left[\sin {\theta }_M\cos \left({\phi }_H-{\phi }_M\right)\right]\\ & +\frac{1}{2}{\mu }_0{M}_S^2{\cos }^2{\theta }_M,\end{array}

where E_{exch} is the exchange energy, E_{ani} is the anisotropy energy, E_{Zee} is the Zeeman energy, and E_{dem} is the magnetostatic energy, K_u is the magnetocrystalline anisotropy constant, respectively. In our model, the azimuth of the in-plane external magnetic field H is (θ_H, φ_H), and the azimuth of the magnetization is (θ_M, φ_M), which are shown in Fig.1(b). As the magnetization is assumed to rotate coherently with the external magnetic field, E_{exch} is constant and can be neglected in the following discussion. We assume that the film is very thin and we set the demagnetization factor N_x=N_y=0, N_z=1. The external magnetic field is applied in the film plane ({\theta }_H={90}^{\circ }), and the equilibrium state can be obtained by setting the first-order derivative of the total energy density [Eqs. (A7) and (A8) in Appendix A] to zero:

(3)\begin{array}{rl}& {\theta }_{M0}={90}^{\circ },\\ & 2K_u\sin {\phi }_{M0}\cos {\phi }_{M0}-{\mu }_0{M}_{S}H\mathrm{s}\mathrm{i}\mathrm{n}\left({\phi }_H-{\phi }_{M0}\right)=0,\end{array}

where {\phi }_{M0} and {\theta }_{M0} represent the stable azimuth angle of the magnetization.

Then we can obtain the resonant frequency by solving the second-order derivative [Eqs. (A9)−(A11) in Appendix A]:

(4)\begin{array}{rl}{\omega }_0=& \frac{\gamma }{{\mu }_0{M}_s\sin {\theta }_{M0}}{\left\{{\frac{{\mathrm{\partial }}^2{F}_{sta}}{\mathrm{\partial }{\theta }_M^2}|}_{{\phi }_{M0},{\theta }_{M0}}\cdot {\frac{{\mathrm{\partial }}^2{F}_{sta}}{\mathrm{\partial }{\phi }_M^2}|}_{{\phi }_{M0},{\theta }_{M0}}-{\left({\frac{{\mathrm{\partial }}^2{F}_{sta}}{\mathrm{\partial }{\theta }_M\mathrm{\partial }{\phi }_M}|}_{{\phi }_{M0},{\theta }_{M0}}\right)}^2\right\}}^{\frac{1}{2}}\\ =& \frac{\gamma }{{\mu }_0{M}_s}\{\left[2K_u{\cos }^2\left({\phi }_{M0}-{\phi }_K\right)+{\mu }_0{M}_{S}H\cos \left({\phi }_H-{\phi }_{M0}\right)+{\mu }_0{M}_S^2\right]\\ & \cdot \left[2K_u\cos ({2\phi }_{M0}-{2\phi }_K)+{\mu }_0{M}_{S}H\cos \left({\phi }_H-{\phi }_{M0}\right)\right]{\}}^{1/2}.\end{array}

Next, we will derive the expression of the precessional cone angle. We regard the SAW excitation as a perturbation to the total static energy, and the total free energy density can be written as

(5)\begin{array}{rl}{F}_{total}=& -K_u{[\sin {\theta }_M\cos ({\phi }_M-{\phi }_K\left)\right]}^2\\ & -{\mu }_0{M}_{S}H\left[\sin {\theta }_M\cos \left({\phi }_H-{\phi }_M\right)\right]\\ & +\frac{1}{2}{\mu }_0{M}_S^2{\cos }^2{\theta }_M+b_1{\epsilon }_{xx}{\left(\sin {\theta }_M\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_M\right)}^2,\end{array}

where b_1 is the first magnetoelastic coupling constant, {ϵ}_{xx} is the dynamic strain field generated by SAW and the shear strain component is ignored.

From Eq. (1), transfer the first-order derivative \frac{\mathrm{\partial }F}{\mathrm{\partial }\varphi }, \frac{\mathrm{\partial }F}{\mathrm{\partial }\theta } to the differential form:

(6)\{\begin{array}{l}-\delta \dot{\theta }={\displaystyle \frac{\gamma }{{\mu }_0{M}_s\sin \theta }}\left(F_{\varphi \varphi }\delta \varphi +F_{\varphi \theta }\delta \theta +F_{\varphi \epsilon }\delta \epsilon \right)+\alpha \sin \theta \delta \dot{\varphi },\\ \delta \dot{\varphi }={\displaystyle \frac{\gamma }{{\mu }_0{M}_s\sin \theta }}(F_{\theta \theta }\delta \theta +F_{\theta \varphi }\delta \varphi +F_{\theta \epsilon }\delta \epsilon )+{\displaystyle \frac{\alpha }{\sin \theta }}\delta \dot{\theta }.\end{array}

Here, F_{\varphi \varphi }, F_{\varphi \theta }=F_{\theta \varphi } and F_{\theta \theta } can be gained from second-order derivative of the static energy [Eqs. (A9)−(A11) in Appendix A]. From Eq. (5), strain-related items are easy to solve:

(7)F_{\varphi \epsilon }=-2b_1{\sin }^2{\theta }_M\sin ({2\phi }_M),

(8)F_{\theta \epsilon }=2b_1{\cos }^2{\phi }_M\sin ({2\theta }_M).

Substitute the equilibrium condition in Eq. (3), and we can get

(9)F_{\varphi \epsilon }=-2b_1\sin ({2\phi }_M),

(10)F_{\theta \epsilon }=0.

