Magnonic Band Structure in Vertical Meander-Shaped ${\mathrm{Co}}_{40}$${\mathrm{Fe}}_{40}$${\mathrm{B}}_{20}$ Thin Films

Exploring the third dimension in magnonic systems is essential for the investigation of alternative physical phenomena and for the control of spin-wave propagation at the nanoscale. Here, the characteristics of spin waves in vertical meander-shaped ${\mathrm{Co}}_{40}{\mathrm{Fe}}_{40}{\mathrm{B}}_{20}$ thin films consisting of nanosegments located at 90° angles with respect to each other are investigated by Brillouin-light-scattering spectroscopy over four Brillouin zones in reciprocal space. We reveal the dispersion relations and the periodic character of several dispersive branches as well as alternating frequency bands, where spin waves are allowed or forbidden to propagate. Between each couple of successive modes, frequency band gaps exist only for wave numbers k = 2mπ/a, where m is an integer number and a is the size of the meander unit cell, whereas the spectra show propagating modes in the orthogonal film segments for all the other wave numbers. Micromagnetic simulations and analytical calculations are used to understand and explain the results in terms of the mode spatial localization and symmetry. We show that the width and the center frequency of the magnonic band gaps can be controlled by changing the geometrical parameters of the meander-shaped film. The investigated samples behave as three-dimensional waveguides where spin waves propagate in the film segments located at 90° angles with respect to each other, thus making possible vertical spin-wave transport for multilayer magnonic architectures and signal processing.

Figure 1

(a) Cross-section sketch of the meander-shaped film unit cell with a periodicity of a = L₁+2L₂= 600 nm, where L₁ and L₂ are the lengths of the horizontal nanosegments with thickness d₁. The width of the vertical segments is d₂, the meander depth is h = 50 nm, and the total meander thickness is thus S = h + d₁. The dashed-dotted line represents the coordinate (ξ) along the meander structure. An external magnetic field is applied along the z direction and propagation occurs in the x-y plane. k₁ and k₂ represent the SW wave numbers along the horizontal and vertical segments, respectively. Ψ_n and Ψ_n+1 are the SW amplitudes at the input (x = −a/2) and output (x = +a/2) sections of the meander-structure unit cell. (b), (c) SEM images of the meander-shaped 23-12 $\mathrm{Co}\mathrm{Fe}\mathrm{B}$ film.

Figure 2

BLS spectra at µ₀H₀= 50 mT recorded at different k values within the first two BZs for (a) 23-12 and (b) 15-8 meander-shaped $\mathrm{Co}\mathrm{Fe}\mathrm{B}$ films. Labels of each spectrum correspond to the k value expressed in 10⁷ rad/m. On the anti-Stokes side of the spectra, the two lowest frequency modes are highlighted by the red (R) and blue (B) peak. For the 23-12 sample, a vertical dashed line is included as a guide for the eye to indicate the position of the PSSW mode.

Figure 3

Measured magnonic band structure for the vertical meander-shaped (a) 23-12 and (b) 15-8 $\mathrm{Co}\mathrm{Fe}\mathrm{B}$ films. Red and blue points represent the frequency of the lowest two frequency peaks in Fig. 2 while micromagnetic dispersion data are represented by the green solid lines. The gray regions indicate the BG_m frequency ranges for m = 1 and 2.

Figure 4

Spatial distribution of modes (M_y/M_s) in the meander-shaped 23-12 $\mathrm{Co}\mathrm{Fe}\mathrm{B}$ film calculated by micromagnetic simulations at the edges of BG₁ (f₃ and f₄) and BG₂ (f₅ and f₆).

Figure 5

(a) Dispersion diagram of dipolar SWs in a meander-type structure with thicknesses d₁= 23 and d₂= 12 nm. The position and the width of the first three forbidden zones (BG₁₋₃, Δf₁₋₃) are depicted. The dashed lines indicate the values of the Bloch wave numbers k_m = mπ/a. The characteristic frequencies f₁₋₆ are denoted with red dots. (b) As in (a) for d₁= 15 nm and d₂= 8 nm.

Figure 6

(a)−(c) Scheme for setting the eigenfunction form along the corresponding segments of the structure with coordinate ξ together with the absolute value of the spatial SW mode amplitudes (Ψ) at different frequencies from f₁ to f₆. Notice that in (a) the vertical scale for $|\mathrm{\Psi }(\xi )|$ goes from 0.8 to 1 and not from 0 to 1 like in (b) and (c). (d) Real part of spatial distribution of SW amplitudes at frequencies f₃, f₄ demonstrating the localization either in vertical or in horizontal segments and symmetry with respect to the center of the unit cell (x = 0). (e) As in (d) for modes at frequencies f₅ and f₆.

Figure 7

(a) Dependence of the BG width (Δf_m) and (b) center frequencies f_cm on the width of the vertical segment d₂ of the meander-shaped film with d₁= 23 nm. (c) BG width (Δf_m) and (d) center frequencies f_cm as a function of the meander depth h with d₁= 23 nm and d₂= 12 nm. Data are shown for the three lowest frequencies BG_m (m = 1,2, and 3).