# Magnetic-field driven domain wall evolution in rhombohedral magnetostrictive single crystals: a phase-field simulation

*Microstructures*2024;4:2024052.

## Abstract

Single crystal of Tb_{0.3}Dy_{0.7}Fe_{2} (Terfenol-D) with a composition close to the pre-transitional rhombohedral side of the ferromagnetic morphotropic phase boundary has demonstrated remarkable magnetostrictive properties, stimulating intensive research interest in the field of magneto-mechanical transducers and actuators. The enhanced magnetoelastic response of (Tb-Dy)Fe_{2} single crystals has been extensively linked to the structural phase transition and magnetic domain evolution. This research utilized the micromagnetic microelastic phase-field technique to examine the evolution of domain walls in rhombohedral ferromagnetic single crystals of (Tb-Dy)Fe_{2}, which is essential for understanding the magnetostriction “jump” effect. The study involved simulating the creation and development of domains and domain boundaries under a periodic boundary condition that allows for non-zero strain. It was found that the two typical distinct types of domain walls (i.e., 71° and 109°) exhibited disparate responses to the applied magnetic fields. At magnetic field magnitudes below the coercive field, a domain wall broadening mechanism was detected within the 71° domain wall. However, upon surpassing the coercive field, a process of homogeneous magnetization switching ensued, devoid of evident displacement of the 71° domain walls. The magnetization switching effectively elucidated the magnetostriction “jump” effect of the rhombohedral single crystals. The act of sweeping the 109° domain walls resulted in the occurrence of heterogeneous magnetization switching. This study elucidates the evolutionary mechanism of two typical rhombohedral domain walls in response to applied magnetic fields, potentially offering valuable insights into the future design of excellent magnetostrictive materials through domain engineering.

## Keywords

*,*ferromagnetic MPB

*,*magnetostriction

*,*domain wall

## INTRODUCTION

Giant magnetostrictive materials, i.e., quasi-binary (Tb-Dy)Fe_{2} alloys, have attracted considerable attention due to their potential applications as energy conversion components in magneto-mechanical transducers and actuators^{[1-6]}. During the past decades, efforts have been made to develop an anisotropy compensation system that minimizes magnetic anisotropy in order to take advantage of the giant magnetostriction at low fields^{[7-10]}. Newnham^{[11]} firstly credits the spin reorientation boundary of (Tb-Dy)Fe_{2} as a magnetic equivalent to the morphotropic phase boundary (MPB) of a quasi-binary ferroelectric solid solution, i.e., the ferromagnetic MPB, which has been confirmed by Yang *et al.*^{[12]} and Bergstrom *et al.*^{[13]}. While the ferromagnetic MPB has demonstrated utility as a design approach for achieving highly sensitive magnetoelastic responses, the predominant focus in current research has been on materials exhibiting a rhombohedral structure, exemplified by Tb_{0.3}Dy_{0.7}Fe_{2} (Terfenol-D)^{[13,14]}. This is because of the highly anisotropic magnetostriction λ_{111} in the pre-transitional rhombohedral side of ferromagnetic MPB, in which giant magnetostriction could be induced by movement of non-180° domain walls or rotation of magnetic moments^{[1,14-16]}.

As is well known, the magnetization of the rhombohedral (Tb-Dy)Fe_{2} domains is expected to distribute equally along one of the eight <111> easy axes, as shown in Figure 1. The magnetostriction “jump” effect is primarily attributed to the motions of 109° and 71° domain walls under applied magnetic fields^{[17-19]}, although the underlying internal mechanism remains unclear. Several phenomenological approximations, such as domain wall motion and magnetization rotation models, have been proposed to elucidate the magneto-mechanical behaviors of Terfenol-D^{[8,20-23]}. However, the prevailing mechanisms largely overlook the presence of internal magnetic (stress) fields and assume that each domain evolves independently, which means no magnetic and elastic long-range interactions among domains are considered. Furthermore, the inherent stress resulting from elastic incompatibility in domain walls is of considerable significance, particularly in the case of giant magnetostrictive materials with large intrinsic strain. Since Khachaturyan’s microelasticity theory^{[24]} was introduced into the phase-field method, it has emerged as a robust technique for simulating the mesoscale microstructural evolution of ferroelectric systems^{[25-28]}. Zhang *et al*. have proposed a phase-field model that integrates the Khachaturyan microelasticity theory with micromagnetic simulation, enabling the prediction of domain structure stability and temporal evolution^{[29]}. Subsequently, comparable micromagnetic microelastic phase-field models have been developed to explore domain evolution and magnetoelastic response in proximity to the ferromagnetic MPB^{[15,16,30,31]}.

