Spin-Wave Dispersion Measurement by Variable-Gap Propagating Spin-Wave Spectroscopy

Knowledge of the spin-wave dispersion relation is a prerequisite for the explanation of many magnonic phenomena as well as for the practical design of magnonic devices. Spin-wave dispersion measurement by established optical techniques such as Brillouin light scattering or the magneto-optical Kerr effect at ultralow temperatures is often forbiddingly complicated. By contrast, microwave spectroscopy can be used at all temperatures but it usually lacks spatial and wave-number resolution. Here we develop a variable-gap-propagating-spin-wave-spectroscopy (VGPSWS) method for the deduction of the dispersion relation of spin waves in a wide frequency and wave-number range. The method is based on the phase-resolved analysis of the spin-wave transmission between two antennas with variable spacing, in conjunction with theoretical data treatment. We validate the method for in-plane magnetized $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ and yttrium iron garnet thin films in $\mathbf{k}\perp \mathbf{B}$ and $\mathbf{k}\parallel \mathbf{B}$ geometries by deducing the full set of material and spin-wave parameters, including spin-wave dispersion, hybridization of the fundamental mode with the higher-order perpendicular standing spin-wave modes, and surface spin pinning. The compatibility of microwaves with low temperatures makes this approach attractive for cryogenic magnonics at the nanoscale.

Figure 1

(a) Schematics of the PSWS experiment using a pair of stripline antennas connected to the VNA by microwave probes. (b) Schematic geometry of the three antenna types used in the experiments. GND, ground.

Figure 2

The outcomes of the PSWS measurement for a 30-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ layer in the $\mathbf{k}\perp \mathbf{B}$ geometry and for a gap width of $1.8\mu \mathrm{m}$. Real (a) and imaginary (b) parts of $S_{21}(B,f)$ as measured by the VNA. Real (c) and imaginary (d) parts of $\mathrm{\Delta }{S}_{21}(B,f)$, obtained by subtraction of the nonmagnetic background from $S_{21}(B,f)$. The measurement reveals the nonreciprocity of spin-wave propagation for the $\mathbf{k}\perp \mathbf{B}$ geometry, which is reflected in the larger signal in $+B$ fields than in $-B$ fields. (e) Real and imaginary parts of $\mathrm{\Delta }{S}_{21}(f)$ and (f\!) the unwrapped phase of $\mathrm{\Delta }{S}_{21}(f)$ at fixed magnetic field $B=20\mathrm{mT}$.

Figure 3

Extraction of the dispersion relation from the outcomes of variable-gap-propagating-spin-wave-spectroscopy (VGPSWS) experiment on 30-nm $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ thin film at $B=20\mathrm{mT}$ in the $\mathbf{k}\perp \mathbf{B}$ geometry. (a) The unwrapped $\mathrm{\Delta }{S}_{21}$ phases measured over gap widths ranging from $0.9\mu \mathrm{m}$ (the lowest line) to $2.9\mu \mathrm{m}$ (the steepest line) with a step of 200 nm. (b) The representative fits of the phase at selected frequencies versus gap size where the slope of the fit yields the desired $k$-vector at that frequency. (c) The dispersion relation extracted for all frequencies within the range with sufficient VG PSWS signal (red line with filled-area error bars) compared with dispersion obtained by phase-resolved BLS (blue squares with error bars). (d) The phase measurements obtained by VGPSWS (red) and phase-resolved BLS (blue) at frequency $f=11.5\mathrm{GHz}$.

Figure 4

Examples of VGPSWS dispersion measurements on three different samples. Each panel shows the measured dispersion (red points) compared with the analytical dispersion from the Kalinikos-Slavin model (black lines) together with the normalized excitation spectrum of the antenna used for the measurement (blue lines). For clarity, the data for $\mathbf{k}\perp \mathbf{B}$ are plotted in the domain of positive wave numbers $k$ and the $\mathbf{k}\parallel \mathbf{B}$ data are presented with negative $k$. Panel (a) shows 30-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ measured by the stripline antenna. Panel (b) shows 100-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ measured by the CPW (in negative $k$) and by the stripline antenna (in positive $k$). Panel (c) shows 100-nm-thick YIG measured by the GS antenna. For the Kalinikos-Slavin model we fit the following values of the parameters ($t$ stands for thickness of the layer): for (a) 30-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$, $M_s=1230\mathrm{kA/m}$ (${\mu }_0{M}_s=1.54\mathrm{T}$), $\gamma /2\pi =30.5\mathrm{GHz/T}$, $t=29.0\mathrm{nm}$; (b) for 100-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$, $M_s=1310\mathrm{kA/m}$ (${\mu }_0{M}_s=1.64\mathrm{T}$), $\gamma /2\pi =30.1\mathrm{GHz/T}$; (c) for 100 nm YIG, $M_s=133\mathrm{kA/m}$ (${\mu }_0{M}_s=0.166\mathrm{T}$), $\gamma /2\pi =28.3\mathrm{GHz/T}$. In the fitting procedure, we fix the exchange stiffness $A_{\mathrm{ex}}$ to $14\mathrm{pJ/m}$ for $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ films and to $3.6\mathrm{pJ/m}$ for the YIG film.

Figure 5

(a) Measured dispersion relation (red points) for a 100-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ layer at 260 mT exhibiting hybridized spin-wave modes, visible mainly at the 02 crossing. The dashed lines represent a numerical model of dispersion with asymmetric surface pinning conditions with parameters $K_S=0$ and $1600\mu {\mathrm{J/m}}^2$ on the top face and the bottom face of the layer, respectively. (b) Corresponding magnitude of the transmission parameter $\mathrm{\Delta }{S}_{21}$ exhibiting dips at the crossing positions ($|\mathrm{\Delta }{S}_{21}|$) data shown for a gap width of $1.8\mu \mathrm{m}$. (c)–(e) Details of the 01, 02, and 03 crossings, respectively. Dashed blue lines correspond to the numerical model with symmetric surface pinning conditions ($K_S=700\mu {\mathrm{J/m}}^2$ on both faces).

Figure 6

(a)–(c),(e) Determination of decay length and lifetime from VGPSWS measurements of 30-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ film and and (d) impact of the number of gaps on the quality of the result. (a) The blue line shows the spin-wave decay lengths obtained directly from exponential fits, the orange line shows the decay length calculated by multiplication of the lifetime and the group velocity, and the green line is calculated from the Kalinikos-Slavin model. (b) Representative exponential fits of the $\mathrm{\Delta }{S}_{21}$ magnitude that were used to obtain the spin-wave propagation length. (c) Spin-wave lifetime extracted from the derivative of the field-dependent dispersion with use of $\alpha =0.075$ (orange circles), calculated by our dividing the propagation length (from the $\mathrm{\Delta }{S}_{21}$ magnitude) by the numerically-calculated group velocity (blue diamonds) and calculated from the Kalinikos-Slavin model (green line). (d) Mean frequency difference of the dispersion obtained from the reduced number of gap widths compared with the analytical fit of the dispersion obtained from the complete set of 11 measured points. Blue points represent different combinations of gaps and the red area represents the difference spread. (e) Spin-wave caustics caused by the finite length of the antenna, measured on 100-nm-thick $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ film by BLS with an excitation frequency of 11 GHz in an external magnetic field of 26 mT. Caustics can destroy the phase coherence of propagating spin waves. All data presented are collected at propagation distances (gap widths) between $0.9$ and $2.9\mu \mathrm{m}$ (green region) to avoid the negative effects of the formation of caustics. GND, ground.