Perpendicular magnetic anisotropy in thin films enables extraordinary spin-wave phenomena: anti-Larmor precession, negative reflection and refraction, multi-reflection and multi-refraction

TL;DR

Perpendicular magnetic anisotropy (PMA) fundamentally alters spin-wave dynamics in thin films at low fields, producing sombrero-like isotropic minima in ultrathin layers and cowboy-hat anisotropy in thicker layers. This dispersion topology yields nontrivial interface phenomena, including negative reflection, bireflection, and tri-refraction, while also enabling anti-Larmor precession near the dispersion minimum in thicker films. The authors develop a unified theoretical framework and perform FEM and micromagnetic simulations across diverse PMA materials, demonstrating the universality and experimental accessibility of these effects, with garnet-based systems offering favorable wavelengths and damping for verification. Overall, the work reveals new wave-physics phenomena in PMA systems and points to opportunities to engineer dispersion and explore nonlinear spin-wave dynamics beyond magnonics.

Abstract

We present a theoretical and numerical investigation of the role of perpendicular magnetic anisotropy (PMA) in shaping spin-wave (SW) dynamics under low magnetic fields in thin and ultrathin magnetic films. PMA introduces an in-plane torque that counteracts exchange, dipolar, and Zeeman contributions, fundamentally modifying SW dispersion and inducing a local minimum that, under specific conditions, becomes the lowest frequency across all geometric configurations. This results in a sombrero-shaped dispersion in ultrathin films and a cowboy-hat-like shape in thicker films, where dipolar interactions dominate. Using isofrequency contour (IFC) analysis, we demonstrate that these PMA-induced dispersion shapes enable nontrivial wave phenomena unprecedented in uniform media: bireflection and negative reflection in ultrathin films, and trireflection in thicker films--where a single incident beam splits into three reflected components, two with negative angles. Most remarkably, we predict and demonstrate tri-refraction, where one incident beam generates three refracted beams with two exhibiting negative refraction angles. We further show anti-Larmor precession of magnetization near the dispersion minimum in thicker films, arising from the interplay between PMA-induced and dipolar torques. Systematic simulations across diverse material systems--metallic films, ferrimagnetic garnets, hybrid structures, and multilayers--confirm the universal nature of these phenomena in any PMA system supporting stripe domain transitions. These results open new opportunities to explore wave phenomena beyond magnonics.

Figures (6)