In this work, the micromagnetic microelastic phase-field method was employed to study the formation and evolution of domains in (Tb-Dy)Fe_{2} single crystals within the rhombohedral phase region of the ferromagnetic MPB. Particular emphasis was placed on the two distinct categories of domain walls, namely 71° and 109° domain walls. In the case of the 71° domain wall, the application of an external magnetic field below the coercive field leads to the observation of a domain wall broadening effect, while exceeding the coercive field results in domain switching. Additionally, the applied external magnetic field in the [100] direction leads to 109° domain walls sweeping, which results in heterogeneous magnetization switching.

## MATERIALS AND METHODS

The phase-field model employs the local magnetization vector **M** (*m*_{1}, *m*_{2}, *m*_{3}) as the primary order parameter, with its spatial distribution representing the magnetic domain microstructure. Therefore, the domain structure can be derived from the time evolution of the local magnetization configuration as governed by the Landau-Lifshitz-Gilbert (LLG) equation, denoted as

where *M _{s}* represents the saturation magnetization, and

*α*and

*γ*denote the damping constant and gyromagnetic ratio, respectively.

_{0}The effective magnetic field, denoted as *H** _{eff}*, is calculated by

**H**

*= -(μ*

_{eff}_{0}

*M*)

_{s}^{-1}(

*δE*/

_{tot}*δ*

**m**) with the vacuum permeability

*μ*and the total free energy

_{0}*E*. The total free energy, expressed as

_{tot}^{[1]}

is composed of the magnetocrystalline anisotropy (*E _{ani}*), exchange (

*E*), magnetostatic (

_{exc}*E*), elastic (

_{ms}*E*), and external energy (

_{el}*E*).

_{ext}The magnetocrystalline anisotropy energy can be defined as^{[20,32]}

where *K*_{1} and *K*_{2} represent the magnetocrystalline anisotropy coefficients, and *V* indicates the total volume of the system.

The exchange energy is determined solely by the spatial variation of the magnetization direction and can be written as^{[29]}

where spatial differentiation is denoted by a comma and *A* is the exchange stiffness constant.

The magnetostatic energy of a system can be denoted as^{[29,33]}

where **H**_{d} represents the stray field resulting from the long-range interaction between magnetic moments within the system. In the context of a periodic boundary condition, the simulation system is conceptualized as a repetitive building block within 3-D space, and the stray field can be expressed as

where **N*** _{D}* is the demagnetizing factor dependent solely on the sample’s shape. Additionally,

*ϕ*represents the magnetic scalar potential solved utilizing the Fourier spectral method under periodic boundary conditions.

The elastic energy resulting from local deformation can be given as^{[20,24,34]}

where *c _{ijkl}* represents the elastic stiffness tensor,

*e*denotes the elastic strain,

_{ij}*ε*signifies the total strain, and

_{ij}Where *λ*_{100} and *λ*_{111} are the magnetostrictive constants. Khachaturyan’s elastic theory^{[24]} posits that the total strain *ε _{ij}* can be expressed as the sum of homogeneous strain

where the stress component *σ _{ij}* is calculated using

Given the assumption of elastic equilibrium at each evolutionary step, the strain and stress values are determined by solving the mechanical equilibrium equation ^{[24]}. When a system is exposed to a homogeneous applied stress ^{[35]}.

