# All Acoustical Excitation of Spin Waves in High Overtone Bulk Acoustic Resonator

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. HBAR Structure and Basic Properties

_{3}Fe

_{5}O

_{12}) 15 µm thick were used; in device 2, YIG films substituted with La and Ga with a lower saturation magnetization μ

_{0}M

_{0}~80–90 mT (compared to 175 mT of pure YIG) and with a thickness of 31 µm were used. Device 3 was based on a YIG (100) single crystal. Device parameters are shown in Table 1.

#### 2.2. Theoretical Description

^{ME}of a magnetically ordered cubic crystal and it was shown that the ME interaction leads to a connection between the dynamic equations for the elastic and magnetic subsystems—Newton’s equation for the elastic displacement $\overrightarrow{u}$ in a medium with mass density ρ and Landau–Lifshitz equation for precession of magnetization $\overrightarrow{M}={\overrightarrow{M}}_{0}+\overrightarrow{m}$:

_{0}is the magnetic constant, $\overrightarrow{M}={\overrightarrow{M}}_{0}+\overrightarrow{m}$, where ${\overrightarrow{M}}_{0}$ = (0, 0, M

_{0}) is the saturation magnetization in the constant tangential magnetic field ${\overrightarrow{H}}_{0}$ = (0, 0, H

_{0}), $\overrightarrow{m}=\left({m}_{x},{m}_{y},0\right)$ is the variable magnetization; ${\overrightarrow{H}}_{\mathrm{ef}}$ and ${T}_{ij}$ are the effective magnetic field and elastic stress tensor, which are found through the variational derivatives of the total energy, taking into account W

^{ME}. The coordinate system employed corresponds to those in Figure 1a.

^{(i)}x − ωt)], where $j=\sqrt{-1}$, ω = 2πf, k

^{(i)}is a wave number for the (i)-th layer. Then, the ME contributions to the effective field and the stress tensor are:

_{2}is one of two ME constants for cubic symmetry crystals [9].

^{(i)}= ω/V

^{(i)}. Here, ${V}^{\left(i\right)}=\sqrt{{C}^{\left(i\right)}/{\rho}^{\left(i\right)}}$ is a shear AW velocity, ρ

^{(i)}is the mass density, and C

^{(i)}is the effective elastic modulus for the (i)-th layer. As for magnetic layers the appropriate wave numbers are found form the secular equation, obtained from (1) and (2) together with Maxwell equations, in the following form:

_{ef}(k

^{2})= H

_{0}+ Dk

^{2}and M

_{ef}≈ M

_{0}are the uniform effective magnetic field and magnetization, D is the exchange stiffness [9,34]. Equation (3) is biquadratic with respect to ω and is easily solved. The crossover of two independent solutions (3) ${\omega}_{1,2}^{2}\left({k}^{2}\right)$ in the case of ξ = 0 determines MER frequency and wave number:

_{MER}, k

_{MER}) take place. The frequency width of the MER region is determined as a minimal repulsion of coupled ME waves branches Δω

_{MER}= (ξω

_{H}ω

_{M})

^{1/2}. As one can see, for a given real positive ω, the secular Equation (3) becomes bicubic with respect to k

^{2}and has three real roots ${k}_{p}^{2}\left(\omega \right)$ (p = 1, 2, 3). Using six roots ±k

_{1,2,3}, one can construct the general solutions for u = u

_{z}, m

_{x}

_{,y}, and for normal stress component T = T

_{zx}= C(∂u/∂x) + bm

_{x}/M

_{0}.

^{(i)}should satisfy the elastic and electrodynamic boundary conditions with the additional conditions for m

_{x}

_{,y}at the magnetic layer boundaries. Here, we used the so-called free spin conditions: ∂m

_{x}

_{,y}/∂x = 0. The set of equations obtained provides the expression for the complex electric impedance of the piezoelectric transducer:

^{(1)}tan(θ

^{(1)}/2)]/(z

^{(1)}tan θ

^{(1)}−jz) is the function of phase θ

^{(1)}, the acoustic impedance ${z}^{\left(1\right)}={({\rho}^{\left(1\right)}{C}^{\left(1\right)})}^{1/2}$ of the piezoelectric layer and the acoustic impedance of the adjacent load z(ω, H

_{0}), i.e., the GGG substrate and the YIG layers; K

_{t}, and C

_{0}are the effective piezoelectric constant, and the capacity of the piezoelectric layer. The impedance z(ω, H

_{0}) contains all the information about ME interactions in FM media and the resonant rearrangement of the HBAR spectrum in a magnetic field. Note that the eigenfrequencies of the structure correspond to the poles of the function F:

^{(1)}tan θ

^{(1)}−jz(ω, H

_{0}) = 0.

