Introduction

The insulating ferrimagnet yttrium iron garnet (YIG) has recently attracted considerable interest in both spintronics1,2,3 and spin caloritronics4,5,6,7. One reason is that bulk and micrometer-thick YIG offers the smallest possible damping parameter α for spin waves (SWs) in the GHz frequency regime. The value of α amounts to about 5 × 10−58 and is two orders of magnitude smaller compared to the best metallic magnets9,10. At the same time the research field of magnonics has evolved rapidly based on nanostructured metals11. In magnonics one aims at the transmission, storage and processing of information using propagating and interfering spin waves (SWs)12. A small damping parameter α is key for magnonic crystals13 and spin wave logic devices based on interference effects14,15. Advanced concepts are based on coherent SW processing in cellular non-linear networks and magnonic holography memory14,16 creating also a demand for SWs that can propagate over large distances i.e., possessing large attenuation (decay) lengths. Nonreciprocal SW properties17,18,19 thoroughly studied in YIG and the metal Permalloy (Ni80Fe20) allow one to implement alternative ways for spin wave logic applications20,21. The relatively small attenuation length ld of only a few 10 μm found in metallic ferromagnets22 hinders however such developments. Magnonic nanodevices from the insulator YIG would be ground-breaking but have yet been elusive as relevant films with low damping needed to be micrometer thick12. Very recently, Liu et al.23 studied ferromagnetic resonance of YIG films with a thickness of 7 to 26 nm finding a promising damping constant α of 0.001. This value is already a factor of about three better than metallic ferromagnets used in magnonics. Hahn et al. showed that the damping constant is unaffected when nanostructuring ultra-thin YIG films24. Pirro et al.25 studied a 100 nm thick YIG thin film and observed a spin-wave decay length ld of 31 μm. Here we report propagating spin waves in a 20 nm-thick YIG film exhibiting a damping parameter of 2.3 × 10−4. We estimate a decay length of 580 μm from our data surpassing ld of metallic ferromagnets by an order of magnitude or even more. We explore spin wave modes with nine different wavelengths λ down to 560 nm. The group velocities vg are found to vary between 0.6 km/s and 1.2 km/s depending on the wave vector and applied magnetic field H. The nonreciprocity is found to increase with increasing wavevector k = 2π/λ and is thus promising in view of nanocale spin-wave logic devices. Our experimental work demonstrates that the thin-film magnetic insulator YIG allows one to overcome the damping problem for nanomagnonics and to boost further developments in magnonic circuit technology for information processing at GHz operational frequencies.

Results

Spin-wave propagation in 20-nm-thick YIG

Figure 1a shows the sketch of a thin-film YIG mesa with two integrated coplanar waveguides (CPWs).

Figure 1
figure 1

(a) Sketch of the YIG film with integrated CPWs. The microwave magnetic field hrf (circles) and the different spin-wave propagation directions are indicated. (b) Excitation spectrum I(k) obtained by Fourier transformation of irf (in-plane component). The inset shows the same chart with a logarithmic y-axis scale in order to highlight higher-order modes. (c) Spin wave spectrum of S12 (transmission), meaning spin waves were excited at CPW2 and detected at CPW1. The applied in-plane field amounted to −2 mT (θ = 0, parallel to CPW lines). In the inset we compare spectra S12 (red line) and S21 (black line) taken at −2 mT. (d) Color-coded spectra of spin-wave propagation data S12 as a function of H (θ = 0). The spectrum shown in (c) is extracted from the white dotted line in (d). The coercive (reversal) field is about 0.5 mT.

