Influence of stripline coupling on the magnetostatic mode line width of an yttrium-iron-garnet sphere

The damping of magnetic materials, which characterizes the decay of non-equilibrium magnetization, plays a crucial role in traditional data storage technologies. During the 1950s and 1960s several physical damping mechanisms, such as magnon scattering and the Kasuya-LeCraw mechanism, were identified.^1–4 More recently investigations of the spin pumping mechanism⁵ and strong magnon-photon coupling⁶ have sparked renewed interest in the damping of yttrium-iron-garnet (YIG). For example, in a YIG-based photon-magnon system with possible applications in coherent information processing, the coherent coupling frequencies can be orders of magnitude larger than the dissipative loss rates.^7–10 In this situation both resonant and non-resonant magnon-photon coupling result in significant line width broadening of the magnon mode as elucidated in Ref. 6. These observations raise an important question as to whether or not similar non-resonant magnon-coupling will impact standard damping measurements.

Current damping measurement techniques differ from the early fixed frequency investigations, using instead a broadband planar waveguide.^11–13 In order to effectively excite the magnetization precession, the space between the microwave line and the sample is often very small, resulting in a strong coupling between the magnon and microwave photon. Such broadband measurements have found an important role in the characterization of magnetic materials, and are widely used to separate different damping mechanisms.^14–16 

In this work we present a systematic study of the damping in a wide frequency range for different magnon modes by varying the separation between the microwave stripline and a YIG sphere and hence varying the coupling between the magnon and microwave photon systems. We find that this non-resonant coupling indeed plays an important role and must be carefully characterized in order to determine the intrinsic Gilbert damping and inhomogeneous broadening. We also observe that coupling to the stripline results in a much greater broadening for the uniformly precessing mode than other higher order modes. With sufficient YIG-stripline separation, a broadband measurement allows us to determine the intrinsic Gilbert damping parameter α = (6.5 ± 0.5) × 10⁻⁵ and an inhomogeneous broadening which is mode dependent and as small as 0.075 MHz.

Figure 1(a) shows the measurement setup. A 1-mm diameter YIG (Y₃Fe₅O₁₂) sphere,¹⁷ chosen for its low Gilbert damping and shape which enables uniform precession,^18,19 is mounted on a Teflon sample holder and fixed on the left pole of an electromagnet without consideration of its direction, while a microwave stripline is fixed on an x − y − z stage and connected to a vector network analyzer (VNA). This setup allows us to continuously tune the separation D along the z-axis, between the YIG sphere and the stripline, and hence to change the local microwave field while the local static magnetic field at the YIG sphere remains unchanged. The diameter of the electromagnets is 10 cm with a separation of 5 cm to ensure a sufficient homogeneity of the static magnetic field. The 50 Ω microwave stripline with a length of 8 cm was fabricated on a 0.813 mm thick RO4003 substrate.

The broadband ferromagnetic resonance measurement was performed by sweeping the microwave frequency at fixed static magnetic fields. Figure 1(b) shows a typical spectrum measured at μ₀H = 150 mT and D = 1 mm, where several magnetostatic modes of the YIG sphere are excited. The measured dispersion (symbols) of these modes are summarized in Fig. 1(d). To make the dispersion clearer the resonance frequency of the (110) mode, ω₁₁₀, is subtracted from all frequencies. The saturation magnetization μ₀M_s in our YIG sphere is 178 mT. The gyromagnetic ratio γ = 26.5 × 2π μ₀GHz/T and the anisotropy field μ₀H₀ = 6.8 mT were determined from the dispersion of the (110) mode. Here the mode numbers are labelled according to Walker’s convention based on the magnetization symmetry^20,21 and solid lines in Fig. 1(d) are the dispersion calculations of the modes in our YIG sphere.^22,23

Interestingly, some modes that cannot be directly excited by the microwave stripline, such as the (310) mode, still appear when their frequencies are close to other excited modes such as the (110) mode. This effect can be attributed to the dipole-dipole and exchange interactions between propagating modes, which lifts the mode orthogonality leading to inter-mode coupling.^24–26 The strong coupling effect between (110) and (310) modes is shown in Fig. 1(c), from which a coupling strength of 6.9 MHz is deduced. The fact that the impact of the microwave stripline is significantly different for different modes allows us to quantitatively study the coupling of the microwave stripline with the YIG sphere as discussed in the following experiments.

