Controlling Selection Rules for Magnon Scattering in Nanomagnets by Spatial Symmetry Breaking

Nanomagnets are the building blocks of many existing and emergent spintronic technologies. The magnetization dynamics of nanomagnets is often dominated by nonlinear processes, which have been recently shown to have many surprising features and far-reaching implications for applications. Here we develop a theoretical framework uncovering the selection rules for multimagnon processes and discuss their underlying mechanisms. For its technological relevance, we focus on the degenerate three-magnon process in thin elliptical nanodisks to illustrate our findings. We parameterize the selection rules through a set of magnon interaction coefficients which we calculate using micromagnetic simulations. We postulate the selection rules and investigate how they are altered by perturbations that break the symmetry of static magnetization configuration and spatial spin-wave profiles and that can be realized by applying off-symmetry-axis or nonuniform magnetic fields. Our work provides the phenomenological understanding of the mechanics of magnon interaction as well as the formalism for determining the interaction coefficients from simulations and experimental data. Our results serve as a guide to analyze the magnon processes inherently present in spin-torque devices in order to boost their performance or to engineer a specific nonlinear response in a nanomagnet used in a neuromorphic or quantum magnonic application.

Figure 1

(a) The sample model is a thin elliptical disk in a bias magnetic field ${\mathbf{B}}_e$. (b) Bias field dependence of the first six spin-wave modes’ eigenfrequencies for ${\mathbf{B}}_e\parallel {\mathbf{e}}_x$. The dashed line shows the double frequency of the lowest mode (quasiuniform, $\nu =1$). (c) Spin-wave profiles of the spin-wave modes at $B_e=10$ mT.

Figure 2

Amplitudes of the first and second harmonics of magnetization oscillations $m_z(t)$ excited by a uniform microwave field $b_z=1$ mT at the eigenfrequency of mode 1. The bias field is applied parallel to the long axis of the disk. Vertical dashed lines indicate the fields of the three-magnon resonances $2{\omega }_1={\omega }_{\eta }$. The left inset shows the dependence of the second-harmonic amplitude on the first-harmonic amplitude at $B_e=241$ mT, which is the resonance point for the $1+1\to 5$ process. The dependence is parabolic at low oscillation amplitudes. The spatial map of magnetization oscillations at the second harmonic is shown; it corresponds to the profile of mode 5.

Figure 3

Dependence of three-magnon interaction efficiency $V_{11,\eta }$ on the magnetic field’s in-plane tilt. Note the $y$-axis break. Spin-wave profiles are shown for $\phi ={10}^{\circ }$.

Figure 4

Amplitudes of the first and second harmonic of magnetization oscillations $m_z(t)$ excited by microwave field $b_z=1$ mT at the eigenfrequency of mode 1. The bias field is applied with an in-plane tilt of $\phi ={10}^{\circ }$. Vertical dashed lines indicate fields of three-magnon resonance condition. Insets show spatial distribution of the magnetization oscillations at the second harmonic.

Figure 5

Dependence of three-magnon interaction coefficient $V_{11,\eta }$ on the out-of-plane tilt of the bias magnetic field.

Figure 6

Effect of the spatially nonuniform field with $B_x$-gradient along the $x$ axis. (a) Three-magnon interaction coefficients are shown as functions of the gradient. The bias field $B_x(x)$ varies linearly along the long axis of the disk, as shown in panel (b). The averaged field ${\overline{B}}_x$ is adjusted so that the three-magnon resonance condition is satisfied. (c) Mode profiles are calculated for $\mathrm{\Delta }B=20$ mT.

Figure 7

Effect of antisymmetric bias tilt field on three-magnon interaction coefficients. The field is ${\mathbf{B}}_e=B_x{\mathbf{e}}_x+\mathrm{\Delta }\mathbf{B}$, where $\mathrm{\Delta }\mathbf{B}$ varies linearly with the $x$ or $y$ axis: (a) $\mathrm{\Delta }\mathbf{B}=B_y(x){\mathbf{e}}_y$, (b) $\mathrm{\Delta }\mathbf{B}=B_y(y){\mathbf{e}}_y$, (c) $\mathrm{\Delta }\mathbf{B}=B_z(x){\mathbf{e}}_z$, (d) $\mathrm{\Delta }\mathbf{B}=B_z(y){\mathbf{e}}_z$. Field nonuniformity is characterized by the maximum tilt angle $\phi$ (a),(b) or $\theta$ (c),(d) at the edge of the sample. The coefficient $V_{11,3}$ is not shown since it is not affected by the fields shown and remains negligibly small.

Figure 8

(a) The magnetic disk (FL, free layer) experiences the stray field (b) from the auxiliary (CL, control layer) disk.