Thus Eq. (6) can be simplified as

(11)\{\begin{array}{l}-\dot{\delta \theta }={\displaystyle \frac{\gamma }{{\mu }_0{M}_s}}\left(F_{\varphi \varphi }\delta \varphi +F_{\varphi \epsilon }\delta \epsilon \right)+\alpha \delta \dot{\varphi },\\ \dot{\delta \varphi }={\displaystyle \frac{\gamma }{{\mu }_0{M}_s}}\left(F_{\theta \theta }\delta \theta \right)+\alpha \delta \dot{\theta }.\end{array}

Assume \delta \theta and \delta \varphi can be divided into two parts, a product of a time-dependent term and a time-independent term, \delta \varphi =\delta {\varphi }_0{\mathrm{e}}^{\mathrm{i}\omega t}, \delta \theta =\delta {\theta }_0{\mathrm{e}}^{\mathrm{i}\omega t}. Then we substitute the time derivatives, \dot{\delta \theta }=\mathrm{i}\omega \delta \theta and \delta \dot{\varphi }=\mathrm{i}\omega \delta \varphi, into Eq. (11):

(12)\{\begin{array}{l}-\mathrm{i}\omega \delta \theta ={\displaystyle \frac{\gamma }{{\mu }_0{M}_s}}\left(F_{\varphi \varphi }\delta \varphi +F_{\varphi \epsilon }\delta \epsilon \right)+\mathrm{i}\alpha \omega \delta \varphi ,\\ \mathrm{i}\omega \delta \varphi ={\displaystyle \frac{\gamma }{{\mu }_0{M}_s}}\left(F_{\theta \theta }\delta \theta \right)+\mathrm{i}\alpha \omega \delta \theta .\end{array}

Simultaneous the equations and we can get (the second-order term about \alpha is neglected)

(13)\begin{array}{rl}& \left[\left(-\omega +\frac{{\omega }_0^2}{\omega }\right)+\mathrm{i}\frac{\alpha \gamma }{{\mu }_0{M}_s}\left(F_{\theta \theta }+F_{\varphi \varphi }\right)\right]\delta \theta \\ & =\mathrm{i}\frac{2b_1\gamma }{{\mu }_0{M}_s}\sin ({2\phi }_M)\delta \epsilon ,\end{array}

where {\omega }_0=\frac{\gamma }{{\mu }_0{M}_s}\sqrt{F_{\theta \theta }{F}_{\varphi \varphi }} is the resonant circular frequency acquired by Eq. (4). Neglecting the phase, the amplitude of the precession can be expressed as

(14)\begin{array}{rl}\left|\delta \theta \right|=& \frac{2b_1\gamma }{{\mu }_0{M}_s}\left|\begin{array}{r}\frac{\sin ({2\phi }_M)}{\sqrt{{(-\omega +{\displaystyle \frac{{\omega }_0^2}{\omega }})}^2+{\left({\displaystyle \frac{\alpha \gamma }{{\mu }_0{M}_s}}\right)}^2{(F_{\theta \theta }+F_{\varphi \varphi })}^2}}\end{array}\right|\left|\delta \epsilon \right|\\ =& \frac{2b_1\gamma }{{\mu }_0{M}_s}\left|\begin{array}{r}\frac{\sin ({2\phi }_M)}{|-\omega +{\displaystyle \frac{{\omega }_0^2}{\omega }}|+{\displaystyle \frac{{\left({\displaystyle \frac{\alpha \gamma }{{\mu }_0{M}_s}}\right)}^2{(F_{\theta \theta }+F_{\varphi \varphi })}^2}{2|-\omega +{\displaystyle \frac{{\omega }_0^2}{\omega }}|}}}\end{array}\right|\left|\delta \epsilon \right|,\end{array}

where \left|\delta \theta \right|, \left|\delta \epsilon \right| mean the amplitude of \delta \theta and \delta \epsilon. First-order Taylor expansion is utilized in the denominator to remove the radical sign. The damping term \frac{{\left(\frac{\alpha \gamma }{{\mu }_0{M}_s}\right)}^2{(F_{\theta \theta }+F_{\varphi \varphi })}^2}{2|-\omega +\frac{{\omega }_0^2}{\omega }|} is relatively small and we can simplify it as a constant term {\omega }_{\alpha }({\omega }_{\alpha }>0):

(15)\begin{array}{rl}{\left|\delta \theta \right|}_{SAW}=& \frac{2b_1\gamma }{{\mu }_0{M}_s}\frac{\left|\sin ({2\phi }_M)\right|}{\left|-\omega +{\displaystyle \frac{{\omega }_0^2}{\omega }}\right|+{\omega }_{\alpha }}\left|\delta \epsilon \right|\\ =& \frac{2b_1\gamma \left|\delta \epsilon \right|}{{\mu }_0{M}_s}\frac{1}{\left|-\omega +{\displaystyle \frac{{\omega }_0^2}{\omega }}\right|+{\omega }_{\alpha }}|\sin ({2\phi }_M)|.\end{array}

Note that the simplification of {\omega }_{\alpha } may bring some mistakes in the calculation of precessional cone angle, even when the SAW frequency is close to the resonant frequency or the system has a large damping constant. However, this treatment does not change the resonant magnetic field, so it is valid to describe the fundamental rules of SAW-FMR. If the accurate amplitude of FMR is desired, the exact expression [the first line of Eq. (14)] can be used to calculate it.