In a second experiment we studied the D dependence of the (110) and (310) modes. Here the static magnetic field was fixed at μ₀H = 138.5 mT by applying a constant current, which ensures the fluctuation of μ₀H was less than 0.002 mT during the measurement. At this particular static magnetic field the energy exchange between the (110) and (310) modes due to their strong coupling produces a measurable (310) mode signal. However, because the frequency separation between (110) and (310) modes is much larger than their coupling strength, the coupling induced line width change is negligible.

By continuously tuning D, an ω − D transmission mapping of the (110) and (310) modes was measured, as shown in Fig. 2(a). Two striking differences between (110) and (310) modes are revealed. Firstly, as D increases, the (110) frequency drops from 3.920 GHz to 3.906 GHz whereas the (310) resonance frequency remains unchanged at about 3.888 GHz. Secondly, the line width of the (110) mode decreases rapidly, while the (310) mode displays only a weak distance dependence. These same features are reproducibly observed for excitation powers from 10 dbm to -20 dbm, excluding non-linearities as their origin. To quantitatively study their evolution, line widths were obtained through an empirical procedure in which the data were fit to a modified susceptibility response function which includes a phase shift and a background correction as discussed in detail in Ref. 27. As shown in Fig. 2(b), the experimental data (symbols) at D = 0.9 mm are well fit by this procedure (solid lines). The full width at half maximum (FWHM), corresponding to twice the line width Δω′, is 2.316 MHz for the (110) mode, much greater than the 0.866 MHz obtained for the (310) mode.

The resonance frequency and line width are plotted as a function of 1/D² in Fig. 2(c) and (d), respectively, from which a linear dependence is revealed. To explain this striking feature we adapted the model developed in Refs. 28 and 29 by considering the coupling between the YIG sphere and microwave stripline through the two coupled dispersion equations,

where the first and second diagonal terms represent the magnon modes and the microwave stripline, respectively. Here Δω′ is the line width of the magnon modes and β describes the dissipation of energy due to the microwave stripline. The off-diagonal terms in Eq. (1) describe the magnon-stripline coupling with coupling strength κ, which should be inversely proportional to the distance D according to Gauss’ Law for an infinitely long stripline and YIG sphere of negligible size. When the separation between the YIG sphere and the stripline is comparable to the actual size of the YIG sphere, there exists a small deviation from the ideal inversely proportional relation.²⁹ 

The complex resonance frequency $ω̃=ωm′+iΔωm′$ is determined by the condition $det∏=0$⁠, and therefore from Eq. (1),

and

The second term in both Eqs. (2) and (3), due to coupling, produces a resonance frequency shift and line width broadening proportional to κ² (and thus proportional to 1/D², according to the known coupling strength distance dependence, κ = C/D²⁹). As shown in Fig. 2(c) and (d) the measurement results (symbols) are described well by this model. The line width varies by 1.73 MHz (300%) for the (110) mode, 0.10 MHz (27%) for the (210) mode, 0.03 MHz (9%) for the (220) mode and 0.04 MHz (10%) for the (310) mode. These differences can be explained by the rf magnetization pattern,²³ because integration over the (110) mode profile is much larger than other modes. Details of the mode profiles can be found in the supplementary material.

In our third experiment, shown in Fig. 3, the evolution of the resonance frequency and line width for the (210) mode was measured from 5 to 20 GHz. Compared to the 300% line width change observed for the (110) mode, the line width of the (210) mode over a similar distance range only changes by < 30%. This difference is due to the fact that the mode-dependent coupling strength κ is related to the mode profile, as characterized by the mode conductance.¹⁴ Fitting our data to a linear curve in D⁻² we obtain the solid black lines shown in Fig. 3, where the different slopes for the line width change are mainly due to the field dependence of ω_m. The ratio of resonance frequency and line width slopes determines β = (7 ± 0.5) × 10⁻³, which, as expected, is constant for all fields. From our fits the coupling strength parameter C was determined to be 0.037 mm, 0.03 mm, 0.025 mm, 0.026 mm, 0.019 mm and 0.025 mm for μ₀H = 240 mT, 325 mT, 470 mT, 545 mT, 640 mT and 740 mT, respectively.

