Investigation of spin wave damping in three-dimensional magnonic crystals using the plane wave method

The Landau-Lifshitz equation with a scalar damping constant predicts that the damping of spin waves propagating in an infinite homogeneous magnetic medium does not depend on the direction of propagation. This is not the case in materials with a periodic arrangement of magnetic constituents (known as magnonic crystals). In this paper, the plane wave method is extended to include damping in the calculation of the dispersion and relaxation of spin waves in three-dimensional magnonic crystals. A model material system is introduced and calculations are then presented for magnonic crystals realized in the direct and inverted structure and for two different filling fractions. The ability of magnonic crystals to support the propagation of spin waves is characterized in terms of a figure of merit, defined as the ratio of the spin wave frequency to the decay constant. The calculations reveal that in magnonic crystals with a modulated value of the relaxation constant, the figure of merit depends strongly on the frequency and wave vector of the spin waves, with the dependence determined by the spatial distribution of the spin wave amplitude within the unit cell of the magnonic crystal. Bands and directions of exceptionally long spin wave propagation have been identified. The results are also discussed in terms of the use of magnonic crystals as metamaterials with designed magnetic permeability.

Figure 1

(a) The sc structure of the considered MC (direct crystal). The MC consists of spherical scattering centers (material A of radius $R$) immersed in the host matrix (material B). The lattice constant is $a$ and the external static magnetic field, $H_0$ is directed along the $z$ axis. (b) The first BZ of the sc lattice. The dark (orange) color shows the part of the BZ over which the calculations of the magnonic structure are performed.

Figure 2

The frequency and decay rate of the SWs are shown for the direct MC (spheres of material A in matrix B) in (a) and (b), respectively. A filling fraction of 0.2 was assumed in the calculations. (c) The figure of merit (FOM) is shown for the same structure.

Figure 3

The frequency and decay rate of SWs in the first BZ for a direct MC (spheres of material A in a matrix of material B) are shown in (a) and (b), respectively. A filling fraction 0.5 was assumed in calculations. (c) Figure of merit (FOM) for the same structure.

Figure 4

The frequency and decay rate of SWs in the first BZ for an inverted MC (spheres of material B in a matrix of material A) are shown in (a) and (b), respectively. A filling fraction of 0.5 was assumed in the calculations. (c) The FOM is shown for the same structure.

Figure 5

The real and imaginary parts of the frequency in the first BZ for inverted MC (spheres of material B in matrix of material A) in (a) and (b), respectively. A filling fraction of 0.2 was assumed in the calculations. (c) The FOM is shown for the same structure.

Figure 6

Decay rate of SWs versus its frequency for wave vectors randomly chosen from the first BZ. Direct crystals (A spheres in the B matrix) are shown in (a) and (b), inverted crystals (B spheres in the A matrix) in (c) and (d). In (a) and (c) the results for $f$ $=$ 0.2 and in (b) and (d) for $f$ $=$ 0.5 are shown. Magnonic gaps are colored in yellow color. In (c) the second band with the moonlike shape is marked by blue ellipse, as it is expected to have strong anisotropy in damping.

Figure 7

The amplitude of the dynamical components of the magnetization vector across the planes perpendicular to the $z$ axis and crossing it at 0 [plane (001)] and at $a$/2 [plane (002)]. The profiles from the first and the second band in $\Gamma$ and $R$ point are shown for the direct crystal with $f$ $=$ 0.5 (left columns) and for the inverted crystal and $f$ $=$ 0.2 (right columns). The estimated value of the damping constant of the related mode [calculated according to Eq. (3)] is also given for each profile.

Figure 8

(a) Frequency and (b) decay rate of the SWs are shown as a function of the RLP for the inverted crystal with filling fraction 0.2 calculated with PWM. The frequencies from the second band for the $\Gamma$, $R$, and $X$ points in the BZ are shown. (c) Estimated values of the damping coefficient are shown in dependence on RLP for the $\Gamma$, $X$, and $R$ points for the second band. ${\alpha }_{\text{est}}$ are calculated according to Eq. (3). (d) The product of ${\alpha }_{\text{est}}$ and ${\Omega }^{\prime }$ is shown for the $\Gamma$, $X$, and $R$ points from the second band. There is a close relation between ${\alpha }_{\text{est}}\times {\Omega }^{\prime }$ [shown in (d)] and ${\Omega }^{\prime \prime }$ [shown in (b)].

Figure 9

The absolute value of the product of a group velocity ($v_{\text{g}}$) and FOM for the second band for the inverted crystal (with $f$ $=$ 0.2) along the path in the first BZ.