Identification and selection rules of the spin-wave eigenmodes in a normally magnetized nanopillar

We report on a spectroscopic study of the spin-wave eigenmodes inside an individual normally magnetized two-layer circular nanopillar (permalloy$|$copper$|$permalloy) by means of a magnetic resonance force microscope. We demonstrate that the observed spin-wave spectrum critically depends on the method of excitation. While the spatially uniform radio-frequency (rf) magnetic field excites only the axially symmetric modes having azimuthal index $\ell =0$, the rf current flowing through the nanopillar, creating a circular rf Oersted field, excites only the modes having azimuthal index $\ell =+1$. Breaking the axial symmetry of the nanopillar, either by tilting the bias magnetic field or by making the pillar shape elliptical, mixes different $\ell$-index symmetries, which can be excited simultaneously by the rf current. Experimental spectra are compared to theoretical prediction using both analytical and numerical calculations. An analysis of the influence of the static and dynamic dipolar coupling between the nanopillar magnetic layers on the mode spectrum is performed.

Figure 1

Schematic representation of the experimental setup used for this comparative SW spectroscopic study. The magnetic sample is a circular nanopillar comprising a thin Py${}_a$ layer and a thick Py${}_b$ magnetic layer separated by a Cu spacer. It is saturated by a large magnetic field ${\mathit{H}}_{\mathrm{ext}}$ applied along its normal axis. A cantilever with a magnetic sphere attached at its tip monitors the magnetization dynamics inside the buried structure. The inset is a microscopy image (top view) of the two independent excitation circuits: In red is the circuit allowing the injection of an rf current perpendicular to plane through the nanopillar ($i_{\mathit{rf}}$, red arrow); in blue is the circuit allowing the generation of an rf in-plane magnetic field ($h_{\mathit{rf}}$, blue arrow). The nanopillar is at the center of the yellow cross-hair. The main figure is a section along the $A-A$ direction.

Figure 2

Comparative spectroscopic study performed by mechanical FMR at $f_{\mathrm{fix}}=8.1$ GHz, demonstrating that distinct SW spectra are excited by a uniform in-plane rf magnetic field (a) and by an rf current flowing perpendicularly through the layers (b). The positions of the peaks are reported in Table .

Figure 3

Evolution of the SW spectra measured at $f_{\mathrm{fix}}=8.1$ GHz by mechanical FMR for different values of the continuous current $I_{\mathit{dc}}$ flowing through the nanopillar. Panel (a) corresponds to excitation by a uniform rf magnetic field, and panel (b) to excitation by an rf current through the sample.

Figure 4

Frequency-field dispersion relation: the top spectrum (a) is measured at fixed-bias field $H_{\mathrm{fix}}=10$ kOe by sweeping the frequency, $f$, of the rf current $i_{\mathit{rf}}$ through the nanopillar. The bottom spectrum (b) is the same as in Fig. 2b, and is obtained by sweeping the external magnetic field,$H_{\mathrm{ext}}$, at fixed frequency $f_{\mathrm{fix}}=8.1$ GHz of $i_{\mathit{rf}}$. The top and bottom scales are in correspondence through the affine transformation $H_{\mathrm{ext}}-H_{\mathrm{fix}}=2\pi (f-f_{\mathrm{fix}})/\gamma$.

Figure 5

Schematic representation of the magnetization dynamics under continuous rf excitation. In the steady state, the torque exerted by the rf perturbation field ${\mathit{h}}_1$ (orange arrow) compensates the torque exerted by the damping (green), and the local magnetization vector $\mathit{M}\left(t\right)$ (purple) precesses at the Larmor frequency on a circular orbit[62] around the local equilibrium direction (unit vector $\widehat{\mathit{u}}$). $\mathit{M}\left(t\right)$ is the vector sum of a small oscillating component $M_s\mathit{m}$ and a large static component $M_s\left(1-{\left|\mathit{m}\right|}^2/2\right)$, respectively, transverse and parallel to $\widehat{\mathit{u}}$. The inset shows the simulated spatial distribution of $\widehat{\mathit{u}}$ inside the nanopillar at ${\mathbf{H}}_{\mathrm{ext}}=10$ kOe (see Sec. 4c). In the white regions, the magnetization is aligned along the normal $\widehat{\mathit{z}}$ within 0.05${}^{\circ }$. In the colored regions, $\widehat{\mathit{u}}$ is flaring ($<$$5^{\circ }$) in the radial direction (the hue indicates the direction of $\widehat{\mathit{u}}-\widehat{\mathit{z}}$ according to the color code defined in Fig. 6).

