Magnetoelastic conversion in integrated YIG nanostructures

Abstract

Motivated by the recent proposal of two-step transduction from microwave to optical domain using magnetic and elastic intermediate stages arXiv:2205.05088, we consider the coupling between resonant magnetic and elastic modes within a simple axially-symmetric nanodevice designed to host high-quality-factor acoustic modes: A suspended YIG ring structure supported by a central stem, fabricated from a continuous single-crystal film. We study the modes of the system with our custom finite element solvers. We identify the lowest order ``breathing'' mode of a magnetic vortex and the lowest order elastic breathing mode as having the largest mode overlap. For this pair of modes, the external out-of-plane magnetic bias field is critical for bringing them into resonance; however, we show that at the same time it also affects the strength of the coupling. To counteract this, we optimize the radius of the ring at fixed thickness. For the 100 nm-thick film the resonant coupling is maximized at $g/2π= 8\text{MHz}$ at $R\approx1.7μ\text{m}$, indicating that the overlap integral approaches the idealized limit assumed in previous order-of-magnitude estimates. Our results pave the way for the design of tunable frequency-conversion devices based on magnetoelastics.

Figures (9)

• Figure 1: Scanning Electron Microscope (SEM) image of a suspended magnetic YIG (yttrium iron garnet) ring mounted on a supporting disk stem. The structure is a single crystal oriented in the (111) direction. Image courtesy of the G. Schmidt group from the University of Halle Heyroth2019. The top inset presents a schematic cross-sectional view illustrating the nanostructure’s profile used in the simulations by finite element methods.
• Figure 2: EIGENELAAxi FEM simulation of the $n_r=0$ elastic breathing mode (a) and the transverse mode closest to it in frequency (b) in the structure from Fig. \ref{['fig:structure']}. The structural central attachment stem causes the longitudinal oscillation to have a significant out-of-plane component compared to the idealized planar structure. The cross-section of the (exaggerated) deformation is compared to the non-deformed profile (outlined with the dashed line). The color represents a hydrostatic strain $\mathop{\mathrm{Tr}}\nolimits \varepsilon\equiv\nabla \mathbf{u}$ due to the local compression and expansion. Globally the structure is depicted in the extension phase of the breathing oscillation. For a fundamental purely longitudinal mode, we would only have tensile stress in this phase, but we also see small pockets of compressive stress because the longitudinal mode is coupled to the transverse mode.
• Figure 3: Frequencies of the mechanical modes of the structure shown in Fig. \ref{['fig:structure']}. Torsion $u_\varphi\ne0$ modes have been omitted. The color gradient in green and red denotes the ratio between the average in- and out-of-plane displacements $\sqrt{\int u_r^\ast u_r dV} / \sqrt{\int u_z^\ast u_z dV}$; the ratio is used to describe the modes in terms of familiar longitudinal and transverse modes observed in planar geometries. The dashed line shows the analytical behavior of the fundamental $n_r=1$ longitudinal mode for a disk with a 300 diameter hole in the middle for comparison.
• Figure 4: Simulated evolution of the ferromagnetic resonance (FMR) modes as a function of the external magnetic field $B$ applied perpendicularly to the plane of the disk, computed using a custom-developed axisymmetric finite element eigensolver. The device corresponds to the suspended YIG disk shown in Fig. \ref{['fig:structure']}. Modes obtained via FEM are classified based on their excitation symmetry: modes predominantly coupled to an in-plane AC field (azimuthal symmetry $m = 1$) are shown in green, while those coupled to an out-of-plane AC field ($m = 0$) are shown in red. The line opacity shows the magnitude of the total dynamic dipolar moment of the mode $\left|\int dV\, \bm{m}^{(d)}\right|$, which is directly proportional to the coupling strength of a given mode to a microwave cavity; modes that lose their coupling are continued with dashed lines. As the external field increases, the magnetic texture continuously deforms from the vortex state through the cone state and into the saturated uniform state. The common mode softening point indicates the transition field $B_s \approx \qty{1650}{Oe}$, at which the magnetization switches from a cone state to a uniform state. Notably, for certain disk radii, such as the one presented here, an additional transition is observed at a slightly lower field than $B_s$, manifested by the softening of only a limited number of modes. This transition corresponds to an independent saturation of the stem and affects modes that are effectively pinned to it.
• Figure 5: Calculated spatial distribution of the vortex breathing mode intensity $\left[\mathbf{m}_0\times\mathbf{m}^{(d)\ast}\right]\mathbf{m}^{(d)}$ using the SLLGAxi FEM solver with external field $B=0.1$ T. The stem defines three regions, viz. the suspended ring, the joint, and the circular base of the structure.