^{(3)}+ arctan(r

_{4,3}tan θ

^{(4)}) + arctan[r

_{2,3}tan(θ

^{(2)}+ δ)] = πn.

_{1,2}tan θ

^{(1)}) is the contribution of the piezoelectric layer to the total phase shift; r

_{i}

_{,k}= z

^{(i)}/z

^{(k)}; θ

^{(3)}= k

^{(3)}l

^{(3)}= ωl

^{(3)}/V

^{(3)}and z

^{(3)}—phase incursion in the substrate and its material acoustic impedance; effective parameters θ

^{(2,4)}=K(H

_{0})

^{(2,4)}l

^{(2,4)}, z(H

_{0})

^{(2,4)}and K(H

_{0})

^{(2,4)}—phase shifts, acoustic impedances and wavenumbers for FM layers. The last values depend on roots k

_{1,2,3}of Equation (3). For more details, see [25,28,30].

_{11}and the electrical impedance Z

_{e}= Z

_{0}(1 + S

_{11})/(1 − S

_{11}) where Z

_{0}is the characteristic impedance (50 Ω) of the RF probes. The losses in the system are taken into account by replacing H

_{0}→ H

_{0}+ jΔH

_{0}, where ΔH

_{0}is FMR line width and C

^{(i)}→ C

^{(i)}+ jωη

^{(i)}, where η

^{(i)}is viscosity factor.

#### 2.3. Experimental Methods

_{11}in a wide range of frequencies. The frequency dependence of |S

_{11}| (spectrum) of the HBAR in the absence of the external magnetic field looks similar to a comb made of evenly distributed resonance dips (harmonics) at frequency f

_{n}, where n is the harmonic number. The distance between adjacent harmonics Δ f

_{n}= f

_{n}− f

_{n−}

_{1}in accordance with the ratio (6) slightly oscillates around the mean value with the change of n [32,35,36]. The spectrum of device 1 is shown in Figure 2. The darker areas represent frequency regions of effective shear wave excitation since the shear resonances are more densely placed on the frequency axis than the longitudinal ones. The lower inset corresponds to the frequency at which the magnitudes of the two series of dips (longitudinal and transverse) are close. The top inset represents the frequency at which longitudinal resonances are suppressed, which is convenient for analyzing the results.

_{11}(f, H

_{0})| we placed the studied HBAR in the electromagnet controlled by the computer. The external magnetic field H

_{0}was oriented in the plane of the structure and magnetized YIG films up to uniform saturation magnetization M

_{0}. The VNA was also controlled by that computer. Before the start of the experiment, a set of frequencies {F} and a set of magnetic fields {H} were selected. At each step of the procedure, the next value of H

_{0}from {H} was set on the electromagnet, and the |S

_{11}(f, H

_{0})| spectrum was recorded for each f from {F}.

## 3. Results

_{MER}(H

_{0}) lines. The position of the f

_{MER}(H

_{0}) line corresponds to the conditions for the intersection of the dispersion branches of noninteracting AW and SW (see (4), Section 2.2). The detuning of resonant frequencies of HBAR is a consequence of the inverse effect of resonant excitation of spin-wave dynamics, ADSW, in magnetic films [23,26,28]. Figure 4b shows that for device 2, in addition to the excitation of resonant ADSWs, there is also the excitation of magnetic oscillations propagating in the plane, such as surface and bulk magnetostatic waves (MSWs). These excitations form a distinct quasi-vertical relief of dark and light bands. The fact that the nature of these features is associated with the MSW spectrum is confirmed by the correspondence of these bands slopes to the slope of the Kittel line f

_{MER}(H

_{0}) = (γμ

_{0}/2π)[H

_{0}(H

_{0}+ M

_{0})]

^{1/2}. The reason for the MSW excitation in device 2 is the specific elongated shape of the transducer electrodes (Figure 1a), which creates an alternating magnetic field around them. In a structure with a continuous bottom electrode on the YIG surface (Figure 1b), it is possible to suppress such inductive excitation of the MSW.

_{11}(f, H

_{0})| for device 3, which consists of a substrate in the form of a YIG (100) single crystal plate, on which a transducer of the first type is deposited (see Figure 1b). Since the substrate of device 3 was about two times thicker than that of devices 1 and 2, the intermodal distance Δf

_{n}is to be half that of other devices.

_{0}and the magnetoelasticity constant, are given in Table 1.