Full size image

To measure spin-wave propagation, we connected a vector network analyzer (VNA) to the CPWs that provided a microwave current irf at GHz frequencies f. The inhomogeneous microwave magnetic field hrf exerted a torque on the magnetic moments generating spin waves with a characteristic wave-vector distribution I(k)26. I(k) calculated from irf (see section ‘Methods’) is shown in Fig. 1b and contains a main excitation at k1 and several higher-order excitations at k2, k3, k4, k5 and k6. The wave vectors ki are perpendicular to the CPW (i = 1, 2, 3…). We measured both reflection (S11, S22) and transmission signals with spin waves propagating in the opposing directions, namely S12 and S2127. In Fig. 1c we show a spectrum S12 taken on a 20 nm thick YIG film subject to an in-plane field H of μ0H = −2 mT. The field was collinear with the CPWs and larger than the reversal field (see below). Thus Damon-Eshbach-type (DE) modes were excited with k being perpendicular to the magnetization M. To extract the transmitted spin-wave contribution we subtracted a spectrum S12 from all data that was taken at −100 mT under an angle θ of 0 deg. We depict the imaginary part of the transmission signal. As a function of frequency f we observe a series of oscillating signatures indicating spin-wave propagation between CPW1 and CPW2. The most prominent transmission signal is around 0.76 GHz, a second mode at about 1.05 GHz, a third mode at about 1.32 GHz, etc. These values of f are extracted from the extrema of envelope functions that connect local maxima of the oscillations labeled by k1, k2, …. With increasing eigenfrequency the signal strengths are found to diminish in a similar manner as expected from the distribution of excitations strengths I(k) shown in Fig. 1b. The highest k vector we detected out of the noise level is the mode k7 (not shown). When we vary the field strength H the eigenfrequencies f shift with H (Fig. 1d) as one would expect from the Landau-Lifshitz equation. The thin YIG film is a soft magnetic material exhibiting a reversal field of about 0.5 mT in Fig. 1d.

We find that signal strengths of the oscillating signatures are different for positive and negative fields. The discrepancy between +H and −H spectra depends on the frequency. A similar signal-strength discrepancy is found between spectra S12 and S21 taken at one-and-the-same magnetic field. This is highlighted in the inset of Fig. 1c where spectra S12 (red line) and S21 (black line) are compared for an applied field of −2 mT. We attribute the different signal strengths to a wave-vector dependent nonreciprocity that will be detailed below.

The modes observed in S12 of Fig. 1c and 1d suggest a one-to-one correspondence with different k vectors of the CPW excitation distribution I(k) shown in Fig. 1b. This is substantiated in Fig. 2a. Here we show the resonance frequencies f extracted at 5 mT from extrema of the relevant envelope functions as a function of wave vector k. For the square symbols we used Fig. 1b to attribute the eigenfrequencies f to specific values of k. For the two further resonances marked by k2′ and k4′ we used I(k) provided by the out-of-plane component of hrf (shown in the section ‘Methods’). The series of measured spin-wave frequencies follows a characteristic wave-vector dependence. Using the formalism of Kalinikos and Slavin28 we have calculated a spin-wave dispersion relation f(k) using the following parameters for the 20 nm thick YIG film: saturation magnetization Ms = 0.176 T taken from Ref. 29, exchange constant 0.192 × 10−6 erg/cm30, μ0H = 5 mT and θ = 0 deg. The theoretical curve (solid line) quantitatively remodels the measured data substantiating our mode allocation. We conclude that spin waves are excited by both in-plane and out-of-plane components of the CPW exhibiting wave vectors ranging from about 1.0 to 11.2 rad/μm. One-and-the-same CPW excites k values that vary by one order of magnitude. The shortest wavelength amounts to λ = 2π/k = 560 nm, coming close to the minimum wavelength of 365 nm that was reported recently for a nanostructured magnonic grating coupler based on metallic ferromagnets31. Note however that we use a conventional CPW to generate such short-wavelength SWs. We now investigate in detail the signal-strength discrepancies observed for positive and negative fields.

Figure 2
figure 2

(a) Calculated dispersion relation f(k) (line) compared to experimentally observed eigen-frequencies at 5 mT (symbols). Squares (circles) indicate eigenfrequencies attributed to wave vectors from k1 to k7 ( and ). (b) Spin wave spectra of mode k1 extracted from transmission data S21 (black squares) and S12 (red circles) taken at 5 mT. Amplitudes are different attributed to nonreciprocity. From the frequency separation Δf we calculate the group velocity. (c) Nonreciprocity parameter β as a function of applied field calculated for the mode k1 from transmission data of S12 and S21. The black squares represent the DE mode (θ = 0) and the open blue dots are for the BV mode (θ = 90 deg). (d) Spin wave spectra of k2 and k3 modes extracted from S21 (red circles) and S12 (black squares) transmission data at 5 mT. Modes attributed to the out-of-plane field component of hrf are marked by a prime. The inset enlarges the signal near .

