Fast long-wavelength exchange spin waves in partially compensated Ga:YIG

Wave-based logic concepts^1–3 are expected to come along with major advantages over information processing based on current CMOS-based information technology.^4,5 In particular, coherent spin waves⁶ are envisaged to allow for the realization of efficient wave-based logic devices.^3,7–12 However, progress in this field places high demands on the materials used. Specifically, spin-wave elements operating at large clock frequencies demand materials that exhibit a small Gilbert damping constant,¹³ large spin-wave velocities, and good processing properties. Yttrium iron garnet (YIG) has a very small damping constant,¹⁴ and nano-structures of 50 nm lateral sizes have been demonstrated recently.^15–17 However, dipolar spin waves in YIG waveguides feature velocities that are significantly reduced compared to plane YIG films,¹⁶ which are caused by the flattening of the dispersion curve. The fastest dipolar waves in these structures are magnetostatic surface waves, which offer a group velocity of around 0.2 μm/ns.¹⁷ Exchange spin waves with wavelengths in the range of 100 nm or shorter are faster,¹⁸ but the excitation of such short wavelengths is a separate challenge.¹⁶ It can be addressed by the utilization of nanoscopic antennas with increased Ohmic loss, strongly non-uniform magnetic patterns,¹⁹ hybrid nanostructures utilizing different magnetic materials,^20–22 or more complex physical phenomena like magnon Cherenkov radiation.²³ 

To operate with waves of maximized speed in nano-structures, materials with large spin-wave exchange stiffness $λex=2A/(μ0MS)$ are mandatory since the exchange contribution to the group velocity is directly proportional to $λex$⁠. Here, A is the Heisenberg exchange constant, $MS$ is the saturation magnetization, and μ₀ is the permeability of the vacuum. In addition, the possibility to operate with fast exchange-dominated spin waves of larger wavelengths would not only allow for the operations with “standard” micro-scaled antennas,^11,16 but also would give the freedom required for engineering of data-processing units^1,24 since the exchange-dominated dispersion relation is highly isotropic.

In this context, it is promising to study ferrimagnetic insulators that are close to the magnetic compensation since low $MS$ tends to increase $λex$⁠. For this, liquid phase epitaxy (LPE) based¹⁴ growth of Ga-substituted YIG,^25–29 which has been adopted for deposition of $Y3Fe5−xGaxO12$ single crystalline films of sub-100 nm thicknesses, is very interesting. In these films, non-magnetic $Ga3+$ ions preferentially substitute the magnetic $Fe3+$ ions in the tetrahedral coordinated magnetic sub-lattice (⁠$0<x<1.5$⁠), decreasing the $MS$ of this ferrimagnetic material down to the fully compensated antiferromagnetic state (for single crystals grown from high-temperature solutions at contents x of about 1.27 formula units at room temperature $T=295K$⁠).^25,26 Aside from that Ga substitution induces a strong out-of-plane uniaxial anisotropy, enabling the easy axis of the thin film perpendicular to the film surface.

Here, we report on the investigation of the spin waves in Ga:YIG films of sub-100 nm thickness. Ga substitution $x≈1$ was chosen so that the magnetization of the sample is decreased to one tenth of the pure YIG value. This was done in order to change the easy magnetization orientation from the in-plane to out-of-plane, simultaneously avoiding an excessive increase in the magnetic damping. First, a thorough characterization is performed using ferromagnetic resonance spectroscopy (FMR) in combination with vibrating-sample magnetometry (VSM) to obtain saturation magnetization, anisotropy constants, and damping parameters. Afterwards, wave vector resolved^29,30 Brillouin light scattering (BLS) spectroscopy is used to measure the dispersion relation of thermal spin waves $ω(k)$ and the spin-wave exchange stiffness $λex$ directly. The resulting group velocity $dω/dk$ is compared to the velocities of spin waves in pure YIG films. Our presented results are complementary and in good agreement with recent results from an indirect measurement of the dispersion relation at low wave vectors using electrical spectroscopy in Ga:YIG reported by Carmiggelt et al.³¹ 