From Eq. (15), it is obvious that the precessional cone angle can be divided into three parts: a constant coefficient term \frac{2b_1\gamma \left|\delta \epsilon \right|}{{\mu }_0{M}_s}, a term about frequency \frac{1}{\left|-\omega +\frac{{\omega }_0^2}{\omega }\right|+{\omega }_{\alpha }}, and a SAW equivalent field term \left|\sin ({2\phi }_M)\right|. Actually, the SAW equivalent field should be expressed as \frac{2b_1}{{\mu }_0{M}_s}\left|\sin ({2\phi }_M)\right|. The constant term \frac{2b_1}{{\mu }_0{M}_s} only impacts the value of precessional cone angle without influencing the SAW-FMR pattern, while the \left|\sin ({2\phi }_M)\right| term can determine which type the pattern is. Thus, \left|\sin ({2\phi }_M)\right| is listed as a SAW equivalent field term separately in order to emphasize it is the effective term for SAW to influence the SAW-FMR pattern. From this \left|\sin ({2\phi }_M)\right| term, we can conclude that the SAW will become invalid if the magnetization is parallel or perpendicular to the SAW propagation direction.

3.2 Power absorption of SAW in micromagnetics

The precessional cone angle, a concept in micromagnetics, cannot be measured directly in the experiments. In practice, power attenuation is typically used to indirectly reflect the precessional intensity. Therefore, it is essential to establish a combination between the precessional cone angle and power attenuation.

Before deriving the power absorption in SAW-induced FMR, we should make an assumption that magnetic moments precess as a circular motion coherently with the effect of SAW. As shown in the inset of Fig.2(a), the blue arrow depicts the precessional term torque and the green arrow indicates the damping term torque in the LLG equation. During the precession, the precessional term torque (the blue arrow) does not change the precessional cone angle; thus, the Zeeman energy is not changed. But the damping torque drives the magnetization towards the effective field, which reduces the Zeeman energy. When SAW is applied to the film, the SAW equivalent field counteracts the damping effect, i.e., the SAW transfers mechanical energy to the magnetic moments, and the damping term dissipates the energy in the form of heat. So in our model, we consider that the SAW attenuation energy equals the damping consumption, as is depicted in Fig.2(b).

We begin with the regular LLG equation with the normalized magnetization as follow:

(16){\displaystyle \frac{\mathrm{d}\mathit{m}}{\mathrm{d}t}}=-\gamma \mathit{m}\times {\mathit{H}}_{eff}+\alpha \left(\mathit{m}\times {\displaystyle \frac{\mathrm{d}\mathit{m}}{\mathrm{d}t}}\right).

Assume that the initial magnetization {\mathit{m}}_0 rotates towards \mathit{m} due to the damping torque, as shown in Fig.2(b) (Actually, SAW neutralizes this effect). The corresponding energy differential of \mathrm{d}\mathit{m} can be written as

(17)\begin{array}{rl}\mathrm{d}E=& -{\mu }_0\mathit{M}\cdot \mathit{H}-\left(-{\mu }_0{\mathit{M}}_0\cdot \mathit{H}\right)\\ =& -{\mu }_0\mathrm{d}\mathit{M}\cdot \mathit{H}=-{\mu }_0{M}_s\mathrm{d}\mathit{m}\cdot \mathit{H}.\end{array}

Divided by time differential \mathrm{d}t:

(18){\displaystyle \frac{\mathrm{d}E}{\mathrm{d}t}}=-{\mu }_0{M}_s{\displaystyle \frac{\mathrm{d}\mathit{m}}{\mathrm{d}t}}\cdot \mathit{H}=-{\mu }_0{M}_s\alpha \left(\mathit{m}\times {\displaystyle \frac{\mathrm{d}\mathit{m}}{\mathrm{d}t}}\right)\cdot \mathit{H},

where the precessional term is perpendicular to the \mathit{H} and does not distribute to the total energy.

According to Eq. (13), when applying an alternating excitation at a frequency of \omega, the magnetization will precess at the same frequency. The precession angular velocity is proportional to the frequency of SAW, thus the precessional term:

(19){\displaystyle \frac{\mathrm{d}\mathit{m}}{\mathrm{d}t}}=-\gamma \mathit{m}\times {\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}=-\gamma \mathit{m}\times {\displaystyle \frac{H}{H_0}}{\mathit{H}}_0=-\gamma {\displaystyle \frac{\omega }{{\omega }_0}}\mathit{m}\times {\mathit{H}}_0,

where {\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}} is the total effective field and {\mathit{H}}_0 is the static effective field. Thus the power consumption can be obtained by Eqs. (18) and (19):

(20)\begin{array}{rl}\mathrm{\Delta }P=& \frac{\mathrm{d}E}{\mathrm{d}t}={\mu }_0\alpha \gamma M_S\frac{\omega }{{\omega }_0}\left(\mathit{m}\times (\mathit{m}\times {\mathit{H}}_0)\right)\cdot {\mathit{H}}_0\\ =& {\mu }_0\alpha \gamma M_S\frac{\omega }{{\omega }_0}\left(\left(\mathit{m}\cdot {\mathit{H}}_0\right)\mathit{m}-(\mathit{m}\cdot \mathit{m}){\mathit{H}}_0\right)\cdot {\mathit{H}}_0\\ =& {\mu }_0\alpha \gamma M_S\frac{\omega }{{\omega }_0}\left({\left(\mathit{m}\cdot {\mathit{H}}_0\right)}^2-{\left(m{H}_0\right)}^2\right)\\ =& -{\mu }_0\alpha \gamma M_S\frac{\omega }{{\omega }_0}{H}_0^2{\sin }^2\left|\delta \theta \right|,\end{array}

where \left|\delta \theta \right| is the precessional cone angle calculated by Eq. (15) (or simulated by the micromagnetic software), {\omega }_0=\frac{\gamma }{{\mu }_0{M}_s}\sqrt{F_{\theta \theta }{F}_{\varphi \varphi }} is the resonant circular frequency, and H_0 is the static effective field calculated by {\mu }_0{\mathit{H}}_0=-{\mathrm{\nabla }}_{\mathit{M}}{F}_{sta}:

(21)\begin{array}{rl}{H}_0^2=& H_{0x}^2+H_{0y}^2+H_{0z}^2\\ =& H^2+{\left(\frac{2K_u}{{\mu }_0{M}_s}\right)}^2{\cos }^2\left({\phi }_{M0}-{\phi }_K\right)\\ & +\left(\frac{4K_u}{{\mu }_0{M}_s}\right)\cos \left({\phi }_{M0}-{\phi }_K\right)\cdot \cos \left({\phi }_H-{\phi }_K\right).\end{array}

The SAW power attenuation can now be successfully calculated. This establishes a connection between experiments and calculations/simulations.

