Standing spin waves in perpendicularly magnetized triangular dots

Standing spin waves in triangular dots (truncated pyramids) were investigated both experimentally and theoretically. Arrays of nickel triangular pyramids with the base side of 270 nm and height of 70 nm were deposited on Si (111) substrate. The spectra of ferromagnetic resonance were obtained at room temperature with an external saturating magnetic field directed perpendicular to the array plane. The theoretical approach, which allows to describe standing spin wave modes both in perpendicular magnetized regular prisms and in close to prisms truncated pyramids, was developed. Theoretically calculated resonance fields for the observed modes are in a good correlation with the experiment.

Figure 1

A schematic presentation of a formation of the triangular dots array using the nanosphere lithography method (left). AFM image of the periodic array of nickel islands used in the experiments (middle). A schematic presentation of the triangular dots array (right). The hexagonal array is formed by nested regular triangular lattices with basis vectors ${\mathit{\zeta }}_1$, ${\mathit{\eta }}_1$ and ${\mathit{\zeta }}_2$, ${\mathit{\eta }}_2$.

Figure 2

The magnetic resonance spectrum of the nickel triangular dots array recorded at room temperature at 9.86 GHz. Magnetic field is perpendicular to the array plane. Resonance lines positions are denoted by arrows and the corresponding values of resonance fields are presented in Table 1.

Figure 3

(a) Coordinate system and its position with respect to the pyramid. Here, $a$ is the base side of the truncated pyramid, $h$ is its height, and $h_{\mathrm{pyr}}$ is the height of the corresponding nontruncated pyramid. (b) The sketch of prism and its transformation into the pyramid.

Figure 4

The in-plane dependence of prism demagnetization field on $xy$ plane calculated using (3a) with the following parameters: $a=270\mathrm{nm}$, $h=70\mathrm{nm}$, and $M_0=484\mathrm{emu}/\mathrm{c}{\mathrm{m}}^3$.

Figure 5

The triangular membrane oscillations eigenfunction ${\psi }_{21}^{\left(\mathrm{tr}\right)}$ profile (a) calculated using (8) with the parameters described in the text. Bright and dark colors correspond to the areas oscillating in antiphase. (b) The same eigenfunction profiles for heights $z=0$, $h/2$, $h$ in truncated pyramid.

Figure 6

Magnetic resonance spectrum of nickel dots: the experiment (dotted line) and the theory for frustrum (solid line). Theoretical lines positions (see bottom inset) for the cases of regular prisms (circles) and frustrums (stars) were obtained solving Eqs. (6) and (9) correspondingly. The intensities of the lines were calculated using standard approach $I_j\sim {\left(\int {\mathrm{\Psi }}_j^{}dV\right)}^2$. The theoretical spectrum was plotted in supposition of Gaussian shape of the lines. The dispersion was evaluated from the experiment.