_{G}depends on the filling factor of the FM medium Φ = (l

^{(2)}+ l

^{(4)})/D

_{tot}(D

_{tot}is the total thickness of the structure) and is maximal for a monolithic FM resonator (c). In this case, the frequency tuning exceeds several intermodal distances. In Figure 4a,b, the ME gap decreases significantly and the spectral branches on both sides of the MER gap approach the MER line but do not pass into each other.

_{G}between resonant branches with numbers n and n + 1 located on opposite sides of the MER line, where all modes completely disappear. As the resonances approach the MER line, they lose their quality factor and their position is less well-defined. Therefore, the ascending and descending branches, n and n + 1, seem to transit into each other (see Figure 4), although this does not in fact happen. The theoretical red lines show that resonances can move even closer to the MER line and the minimum distance (δH

_{G})

_{theory}< δH

_{G}will then take place for branches with numbers n − 1 and n + 1.

_{tot}and Φ, but with only one double thickness YIG layer on the opposite side from the transducer. The behavior of the resonant branches outside the MEP region is of the same character; however, the advance into the MEP region becomes slower: the frequency shift does not even reach half the intermodal distance. This can be explained by the fact that the behavior of the resonant frequencies depends not only on the change in the phase shift over the total thickness of the FM medium, caused by the rearrangement of the dispersion dependences, but also on the change in the boundary conditions due to the ME contribution to the elastic stress tensor (see Section 2.2). This contribution enters (6) through the functions r

_{i}

_{,k}, namely through the impedances z(H

_{0})

^{(2,4)}. Note also that even far from the MER, the resonant frequencies of the two structures in Figure 5 are slightly shifted because 2 arctan x ≠ arctan 2x. This difference becomes significant in the MER region.

## 4. Discussion

^{i}

^{,k}=z

^{(i)}/z

^{(k)}= 1 in (6), then we obtain a simplified equation θ

^{(3)}+ θ

^{(4)}+ θ

^{(2)}+ θ

^{(1)}= πn to find resonant frequencies. In this case, the HBAR spectrum is characterized by a comb of equidistant resonant frequencies f

_{n}separated by Δf

_{n}= V

_{tot}/(2D

_{tot}), ≈3 MHz, where V

_{tot}is the effective velocity of AW front propagation in the structure. For comparison, the ME splitting of dispersion branches in FM media (see, Section 2.2) is Δf

_{MER}= Δω

_{MER}/2π, ≈30 MHz. Thus, more than 10 HBAR resonant frequencies appear directly in the MER region, which is seen in Figure 3. In this case, the region of magnetic fields near H

_{MER}, in which resonant frequencies are rearranged, is found to be μ

_{0}ΔH

_{MER}= 2πΔf

_{MER}/γ, ≈6 mT.

^{(2,4)}(0)[1 + 0.5ξω

_{M}ω

_{H}/(ω

^{2}− ω

_{0}

^{2})]. The change of the velocities in the YIG layers will lead to the change in V

_{tot}. Therefore, the resonance condition requires the shift of the resonant frequency δf

_{n}(H) = f

_{n}(H) − f

_{n}(0) ≈ f

_{n}(0)ΦξRe[f

_{M}f

_{H}/(f

_{n}(0)

^{2}− f

_{FMR}

^{2})]. This expression qualitatively explains the slower change of f

_{n}(H) in film structures, for which the filling factor Φ ≪ 1 (device 1 and device 2) compared to the monolithic one with Φ ≈ 1 (device 3). The latter also applies to a decrease in Q-factor of the resonances. For the same value of the magnetic damping parameter ΔH

_{0}, a decrease in the Q-factor of the HBAR resonances and further to complete disappearance of the regular resonance structure occurs in the field gap δH

_{G}= δH

_{G}(ΔH

_{0}, Φ, ξ) proportional to Φ. Summing up, we present the following relations for frequency and field intervals:

_{n}≫ Δf

_{MER}≫ Δf

_{n}, δf

_{n}(H

_{0}),

_{FMR}≫ H

_{MER}≫ ΔH

_{MER}> δH

_{G}> ΔH

_{0}.

_{0})

_{FMR}and f(H

_{0})

_{MER}, represented by dash-dotted and dotted lines are shown in Figure 5. At a given frequency, the fields of the corresponding resonances are related as H

_{MER}= H

_{FMR}− H

_{ex}, where μ

_{0}H

_{ex}= μ

_{0}Dk

^{2}≈ 1.8 mT is the field of inhomogeneous exchange, k = k

_{MER}= 2πf

_{n}/V

^{(2,4)}≈ 6 × 10

^{6}rad/m. Thus, we can conclude that as the external field decreases in the range H

_{FMR}> H

_{0}> H

_{MER}, the ADSW wave numbers change in the range from zero to k

_{MER}and exceed k

_{MER}in the field region H

_{MER}> H

_{0}> H

_{MER}+ ΔH

_{MER}/2.

