# Periodic boundary conditions for the simulation of 3D domain patterns in tetragonal ferroelectric material

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## Abstract

Periodic domain patterns in tetragonal ferroelectrics are explored using a phase field model calibrated for barium titanate. In this context, we discuss the standard periodic boundary condition and introduce the concept of reverse periodic boundary conditions. Both concepts allow the assembly of cubic cells in accordance with mechanical and electrical conditions. However, application of the reverse periodic boundary condition is due to an increased size of the RVE and enforces more complex structures compared to the standard condition. This may be of particular interest for other multiphysics simulations. Additionally, we formulate mechanical side conditions with minimal spherical (hydrostatic) stress, or conditions with controlled average strain. It is found that in sufficiently small periodic cells, only a uniform single domain, or the simplest stripe domains constitute equilibrium states. However, once the periodic cells are of order 20 domain wall widths in size, more complex, 3-dimensional patterns emerge. Some of these patterns are known from prior studies, but we also identify other domain patterns with long, ribbon-like domains threaded through them and some vortex-like structures.

## Keywords

Periodic boundary conditions RVE simulation Tetragonal ferroelectrics Polarization patterns Polarization vortex Multiphysics simulations## 1 Introduction

Tetragonal ferroelectrics such as barium titanate, \(\hbox {BaTiO}_{3}\), are characterized by a spontaneous electric polarization lying parallel to the pseudo-cubic axes of the crystal. On cooling \(\hbox {BaTiO}_{3}\) through the Curie temperature, \(T_{c}\cong 120\,^{\circ }\)C, the material gains a spontaneous polarization and transforms its crystal structure from the cubic, paraelectric to the tetragonal, ferroelectric state. This phase transformation induces lattice strains and, depending on the surrounding constraints, it causes deformation or internal mechanical stress [1, 2]. During the phase transformation, regions with uniform electric polarization group together, forming domains separated by narrow domain walls.

According to Maxwell’s equations, domains usually arrange with electrically compatible ‘head-to-tail’ polarization, which minimizes energy due to charged domain walls or strong compensating electric fields [3, 4]. Similarly, domain patterns minimize internal mechanical stress through compatible arrangements of spontaneous strain [5]. This results in the well-known \(90^{\circ }\) and \(180^{\circ }\) domain walls in tetragonal ferroelectrics.

Energy minimization provides a powerful method for understanding how domain patterns form. Since the domain walls are typically less than a few nm in thickness, a sharp interface approximation is attractive [6]. From this approach and from observation of micro- or nano-structure, many domain patterns have been identified [3, 5, 7, 8, 9, 10]. The simplest known arrangements are bundles of \(90^{\circ }\) and \(180^{\circ }\) domains forming laminates of alternating stripes. In thin films or lamellae, the domain bundles are commonly polarized in-plane [11]. However, thicker plates or bulk materials can form energy minimizing patterns with domains polarized in all three axial directions [3, 12]. The formation of periodic or nearly periodic patterns of domains is commonly observed; the effect of specimen size and mechanical strain on the formation of such periodic domain patterns has also been widely reported [4, 12, 13, 14, 15]. The spatial distribution of domains affects ferroelectric properties at both fine scale and macroscale [8, 11, 16, 17, 18, 19, 20, 21]. Rödel [21] discussed the dependence of piezoelectric coefficients on the volume fractions of domains, while Weng and Wong [22] showed that the macroscale ferroelectric behaviour is enhanced in certain laminated domain patterns. There is great interest in the domain patterns because of the potential to exploit engineered configurations of domains for energy harvesting, actuation, sensing, and memory devices [23, 24, 25, 26, 27].

Significant progress in understanding domains has been made with relatively simple models. However, results based on a sharp interface approach neglect the details of domain walls and their interactions, which become important when the size of domains is comparable to the domain wall width. In such cases, the formation of domain patterns and their stability has not been fully explored. Phase field models, with an order parameter that varies across the domain walls, offer an opportunity to study nanoscale domain patterns with a physically realistic representation of domain wall energy, bulk electromechanical energy, and the effects of external electromechanical loading. The use of phase field models for ferroelectric domains is well-established [28, 29, 30, 31, 32, 33]. Several studies explore the evolution of polarized domains in thin films [34, 35, 36, 37], under the effect of electromechanical loads. These studies explore the local effects of substrate strain on domain shapes [36], domain wall movement [34] and the coercive field [37]. In ferroelectric-bulk, the 3-dimensional effect of the electromechanical boundary conditions influences the formation of equilibrium domain patterns. 3D periodic laminates with microstructural features such as curved domain walls [5, 15] and ribbon-shaped domains [9] have been experimentally observed. These 3D microstructural features evolve under external electromechanical loads and influence the nanoscale behaviour of ferroelectrics [9]. Exploring the spatiotemporal evolution of periodic polarization patterns in a theoretical framework would indicate the complexities in pattern formation and provide initial steps towards nanoscale experimentation. Recently, we used the same phase field model as in this work to explore nanoscale domain patterns [14]. Starting from the patterns predicted by energy minimization with sharp interfaces [5], it was found that several of the patterns dissolve into simpler, more stable arrangements such as single domains or alternating stripes. For computational speed, the calculations were mainly carried out using 2D models that limit the freedom of the system to relax into fully 3D domain patterns. Extending the study to consider periodic volumes large enough to support domain patterns makes heavy demands on computation, and the calculations reported in the present work typically took days to weeks each, running as parallelized code using 32 cores on an Intel Xeon CPU at 3.2 GHz.

