Influence of structural changes in a periodic antidot waveguide on the spin-wave spectra

We demonstrate that the magnonic band structure, including the band gap of a ferromagnetic antidot waveguide, can be significantly tuned by a relatively weak modulation of its structural parameters. We study the magnonic band structure in nanoscale spin-wave waveguides with periodically distributed small antidots along their central line by two independent computational methods, namely, a micromagnetic simulation and a plane-wave method. The calculations were performed with consideration of both the exchange and dipolar interactions. For the exchange dominated regime, we discuss, in details, the impact of the changes of the lattice constant, size, and shape of the antidots on the spin-wave spectra. We have shown that a precise choice of these parameters is crucial for achieving desired properties of antidot waveguides, i.e., a large group velocity and filtering properties due to existence of magnonic band gaps. We discuss different mechanisms of magnonic gap opening resulting from Bragg scattering or anticrossing of modes. We have shown that the dipolar interactions start to assert their role in the spin-wave spectrum when the waveguide is scaled up, but even for a period of few hundreds of nanometers, the magnonic band structure preserves qualitatively the properties found in the exchange dominating regime. The obtained results are important for future development of magnonic crystal based devices.

Figure 1

The structure of the antidot waveguide, where the row of the equidistant square holes was placed in its center. The size $s$ and the distance between antidots (i.e., the period of the structure $a$) are 6 and 15 nm, respectively. The thickness of the waveguide is 1 nm. The sketch below the waveguide structure depicts the precession of magnetization around the direction of external magnetic field $H_0$.

Figure 2

The dependence of size of antidots on SW spectra of ADW. The inset above the central part of the figure shows the system under investigation: 1-nm thick, 45-nm wide, infinitely long Py stripe with a periodic series of square antidots of size $s\times s$, where $s=$ 4, 6, and 8 nm, disposed along the waveguide with a period of $a$ = 20 nm. A bias magnetic field ${\mu }_0{H}_0$ = 1 T is oriented along the waveguide. The row of antidots divides the waveguide into two subwaveguides. The coupling between subwaveguides is controlled by the size of antidots with small antidots resulting in strong coupling ($s$ = 4 nm) and big antidots in weak coupling ($s=$ 8 nm). Red dashed lines show the dispersion for a homogeneous waveguide of width $w$ = 19.5 nm with artificial folding-back of the dispersion to the first BZ. The colored maps present the squared amplitude of the out-of-plane component of dynamical magnetization for bands marked by letters from (a) to (i) in the SW spectra.

Figure 3

The dependence of SW spectra of ADW on the waveguide period. The size of the antidots is kept constant, $s=$ 6 nm. The increase of the period (from $a=$ 15 to 21 and 30 nm) leads to increase of the coupling between SWs in the subwavegudies. Note the change in location of the BZ edges marked by blue vertical lines. In the first row, schematic plots of the ADW are shown, in the second and third row, the dispersion of SWs calculated by micromagnetic simulations (oommf) and PWM are presented, respectively. Together with the PWM results, the dispersion for homogeneous waveguides of width $w$ = 19.5 nm (for $a$ = 15 and 21 nm) and $w$ = 45 nm (for $a$ = 30 nm) with artificial folding back of the dispersion to the first BZ is shown with dashed (red online) lines. The colored maps on the bottom of the figure show the squared amplitude of the out-of-plane component of magnetization calculated with PWM for points of the magnonic band structure labeled by (a)–(f).

Figure 4

The SW spectra for the ADW with square (black solid line) and circular (green dashed lines) antidots. The lattice constant is fixed ($a=15$ nm) and areas of square and circular antidots are the same (36 nm${}^2$). The maps in the two columns on the right present the out-of-plane components of dynamical magnetization for selected modes in the center and the edge of the BZ for square and circular antidots, on the left and right, respectively.

Figure 5

The PWM calculations of SW spectra for the ADW magnified by the factor of 6 in reference to the structure presented in Fig. 4 with square holes showing the crossover of exchange and dipolar effects related to the stronger manifestation of dipolar interactions. The structural parameters are the lattice constant, $a=90$ nm, antidot size, $s=36$ nm, and thickness and width, 6 and 270 nm, respectively. Red dashed lines show the dispersion for a homogeneous waveguide of width 135 nm [i.e., half of the total ADW width]. The maps (a) and (b) present the out-of-plane components of dynamical magnetization for two modes in the center of the BZ. (c) The map of the static demagnetizing field (its component along the waveguide, $H_{\text{dm}}$). The peaks of the static demagnetizing field are significantly smaller than the value of the external magnetic field ${\mu }_0{H}_0=1$ T.

Figure 6

The computational scheme leading to dispersion relation ${\widetilde{\stackrel{̃}{M}}}_x^{y_0}(f,k_z)$ (red box) and spatial distribution of out-of-plane component of dynamical magnetizations for the selected eigenmode ($k_0,f_0$), ${\tilde{M}}_x^{k_0,f_0}(y,z)$ (green box) based on MS. $f$ is a frequency of SW.