Origin of sawtooth domain walls in ferroelectrics

TL;DR

This paper investigates the origin of sawtooth-shaped domain walls in ferroelectric materials like BiFeO3 and PbTiO3, revealing that the interplay between Coulomb and short-range interactions causes their formation and influences their periodicity.

Contribution

A minimal model combined with Monte-Carlo simulations explains the formation and periodicity of sawtooth domain walls in ferroelectrics, highlighting the role of competing interactions.

Findings

Competition between Coulomb and short-range interactions causes sawtooth domain walls.

Relative strength of interactions determines the periodicity of the domain walls.

Conditions for the formation of such domain walls are identified.

Abstract

Domains and domain walls are among the key factors that determine the performance of ferroelectric materials. In recent years, a unique type of domain walls, i.e., the sawtooth-shaped domain walls, has been observed in BiFeO$_{3}$ and PbTiO$_{3}$. Here, we build a minimal model to reveal the origin of these sawtooth-shaped domain walls. Incorporating this model into Monte-Carlo simulations shows that (i) the competition between the long-range Coulomb interaction (due to bound charges) and short-range interaction (due to opposite dipoles) is responsible for the formation of these peculiar domain walls and (ii) their relative strength is critical in determining the periodicity of these sawtooth-shaped domain walls. Necessary conditions to form such domain walls are also discussed.

Full text

Supplemental Material

J. Zhang

School of Microelectronics & State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China

  

Y.-J. Wang

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Wenhua Road 72, 110016 Shenyang, China

  

J. Liu

State Key Laboratory for Mechanical Behavior of Materials, School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China

  

J. Xu

School of Microelectronics & State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China

  

D. Wang

dawei.wang@xjtu.edu.cn

School of Microelectronics & State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China

  

L. Wang

Electronic Materials Research Laboratory, School of the Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China

  

X.-L. Ma

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Wenhua Road 72, 110016 Shenyang, China

  

C.-L. Jia

School of Microelectronics & State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China

Ernst Ruska Center for Microscopy and Spectroscopy with Electrons, Research Center Jülich, D-52425 Jülich, Germany

  

L. Bellaiche

Department of Physics and Institute for Nanoscience and Engineering, University of Arkansas Fayetteville, Arkansas 72701, USA

(March 13, 2024)

In this supplemental material we show that adding another factor to the minimal model will make our results compare even better to experiments. This further confirms that our model indeed captures the most important factors that give rise to the sawtooth domain walls.

We have shown that the the minimal model shows that inclined domain walls can develop in ferroelectrics, which can turn into sawtooth domain walls when they encounter interfaces or interacting with other domain walls. The reason is that the inclined flat domain wall will eventually contact with the interface resulting in effective charges accumulating along a straight line, which has higher energy than the swatooth domain wall. However, in the extreme case that the thickness is much larger than its width (which is not very likely ), the inclined flat domain wall can indeed be seen in the 2D simulations as seen in Fig. 1.

To address this issue, other factors need to be considered and added on top of those adopted in the minimal model. The most likely factor, in our opinion, is that the Coulomb interaction range is limited due to screening imposed by the dipoles in the background. In fact, Zhou et al indicate that the locking of the inclined flat domain wall may contribute to the formation of the zig-zag domain wall (Zou2018, ), which hints toward the fact that far-away domain walls develop independently of each other and become zig-zag when they eventually meet and interlock. This observation supports the idea to use a Coulomb interaction of limited range.

As a matter of fact, while the minimal model has absorbed dipoles to the background, they still exist in the real system and reorient slightly in accordance with the electric field they experience. Such dipole dynamics, while small, could effectively screen the interaction between bound charges, resulting in short-range Coulomb interaction (e.g., like the Yukawa potential). Furthermore, bound charges can attract free charge carriers or charge defects, which can compensate the bound charge, effectively reducing the interaction between the bound charges (Rojac2017, ). To address such effects, we can add additional a factor to our model, i.e., using a finite Coulomb interaction range.

Figure 2 the sawtooth domain walls in 2D after limiting Coulomb interaction range. In this simulation, we have used a $500\times 200$ simulation box, and carry out 320,000 sweeps of MC simulation at 300 K. Here we still use the parameters in the manuscript for the simulation. The short-range interaction parameter is taken to be $J=0.00586$  Hartree. The effective bound change $Z=1.16\left|e_{0}\right|$ and the relative permittivity $\varepsilon_{r}=7.164$ are used in this simulation, similar to those used in the main text. However, in this simulation, we have limit the Coulomb between two bound charges to 15$a_{0}$ With such a model, The simulation results clearly show that the sawtooth domain wall can exist stably in the system, not affected by the size of the simulation box. However, we have verified that the domain wall morphology, and the period in particular, is affected by the interaction strength and the range of the Coulomb interaction specified.

For the 3D case, we empoly the Ewald method and periodic boundary conditions to perform the simulation in a $40\times 40\times 40$ box with the same parameters as in the 2D case. In addition, we also set the limitation on the Coulomb interaction range to 10$a_{0}$ (since the simulation box size is smaller than the 2D case). After 30,000 Monte Carlo sweeps (i.e., $30000\times 40^{2}$ attempts to move the domain wall), the simulation results show domain walls converges to a series of conical peaks as shown in Fig. 3(a). Figure 3(b) shows a cross section of the domain wall in the $y$-$z$ plane at $x=23$ and Fig. 3(c) shows a cross section o shows a projection of 10 $y$-$z$ layers along the $x$ direction (to compare to the results shown in the main text) where the position of the domain wall is determined by averaging the positions in those 10 layers. Both of Fig. 3(b) and (c) clearly show sawtooth domain walls.

An important feature for the 3D case is that, whether we cut the sample along $x$ or $y$, the sawtooth wall will always appear. As a matter of fact, Jia et al have indicated that the striped domains are filled with sawtooth domain walls that can be seen along different projection directions (Jia2015, ), supporting our simulation results. In addition, Fig. 4 shows a similar projection as Fig. 3(c), but also records the averaged dipole values with negative (positive) values indicate dipoles pointing upward (downward), which may be used to explain the smaller dipoles seen on the domain wall as seen in experiments (Jia2015, ). Finally, we have also tried simulation boxes with different sizes, obtaining qualitatively similar results.