Figure 6

Color representation of the Bessel spatial patterns for different values of the azimuthal mode index $\ell$ (by row) and radial mode index $n$ (by column). The arrows are a snapshot of the transverse magnetization ${\mathit{m}}_{\nu }$, labeled by the index $\nu =\ell ,n$. All arrows are rotating synchronously in plane at the SW eigenfrequency. In our coding scheme, the hue indicates the phase $\phi =\arg \left({\mathit{m}}_{\nu }\right)$ (or direction) of ${\mathit{m}}_{\nu }$, and the brightness the amplitude of $|{\mathit{m}}_{\nu }{|}^2$. The nodal positions ($|{\mathit{m}}_{\nu }|=0$) are marked in white.

Figure 7

Analytically calculated spectra at $H_{\mathrm{fix}}=10$ kOe using the set of Bessel functions (see Fig. 6) as the trial eigenvectors. Panel (a) shows the linear response to a uniform excitation field ${\widehat{\mathit{h}}}_1=\widehat{\mathit{x}}$ and panel (b) to an orthoradial excitation field ${\widehat{\mathit{h}}}_1=-\sin \phi \widehat{\mathit{x}}+\cos \phi \widehat{\mathit{y}}$. A light (dark) color is used to indicate the energy stored Eq. (12) in the thin Py${}_a$ and thick Py${}_b$ layers.

Figure 8

Schematic representation of the coupled dynamics between two different magnetic disks. Here, ${\omega }_b$, the eigenfrequency of the lowest energy precession mode in the thick layer (the thin layer being fixed at equilibrium) is smaller than ${\omega }_a$, the one in the thin layer (the thick layer being fixed at equilibrium). When the two disks are dynamically coupled through the dipolar interaction, the binding state $B$ corresponds to the two layers oscillating in antiphase at ${\omega }_B$, with the precession occurring mostly in the thick layer, whereas the antibinding state $A$ corresponds to the layers oscillating in phase at ${\omega }_A$, with the precession mostly in the thin layer. This is shown by displaying the dipolar charges and the precession profile $\mathit{m}\left(\rho \right)$ in each layer using a light (dark) color to represent the contribution of the thin (thick) layer.

Figure 9

Panel (a) is the numerically calculated spectral response to a uniform excitation field ${\mathit{h}}_1\propto \widehat{\mathit{x}}$, from a 3D micromagnetic simulation performed at $H_{\mathrm{fix}}=10$ kOe. The peaks are labeled according to their precession profiles shown in Fig. 11. A light (dark) color is used to indicate the energy stored in the thin (thick) layer. Panel (b) recalls the experimental spectrum measured by mechanical FMR when exciting the nanopillar by a homogeneous rf magnetic field at $f_{\mathrm{fix}}=8.1$ GHz.

Figure 10

Panel (a) is the simulated spectral response to an orthoradial excitation field ${\mathit{h}}_1\propto -\sin \phi \widehat{\mathit{x}}+\cos \phi \widehat{\mathit{y}}$. Panel (b) recalls the experimental spectrum measured by mechanical-FMR for an rf current excitation.

Figure 11

Simulated precession patterns of the eigenvectors. Column (a) shows the precession profiles across the thin (light color) and thick (dark color) layers. Columns (b) and (c) show the dynamics in the thin Py${}_a$ and thick Py${}_b$ layers, respectively, with the color code defined in Fig. 6.

Figure 12

Dependence of the mechanical-FMR spectra excited by a uniform rf magnetic field (a) and by a rf current flowing through the nanopillar (b) on the polar angle ${\theta }_H$ between the applied field and the normal to the layers. Superposed (in purple) is the behavior of the high field tail at larger power.

Figure 13

Simulated SW spectra for a nanopillar with an elliptical section (see text). Linear response to a homogeneous rf magnetic field excitation (a) and to an orthoradial rf Oersted field excitation (b). The precession patterns of the lowest energy modes are shown in the insets.