Influence of nonmagnetic dielectric spacers on the spin-wave response of one-dimensional planar magnonic crystals

One-dimensional planar magnonic crystals are usually fabricated as a sequence of stripes intentionally or accidentally separated by nonmagnetic spacers. The influence of spacers on shaping the spin-wave spectra is complex and still not completely clarified. We perform detailed numerical studies of the one-dimensional single- and bicomponent magnonic crystals comprised of a periodic array of thin ferromagnetic stripes separated by nonmagnetic spacers. We show that the dynamic dipolar interactions between the stripes, mediated even by ultranarrow nonmagnetic spacers, lead to a significant increase in the frequency of the ferromagnetic resonance mode, while simultaneously reducing the spin-wave group velocity. We attribute these spectral deformations to the modifications of dipolar pinning and shape anisotropy, both of which are dependent on the width of the spacers and the thickness of the stripes, as well as changes with the dynamical magnetic volume charges formed due to inhomogeneous spin-wave amplitude.

Figure 1

(a) Geometry of a single-component magnonic crystal (MC) in the form of a 1D array of Py stripes, where the stripes' width is $a_{\text{Py}}=250$ nm, and the lattice constant is $a=a_{\text{Py}}+l$. (b) Geometry of a bicomponent MC in the form of a 1D periodic array of Co and Py stripes, where the stripes' width is universally $a_{\text{Py}}=a_{\text{Co}}=250$ nm, and the lattice constant is $a=a_{\text{Py}}+a_{\text{Co}}+2l$. In both (a) and (b), stripe thickness is $d$, and the separation between the stripes by nonmagnetic spacers (NMSs) is $l$. The stripes are magnetically saturated along their axes. Spin waves (SWs) propagate within the plane of the MC, perpendicular to the magnetization. The unit cell used in calculations (periodically repeated along the $y$ axis) is marked by the semitransparent box.

Figure 2

(a) The dependence of the frequency of fundamental SW mode $\left(k_y=0\right)$ on the thickness $d$ of Py stripes in a single-component MC. The width of the NMSs between the stripes $l$ is fixed to $1,5$, or 50 nm. The width of the stripes $a_{\mathrm{Py}}$ is equal to 250 nm. The calculation was done using the FEM for $H_0=0$ in the absence of the surface anisotropy (i.e., absence of exchange pinning). To illustrate the impact of the thickness $d$ on the SW pinning, we show selected profiles of the in-plane component of the SW amplitude $m_{\mathrm{y}}$ in the cross section of one stripe. (b) Pinning parameter $p$ for $K_s=0$ (in the limit of an isolated stripe: $l\to \infty )$ as a function of the thickness $d$. In the realistic range of $d$, considered in (a), an increase of $d$ makes the SW gradually unpinned, as also illustrated in the SW profiles shown in (a). The dashed vertical line in (a) marks the value of $d=30$ nm for which the further results in Fig. 3 are shown. The inset in (b) presents the dependence of the frequency of fundamental mode on demagnetizing factors.

Figure 3

The dependence of the frequency of the fundamental SW mode $\left(k_y=0\right)$ on the separation between Py stripes, $l$, for $H_0=0$. The stripe separation is scaled logarithmically. The red full and empty dots are the results of FEM calculations, the blue squares are the results of FDTD calculations, and the lines correspond to the PWM results for two limiting cases which concern the SW pinning at the edges of stripes: absence of exchange pinning $\left(K_{S,\mathrm{Py}}=0\right)$, solid line, and strong exchange pinning $\left(K_{S,\mathrm{Py}}\gg {\mu }_0{M}_{S,\mathrm{Py}}^{2}d/4\right)$, dashed line. The insets show the distribution of the in-plane component of SW amplitude $m_{\mathrm{y}}$ in the cross section of one stripe for $l=1$, 5, and 50 nm. The calculations were done for the stripes' thickness $d$ and width $a_{\mathrm{Py}}$ equal to 30 and 250 nm, respectively.

Figure 4

(a),(b) The amplitude of the $m_y$ and $m_x$ components of the dynamic magnetizations across the width of the stripe, respectively, calculated at the FMR frequency as identified from Fig. 3. These profiles were obtained from the FDTD calculations (in the absence of exchange pinning and for $d=30$ nm, $a_{\mathrm{Py}}=250$ nm). The magnetization profiles shown correspond to different separation $l$ values, as indicated, ranging from $l=5$ nm to $l=\infty$.

Figure 5

(a) The dependence of the frequency of the fundamental mode (FMR frequency) in the 1D bicomponent (Co/Py) MC on the separation between the Co and Py stripes, $l$, calculated with the FEM (solid red line). For comparison, the frequencies of the FMR modes of the single component MC composed of Py stripes in the absence of exchange pinning $\left(K_s=0\right)$ and with exchange pinned SWs at the edges are also shown with dashed black and solid black lines, respectively. The amplitude profile of the fundamental SW mode in the Co/Py MC with a 2-nm-wide NMS, for $H_0=0$, is shown in the inset. (b) The dispersion relations characterizing the two lowest bands for the bicomponent MC (red lines) and the lowest band for single-component MC (black line) in the absence of exchange pinning. The dispersion relation for the single-component MC was folded into the first Brillouin zone of the bicomponent MC due to our selection of a unit cell which was composed of two Py stripes. We fixed the width of the NMSs to $l=2$ nm. The thickness $d$ and the widths of stripes $a_{\mathrm{Py}}=a_{\mathrm{Co}}$ are equal to 30 and 250 nm, respectively, both in (a) and (b).

Figure 6

(a,b) Dispersion relation of SWs in the first Brillouin zone in a bicomponent MC with $H_0=0$: (a) for fixed thickness $d=30$ nm and three separations between the Co and Py stripes—the solid, dashed, and dotted lines correspond to $l=0$, 10, and 200 nm, respectively; (b) for a fixed NMSs' width of $l=10$ nm, the solid, dashed, and dotted lines correspond to the MC's thicknesses of $d=10$, 20, and 30 nm, respectively. For every separation value in (a), the respective MC has a different lattice constant, and so, the Brillouin-zone boundary also appears at a different wave number. (c) The fundamental mode frequency at $k=0$ in dependence on the external magnetic field for the three NMSs' widths $l=0$, 10, and 200 nm (solid, dotted, and dashed lines, respectively).