# Nano-Magnonic Crystals by Periodic Modulation of Magnetic Parameters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Analytical Derivation

#### 2.1.1. Exchange Constant Modulation

#### 2.1.2. DMI Modulation

#### 2.2. Micromagnetic Simulations

^{−1}. Nominal parameters for permalloy were used, namely: M

_{s}= 790 kA/m, A = 10 pJ/m, and $\alpha =0.01$.

#### 2.2.1. Sinc Excitation

^{−1}, i.e., a wavelength of twice the cell size. The periodicity of the sinc function only extends along the x direction so that the excited magnons follow a well-defined wavevector. Here, the thin film was set with dimensions 1000 nm $\times 1000$ nm $\times 5$ nm, a cell size of 4 nm $\times 4$ nm $\times 5$ nm, and periodic boundary conditions set in the x and y directions.

#### 2.2.2. Local Microwave Field

^{−1}. Because the wave decayed in simulations due to dissipation, and to reduce the effect of non-periodic edges, we used a Hann window for each snapshot. Similar to our sinc function approach, the data of interest were the magnetization variation along the x axis, so that here the data were effectively one-dimensional. In other words, the wavenumber error estimated directly from the spatial region was considered to be a good metric.

## 3. Results

#### 3.1. Dispersion Relations

^{−1}, leading to apparent bandgaps. The evanescent nature of the waves in between the bandgap in the SW configuration are shown in Section 3.1.1 These results demonstrate that the modulation of a continuous film can establish a band structure that is mediated by spin waves at the nanoscale.

^{2}as the average value, consistent with experimentally reported values for the interfacial DMI in CoFeB/Pt bilayers [45]. This is a relatively strong DMI interaction, but it is used here to showcase the effect of its spatial modulation. The numerical results are also displayed as colorscale maps in Figure 3, with panels (a) and (b) showing the FVW and SW configurations, respectively, and analytical calculations shown by red curves. In this case, we find more striking differences between the configurations. For the FVW configuration, we find only one dominant dispersion curve. This was expected from the analytical derivation, since DMI was not active in this case. However, some additional bands are visible. This was due to nonlinearities in the Landau–Lifshitz equation and can be considered negligibly small.

#### 3.1.1. Local Microwave Field

#### 3.2. Square Modulations

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic illustration of the investigated configurations. The gradient red regions display the periodicity of the magnetic parameters with wavenumber q. The magnons propagated along the modulated parameters with wavenumber k. The (

**a**) forward volume wave (FWV) and (

**b**) surface wave (SW) configurations were achieved by using a saturating magnetic field normal to the plane and a small magnetic field in-plane and perpendicular to k, respectively.

**Figure 2.**Numerical dispersion relations when the exchange constant is modulated with $\beta =0.4$ and $q=0.2$ nm

^{−1}for (

**a**) FVW and (

**b**) SW configurations. The analytical equations from Equations (3) and (4) are superimposed with red curves. The solid curve was calculated with $N=0$, and the dashed curves were computed with $N=\pm 1$. The black dashed lines at $k=-0.2$ and $0.2$ nm

^{−1}represent the modulation period. The semi-transparent region in (

**b**) displays the frequency gap.

**Figure 3.**Numerical dispersion relations when the DMI constant was modulated with $\beta =0.4$ and $q=0.2$ nm

^{−1}for the (

**a**) FVW and (

**b**) SW configurations. The analytical equations from Equations (3) and (4) are superimposed with red curves. The solid curve was calculated with $N=0$, and the dashed curves were computed with $N=\pm 1$. The black dashed lines at $k=-0.2$ and $0.2$ nm

^{−1}represent the modulation period.

**Figure 4.**Numerical dispersion relations for the SW configuration when the DMI constant was modulated with $\beta =0.4$ and $q=0.2$ nm

^{−1}. The non-local dipole contribution was removed from these simulations and replaced by a negative uniaxial anisotropy that mimicked the thin film approximation used in the analytical calculations. Because the magnetostatic waves were not available, the derived dispersions accurately predicted the numerical results. The black dashed lines at $k=-0.2$ and $0.2$ nm

^{−1}represent the modulation period.

**Figure 5.**Numerically determined wavenumber values for a range of microwave frequencies, shown by black circles. These are compared with analytical calculations shown in solid red curves with modulations in the (

**a**) exchange constant and (

**b**) the DMI constant. The black dashed lines at $k=-0.2$ and $0.2$ nm

^{−1}represent the modulation period.

**Figure 6.**Wave profiles from the ${m}_{z}$ component excited by a microwave field. The solid black curves are propagating waves, while the solid gray curves are evanescent waves. The simulations were run for 400 microwave cycles to ensure the same phase was obtained. The profiles are vertically shifted for clarity.

**Figure 7.**Numerical dispersion relations in the SW configuration using a square modulation of (

**a**) exchange and (

**b**) DMI parameters. The dispersion relations had an increased spectral content because of the harmonics enabled by the square modulation. The black dashed lines at $k=-0.2$ and $0.2$ nm

^{−1}represent the modulation period.

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Roxburgh, A.; Iacocca, E.
Nano-Magnonic Crystals by Periodic Modulation of Magnetic Parameters. *Magnetochemistry* **2024**, *10*, 14.
https://doi.org/10.3390/magnetochemistry10030014

**AMA Style**

Roxburgh A, Iacocca E.
Nano-Magnonic Crystals by Periodic Modulation of Magnetic Parameters. *Magnetochemistry*. 2024; 10(3):14.
https://doi.org/10.3390/magnetochemistry10030014

**Chicago/Turabian Style**

Roxburgh, Alison, and Ezio Iacocca.
2024. "Nano-Magnonic Crystals by Periodic Modulation of Magnetic Parameters" *Magnetochemistry* 10, no. 3: 14.
https://doi.org/10.3390/magnetochemistry10030014