Soft-X-ray momentum microscopy of nonlinear magnon interactions below 100-nm wavelength

TL;DR

The paper introduces Magnon Momentum Microscopy (MMM), a soft-X-ray scattering technique that images magnon populations directly in two-dimensional momentum space with high sensitivity, enabling access to sub-100-nm spin waves in YIG. By mapping 2D magnon distributions, MMM reveals nonlinear spin-wave interactions, including a four-magnon parametric mechanism that redistributes energy omnidirectionally into an elliptical ring in momentum space. The authors derive a spin-wave model (SWM) to explain the observed threshold for parametric excitation and validate it against experimental dispersion data, demonstrating agreement for fundamental and harmonic spin-wave modes. MMM's combination of element specificity, bulk sensitivity, and 2D momentum-space access promises a versatile platform for exploring short-wavelength and nonlinear magnonics, with potential extensions to time-resolved studies and other magnetic materials and geometries.

Abstract

Magnons are quantised collective excitations of long-range ordered spins. At nanometre wavelengths, exchange interactions increasingly govern their dynamics, giving rise to a largely unexplored regime of couplings between magnons and other quasiparticles. Yet, detecting such short-wavelength spin waves has remained a key experimental challenge. Here, we introduce Magnon Momentum Microscopy (MMM) -- a quasi-elastic, resonant magnetic soft-X-ray scattering technique that directly images magnon populations across two-dimensional momentum space. Owing to its remarkable sensitivity, MMM can capture nonlinear magnon-magnon interactions over large regions of the dispersion plane. Applying MMM to the prototypical magnonic material yttrium iron garnet (YIG), we uncover a rich variety of previously unobserved nonlinear magnon interactions. With its element specificity, bulk sensitivity, as well as intrinsic access to nanometre-scale wavelengths without frequency limitation, soft-X-ray MMM establishes a powerful and versatile platform for exploring short-wavelength and nonlinear magnonics.

Figures (8)

• Figure 1: Soft-X-ray magnon momentum microscopy.a, Schematics of the soft-X-ray scattering geometry. Plane-wave magnons in DE configuration ($\mu_0 \mathbf{H} \perp \mathbf{k}_\mathrm{SW}$) propagate away from the spin-wave emitter, as indicated by the bluish, out-of-plane magnetisation contrast. Quasi-elastic, resonant magnetic scattering with the magnons results in $+1$st and $-1$st-order diffraction peaks on the detector, revealing the magnon wave vector, $\mathbf{k}_\mathrm{SW}$, directly in momentum space. b, SEM image of the spin-wave emitter, consisting of a CPW and thin permalloy stripes, forming the GC, as indicated by the blue areas. Scalebar, 2.
• Figure 2: Imaging nonlinear magnon processes in momentum space.a, Elliptical dispersion ring resulting from nonlinear magnon--magnon scattering of the directly excited DE spin waves into modes of arbitrary propagation direction at $f_\mathrm{RF}=\qty{9.00}{GHz}$. b, Higher harmonics and mode redistribution at $f_\mathrm{RF}=\qty{2.38}{GHz}$; red dashed lines indicate theoretical dispersion curves for $f_\mathrm{SW} = n\, f_\mathrm{RF}$. c, Calculated RF critical field for nonlinear scattering from the SWM, evaluated for the excitation power corresponding to -5.0dBm used in a and b. d--f, Power-dependent transition at $f_\mathrm{RF}=\qty{8.84}{GHz}$ (bias field $\mu_0 H=\qty{20}{mT}$ deviating from other shown measurements) from linear excitation (d) to a nonlinearly excited elliptical ring (e) and higher/fractional harmonics as indicated (f). Red dashed lines represent theoretical dispersion iso-frequency curves for the most prominent harmonics. Panels a, b, d--f share the intensity scale.
• Figure 3: Extracted magnon dispersions.a, Dispersion along $q_{\parallel}$ and $q_{\perp}$, corresponding to the DE and the BWV spin-wave modes, respectively. The BWV mode arises from nonlinear scattering. Red dashed lines indicate theoretical dispersions for $f_{\mathrm{SW}} = n\,f_{\mathrm{RF}}$ with $n=1, 2, 3$. b, Zoom into the DE dispersion at high frequencies (region marked in a), showing intensity maxima and discrete wave vector jumps due to the grating coupler geometry. Lines are guides to the eye.
• Figure E1: Normalised intensity of the $4G$-scattering peak at 9.00GHz for increasing excitation power, $P_\mathrm{RF}$, benchmarking the sensitivity of the MMM technique. The SNR enables detection of the scattering signal down to $P_\mathrm{RF} = \qty{-34}{dBm}$ at reasonable acquisition times of 30s for a single MMM image. For $P_\mathrm{RF} > \qty{-20}{dBm}$, the integration time is reduced to only 5s. The signal follows the expected exponential behaviour (solid blue line).
• Figure E2: Calculated magnon excitation of the CPW and GC. a Excitation efficiency of the CPW and GC. The calculation is performed by Fourier transformation of the simulated RF in-plane Oersted fields upon microwave excitation. The CPW efficiently transfers a wave vector $k_1 = \qty{0.85}{\per\micro\meter}$ to the YIG film Baumgaertl2020. The efficiently transferred larger wave vectors mainly result from the GC pattern. Its natural wave vector is $G = 2\pi/a = \qty{15.7}{\per\micro\meter}$, along with the higher orders $2G$, $3G$, and $4G$. b Magnetic excitation field distribution below the CPW (red areas) and GC (grey areas).