In the following, we present exemplary broadband ferromagnetic resonance-vector network analyzer (FMR-VNA) spectroscopy data for a LPE-grown 59 nm thick film of Ga:YIG/GGG(111). Data for the other films' thicknesses of 105 nm-thick Ga:YIG/GGG(111), 95 nm-thick Ga:YIG/GGG(001), and a reference film of 97 nm-thick YIG/GGG(111) are provided in the supplementary material. The films grown on GGG(111) were cut into square specimens with the edges oriented along $[112¯]$ and $[11¯0]$ crystallographic directions and samples on GGG (001)—along $[100]$ and $[010]$ (see Figs. S1 and S5 in the supplementary material). For magnetic characterization, FMR-VNA was performed in the frequency range up to 20 GHz and at the rf power of 0 dBm. To define the crystallographic parameters of the samples, the theoretical model of Bobkov and Zavislyak was used.³² The model differentiates three main anisotropy fields—a cubic field $Hc$⁠, a uniaxial anisotropy field of the first order $Hu1$⁠, and a uniaxial anisotropy field of the second order $Hu2$⁠. The cubic anisotropy originates from the magnetization along the preferred crystallographic directions in the garnet lattice,^14,32 while the uniaxial anisotropy consists of cubic, growth-induced, and strain-induced contributions resulting in an effective uniaxial anisotropy.^14,33,34 In addition to the direction of a magnetic field H, the FMR frequency also depends on the crystallographic orientation of the GGG substrate.³² To define all the anisotropy fields experimentally, a magnetic field was applied in-plane (IP) along the two orthogonal crystallographic axes $[112¯]$ and $[11¯0]$ (corresponding to the sample's edges) and out-of-plane (OOP) along the $[111]$ direction normal to the film plane. The saturation magnetization $Ms$ was measured by VSM. Detailed descriptions of the employed theoretical model, the measurement procedure, and the FMR analyses for the film under investigation are given in the supplementary material.

The dependence of the FMR frequency on the in-plane magnetic field is shown in Fig. 1(a). The Gilbert damping parameter α and the inhomogeneous linewidth broadening $ΔH(0)$ were found according to the standard approach described in Ref. 35 [see Fig. 1(b)], taking into consideration that $Hu1>Ms$⁠. All obtained parameters are summarized in Fig. 1(c) along with the results of a reference 97 nm-thick YIG/GGG(111) film. A comparison of these values demonstrates that Ga doping led to a $≃9$ times reduction in the saturation magnetization $Ms$⁠, a $≃27$ times increase in the uniaxial anisotropy $Hu1$⁠, and a $≃4.7$ times increase in the Gilbert damping α. A strong increase in the uniaxial anisotropy indicates the out-of-plane easy axis of a Ga:YIG thin film. While an increase in α was observed, it is still significantly lower compared to metallic compounds² and better³⁶ or comparable³⁷ to other doped YIG films, e.g., using bismuth-doping.^37,38 A detailed comparison of the material parameters of our Ga:YIG with those alternative materials including Bi:YIG is given in the supplementary material. The obtained values for Ga:YIG are also in good agreement with those reported in Ref. 31.

After the characterization via FMR, the dispersion relation of thermally excited, magnetostatic surface spin waves propagating perpendicularly to the applied field is probed by BLS spectroscopy³⁰ to obtain the exchange stiffness $λex$⁠. An external field of $μ0H=$ 300 mT is applied in the film plane along the sample edge, which ensures an in-plane magnetization. For the spectral analysis of the scattered light, a six-pass tandem Fabry–Pérot interferometer is used.³⁹ In all measurements presented here, a laser with a wavelength of $λLaser=491 nm$ is used. The in-plane component $kζ$ of the wave vector of the probed spin wave is varied by changing the angle of light incidence Θ, $kζ=4π sin (Θ)/λLaser$—see Fig. 2(a). The results for the 59 nm thick Ga:YIG film are presented in Fig. 2. Figure 2(b) shows three exemplary BLS spectra (anti-stokes part). Data for three different in-plane wave vectors are presented. Aside from the quite strong phonon signal, one can distinguish the fundamental spin-wave mode and the first perpendicular standing spin-wave mode (PSSW). For wave vectors between 12 and 20 rad μm⁻¹, no fundamental mode could be observed because of the strong signal of the phonon mode that crosses the fundamental spin-wave mode in this area (see, e.g., the spectrum for $kζ=14.7 rad μm−1$⁠).

