# Dynamic Footprints of the Specific Artificial Spin Ice Microstate on Its Spin Waves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### Micromagnetic Framework and Simulations

## 3. Results

#### 3.1. Footprints of the Microstate on the Fundamental Mode

#### 3.2. Main Effects in the Spectra Due to the Different Vertex Configurations

#### 3.2.1. Lattice Interaction and SW Profile

#### 3.2.2. Macrospin Size

#### 3.2.3. Aspect Ratio

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The three microstates in the square ASI: (

**a**) 4N, which contains the vertex types 4N, 4S, 2N2S, 2S2N; (

**b**) 3N1S, which contains also 3S1N; (

**c**) 2N1S, which is reproduced at all ASI vertices, since the geometric primitive cell matches with the magnetic one; (

**d**) which displays a closure distribution (Vortex). Note that we refer to spin or macrospin, but the actual physical quantity that is shown through the arrows is the magnetic moment. The red square frame encloses the primitive cell.

**Figure 2.**(

**Upper panels**) SW spectra and (

**lower panels**) mode phase profiles for the 4N microstate at the three different separation values between the macrospins: s = 96 nm (

**panel i**), s = 64 nm (

**panel ii**), s = 40 nm (

**panel iii**). Each peak has a label corresponding to the calculated mode profiles, in particular (

**a**) is the EM, (

**b**) is the FM, (

**c**,

**d**) are higher order modes, with 1 and 3 nodal lines, respectively. In panel (

**i**) the inset features a semi-log plot in correspondence to the EM frequency and an arrow to highlight the position of the EM peak (

**a**). In this microstate, EM and FM frequencies have opposite behavior as a function of macrospin separation.

**Figure 3.**Fundamental mode profile for the different ASI microstates: (

**a**) 4N, (

**b**) 3N1S, (

**c**) 2N2S, (

**d**) Vortex. Note how modes (

**c**,

**d**) differ only at the edges of the ASI elements, as the static magnetization does, i.e., curled in (

**c**) and straight in (

**d**).

**Figure 4.**(

**Upper panels**) Magnetization distributions at remanence of microstate (

**a**) 4N, (

**b**) 3N1S, (

**c**) 2N2S, (

**d**) Vortex. The arrows represent the average direction of the magnetization in 5 micromagnetic cells, while the green background marks the effective area of the macrospins in the micromagnetic representation. (Lower panels) Magnitude of internal ${B}_{\mathrm{eff}}$ in unit [T] for microstate (

**e**) 4N, (

**f**) 3N1S, (

**g**) 2N2S, (

**h**) Vortex. Note that the minima at the edges host the EM oscillation, while the central region in the bulk hosts the FM oscillation: these confinements play a crucial role in determining the ultimate dynamics of the modes.

**Figure 5.**Evolution of the fundamental mode phase profile (amplitude in arb. units), as appearing in a single macrospin belonging to a square–ASI in the microstate 4N, as the spacer between adjacent elements is increased: (

**a**) $s=96$ nm, (

**b**) $s=128$ nm, (

**c**) $s=256$ nm, (

**d**) $s=512$ nm, (

**e**) $s\to \infty $.

**Figure 6.**(

**Upper panels**) SW spectra and (

**lower panels**) mode phase profiles for the 3N1S microstate at the three different separation values between the macrospins: s = 96 nm (

**panel i**), s = 64 nm (

**panel ii**), s = 40 nm (

**panel iii**). Each peak has a label corresponding to the calculated mode profiles, in particular (

**a**) is the EM, (

**b**) is the FM. We show also mixed modes: (

**c**) is a FM-like mode in the horizontal macrospins and a mode with 1 node in the vertical ones; (

**d**) a mode with 2 nodes in the horizontal macrospins and 1 node in the vertical ones; (

**e**) a mode localized in the vertical macrospins only, with 3 nodes. In this microstate, EM and FM frequencies have opposite behavior as a function of macrospin separation.

**Figure 7.**(

**Upper panels**) SW spectra and (

**lower panels**) mode phase profiles for the 2N2S microstate at the three different separation values between the macrospins: s = 96 nm (

**panel i**), s = 64 nm (

**panel ii**), s = 40 nm (

**panel iii**). Each peak has a label corresponding to the calculated mode profiles, in particular (

**a**) is the EM, (

**b**) is the FM, (

**c**) is a hybrid of FM and a mode with two nodal lines perpendicular to the magnetization. In this microstate, EM and FM frequencies have the same behavior as a function of macrospin separation.