4 Results and discussion

In this part, we first compare the character of SAW-FMR with EMW-FMR based on the above model by discussing the expression of precessional cone angle respectively. The distinct “butterfly” pattern spectrum similar to the experimental results [14, 17] is obtained in Section 4.1. Then we further list different types of field-dependent attenuation spectrum patterns and the corresponding rules in Section 4.2. In Section 4.3, the SAW frequency is changed continuously to find the dependence of the acoustic resonant frequency on the external magnetic field. The anisotropy direction, as well as distribution, is discussed finally in Section 4.4.

4.1 The character of SAW induced-FMR compared with electromagnetic wave (EMW) induced-FMR

In the above text, we have derived the expression of the precessional cone angle to describe the SAW-FMR. The parameters can be broadly categorized into two parts: the frequency-dependent part and the SAW equivalent field part. However, the mechanisms by which various parameters influence SAW-FMR, as well as the factor that makes SAW attenuation spectrums a unique “butterfly” pattern, remain elusive and await further exploration. Hence, we compare SAW-FMR with the EMW-FMR, aiming to identify the reason that differentiates them.

Before we compare them, it is necessary to solve the expression of precessional cone angle of EMW-FMR. For an EMW with frequency \omega and amplitude \left|h\right|, the EMW-FMR precessional cone angle can be calculated using the same method in Section 3.1:

(22){\left|\delta \theta \right|}_{EMW}=\gamma \left|h\right|{\displaystyle \frac{1}{\left|-\omega +{\displaystyle \frac{{\omega }_0^2}{\omega }}\right|+{\omega }_{\alpha }}}

and the detailed deduction is shown in Appendix B.

Comparing the SAW precessional cone angle [Eq. (15)] with that of EMW [Eq. (22)], the constant term influences the magnitude of the cone angle but does not change the fundamental rules. The expressions have the same part about the frequency, while the SAW precessional angle has an extra term \left|\sin ({2\phi }_M)\right|. This extra term is the origin of the unique SAW attenuation spectra.

To display the difference between SAW-FMR and EMW-FMR, we calculate the magnetic field direction-dependent field sweeping spectrum using the \left|\delta \theta \right| expressions [Eqs. (15) and (22)]. In the case of the Ni film system with an anisotropic easy axis along the x-axis, SAW and EMW with the same frequency 2 GHz are respectively applied along the x-axis. The amplitude of strain caused by SAW is 10⁻⁵, and the amplitude of EMW is 0.5 mT (the maximum amplitude of the SAW equivalent field is about 0.5 mT). The angle between the direction of the external magnetic field and the +x-axis varies from 0 to 180° in the step of 2°. The external magnetic field is downswept from 0.1 T to −0.1 T in the step of 2 mT. The field direction-dependent field sweeping \left|\delta \theta \right| spectrum results are shown in Fig.3. The SAW spectrum is a typical “butterfly” shape as is observed in previous experiments, while the EMW spectrum shows two “O” shapes. A sudden increase of precession angle is discovered near the negative coercivity in the field-sweeping spectrum, and this type of peak is called the “reversal peak”. Both patterns are symmetric about the direction of the hard axis, {\phi }_H={90}^{\circ }. If the reversal peak and nonreciprocal propagation are neglected, the zero external magnetic field line is another axis of symmetry. This suggests that the anisotropy is the origin of the symmetry of the pattern, and the system has a field-dependent reciprocation regardless of shear strain. Comparing Fig.3(a) with Fig.3(b), it is obvious that the SAW-FMR fades away when {\phi }_H={90}^{\circ } (i.e., {\phi }_M={90}^{\circ }) or {\mu }_{0}H=0\mathrm{T} (i.e., {\phi }_M=0^{\circ }), where the magnetization is perpendicular or parallel to the SAW propagation direction ({\phi }_{SAW}=0^{\circ }). In these two cases, the SAW equivalent field is close to zero and the corresponding color blocks show white. As a result, the two “O” patterns are “cut” into a “butterfly” pattern.

We can conclude from Eq. (15) that the SAW provides an anisotropic equivalent field term \left|\sin ({2\phi }_M)\right| depending on the angle between the magnetization and the SAW propagation direction. The SAW equivalent field is maximum when {\phi }_M={45}^{\circ } and disappears when the magnetization is parallel or perpendicular to the SAW propagation direction, which agrees well with the results in Fig.3(a) and Ref. [14]. However, the electromagnetic wave (EMW) provides an isotropic equivalent field no matter which direction the magnetization is. This is the difference between SAW and EMW.

In order to deeply explain the essence of the patterns, we calculate the resonant frequency f_0 and the SAW equivalent field factor \left|\sin ({2\phi }_M)\right| as is exhibited in Fig.4. Fig.4(a) is the resonant frequency color map and different colors represent f_0 in various fields for both SAW-FMR and EMW-FMR. The resonant frequency is the property of the magnetic film, no matter what kind of motivation. The external magnetic field has two degrees of freedom: the magnitude and the direction. Both of them can lead to the change in resonant frequency and the magnetization direction based on Eqs. (3) and (4), as demonstrated in Fig.4(a). In general, both a larger value of external magnetic field or a field with its direction closer to the easy axis can raise the resonant frequency. However, this rule becomes ineffective in the small field region around the hard axis because of hysteresis. The resonant frequency plummets when the external magnetic field is near the coercivity, which is the cause of the reversal peak.