The corresponding analytical description of the spin-wave dispersion relation for the case of a ferromagnetic film in the (111) orientation having uniaxial and cubic anisotropy with unpinned surface spins has been obtained by Kalinikos et al.⁴⁰ The wave vector quantization along the film normal, which results in the appearance of the PSSWs,⁴¹ is described by the index n such that the dispersion relation is given by

where $kn=kζ2+κn2$ is the total spin wave vector consisting of the in-plane spin wave vector $kζ$ and the out-of-plane spin wave vector κ_n with $κn=nπ/t, n=0,1,2,…$⁠. Here, t is the thickness of the film, $ϕ$ is the angle between the static magnetization and the in-plane wave vector $kζ, ϕM$ is the angle between the static magnetization and the [$11¯0$] axis, H is the applied magnetic field, and γ is the gyromagnetic ratio that we take from the FMR measurements. The matrix element P_nn is a function of $kζt$ (⁠$0≤Pnn<1$ if $0≤kζt<∞$⁠). In the long wavelength limit (⁠$kζt≪1$⁠) and for unpinned surface spins, the following approximations have been obtained: $P00=kζt/2$ for n = 0 and $(kζt/nπ)2$ for $n≠0$⁠.⁴² 

The dispersion relations of the respective modes extracted from the BLS spectra are shown in Fig. 2(c) together with fit curves according to Eq. (1). FMR measurements show that $Hc$ is about one order of magnitude smaller than $Hu1$ [compare Fig. 1(c)]. Consequently, the last term in Eq. (1) that is quadratic in $Hc$ can be safely neglected. For the fits, we have fixed the saturation magnetization to $μ0MS=20.2 mT$ as obtained from VSM and the gyromagnetic ratio to $γ=179 rad T−1ns−1$ as obtained from the FMR measurements. The extracted values from the simultaneous fits of the fundamental mode and the first PSSW mode are exchange stiffness $λex=(13.54 ± 0.07)×10−11 Tm2$ and exchange constant $A=(1.37 ± 0.01) pJm−1$⁠, respectively. The sum of the anisotropy fields is $μ0(Hu+Hc)=(91.3 ± 0.4) mT$⁠, in very good agreement with the values obtained from FMR [compare to Fig. 1(c)].

The exchange stiffness in the film under investigation in this work is about three times as large as the one for pure YIG.⁴³ This results in a much higher group velocity than in pure YIG as can be seen in Fig. 3. There the fitted dispersion relation of the fundamental mode for the investigated Ga:YIG film and the corresponding dispersion relation for a pure YIG film of the same thickness of 59 nm and at the same applied field of 300 mT is also shown. Here, the standard parameters of YIG^14,43 have been used: $γ=177 rad T−1ns−1$ (from FMR), $λex=4.03×10−11 Tm2, and μ0MS=177.2 mT$⁠. (All anisotropy contributions are neglected.) The corresponding group velocities calculated by $vgr=2π∂fn(k)/∂kn$ are plotted in the lower part of Fig. 3(b). The exchange dominated region is characterized by a linear dependence of the group velocity on the wave vector. Thus, spin waves in Ga:YIG can be considered as exchange dominated down to very low wave vectors as it is directly visible from Fig. 3(b). For wave vectors above $kζ>4 rad μm−1$⁠, spin-waves in Ga:YIG are faster compared to pure YIG. For wave vectors $kζ>30 rad μm−1$⁠, both dispersion relations are exchange dominated, and the ratio r of the group velocities is converging to the ratio of the exchange stiffness constants $r≈λex$(Ga:YIG)$/λex$(YIG) $≈3.4$⁠. Please note that this comparison of group velocities strongly depends on the chosen geometry; for propagation along the static magnetization (⁠$ϕ=0$⁠), the dipolar dominated waves in YIG are even slower. As a result, for $ϕ=π/2$⁠, the exponential amplitude decay length $δ=vgr·τ$ (where τ is the lifetime of the magnon, calculated based on the value of α and neglected an potential influence of the inhomogeneous broadening $ΔH0$¹⁶) might be still larger for pure YIG (⁠$δ≈40 μm$⁠, for a wave vector of $kζ=15 rad μm−1$⁠) compared to Ga:YIG (⁠$δ≈35 μm$ for the same $kζ$⁠). However, for propagation with the same wave vector along the static magnetization (⁠$ϕ=0$⁠), the YIG decay length drops drastically (below 10 μm) due to the changed dipolar contribution, whereas Ga:YIG preserves its large $δ≈$ 35 μm due to its isotropic, exchange-dominated group velocity. In addition, Ga:YIG is expected to maintain its decay length also for lower film thicknesses, whereas the decay length in YIG can significantly drop due to changed dipolar contribution.