**Figure 8.**(

**Upper panels**) SW spectra and (

**lower panels**) mode phase profiles for the Vortex microstate at the three different separation values between the macrospins: s = 96 nm (

**panel i**), s = 64 nm (

**panel ii**), s = 40 nm (

**panel iii**). Each peak has a label corresponding to the calculated mode profiles, in particular (

**a**) is the EM, (

**b**) is the FM. Then we show a mode with (

**c**) two and (

**d**) four nodal lines perpendicular to the magnetization, displaying non-physical large intensities, which we attribute to a computational artifact. In this microstate, EM and FM frequencies have the same behavior as a function of macrospin separation.

**Figure 9.**SW spectra (

**panels on the left**) and mode phase profiles (

**panels on the right**) for microstate (

**i**) 4N, (

**ii**) 3N1S, (

**iii**) 2N2S, (

**iv**) Vortex, for the small macrospin ASI. Below 15 GHz only two modes are found with non-negligible intensity: (

**a**) the EM and (

**b**) the FM. Note how, at these small sizes, the gap between the frequencies of these modes greatly decreases with decreasing effective charge at vertex.

**Figure 10.**(

**Upper panels**) SW spectra and (

**lower panels**) mode phase profiles for the 4N microstate at the three different aspect ratios, with fixed length (256 nm) and variable width: $w=96\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$ (

**panel i**), $w=88\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$ (

**panel ii**), $w=80\phantom{\rule{3.33333pt}{0ex}}\mathrm{nm}$ (

**panel iii**). Each peak has a label corresponding to the calculated mode profiles, in particular (

**a**) is the EM, (

**b**) is the FM. Then, we show a mode with (

**c**) one and (

**d**) two nodal lines perpendicular to the magnetization.

**Figure 11.**(

**Upper panels**) SW spectra and (

**lower panels**) mode phase profiles for the 3N1S microstate at the three different aspect ratios, with fixed length (256 nm) and variable width: w = 96 nm (

**panel i**), w = 88 nm (

**panel ii**), w = 80 nm (

**panel iii**). Each peak has a label corresponding to the calculated mode profiles, in particular (

**a**) is the EM, (

**b**) is the FM. Mode (

**c**) is a mixed mode, having one nodal line in the vertial macrospins and two in the horizontal ones; mode (

**d**) shows two nodes in the vertical macrospins and a profile similar to the FM in the horizontal macrospin.

**Figure 12.**(

**Upper panels**) SW spectra and (

**lower panels**) mode phase profiles for the 2N2S microstate at the three different aspect ratios, with fixed length (256 nm) and variable width: w = 96 nm (

**panel i**), w = 88 nm (

**panel ii**), w = 80 nm (

**panel iii**). Each peak has a label corresponding to the calculated mode profiles, in particular (

**a**) is the EM, (

**b**) is the FM. Remarkably, in this case we do not obtain higher order modes with appreciable intensity.

**Figure 13.**(

**Upper panels**) SW spectra and (

**lower panels**) mode phase profiles for the Vortex microstate at the three different aspect ratios, with fixed length (256 nm) and variable width: w = 96 nm (

**panel i**), w = 88 nm (

**panel ii**), w = 80 nm (

**panel iii**). Each peak has a label corresponding to the calculated mode profiles, in particular (

**a**) is the EM, (

**b**) is the FM. Mode (

**c**) has two nodal lines perpendicular to the magnetization, and gets appreciable intensity only in panel (

**iii**).

**Table 1.**Behavior of the EM frequency with decreasing the macrospin separation s in the 4N microstate. A change of behavior occurs in crossing some ${s}_{c}$ intermediate of 56 and 48 nm.

s (nm) | 96 | 64 | 56 | 48 | 40 |
---|---|---|---|---|---|

Frequency (GHz) | 4.5 | 2.6 | 1.0 | 3.2 | 4.6 |

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**MDPI and ACS Style**

Micaletti, P.; Montoncello, F.
Dynamic Footprints of the Specific Artificial Spin Ice Microstate on Its Spin Waves. *Magnetochemistry* **2023**, *9*, 158.
https://doi.org/10.3390/magnetochemistry9060158

**AMA Style**

Micaletti P, Montoncello F.
Dynamic Footprints of the Specific Artificial Spin Ice Microstate on Its Spin Waves. *Magnetochemistry*. 2023; 9(6):158.
https://doi.org/10.3390/magnetochemistry9060158

**Chicago/Turabian Style**

Micaletti, Pietro, and Federico Montoncello.
2023. "Dynamic Footprints of the Specific Artificial Spin Ice Microstate on Its Spin Waves" *Magnetochemistry* 9, no. 6: 158.
https://doi.org/10.3390/magnetochemistry9060158