Imaging Harmonic Generation of Magnons

Abstract

This work combines theory and experiment to examine the mechanisms underlying the harmonic generation of magnons. We develop a nonlinear spin-wave framework that is directly analogous to harmonic generation in nonlinear optics, and combine it with scanning nitrogen-vacancy (NV) center magnetometry to image and quantify magnonic harmonic generation in a Ni$_{81}$Fe$_{19}$/Pt microstripe. Within this framework, the harmonic response arises from nonlinear magnetization dynamics localized at strongly inhomogeneous textures, such as the sample edges and domain walls, that act as anharmonic confining potentials. Scanning probe imaging confirms that the harmonic response is correspondingly nonuniform and concentrated near the sample edges. We measure an expected nonlinear power-law scaling, a systematic shift toward larger wavevector excitations at higher harmonic order, and a spin-selective response indicative of an increasingly chiral harmonic stray field. These results provide a microscopic understanding of magnonic harmonic generation and highlight its potential for engineering nonlinear functionality in magnonic systems.

Figures (10)

• Figure 1: (a) Schematic of the scanning nitrogen-vacancy (NV) microscope. A single NV center near the apex of a diamond probe detects stray magnetic fields from the sample while raster scanning at a distance $d$. The sample is illustrated with a simulated in-plane magnetization texture. (b) Optical microscope image of the Ni$_{81}$Fe$_{19}$ (5 nm) / Pt (5 nm) device investigated in this work (red box). (c) Optically detected magnetic resonance (ODMR) spectra showing the NV spin resonances near $D \approx 2.87$ GHz and harmonic signals appearing when $n f_0 \approx D$. The low and high frequency spectra are acquired in separate measurements and are stitched together here for clarity.
• Figure 2: Results of a wide-field scanning NV measurement. (a) Scanning NV ODMR image showing the stray magnetic field produced by the permalloy stripe under a static magnetic field of $\sim 25~\mathrm{G}$. (b,c) Scanning ODMR measurements of the amplitude of the third-harmonic signal for the $\ket{0} \leftrightarrow \ket{-1}$ and $\ket{0} \leftrightarrow \ket{+1}$ transitions, respectively. The scale bar in (a) is $5~\mu\mathrm{m}$ and applies to all images.
• Figure 3: (a,b) Mean harmonic contrast averaged over the scan region as a function of RMS drive voltage $V_{\mathrm{rms}}$ for harmonic orders $n=3$ (blue), $n=4$ (orange), and $n=5$ (green), measured using the $\ket{0}\leftrightarrow\ket{-1}$ (top) and $\ket{0}\leftrightarrow\ket{+1}$ (bottom) NV spin transitions. Dashed lines indicate fixed-exponent power-law trends with $n=3$, 4, and 5, shown as guides to the eye, while solid lines show power-law fits with the exponent treated as a free parameter using Equation \ref{['eq:powerlawfit']}. (c,d) Mean harmonic contrast as a function of the NV-to-sample separation $d$, where the colors correspond to the same harmonic orders. Solid lines are fits using Equation \ref{['eq:heightfit']}.
• Figure 4: (a) Harmonic contrast maps measured on the $\ket{0} \leftrightarrow \ket{-1}$ (top row) and $\ket{0} \leftrightarrow \ket{+1}$ (middle row) NV spin transitions for harmonic orders $n=3,4,5$. Data are acquired over a $2.5 \times 2.5~\mu\text{m}^2$ region at an excitation power of $16~\mathrm{mW}$ (left) and a $2.0 \times 2.0~\mu\text{m}^2$ region at $4~\mathrm{mW}$ (right). The bottom row shows the pixel-wise difference between the two spin transitions, emphasizing the spin-selective (chiral) nature of the stray magnetic field. Scale bars indicate $1~\mu$m (note that the scan areas differ between the two data sets). (b) Active area and chiral polarization $P_n$ extracted from the harmonic contrast maps, showing a systematic increase in chiral polarization with harmonic order.
• Figure 5: Spin configurations and corresponding effective magnetic potentials for two representative magnetization textures. (a) Néel-type domain wall described by the Walker profile. (b) Hard boundary geometry, where the domain-wall center is effectively pinned at the boundary. (c,d) Corresponding effective magnetic potentials $\left(1 - 2\,\mathrm{sech}^2\eta\right)$ appearing in Equation \ref{['eq:SchrEq']}. In both geometries, the spatial variation of the magnetization produces a localized potential well that can confine spin-wave excitations.