Full size image

Analysis of propagating spin waves: nonreciprocity

By comparing both transmission data S12 at different fields (inset of Fig. 1c) as well as S12 and S21 taken at the same field (Fig. 2b) we clearly observe discrepancies in the mode intensities for both scenarios. Such variations are known as the nonreciprocity effect when exciting spin waves. Two different microscopic origins have been discussed for nonreciprocal properties of spin waves in YIG with k being perpendicular to M, i.e., the intrinsically nonreciprocal character of the DE surface mode12 and the excitation characteristics given by an antenna19,32. Using a very thin YIG film we do not consider the first reason to be relevant and attribute the experimentally observed nonreciprocity to the CPW consistent with Refs. 19, 32. For the following analysis we define and evaluate the nonreciprocity parameter β and decay length ld using the following set of reference33:

where ld is the same for counter-propagating spin waves in a given magnetic state. Following equation (1) the value β = 0.5 indicates reciprocal characteristics. The signal amplitudes auv are extracted from the peak-to-peak variation in the imaginary part of the spectra Suv where u, v = 1, 2 label the CPW and port of the VNA.

In Fig. 2c we show parameters β evaluated for the DE mode labelled with k1 (full symbols). For 1 mT ≤ μ0H ≤ 10 mT we find an almost constant β between about 0.55 and 0.58. These values close to 0.5 indicate a weak nonreciprocity. When the field H is applied at θ = 90 deg to rotate M, the CPWs excite spin-wave modes in the backward volume magnetostatic spin wave mode (BV mode) configuration. The propagation signal (not shown) is weaker due to the small torque by hrf acting on the spins. For such modes, we evaluate the nonreciprocity parameter β as shown by open symbols in Fig. 2c. Despite the larger error bar, β is found to be smaller compared to the DE mode and around 0.5. This suggests reciprocal behavior. The absence of nonreciprocity for the BV mode is expected as M does not possess a component being perpendicular to k32,34.

Spin waves with different wave vectors

Figure 2d shows spin wave transmission spectra for θ = 0 deg with an applied field of 5 mT for larger DE-mode wave vectors k. Compared to the k1 mode shown in Fig. 2b, the difference between S21 (red) and S12 (black) is more prominent and indicates a larger nonreciprocity effect. We find β to vary from mode k1 to . In Fig. 3a, we summarize β for modes with different vectors k observed at 5 mT. We find an increase of β from about 0.56 to 0.79 with increasing k. An analytical model for the spin wave nonreciprocity was given in Ref. 32,

where a± indicates the spin wave amplitude of positive (+) and negative (−) propagating direction. γ is the gyromagnetic ratio. Since YIG has a low saturation magnetization MS, a pronounced nonreciprocity can result from equation (2). Using the dispersion relation f(k) shown in Fig. 2a, we evaluate β in that we consider the relevant spin-precessional frequency f for the different k in equation (2). The evaluation is depicted by the solid line in Fig. 3a. The calculated dependency explains the more pronounced nonreciprocity observed for large k vectors. Thin YIG films thus seem to be particularly interesting for nonreciprocity-based logic20 using spin waves with large values of k, i.e., short wavelengths. Thin YIG thus offers such devices with nanoscale dimensions.

Figure 3
figure 3

(a) Nonreciprocity parameter β calculated for different vectors k from transmission data. Black squares show CPW excitation from k1 to k7 modes. Red dots show out-of-plane component CPW excitations. The black line is the calculated curve using equations (1) and (2) and f(k) from Fig. 2a. (b) Group velocity of DE spin waves with different vectors k. The line indicates the group velocity calculated from the dispersion relation in Fig. 2a. (c) Spin wave relaxation time calculated from equation (4) using experimental data. The arrow indicates the value τ0 calculated from the damping parameter α using equation (5). All data are for μ0H = 5 mT. Black squares show CPW excitation from k1 to k7 modes. Red dots show out-of-plane component CPW excitations.