Please note also that the challenge to achieve the perpendicular magnetic anisotropy while preserving the low magnetic losses was explored in other YIG-based materials^36–38 by using substrates with enlarged lattice parameters to achieve strong tensile stress. However, this approach might fail at larger film thicknesses³⁸ and has no significant influence on the saturation magnetization. Therefore, the exchange stiffness is not significantly increased in those systems leading to much slower spin-waves in the exchange dominated regime compared to Ga:YIG.

In conclusion, we have investigated spin-wave properties in Ga-substituted YIG with significantly decreased saturation magnetization $μ0MS≈20$ mT and increased exchange stiffness $λex=(13.54±0.07)×10−11 T m2$⁠. The saturation magnetization $MS$ was measured using VSM, the three anisotropy constants $Hc,Hu1, and Hu2$ and the gyromagnetic ratio γ were determined using FMR, and the exchange stiffness $λex$ was determined from BLS measurements of the dispersion relation of the fundamental and the first PSSW mode. We find that even spin waves of relatively small wave vector $k≈4$ rad/μm exhibit the exchange nature, and their velocities are higher than in pure YIG, reaching a ratio of approximately 3.4 as defined by the ratio of the individual exchange stiffness constants. As a further consequence, waves in Ga:YIG have a significantly more isotropic dispersion relation than waves of the same wavelength in YIG. Thus, for magnonic waveguides structured from Ga:YIG, only a weak dependence of important parameters, such as the wave velocity and the wave phase accumulation, on the structure sizes and on the magnetization orientation can be expected. The small saturation magnetization and the uniaxial anisotropy lead to an out-of-plane easy axis, which also facilitates the use of the entirely isotropic forward volume waves. Since the relative drop of the exchange constant A with Ga substitution x is weaker than the drop of the saturation magnetization M_s, one can expect that a further reduction of M_s by increased Ga substitution will lead to even faster and more isotropic spin waves. Eventually, a fully compensated Ga:YIG film might serve as a model system for antiferromagnetic magnonics. Thus, Ga:YIG opens access to the operation with fast and isotropic exchange spin waves of variable wavelengths in future magnonics networks.

See the supplementary material for the detailed description of the FMR measurements and of the used model to analyze the FMR data for different directions of the applied bias field. Also, an overview of the measured parameters for other Ga:YIG film thicknesses and substrate orientations is provided. In addition, a comprehensive comparison of the material parameters obtained in this work to other substituted YIG thin films is presented.

This research has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—No. 271741898, by the DFG Collaborative Research Center via No. SFB/TRR 173-268565370 (Project Nos. B01 and B11), by the Austrian Science Fund (FWF) through Project No. I 4696-N, and by the European Research Council project ERC Starting Magnon Circuits (Grant No. 678309). The authors thank Volodymyr Golub (Institute of Magnetism, National Academy of Sciences of Ukraine) for support and valuable discussions, as well as M. Lindner and T. Reimann for production of the YIG reference sample and R. Meyer (INNOVENT e.V.) for the technical assistance.