The external energy resulting from the influence of an externally applied magnetic field, denoted as **H**_{ex}, can be established as^{[36]}

The LLG equation is employed to elucidate the progression of domain microstructure, and its numerical solution is achieved through the Gauss-Seidel projection method^{[37]}. The simulation focuses on the representative rhombohedral ferromagnetic single system, utilizing material parameters derived from a combination of experimental data and theoretical computations conducted previously^{[1,38-41]}, as outlined comprehensively in Table 1. Figure 1 provides a schematic illustration of the unit cell of (Tb-Dy)Fe_{2}, which is cubic. The process of domain formation was simulated using the phase-field method, with a dimension of 512Δ*x* × 512Δ*x* × 1Δ*x*. (The 3D phase-field simulation of the domain formation process can be seen in Supplementary Figure 1 in Supplementary Materials) The simulation grid Δ*x* is 2 nm, smaller than the exchange length ^{[42]}. A periodic boundary condition that accommodates non-zero strain is imposed along the three coordinate axes.

The material parameters of (Tb-Dy)Fe_{2} in the work

Parameters | Value | Units |

Saturation magnetization M_{s} | 8 × 10^{5} | A/m |

First-order anisotropy coefficient K_{1} | -6 × 10^{4} | J/m^{3} |

Second-order anisotropy coefficient K_{2} | -2 × 10^{5} | J/m^{3} |

Exchange constant A | 9 × 10^{-12} | J/m |

Elastic stiffness c_{11} | 1.41 × 10^{11} | N/m^{2} |

Elastic stiffness c_{12} | 6.48 × 10^{10} | N/m^{2} |

Elastic stiffness c_{44} | 4.87 × 10^{10} | N/m^{2} |

Magnetostrictive constant λ_{111} | 1,640 | ppm |

Magnetostrictive constant λ_{100} | 100 | ppm |

## RESULTS AND DISCUSSION

Figure 2 illustrates the evolution of domain formation within a representative volume element of the rhombohedral ferromagnetic single crystal, guided by energy minimization towards equilibrium. The initial state is characterized by a random distribution of magnetization, devoid of any predetermined assumptions. The local energy minimum, in conjunction with the presence of inhomogeneous internal stress, serves as the driving force for the initiation nucleation and subsequent growth of various domains. (The distribution of stress during the domain formation process can be seen in Supplementary Figure 2 in ^{[16,31]}. Subsequently, the tetragonal variants gradually disappear as the studied component is in the rhombohedral phase side. In the rhombohedral variants intermediate stages, as shown in Figure 2D and E, the domains of the rhombohedral phase form twins of either {100} or {110} twin planes, where the twin boundaries are 109° and 71° ferroelectric domain walls, respectively. As the evolution of rhombohedral domains progresses, there is a gradual disappearance of 71° domain walls, ultimately resulting in the survival of two 109° domain walls in the final equilibrium state. It is worth noting that Figure 2H just depicts one of the ideal states of global energy minimization. For the (Tb-Dy)Fe_{2} single crystals, 71° and 109° domain walls usually coexist to form complex multi-domain patterns, which collectively affect the magnetoelastic response. To gain deeper insights into the features of both domain walls and their implications for the properties of rhombohedral ferromagnetic single crystals, the detailed evolution behavior of these domain walls under magnetic fields oriented along the

Figure 2. Domain formation process of the rhombohedral (Tb-Dy)Fe_{2} single crystal. The colors correspond to the intensity of magnetization along the z direction, while the black arrows indicate the orientations of magnetization.

We first study the evolution of the 71° domain wall below the coercive field. As shown in Figure 3, the initial configuration of the simulation is a domain structure containing four 71° domain walls, which consist of [111] and _{[110]} and M_{[001]}, respectively, are also plotted in Figure 3. When the applied magnetic field increases along *t* = 0 to *t* = T/4. When the applied magnetic field increases along [110] direction, which is in the same direction of the magnetization, the 71° domain wall becomes wider, as shown from *t* = T/2 to *t* = 3T/4. The domain walls reach their maximum thickness (*t* = T/4) and minimum thickness (*t* = 3/4 T) as the field approaches the positive and negative peaks, respectively. The magnetic field dependence of M_{[110]} also demonstrates the narrowing and broadening of the 71° domain walls. Specifically, there is no domain switching throughout the whole process. The value of M_{[001]} stays constant, indicating that the center of 71° domain walls remains motionless.

Figure 3. The broadening effect of the 71° domain walls occurs when the applied magnetic field is below the coercive field. The domain wall is magnified to illustrate the fluctuations in its thickness.