Full size image

Spin-wave propagation velocities, linewidth and relaxation

Following Refs. 27, 35, the frequency separation Δf (indicated in Fig. 2b) reflects a change in the phase by 2π which is accumulated by spin waves propagating along s between the inner conductors of CPW1 and CPW2. The group velocity vg is calculated from Δf according to27

where s = 30 μm is the distance between two CPW signal lines as indicated in Fig. 1a. We find that the spin wave group velocity decreases from 1.16 to 0.59 km/s with increasing k. When we calculate the group velocity from the theoretical dispersion relation f(k) using we remodel well the observed large variation (line in Fig. 3b).

We observe a finite broadening δk for each excitation peak in Fig. 1b. E.g., mode k1 has a width δk of around 0.5 rad/μm. From this we estimate the effective linewidth δf = vgδk/2π (in reflection geometry)26 to be about 80 MHz at 5 mT (vg = 1 km/s). This value is close to what we observe in the experiment. It is much larger than the intrinsic linewidth of about 15 MHz calculated from the damping parameter α29. The linewidth δf of the reflection spectrum (see section ‘Methods’) thus does not provide the intrinsic linewidth but indicates the finite k distribution from CPW excitation.

We now investigate the spin wave decay length ld and relaxation time τ. To estimate the decay length of spin waves, we use the last line of equation (1) considering β from the previous section. We obtain a decay length of 580 μm for k1 with μ0H = 5 mT in DE mode. A large decay length is essential for spin wave logic device applications36. Using extracted decay lengths ld (Fig. 4a in section ‘Methods’) and the group velocities vg of Fig. 3b, we calculate spin relaxation times τ by

The results are shown in Fig. 3c. We find that the observed spin relaxation times vary slightly between 480 ns and 620 ns depending on k. These values are compared to the expected relaxation time τ0 estimated from α = 2.3 × 10−4 given in Ref. 29 according to

where f0 is the eigenfrequency of the uniform precession (k = 0)37. f0 is extracted from the calculated dispersion relation shown in Fig. 2a at k = 0. We get τ0 = 840 ns from equation (5). In the denominator of equation (5) we did not introduce the ellipticity factor PA (PA ≤ 1)37. Thereby we obtained the lower limit of τ0. The factor PA was not known as we did not measure field-dependent eigenfrequencies at k = 0 (uniform precession).

Figure 4
figure 4

(a) Decay lengths ld extracted at 5 mT from measured scattering parameters and the data of Fig. 3a according to equation (1). (b) Color-coded plot of spin-wave propagation data of S21 as a function of H (θ = 0). The spin waves are excited at CPW1 and detected at CPW2.

Full size image

Our best decay length of 580 μm is more than 18 times larger than the value of 31 μm reported for 100 nm thick YIG by Pirro et al.25. The authors found a similar velocity vg of ≈1.1 km/s. In their studies the authors deposited however Pt on top of the YIG and thereby increased the damping parameter α by a factor of 5.4. Compared to our best value they performed their experiment at a frequency f0 that was larger by a factor of 3.5 suggesting a further reduction in τ according to equation (5). The total reduction factor of 5.4 × 3.5 ≈ 19 thus explains the smaller value ld = vgτ = 31 μm found in their case. The values τ that we obtained experimentally from the spin-wave transmission between CPW1 and CPW2 were smaller than the expected τ0. We attribute the smaller τ to a reduced spin-wave attenuation length ld provoked partly by the metallic ground leads that reside between the two relevant signal lines for excitation and detection. The ground lines are expected to modify the spin-wave properties locally38 and thereby induce either partial reflection or additional damping. The experimentally extracted values τ of Fig. 3c are therefore expected to underestimate the real relaxation time. Using τ0 = 840 ns and vg = 1.16 km/s the theoretically expected decay length ld = vgτ amounts to 970 μm at k = 1 rad/μm. Transmission lines that in particular avoid the ground lines might be appropriate to observe directly the macroscopic ld of about 1 mm predicted by α = 2.3 × 10−4.

In conclusion, we have shown that SWs excited in nm-thick YIG films exhibit a pronounced wave-vector-dependent nonreciprocity and a record decay length for a thin-film magnonic material. The findings are expected to boost the coherent processing of spin waves in integrated Mach-Zehnder interferometers, magnetic cellular networks and magnonic holographic memories.