As both alternating fields (SAW or EMW) applied in Fig.3 are 2 GHz, we locate the corresponding resonant cases in Fig.4(a) and highlight them with white blocks. A broader resonant frequency range of 1.9−2.1 GHz is selected due to damping effects, which makes the precession occur not only at the resonant frequency, but also at other adjacent frequencies with an attenuated amplitude. Therefore, the darkest blocks in Fig.3(b) are regarded as the resonance cases, which show the same pattern as the white blocks in Fig.4(a). Considering that the lighter blocks are the damping expansion of the resonant peaks, it is demonstrated that the pattern in Fig.3(b) is the production of EMW-FMR.

In the case of SAW, it is necessary to consider both the resonance and the SAW equivalent field at the same time. After obtaining the resonant cases from Fig.4(a), only the cases with larger SAW equivalent field can be seen in Fig.3(a) based on Eq. (15). As is illustrated in Fig.4(b), The red blocks represent a larger SAW equivalent field while the purple blocks signify the weaker ones, and other colors represent the intermediate states. Although the SAW frequency may match the resonant frequency of the film in these purple blocks (with weaker SAW equivalent fields), the magnetization is frozen rather than precessing. Therefore, the positions with very weak equivalent fields, marked by the purple blocks in Fig.4(b), should be removed from the resonance cases shown in Fig.4(a). It just seems like to “cut” the resonant pattern obtained from Fig.4(a) (the white blocks) along the boundary of the purple blocks in Fig.4(b). As a result, the case of {\mu }_{0}H=0.045\mathrm{T} and {\phi }_H={90}^{\circ } is cut off, and an SAW-FMR “butterfly” shape pattern is achieved. Consequently, the EMW-FMR is only influenced by the resonant frequency, while the SAW-FMR depends on the joint action of both the SAW equivalent field and the resonant frequency.

Micromagnetic simulations are also carried out to verify the calculation results. The same parameters as Fig.3 are used, and the simulation results are displayed in Fig.5. The simulation results show the same pattern as the calculation ones, which verifies the correctness of our theory. In addition, Fig.5 depicts a larger precessional cone angle than that in Fig.3 due to different damping considerations (constant {\omega }_{\alpha } in calculation while constant \alpha in simulation).

4.2 The influence of the external magnetic field

In Section 4.1, the classic “butterfly” pattern is reproduced by our theoretical model and is attributed to the influence of the SAW equivalent field. However, SAW-FMR spectrum seems to have other types of patterns in experiments [19, 35, 36]. With a different SAW frequency, the resonant field changes and the pattern also alters to several modes accordingly, which requires further research.

Using the calculation and simulation, we obtain the external magnetic field-dependent precessional cone angle \delta \theta and power attenuation \mathrm{\Delta }P color map in several different SAW frequencies, respectively. The SAW propagates along the +x-axis, and the magnetocrystalline anisotropy is in the direction of the x-axis. The angle between the direction of the external magnetic field and the +x-axis ({\phi }_H) varies from 0° to 180°. The external magnetic field is downswept from 0.1 T to −0.1 T. The spectra at various SAW frequencies (1−4 GHz) are exhibited in Fig.6.

All patterns originate from the center ({\phi }_H={90}^{\circ },{\mu }_{0}H=0\mathrm{T}) and extend towards four directions. For every field direction, the \delta \theta varies with the field sweeping process. Two types of peaks are discovered in the field sweeping process, which are called “resonant peak” and “reversal peak”. The darkest blocks in Fig.6 represent the resonant situations that can be obtained from Fig.4(a), which provides the resonant magnetic field at a certain SAW frequency. The damping brings an expansion to the color map pattern. And the \mathrm{\Delta }P color map shows a similar pattern to the \delta \theta. A little difference is that \mathrm{\Delta }P color map has a narrower damping broadening than the \delta \theta color map.

For different SAW frequencies, the color maps display distinct patterns. For the 1 GHz situation, the pattern is located near the {\phi }_H={90}^{\circ } line and shows a very little “butterfly” as is published in Refs. [14, 44]. We can find an explanation that the situation representing f_0=1 GHz only occupies a small part of the blocks in Fig.4(a). The resonant field increases when {\phi }_H becomes far away from 90°. The resonant peak vanishes after {\phi }_H reaches about 95° upwards (or 85° downwards). When {\phi }_H>{95}^{\circ } or , there is only a reversal peak left. When the SAW frequency increases to 2 GHz, the pattern expands to a pair of “C”. When the {\phi }_H is 80° to 100°, there are two pairs of resonant fields for single {\phi }_H. These two-peaks results can be observed in Refs. [35, 36]. The resonant peak vanishes after {\phi }_H reaches about 120° upwards (or 60° downwards). This situation is an intermediate state where the resonant field increases with raising {\phi }_H in the upper left branch of “C” and decreases in the upper right branch. In the lower-frequency cases mentioned above (1 GHz and 2 GHz), the magnetization is not in the same direction as the external magnetic field for hysteresis under the resonant magnetic field. As for the 3−4 GHz cases, the corresponding resonant field is large enough to neglect the influence of hysteresis, and the magnetization lies near the external magnetic field. The field parallel to K provides a larger effective field than the perpendicular case; thus, the resonant frequency of the parallel situation is higher. As a result, the resonant field decreases when the field direction is closer to the easy axis ({\phi }_H=0^{\circ }\mathrm{o}\mathrm{r}{180}^{\circ }). Furthermore, with the frequency increasing from 1 GHz to 4 GHz, the pattern moves towards the higher magnetic field region. The relevant experiment results of this high frequency-mode can be found in Refs. [19, 24, 45]. The simulation results seem to show a smaller resonant field than the calculation, which will be discussed in the next section.