Magnetization switching is observed upon surpassing the coercive field, as depicted in Figure 4. When the applied magnetic field in the _{[110]} value transitions to a negative state, accompanied by the [111] and *t* = T/2, as the applied field diminishes to zero, the _{[110]} undergoes a positive jump, accompanied by the *t* = T, the [111] and _{[001]} curve indicates the complex domain switching (the domain switching can be seen in Supplementary Figures 3 and 4 in Supplementary Materials), which is different from the homogeneous polarization switching observed in ferroelectric domains^{[43]}. The magnetic domain switching near *t* = T/4 and *t* = 3T/4 additionally induces a sudden alteration in the magnetostriction of the single crystal (the magnetostriction can be seen in Supplementary Figure 5 in Supplementary Materials). Throughout the aforementioned domain switching, the position of the 71° domain walls remains constant.

Figure 4. Magnetization switching occurs upon surpassing the coercive field. The domains before and after switching are depicted on the left.

Figure 5 shows magnetic field dependence of the 109° domain wall, which consists of _{[100]} gradually increases from the beginning to step = 7,000, corresponding to the linear movement of 109° domain walls. After that, the curves of M_{[100]} and M_{[010]} vibrate violently and remain at constant values. This phenomenon arises due to the convergence and subsequent merger of the two 109° domain walls, leading to their eventual disappearance. The ^{[16]} to tailor the magnetic domain morphology with enhanced magnetostrictive properties.

## CONCLUSIONS

This study utilized micromagnetic microelastic modeling through the phase-field method to investigate the process of domain formation and the evolution of domain walls in (Tb-Dy)Fe_{2} single crystals situated in the vicinity of the rhombohedral region of the ferromagnetic MPB. Particular emphasis was placed on the 71° and 109° domain walls, as their alterations in domain structure under applied magnetic field are crucial to the magnetelastic response of giant magnetostrictive materials. In the case of low applied magnetic field, a phenomenon of domain wall broadening was noted on the 71° domain walls, whereas in the case of high applied magnetic field, homogeneous magnetization switching took place without any observable movement of domain walls. The magnetization switching also helps understand the magnetostriction "jump" effect of the rhombohedral single crystal. The act of sweeping the 109° domain walls resulted in the occurrence of heterogeneous magnetization switching through the movement of domain walls. The detailed analysis of the two domain evolution mechanisms provided insightful understanding into the engineered domain structures of rhombohedral (Tb-Dy)Fe_{2} single crystals. Furthermore, these findings offer valuable guidance for the future design of magnetostrictive materials through domain engineering.

## DECLARATIONS

### Authors’ contributions

Conception and design of the study: Hu CC, Zhang Z, Xu YX

Data analysis and interpretation: Xu YX, Hu CC, Dong SZ, Huang HH, Li W

Manuscript writing and revising: Xu YX, Cai TT, Hu CC, Zhang Z, Li W

Simulation guide: Huang HB, Rao WF, Chen LQ

Supervision: Hu CC, Zhang Z, Rao WF

### Availability of data and materials

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### Financial support and sponsorship

This work was financially supported by the National Natural Science Foundation of China (No. 52301088), the Science Foundation of Shandong Province, China (No. ZR2022ME030 and No. ZR2020QE028), the National Natural Science Foundation of China (No. 12204215 and No. 12174210), the Research Foundation of Liaocheng University (No. 318012119). Chen LQ is the owner of Mu-PRO LLC, which licensed the computer codes for generating the phase-field results from the Penn State Research Foundation.

### Conflicts of interest

Chen LQ is the owner of Mu-PRO LLC, which licensed the computer codes for generating the phase-field results from the Penn State Research Foundation, while the other authors have declared that they have no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2024.

### Supplementary Materials

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## Cite This Article

## How to Cite

Xu, Y. X.; Cai T. T.; Hu C. C.; Zhang Z.; Dong S. Z.; Huang H. H.; Li W.; Huang H. B.; Chen L. Q.; Rao W. F. Magnetic-field driven domain wall evolution in rhombohedral magnetostrictive single crystals: a phase-field simulation. *Microstructures.* **2024**, *4*, 2024052. http://dx.doi.org/10.20517/microstructures.2023.104

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