Magnonic band gap and mode hybridization in continuous permalloy films induced by vertical dynamic coupling with an array of permalloy ellipses

We investigate dispersion relation of spin waves in a thin homogeneous permalloy film decorated with a periodic array of elliptically shaped permalloy dots and separated by a nonmagnetic Pt spacer. We demonstrate formation of the magnonic band structure for Damon-Eshbach waves propagating in a permalloy film with the band gaps opened at the Brillouin zone border and at smaller wave numbers, due to the Bragg interference and interaction of propagating waves of the continuous film with a standing resonant mode of the nanoellipses, respectively. We show that the predominant role in the formation of the magnonic band structure and opening band gaps is played by a vertical dynamic coupling between propagating waves and magnetization oscillations in the nanodots. The shape anisotropy of the permalloy nanodots allows us to control the spin wave dynamics through the switch between two states of the magnetization with respect to the underneath film magnetization.

Figure 1

(a) SEM image of the sample with the main in-plane dimensions indicated. The directions of the applied magnetic field $H$ and of the wave vector $k$, together with the Cartesian axes, are also represented by the arrows. Dashed white line marks the unit cell of the structure. (b) Schematic presentation of the unit cell in the cross section along the $(x,z)$ plane.

Figure 2

(a) MOKE loop measured with the magnetic field along the long axis of the ellipses (the $x$ direction). (b) The SW frequency at $k=3.48\pi /a$ in dependence on $H$ oriented in the $x$ direction. The BLS measurements were started from saturation in the high positive values of $H$ and decreasing the field toward negative saturation following the descending branch of the MOKE loop. SW frequencies, derived from the measured BLS spectra, are marked by red triangles. The results of numerical calculations are shown with the blue colored dots, with the intensity proportional to the calculated BLS intensity. The green line with the empty dots shows the BLS measurements on the reference 20 nm thick continuous Py film.

Figure 3

(a) Dispersion relation of SWs in DE configuration measured with BLS (red triangles) and obtained from simulations (blue dots with different intensity) at the magnetic field $H=50$ mT, i.e., in the P state. The intensity of the colors in the numerical data relates to the calculated BLS intensity of the SWs. The vertical dashed lines mark the BZ boundary and center. Green and yellow areas indicate the hybridization and the Bragg band gap regions, respectively. (b) Dispersion relation of SWs measured in the reference sample, 20 nm thick Py film. The green line is to guide experimental points.

Figure 4

(a) The BLS spectra for selected wave vectors at $H=50$ mT from which the dispersion in Fig. 3 is formed. The yellow and green areas in (a) mark the $k$ range of the band gap opening at the BZ border and due to hybridization of the DE wave with the confined modes in elliptical nanodots, respectively [see also Fig. 3]. (b) The amplitude of SWs $m_z^2$ in arbitrary units across the midplanes of the elliptical nanodots and the plain film for the two modes M1 and M2 at various $k$ values to demonstrate the hybridization of SWs near the band gap opening due to hybridization.

Figure 5

The calculated amplitude [(a) and (c)] and the phase [(b) and (d)] of SW at the magnonic band gap at the BZ boundary: at $f=8.73$ GHz [(c) and (d)] and $f=9.46$ GHz [(a) and (b)].

Figure 6

The static magnetic field distribution in the cross sections of the unit cell: (a) along the middle axis (the $y$ axis) of the nanodot (green line) and in the Py film below this line (blue line); (b) in the $(x,y)$ plane in the middle of the film. The field is a superposition of the external magnetic field and the stray magnetostatic field, $H_0+H_{\text{stray}}, H_0=50$ mT.