Subsequently, the reversal peak is worth discussing here. It is obvious that the corresponding field magnitude of the reversal peak varies with {\phi }_H. The corresponding field is the highest in the direction of the easy axis or hard axis {\phi }_H=0^{\circ }\mathrm{o}\mathrm{r}{90}^{\circ }\mathrm{o}\mathrm{r}{180}^{\circ }, while it shows the minimum when {\phi }_H={45}^{\circ }\mathrm{o}\mathrm{r}{135}^{\circ }. This variation can be confirmed by the Stoner−Wohlfarth (SW) model. The reversal peak can exist only if the SAW frequency is not very high. When f_{SAW}=1 GHz, the reversal peak has a positive influence on the precessional angle and power attenuation, which is even more obvious than the resonant peak. The reversal peak is weaker and weaker with the increase of the SAW frequency and turns into a weakening effect at a certain frequency. As the resonant peak shifts far away from the coercivity, the reversal peak disappears. The behavior of the reversal peak can be attributed to the change of the resonant frequency shown in Fig.4(a). With a lower SAW frequency, the resonant frequency in the coercivity is comparable with the SAW frequency and we can see the reversal peak. When f_{SAW} is much higher than the resonant frequency in the coercivity, the reversal peak is too weak to detect. Heretofore, it is clear that the “reversal peak” is closely related to the SAW frequency. When applying an opposite magnetic field to the magnetization (no higher than the coercivity), the potential barrier of anisotropy is influenced, which leads to a decrease in the effective field. As shown in Fig.7, the situations of the “reversal peak” have lower resonant frequencies, and this is the reason for the mutation of precession.

In this section, the field dependence of precessional cone angle and power attenuation is illustrated by the color map. These different types of absorption patterns can be comprehended qualitatively. In general, the resonant magnetic field increases with the growing SAW frequency; thus, the pattern moves to the high field region. When the SAW frequency is very low (0−2 GHz), the SAW-FMR can only occur when applying a magnetic field in the negative direction or the hard axis. These treatments can highly decrease the resonant frequency of the film and better match the lower SAW frequency. Therefore, the resonant pattern occupies a small part of the position in the color map (near the hard axis) and shows an obvious reversal peak with a positive effect on the precession. When the SAW frequency is high enough (> 3 GHz), the resonant field is large enough to keep the magnetization in the direction of the magnetic field. Considering the SAW equivalent field, the maximum absorption happens when {\phi }_H={45}^{\circ }\mathrm{a}\mathrm{n}\mathrm{d}{135}^{\circ }. When rotating the magnetic field from the easy axis to the hard axis, the equivalent field of anisotropy decreases and a higher resonant magnetic field is needed to match the SAW frequency, i.e., the resonant field increases with growing {\phi }_H ().

4.3 The influence of SAW frequency and amplitude

In the application of SAW-FMR, it is a usual need to predict the resonant field with a given SAW frequency or regulate the resonant frequency by an external magnetic field. Therefore, the relationship between resonant frequency and external magnetic field needs to be researched.

To find the relationship, it is a good idea to measure the field-sweeping spectrum at continuously changing SAW frequencies. We sweep the field in the same direction downwards at various SAW frequencies and then paint the frequency-dependent field sweeping color map using Eqs. (15) and (20). Five field directions of 0°, 30°, 60°, 80°, and 90° are considered. Both calculation and simulation results of \delta \theta and \mathrm{\Delta }P color map are listed in Fig.7. The bright line depicts the resonant frequencies under the different external magnetic fields. The color of the map represents the value of the precessional cone angle or power attenuation.

In the case of {\phi }_H=0^{\circ }, the magnetization lies in the x-axis and the SAW equivalent field disappears in this direction. The frequency spectrum, which is supposed to appear, displays no pattern except the reversal peak in simulation. The reason for this is that the SAW equivalent field in the calculation is obtained by the equilibrium magnetization, which is zero constantly; While simulation provides a dynamic SAW equivalent field, that is, magnetization can precess at the reversal peak. Certainly, with a higher external magnetic field, the magnetization is fixed in the direction of the x-axis and precession disappears again. With {\phi }_H={30}^{\circ } and {60}^{\circ }, the resonant frequency increases with the external magnetic field as the Kittel formula predicts. At {\phi }_H={80}^{\circ }, the resonant frequency first decreases with the increasing field (about 0−0.03 T) and then returns to the Kittle formula. As for the {\phi }_H={90}^{\circ } case, the resonant frequency decreases with growing external magnetic field at first and then declines drastically at a certain field (about 0.2 T). This can be ascribed to the decreasing effective field impacted by both the anisotropy and the external magnetic field. When the field is large enough to pull the magnetization to the y-axis (hard axis), the SAW equivalent field returns to zero and the precession disappears. Hence, only a low-field region pattern is reserved.

These frequency-spectrums can also explain the origin of the reversal peak well. We mark the resonant frequencies near the reverse coercivity with the white ellipses in the first column of Fig.7. Before the reverse coercivity, the resonant frequency reduces with the reverse field increasing and abruptly returns to the normal value after the reverse coercivity. For a pair of fields with the opposite directions and same magnitude (smaller than the coercivity), the resonant frequency of the negative field (marked in the white ellipse) is lower than that of the positive direction. If SAW with the same frequency is applied, the resonant frequencies of the two cases with opposite directions have different deviations from the SAW frequency. The case with a smaller deviation between SAW frequency and resonant frequency can yield a higher \delta \theta and \mathrm{\Delta }P, which is the cause of the reversal peak.