• Figure 1: (a) and (d) SW dispersion for a $2$-nm-thick CoFeB film magnetized along the $y$-axis with $Q = 0$ (a) and $Q = 1.2$ (d): $\omega = \sqrt{\omega_x \omega_z}$ (black), $\omega_x$ (red, equation (\ref{['eq:wx']})), and $\omega_z$ (green, equation (\ref{['eq:wz']})). (b) Dispersion for external magnetic fields of $220$ mT (lower pair) and $240$ mT (upper pair) in both BV (red) and DE (dashed black) geometries. (e) Dispersion as a function of $k_x$ and $k_y$ for ultrathin film uniformly magnetized along the $x$-axis; red and black lines show the cross-sections for $k_y = 0$ (BV configuration) and $k_x = 0$ (DE configuration), with circles indicating examples of IFCs for different frequencies. (c) IFC at $1.5$ GHz and the magnetic field of value 380 mT applied along the $x$-axis computed using mumax3 showing incident and reflected wavevectors $\mathbf{k_\mathrm{i}}$, $\mathbf{k_\mathrm{r1}}$, and $\mathbf{k}_\mathrm{r2}$; the group velocity direction for $\mathbf{k_\mathrm{r2}}$ is indicated by the white arrow. (f) Micromagnetic simulation of the reflection of an SW beam in a 2 nm CoFeB film, incident at a $25^\circ$ angle on an interface separating regions with different $Q$ values. Wavevectors (parallel to phase velocities) and group velocity directions from (e) are marked by arrows.
• Figure 2: (a) Dispersion of SWs propagating along the $x$-axis, originating from the first (bold solid lines) and second (narrow dashed lines) bands of a 20 nm CoFeB film with $Q = 0.6$, under a bias magnetic field $H$ applied along the $y$-axis with values of 250 mT (green), 500 mT (red), and 800 mT (orange). Letters (b)-(j) correspond to the $k$ and field values marked by dots and stars in (a). The thick magenta line for 250 mT shows the range where anti-Larmor precession occurs. (b)-(d) $x$-component of the torque ($\tau_x$) across the film thickness for wavevectors marked by stars in (a), corresponding to wavevectors and fields of $80$ rad/µ m at 800 mT, $93$ rad/µ m at 500 mT, and $106$ rad/µ m at 250 mT. The dashed blue, green, red, and orange lines show the exchange, dipolar, PMA, and Zeeman components of $\tau_x$, respectively, with the solid black line indicating the effective torque. (e)-(j) Precession orbits of the magnetic moments at $z = 0, 5, 10, 15,$ and $20$ nm. Colors represent phase, while black and red arrows indicate Larmor and anti-Larmor precession, respectively. Panel (e) corresponds to 800 mT, panel (f) to 500 mT, and panels (g-j) to 250 mT. $k$ values are shown at the top of each subplot. See SI for the animated version of (e)-(j).
• Figure 3: (a) Dispersion $f(k_x, k_y)$ for a 20 nm CoFeB film magnetized along the $x$-axis ($Q = 0.6$, $H_0 = 250$ mT), calculated using the finite element method. The red and black curves show cross-sections for $k_x = 0$ (BV) and $k_y = 0$ (DE) configurations. Contours at the bottom represent IFCs at different frequencies, with the $9$ GHz contour highlighted in cyan. (b) IFC at 9 GHz (cyan dashed lines), extracted from (a). The colormap in the background (representing the 2D Fourier transform of (c)) shows small bright spots (see SI for a larger version of the colormap). (c) Reflection of an SW beam in a 20 nm CoFeB film, incident at $24.5^\circ$ from the $Q = 0.6$ region to the $Q = 0$ region at a sharp interface. The semi-transparent white area marks the excitation region. Wavevector directions in (b) and (c) are shown by green ($\mathbf{k_\mathrm{i}}$, $\mathbf{k_\mathrm{r1}}$), cyan ($\mathbf{k_\mathrm{r2}}$), and red ($\mathbf{k_\mathrm{r3}}$) arrows, with white arrows indicating group velocity directions for nonspecular reflections associated to $\mathbf{k_\mathrm{r2}}$ and $\mathbf{k_\mathrm{r3}}$.
• Figure 4: IFCs extracted from the dispersion relation presented in Figure3 for (a) $6.1$ GHz, (b) $7.7$ GHz, (c) $8.3$ GHz, (d) $8.5$ GHz, (e) $11$ GHz, (f) $13$ GHz, (g) $16$ GHz, and (h) $17$ GHz calculated for SWs in a 20 nm-thin CoFeB film with PMA ($Q=0.6$, $\mu_0 H_0=250$ mT). The arrow in (a) shows the direction of the external magnetic field $H_0$ in all the subsequent plots. An animated version of this plot can be found in the SI.
• Figure 5: Reflection and refraction of a SW beam at frequency 9 GHz in a 20 nm thick CoFeB film with $Q=0.6$, incident at $24.5^\circ$ at a 10-nm-deep and 20-nm-wide groove (see schematic representation at top of (b)). The magnetic field $H_0$ of 250 mT is applied along the interface ($y$-axis). (b) Steady-state spatial distribution of $|m_z|$ showing one incident beam with wave vector $\mathbf{k}_\mathrm{i}$, three reflected beams ($r1$, $r2$, $r3$) with wave vectors $\mathbf{k}_\mathrm{r1}$, $\mathbf{k}_\mathrm{r2}$, $\mathbf{k}_\mathrm{r3}$ (scenario discussed in Fig. \ref{['fig:optics_thick']}), and three transmitted (refracted) beams ($t1$, $t2$, $t3$) with wave vectors $\mathbf{k}_\mathrm{t1}$, $\mathbf{k}_\mathrm{t2}$, and $\mathbf{k}_\mathrm{t3}$. The groove interface is marked by a white line at $x=0$. (a, c)$|m_z|$ profiles as a function of $y$ at $x = -3$ μm (a, incident beam region with beams labeled $r1$, $r2$, $r3$) and at $x = 3$ μm (c, transmitted beam region with beams labeled $t1$, $t2$, $t3$). (d, e) Two-dimensional Fourier transforms of the magnetization distribution from panel b for the regions where $x < 0$ (d, incident/reflected region) and $x > 0$ (e, transmitted region). Cyan contours represent isofrequency curves at 9 GHz. Arrows indicate the wave vectors of the incident beam ($\mathbf{k}_\mathrm{i}$) and transmitted beams ($\mathbf{k}_\mathrm{t1}$, $\mathbf{k}_\mathrm{t2}$, $\mathbf{k}_\mathrm{t3}$). White arrows show the group velocity directions $\mathbf{v}_\mathrm{t2}$ and $\mathbf{v}_\mathrm{t3}$ (normal to the isofrequency contours at the corresponding wave vector values), indicating the directions of energy transfer that in these cases do not coincide with the corresponding wavevector direction.