Methods

Sample fabrication

We grow thin YIG films on Gadolinium Gallium Garnet (111) substrates using Pulsed Laser Deposition. Growth temperature is set to be 650°C and oxygen pressure to 0.25 mbar. The YIG thickness is measured using X-ray reflectometry. RMS roughness is better than 0.5 nm. The Gilbert damping parameter is as low as 2.3 × 10−429. We used e-beam lithography to form a mesa with 45 degree edges as shown in Fig. 1a to avoid reflection of the spin waves. The width of the mesa was 300 μm. We used Ar milling to etch away the YIG film around the mesa area. After etching, we measured the step at the edge of the mesa being 40 nm and all the YIG film (20 nm thick) have been therefore etched away. Using again e-beam lithography, coplanar wave guides (CPWs) were integrated on the YIG mesa to perform broadband spin-wave spectroscopy (Fig. 4b). Figure 5 shows a Scanning Electron Microscope (SEM) image of two CPWs with a YIG thin film underneath. These CPWs were designed identically having 2.1 μm wide signal lines and ground lines and 1.4 μm wide gaps between them. The distance s between two central conductors of adjacent CPWs was 30 μm.

Figure 5
figure 5

SEM image of the integrated CPWs on the YIG thin film.

The parameter s indicates the propagation distance being 30 μm. The propagation directions of S12 and S21 are indicated. The angle θ of the applied field is defined.

Full size image

Spin-wave absorption spectrum

We measured absorption spectra in reflection configuration using the scattering parameters S11 or S22. Figure 6 shows a typical spectrum S11. The most prominent signal originates from the main excitation of the CPW, i.e. mode k1. The two weaker ones result from k2 and k3. The linewidth of the signal k1 is mainly caused by the broadening δk seen in I(k) of Fig. 1b near k1. The linewidth (full width at half maximum) δf of the prominent resonance amounts to about 60 MHz in Fig. 6.

Figure 6
figure 6

Spectrum S11 taken in reflection configuration using one-and-the-same CPW at 30 mT (θ = 0).

The imaginary part of the signal is shown.

Full size image

Spin wave propagation

Figure 4b shows the spin wave transmission data for S21. We used measured scattering parameters and evaluated the decay length ld following equation (1) (Fig. 4a). These data entered the calculation of τ (Fig. 3c) according to equation (4).

The wave vector distribution I(k) for the in-plane magnetic field component of hrf was extracted following Refs. 39, 40. The authors assumed a homogeneous current density in the signal line and counter-flowing shielding currents in the two ground lines of half the amplitude. I(k) was extracted from the Fourier transformation of this current profile. In our work the microwave frequencies were relatively small and below 2.5 GHz. The Au leads were only 2.1 μm wide, i.e., on the order of the skin depth at small frequencies. As a consequence we assumed the skin effect to be weak40 and a homogeneous current density to be a reasonable assumption. The authors of Refs. 39, 40 did not provide a formalism for out-of-plane components. Hence for I(k′) representing the out-of-plane component (Fig. 7) we considered the Fourier transformation of a magnetic field distribution provided by the electromagnetic simulation code CST (Computer Simulation Technology for 3-D electromagnetic simulation) used in Ref. 40. We found that some excitation peaks (here at and ) did not coincide with the calculated in-plane wave vectors ki. Considering that the out-of-plane fields yield a smaller excitation strength compared to in-plane components39,40 we interpreted the two significantly smaller signals lying between features attributed to ki in Fig. 2d as excitations at and . The discrepancy for k1 between the approximation and electromagnetic simulations is reported to be about 10–15% for a small inner conductor width of the CPW39,40. Further simulations suggest that the maximum discrepancy occurs for k3 and is up to about 37%. To quantify wave vectors in the experiment time and spatially resolved magneto-optical microscopy would be relevant22.

Figure 7
figure 7

Red vertical arrows indicate in-plane k vectors that are relevant for excitations k1 to k6 and have been calculated following Refs. 39, 40.

The two tilted blue arrows indicate an appreciable excitation strength at wave vectors and provided by the simulated out-of-plane component of the radiofrequency field hrf.

Full size image