Comparing the theoretical calculation results, the simulation results have a higher resonant frequency. To demonstrate the difference, we draw the calculation resonant frequency by a white dashed line in the \mathrm{\Delta }P simulation color map of 30° and 80° (shown in Fig.8). Except for the value of resonant frequency, the simulation results display the same trend with the calculation. This difference can be attributed to the different treatment of SAW in calculation and simulation. The resonant frequency is influenced by both the effective static field and the SAW. We consider both of them in the simulation, while SAW is regarded as a perturbation and only the effective static field is considered in the calculation.

To confirm the effect of amplitude, SAWs with different amplitudes are employed in the power attenuation simulation. The SAW frequency is 1 GHz and the field direction {\phi }_H is 80°. These parameters are chosen for the simultaneous existence of the resonant peak and the reversal peak. The external magnetic field is swept downwards from 50 mT to −50 mT and the amplitudes of strain generated by SAW varies from 2 × 10⁻⁵ to 1 × 10⁻⁴ in the step of 2 × 10⁻⁵ in Fig.9(a). Both reversal peak and resonant peak move to the lower field region and absorb more SAW power with a higher SAW amplitude. It is easy to understand that a larger SAW amplitude will give a stronger driving force to the magnetization precession. Therefore, the precession becomes stronger and absorbs more power with the increase of the SAW amplitude. To better quantitatively display the influence of SAW amplitude, the positions of reversal peaks and resonant peaks under SAW with different amplitudes are shown in Fig.9(b). With the increase of the SAW amplitude, both the magnetic fields corresponding to the reversal peak and the resonant peak decline gradually. For the calculation, it does not consider the influence of SAW on the frequency (i.e., |ε_xx| = 0). Therefore, the reversal field and resonant field in calculation are higher than those in simulation, which validates the rationality of Fig.8.

The magnetization reversal is assisted by SAW and the coercivity decreases with a larger SAW amplitude, which is confirmed by Refs. [8, 46]. This is the reason why the reversal peak shifts to a lower field. On the other hand, the static effective field is constant under the resonant field. The SAW with a larger amplitude provides a stronger SAW equivalent field, thus the resonant magnetic field reduces. The calculation considers SAW as a perturbation and the largest resonant magnetic field is desired. This is the reason for the different resonant frequencies of calculation and simulation in Fig.8.

4.4 The influence of magnetocrystalline anisotropy orientation and distribution

In the above discussions, the magnetocrystalline anisotropy (K) is set in direction of the SAW propagation. The magnetocrystalline anisotropy direction determines the distribution of the resonant frequencies [as Fig.4(a)], and the SAW propagation direction determines the equivalent field distribution. Therefore, the angle between K and the SAW propagation direction has an impact on the combination of SAW equivalent field and resonant frequency distributions.

In order to discuss the influence of magnetocrystalline anisotropy orientation, the SAW propagation direction is assumed to be fixed in the +x-axis. The anisotropic easy-axis is set in the film plane, and the angle between the +x-axis and the easy-axis ({\phi }_K) varies as 0°, 30°, 45°, 60°, 90°. Two SAW frequencies of 2 GHz and 3 GHz are considered. The field-dependent power attenuation color maps for every situation are listed in Fig.10.

For the 2 GHz case, the resonant pattern shifts upwards with the increase of {\phi }_K. The pattern seems to be located near the easy-axis ({\phi }_K) and has a smaller pattern when . The dark color line shows an inverse relationship between {\phi }_H and external magnetic field in the {\phi }_K={30}^{\circ } and {\phi }_K={60}^{\circ } situation. When {\phi }_K={90}^{\circ }, an additional part emerges from the lower edge of the picture. As for the 3 GHz case, the upper part of the pattern follows the same law as the 2 GHz: moves and changes pattern with {\phi }_K. But instead of disappearing, the lower part of the pattern transfers to the low field region. The lower pattern has a weaker resonant peak value than that of the upper pattern and has an inverse {\phi }_H\sim H relationship compared with the upper pattern. At both frequencies, the {\phi }_K=0^{\circ } and {\phi }_K={90}^{\circ } situations provide the same pattern. If the range of {\phi }_H is broadened, the {\phi }_K={90}^{\circ } pattern can be acquired by translating the {\phi }_K=0^{\circ } pattern upward by 90°. This result confirms the four-fold symmetry of the SAW equivalent field: when we rotate the SAW propagation direction by 90°, the SAW equivalent field distribution does not change. When we compare the simulation results with the calculation results, the patterns are essentially matched. The simulation patterns have a higher expansion for different damping treatments. Moreover, the patterns are in different positions as a result of the different resonant frequencies, which has been discussed in Fig.8.

For all patterns in Fig.10, the SAW propagates along the +x-axis. It provides a maximum equivalent field when {\phi }_M={45}^{\circ }\mathrm{o}\mathrm{r}{135}^{\circ }, and shows no effect when magnetization is parallel or perpendicular to the SAW propagation direction. If the external magnetic field is large enough, the magnetization will lie around the external magnetic field. In the high field region, the power attenuation reaches a maximum at 45° and 135°, and the pattern disappears at 0°, 90°, and 180°. In respect of a little magnetic field, the magnetization has a different direction with the external magnetic field due to the hysteresis. The 3 GHz results with various anisotropy directions can be compared with our previous experiment results [37], which have the same positions for both resonant peak and reversal peak. They have some differences in the damping and marked colors. If we set a higher damping constant and use the same mapping colors, similar results can be acquired in Appendix C.