_{0}near H

_{FMR}and it makes sense to speak of a continuous SW spectrum. In contrast, at lower fields, in particular, at H

_{0}≈ H

_{MER}, an AW resonance with number n interacts with a separate high-order SWR with number s = k

_{MER}l

^{(2,4)}/π ≈ 30. The developed theoretical approach correctly takes into account all solutions of (3) and boundary conditions (of unpinned spins) and is applicable to ME interactions of AW modes with both FMR (s = 0) and SWR (s = 1, 2, 3…). For micron YIG films and thinner, the excitation of individual SWRs is accompanied by a slight shift of only one resonator mode, as was theoretically shown in [25]. To detect such acoustically driven SWR, it is efficient to use acoustic spin pumping into a heavy metal film in contact with YIG(4) [37].

## 5. Conclusions

_{n}(H) higher than the MER line, the quasi-magnetic modes with wavenumbers more than 6 × 10

^{6}rad/m are excited, whereas at f

_{n}(H) lower than the MER line frequency, the wavenumbers of excited modes are essentially smaller. The resonant excitation of ADSW can be accompanied by an additional inductive excitation of the magnetic dynamics due to the transducer electrodes. It is shown that the design of the transducer with a continuous bottom electrode makes it possible to acoustically excite SWs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schemes of the HBAR consisting of YIG(2)-GGG(3)-YIG(4) and piezoelectric transducers based on ZnO films (1) with different shapes of Al electrodes and different types of contact with the bottom electrode (

**a**) with ohmic contact (

**b**) with capacitive contact.

**Figure 2.**Frequency dependence (spectrum) of the microwave reflection coefficient |S

_{11}(f)| in the absence of a magnetic field. The insets show two enlarged fragments of the spectrum. The dips in the frequency characteristics correspond to the resonant frequencies of the shear AW (not marked) and longitudinal AW (L) thickness modes; the intermodal distance of the longitudinal modes is approximately two times greater than that of the shear ones.

**Figure 3.**Frequency and magnetic field dependencies of |S

_{11}(f, H)| for device 1. The arrows show the positions of the resonant frequencies of the longitudinal (L) AW, three of which correspond to the frequencies in the inset in Figure 2.

**Figure 4.**Frequency and magnetic field dependencies of |S

_{11}(f, H

_{0})| for the devices under study: (

**a**) for device 1; (

**b**) for device 2; (

**c**) for device 3. The red lines are the results of the theoretical calculation with the fitting parameters of Table 1. The dashed lines are the positions of the lines f

_{MER}(H

_{0}).

**Figure 5.**Magnetic field dependencies of resonance frequencies for device 1. The circles indicate the positions of the resonances in the experimental data from Figure 4a; red lines—approximation of these data by a model with two 15 µm YIG layers; blue lines—the result of the calculation of the model with only one YIG layer of 30 μm; the green dotted line is the MER line; the green dash-dotted line is the FMR line. The calculations were performed for D = 4.5 × 10

^{−14}mT·m

^{2}.

No. | Transducers, Figure 1 | YIG | Thickness, of YIG, μm | Thickness, of GGG, μm | μ_{0}M_{0},mT | b, MJ/m^{3} |
---|---|---|---|---|---|---|

1 | (b) | 2 films, pure | 15 | 500 | 175 | 0.38 |

2 | (a) | 2 films, doped | 31 | 500 | 86 | 0.31 |

3 | (a) | 1 plate, pure | 1180 | 0 | 151 | 0.48 |

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**MDPI and ACS Style**

Alekseev, S.; Polzikova, N.; Luzanov, V.
All Acoustical Excitation of Spin Waves in High Overtone Bulk Acoustic Resonator. *Acoustics* **2023**, *5*, 268-279.
https://doi.org/10.3390/acoustics5010016

**AMA Style**

Alekseev S, Polzikova N, Luzanov V.
All Acoustical Excitation of Spin Waves in High Overtone Bulk Acoustic Resonator. *Acoustics*. 2023; 5(1):268-279.
https://doi.org/10.3390/acoustics5010016

**Chicago/Turabian Style**

Alekseev, Sergey, Natalia Polzikova, and Valery Luzanov.
2023. "All Acoustical Excitation of Spin Waves in High Overtone Bulk Acoustic Resonator" *Acoustics* 5, no. 1: 268-279.
https://doi.org/10.3390/acoustics5010016