# Influence of Physical Symmetries on the Magnetization Dynamics in Magnetic Fibers

^{1}

^{2}

^{3}

^{*}

*Symmetry*and Spin Dynamics)

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}) crystal. This material exhibited ferromagnetism, resulting from the existence of four sublattices—each with its own arrangement of spins/magnetic moments—in which the spin moments were reoriented with respect to the rhombohedral direction [111]. Going beyond such a local view, the bulk phase of the α-Fe

_{2}O

_{3}crystal, indispensable for the occurrence of the DMI, belongs to the trigonal system—the system separated from the hexagonal one—and possesses the R3c space group. This means that the essential feature of the trigonal system is that three of the four crystallographic axes of the system lie in one plane and have the same length. The fourth axis has a different unit length, is a threefold-symmetry axis, and, importantly, is oriented perpendicular to the plane.

## 2. Materials and Methods

^{−3}J/m

^{2}, as well as the possibilities to apply external magnetic fields with freely definable time-dependence at defined positions and collect data from specific volumes were implemented in our previous studies [25,26,27].

_{0}(red and blue) and thus in the nucleation of domain walls. The lengths of both wires are 1570 nm, their cross-section is 10 nm × 60 nm.

_{S}= 1.005 T, exchange constant A = 1.3·10

^{−11}J/T, anisotropy constant zero as usual for permalloy (Py), and damping constant α = 0.02. The magnetoelastic anisotropy was neglected.

## 3. Results

_{i}and p

_{j}, and the Dzyaloshinskii–Moriya interaction energy as a result of broken inversion in crystallographic symmetry

_{0}, P

_{1}, and P

_{2}, as defined in Figure 1. It should be mentioned that simulations of domain walls inserted in magnetic nanowires usually apply defined areas of specific shapes, typically much broader or thicker than the nanowire itself, to insert the domain walls [4,32,33,34]. Here, the nucleation of domain walls is oppositely performed directly inside the nanowire, which can be assumed to be less reliable than the aforementioned common method.

_{0}of 80 nm, as opposed to the much shorter length of the input area of 1 nm. On the other hand, time-asymmetries are visible in the simulations taking into account the DMI as an antisymmetric exchange.

_{1}nor P

_{2}shows significant oscillations due to the small input area of the applied rotating magnetic field, as depicted in Figure 3 and Figure 4.

_{y}in Figure 5. Here, it is visible that the rotating magnetic field with a length s = 1 nm can change the magnetization in the middle of the straight nanofiber, as visible by the large oscillations in Figure 2 (no DMI, straight fiber); however, the induced domains are not stable, but lose their magnetic orientation during propagation to the ends of the fiber. In this way, no stable domain walls are introduced into the system.

_{0}, as depicted in Figure 6. For the curved nanofiber, simulated without DMI, now oscillations become visible, starting around 17 ns after the introduction of the external magnetic field in P

_{1}(Figure 7) and after around 11 ns in P

_{2}(Figure 8).

_{y}in the curved sample without DMI for the case s = 5 nm. As visible here, domain walls propagate to both ends, P1 and P2, of the curved nanowire, while reaching the left end P2 earlier than the right end P1, as also visible in the comparison of Figure 7 and Figure 8.

_{0}is equal to zero after an initial peak in the case of simulations without DMI, while all simulations with DMI show a small z-oscillation in position P

_{0}which can also be attributed to “echoes” of domain wall annihilations near this point. For a better visualization of these effects, movies of the magnetization dynamics of all four cases (with and without DMI, straight and curved samples) for s = 10 nm, 60 nm, and 500 nm are presented in the Supplementary Materials.

## 4. Conclusions and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The external rotating field was applied in rectangular regions of width 200 nm, height 10 nm, and adjustable lengths s of 1 nm, 5 nm, 10 nm, 20 nm, 30 nm, 40 nm, 50 nm, 60 nm, 100 nm, 200 nm, and 500 nm. At the square locations P

_{0}, P

_{1}, and P

_{2}with lateral dimensions 60 nm × 60 nm, the magnetization was detected. P

_{0}senses the external field directly. (

**a**) Straight nanofiber; (

**b**) bent nanofiber.

**Figure 2.**Magnetization dynamics in position P

_{0}, i.e., at the position of the applied external magnetic field, for a straight and a curved nanofiber, simulated without and with DMI, for s = 1 nm.

**Figure 3.**Magnetization dynamics in position P

_{1}, i.e., at the right end of the fiber, for a straight and a curved nanofiber, simulated without and with DMI, for s = 1 nm.

**Figure 4.**Magnetization dynamics in position P

_{2}, i.e., at the left end of the fiber, for a straight and a curved nanofiber, simulated without and with DMI, for s = 1 nm.

**Figure 5.**Magnetization dynamics of a straight nanofiber, simulated without DMI, for s = 1 nm. The figure shows the evolution in time of the magnetization component M

_{y}(from top to bottom).

**Figure 6.**Magnetization dynamics in position P

_{0}, i.e., at the position of the applied external magnetic field, for a straight and a curved nanofiber, simulated without and with DMI, for s = 5 nm.

**Figure 7.**Magnetization dynamics in position P

_{1}, i.e., at the right end of the fiber, for a straight and a curved nanofiber, simulated without and with DMI, for s = 5 nm.

**Figure 8.**Magnetization dynamics in position P

_{2}, i.e., at the left end of the fiber, for a straight and a curved nanofiber, simulated without and with DMI, for s = 5 nm.

**Figure 9.**Magnetization dynamics of a curved nanofiber, simulated without DMI, for s = 5 nm. The figure shows the evolution in time of the magnetization component M

_{y}(from left to right and top to bottom).

**Figure 10.**Magnetization dynamics in position P

_{1}, i.e., at the right end of the fiber, for a straight and a curved nanofiber, simulated without and with DMI, for s = 20 nm.

**Figure 11.**Magnetization dynamics in position P

_{1}, i.e., at the right end of the fiber, for a straight and a curved nanofiber, simulated without and with DMI, for s = 50 nm.

**Figure 12.**Magnetization dynamics in position P

_{2}, i.e., at the left end of the fiber, for a straight and a curved nanofiber, simulated without and with DMI, for s = 50 nm.

**Figure 13.**Magnetization dynamics of a curved nanofiber, simulated with DMI, for s = 500 nm. The figure shows the evolution in time of the magnetization component M

_{y}(from left to right and top to bottom).

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**MDPI and ACS Style**

Blachowicz, T.; Steblinski, P.; Ehrmann, A.
Influence of Physical Symmetries on the Magnetization Dynamics in Magnetic Fibers. *Symmetry* **2023**, *15*, 234.
https://doi.org/10.3390/sym15010234

**AMA Style**

Blachowicz T, Steblinski P, Ehrmann A.
Influence of Physical Symmetries on the Magnetization Dynamics in Magnetic Fibers. *Symmetry*. 2023; 15(1):234.
https://doi.org/10.3390/sym15010234

**Chicago/Turabian Style**

Blachowicz, Tomasz, Pawel Steblinski, and Andrea Ehrmann.
2023. "Influence of Physical Symmetries on the Magnetization Dynamics in Magnetic Fibers" *Symmetry* 15, no. 1: 234.
https://doi.org/10.3390/sym15010234