In Section 4.1, we have explained the field-dependent color map as the combination of resonant frequency and SAW equivalent field. Following the same approach, we calculate the field-dependent f_0 and \left|\sin ({2\phi }_M)\right| color maps for various magnetocrystalline anisotropy orientations in Fig.11. Every pattern in Fig.10 can find its origin from the f_0 and \left|\sin ({2\phi }_M)\right| color maps. The f_0 distribution moves upwards with the {\phi }_K, while the pattern shape remains unchanged. It is demonstrated that the SAW direction does not influence the resonant frequency. As for the \left|\sin ({2\phi }_M)\right| maps, red represents the strong SAW equivalent field while purple represents the opposite. Only when {\phi }_K=0^{\circ }\mathrm{o}\mathrm{r}{90}^{\circ }, the commonly mentioned “butterfly” pattern appears. For other K directions, the patterns are very complex. The upper part of the red blocks region becomes smaller while the lower part enlarges. It can play an important reference role in the regulation of FMR by magnetocrystalline anisotropy (or other anisotropy types).

Our theoretical model is based on the single-crystal assumption, but polycrystals are more common in the experiment. Polycrystals have a lot of grains and each grain has distinct anisotropy directions. To fit the property of the polycrystal, we propose a method of averaging the results of single crystals. A polycrystal is assumed to be divided into several small single crystals with different easy axes. Based on this assumption, polycrystal results are obtained by averaging the results for different anisotropy directions in proportion to the distribution density. In our calculation, the anisotropy directions of the single crystals follow a Gaussian distribution, i.e., most anisotropies are supposed to lie around one direction in a range of a certain angle. The direction is acquired from the polycrystalline total anisotropy, and the range of the angle represents the dispersion degree of the distribution.

In order to discuss the impact of the distribution, we select two typical situations and calculate the \mathrm{\Delta }P color maps for different distributions as is depicted in Fig.12. The dispersion degree varies from 0° to 30° in the step of 10°. All of these results have an expansion of the attenuation pattern and keep the fundamental patterns unchanged. The curve-shape pattern becomes rounder and smoother with the increasing dispersion degree, which is closer to the experiment results [14, 17, 19, 35, 36].

5 Outlook

This work focuses on a theoretical model of SAW-induced FMR and its influencing factors (including external magnetic field, SAW amplitude and frequency as well as the magnetocrystalline anisotropy direction and distribution). It may play a promoting role in the following areas.

Our theoretical model provides a quantitative method to describe the SAW-FMR and shows the influence of different parameters explicitly, which paves a new way to understand the experiment results better. In addition, our theory combines the experimental value power attenuation with the theoretical value precessional cone angle. This framework allows for future simulation research on power attenuation in bulk magnets or magnetic domain walls, which cannot be investigated by theoretical calculation.

The investigations of SAW-FMR can provide insights into the effects related to magnetization precession. For the magnetization precessional switching, our model can be used to select the largest precession amplitude and an appropriate SAW frequency. This work focuses on the issue related to the amplitude of SAW-FMR; however, the process to drive magnetization towards the desired target is beyond the scope of our discussion and can be explored in future research. Furthermore, aspects such as acoustic spin pumping, acoustic spin wave resonance, acoustic voltage signal, and other effects associated with SAW-FMR may be regulated based on the findings of this study.

The reversal peak can also be utilized in the device design. One idea is a SAW switch device based on the reversal peak. This switch device consists of a three-layer film structure made up of a piezoelectric substrate layer, a ferromagnetic (FM) layer, and an antiferromagnetic (AFM) layer. The AFM layer provides an exchange bias field to make the magnetization of the FM layer in the reversal peak position (smaller than the coercivity field) without any excitation. Initially, the switch device is in the “OFF” state (In this state, the magnetization is in the opposite direction of the external magnetic field) and the SAW is absorbed by the FM layer. When a field pulse is applied along the exchange bias direction, the magnetization rotates to the opposite direction, leading to an attenuation reduction and switching the device to the “ON” state (In this state, the magnetization has the same direction as the external magnetic field). A negative field pulse can then reset the device to the “OFF” state. The SAW induces two primary effects: strain and thermal effects, which can assist the magnetization reversal. Considering the thermal effect in the environment simultaneously, the bias field should be set to a value smaller than the coercivity in order to avoid the reversal induced by the thermal perturbation or SAW. On the other hand, the SAW power should be limited to a threshold. For better absorption of SAW, the size of the device should be as large as possible and high-quality films are desired to avoid complex magnetic structures. The reversal peak has a characteristic of mutation near the coercivity field, which means that many devices can be created utilizing this advantageous property.

In addition, the different effects of SAW and EMW mentioned in Section 4. A can be employed to filter the EMW in a mixed signal specifically. The abrupt change of attenuation with {\phi }_H also provides an opportunity to develop the magnetic field direction sensor. In summary, the SAW-FMR resonant devices have broad application prospects in the future.

6 Conclusion

We present a new, simplified theoretical model to calculate the precessional cone angle in surface acoustic wave ferromagnetic resonance (SAW-FMR). This model explicitly outlines the factors that influence resonance, allowing us to understand how surface acoustic waves, external magnetic fields, and the intrinsic properties of the film affect precession. Additionally, we provide an expression of power attenuation to align the calculation results with the experiment results.

The SAW-FMR is compared with EMW-FMR. The SAW provides a special equivalent field related to the direction of magnetization and produces an anisotropic equivalent field distribution. In contrast, the EMW offers an isotropic equivalent field. The equivalent field distinctions make their different precessional cone angles and power attenuation.

The factors that influence the SAW-FMR are discussed through calculations and validated by both simulations and previous experiments. The external magnetic field magnitude and direction are confirmed to determine the intrinsic resonant frequency and influence the magnetization precession. As the SAW frequency increases, the resonant peak shifts toward the higher field region. Conversely, both the resonant peak and the reversal peak occur at lower fields with the growing SAW amplitude. The magnetocrystalline anisotropy direction disrupts the distribution of the SAW equivalent field, subsequently altering the distribution of SAW absorption. These findings can explain the previous experimental results excellently.

Our research also provides valuable insights for the design of SAW-FMR devices.

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