The fluctuation we observe in C results in a non-monotonous relationship between the slope and frequency, making it difficult to remove the coupling effect caused by the stripline by simple calculation. However, since our study has found that the coupling induced damping is inversely proportional to the YIG-stripline separation, the coupling effect can be removed by optimizing the experimental setup. To do so we increase the separation D, which still enables a quantitative line shape analysis provided the signal to noise ratio is sufficient.¹⁴ By effectively neglecting the coupling effect of the stripline and using a Gilbert-type damping for all magnon modes in our highly-polished single crystal YIG sphere, the line width of YIG can be written with a linear frequency dependence as

The linear frequency dependence due to the Gilbert damping contribution αω_m is a bulk property mainly determined by the Kasuya-LeCraw mechanism at room temperature,² where α is the phenomenological Gilbert damping parameter³ and is assumed to be frequency-independent, following previous studies.^15,27 The inhomogeneous line shape broadening Δω₀ is due to the inhomogeneous resonance conditions in the YIG sample and is frequency-independent in our highly polished sphere.⁴ 

We can now perform frequency dependent measurements to determine α and Δω₀ by setting D = 5 mm. For such a large separation, the line width broadening due to the coupling effect of the microwave stripline is insignificant. By continuously tuning the field μ₀H from 100 mT to 740 mT, ω_m covers a wide frequency range between 2 and 20 GHz. The deduced line width, Δω′, versus resonance frequency, ω_m, of the (210), (220), (320) and (500) modes are plotted as symbols in Fig. 4. For comparison, Δω′ deduced from the D-dependent measurements in Fig. 3(b) are also plotted as solid squares in Fig. 4(a). The agreement of these two methods illustrates that the coupling effect has been effectively suppressed. Using Eq. (4) the parameters of α and Δω′ can be determined by a linear fit, shown as solid lines in Fig. 4. From these fits we determined that both modes have nearly the same Gilbert damping parameter, α = (6.5 ± 0.5) × 10⁻⁵, which is consistent with the Gilbert damping previously reported for YIG spheres.^30–32 The zero-frequency intercept determines an inhomogeneous broadening of Δω₀ = (0.120 ± 0.010) MHz for the (210), Δω₀ = (0.075 ± 0.005) MHz for the (220) mode, Δω₀ = (0.077 ± 0.010) MHz for the (320) and Δω₀ = (0.125 ± 0.015) MHz for the (500) mode, which is also in agreement with reported values.^15,33,34 We note that an analysis of the (110) mode is not presented here, since its large coupling effect cannot be suppressed within D < 5 mm allowed by our experiment. However, using a 0.3 mm diameter YIG sphere to reduce the coupling strength we observe the same behaviour described above, even for the (110) mode. Therefore the method we present for damping characterization is applicable to all modes in all size spheres, provided the experimental setup allows sufficient sample-stripline separation.

We have identified and separated coupling effects in a YIG-stripline system from damping contributions to magnetostatic modes in a YIG sphere. The former can be attributed to magnon energy losses that, through non-resonant magnon-stripline coupling, result in a well defined distance dependence of the resonance frequency and line width. By minimizing the coupling effect we use a broadband technique to determine the intrinsic Gilbert damping parameter α = 6.5 × 10⁻⁵ and an inhomogeneous broadening less than 0.125 MHz for the (210), (220), (320) and (500) modes using a broadband technique. We find that these modes have similar Gilbert damping, however the inhomogeneous contribution varies between modes. Our versatile method can be applied to explore the influence of the coupling effect on damping in other magnetic systems and may help in the design and adoption of coupled spin-photon systems for practical applications.

See supplementary material for the calculation and description of the spin wave magnetization profiles as well as additional line width data and analysis describing suppression of the radiation damping.

This work has been funded by NSERC and NSFC (11429401) grants (C.-M. H), the China Scholarship Council, and the Science and Technology Commission of Shanghai Municipality (STCSM No. 16ZR1445400). We would like to thank Y. T. Zhao for help with the mode profile calculations, and also acknowledge CMC Microsystems for providing equipment that facilitated this research. We also want to thank the reviewers for their valuable comments on our initial manuscript.