The goal of this work is to establish and explore two kinds of periodic boundary conditions for scalar and vectorial fields at the bound of representative volume elements (RVE). This is motivated by the fact that periodic boundary conditions generally restrict the solution space. Thus, any RVE simulation may benefit from this discussion since our model demonstrates that differing periodic boundary conditions naturally lead to an enriched variety of periodic domain patterns. The principle of the reverse periodic boundary condition is an enhanced assembly rule, where we superpose translation and rotation.

We study a range of periodic cell sizes and then focus on a size just sufficient to allow complex domain patterns to form. Such complex patterns could be useful in memory applications where arrays of writable domains and skyrmion-like topologies [38] are of current interest. Periodic patterns may be artificially enforced by imposing a template or may arise naturally through minimization of energy. Nature, of course, has available any periodic cell size or none. The exact reason why nearly periodic patterns of domains commonly form is not known. It may be that similar conditions of stress or cooling rate during the phase transition favour the nucleation of particular domains that then adjust into a nearly periodic form to minimize energy. More likely, a small region of pattern first forms and then becomes the template for surrounding regions which copy the pattern. We show that cell sizes of at least tens of domain wall widths are necessary in order to support domain patterns other than simple stripes or single domains.

Further, we explore the effect of strained states on the formation of domain patterns. The study is motivated by a classification of low-energy periodic domain patterns given by Tsou et al. [5] in which several families of periodic domain patterns were identified. We employ a 3-D phase field model [39, 40], which is calibrated for barium titanate and reproduces the transition from an initial paraelectric state to the tetragonal ferroelectric state at the Curie point, whereupon domain patterns evolve to minimize internal energy. We do not force any particular pattern by imposing a template or by initial conditions: instead, the simulations take a randomly perturbed initial state and relax this towards equilibrium. Since it is impractical to simulate 3-dimensional regions much larger than a few tens of nm in size, we make use of periodic boundary conditions, which allow the assembly of simulated volumes into larger periodic cells. Additionally, the simulations are carried out with controls on the average strain, introduced as various side conditions. This allows for the effect of a remanent ferroelastic strain, appearing at the phase transition from the paraelectric to the ferroelectric state.

The paper is organized as follows: the underlying model is given in Sect. 2, followed by a discussion of periodic boundary conditions in Sect. 3. The results of our simulations are presented and discussed in Sect. 4, followed by conclusions in Sect. 5.

## 2 The material model

The minimum possible value for \(\Psi \) is given for the tetragonal crystal structure with homogeneous polarization \(P_{0}\) in a state free of stress or electric field. Then, the model yields \(\Psi _{0}=-0.5345\,E_{0}\,P_{0}\), where \(P_{0}=0.26\) Cm\(^{-2}\), is the remanent polarization of the relaxed domain and \(E_{0}=2.182\times 10^{7}\) Vm\(^{-1}\) is the coercive electric field of the single domain (which is much greater than the coercive field of polydomain material). The results given in Sect. 4 are normalized using \(\Psi _{0}\), \(E_{0}\) and \(P_{0}\).

*q*, the electric displacement satisfies

The majority of simulations represent cubic volumes of material, modelled with up to \(20^{3}=8000\) tri-quadratic finite elements of approximately 2 nm element size. This yields of the order of \(10^{6}\) degrees of freedom. Extra degrees of freedom are introduced at domain walls, as they form, by enriching elements with quartic shape functions wherever the gradients of \(\mathbf{{P}}\) exceed a critical magnitude; details of the enrichment method have been given by Muench and Krauß [44, 45]. At the initial stage of the simulation, random values of \(\mathbf{{P}}\) of magnitude \(0.01P_{0}\) are imposed on the system. This gives a perturbed near-zero polarization field corresponding to a state close to the maximum potential energy, representing the cubic state of the crystal. The subsequent evolution of domain patterns is due to the transformation from paraelectric to ferroelectric material at room temperature. Unless otherwise specified, the results show equilibrium states with \(\dot{\mathbf{{P}}}=0\). Equilibrium was checked at the end of each simulation by increasing the time increment \(\Delta t\) of the implicit time integration scheme by a factor of \(10^{6}\) relative to the time steps used during the simulation, and checking for any change in polarization.

## 3 Periodic boundary conditions

### 3.1 Standard periodic boundary condition

*I*and \(I+3\), the periodic boundary conditions on opposite faces \(\partial {{\mathcal {B}}}_{c}^{I}\) and \(\partial {{\mathcal {B}}}_{c}^{I+3}\) are then given by

### 3.2 Reverse periodic boundary condition

As with the standard periodic boundary condition, the reverse periodic boundary condition yields solenoidal polarization averaged over \({{\mathcal {B}}}_{c}\). Similarly, this boundary condition implies that the mean strain and electric field vanish over the extended unit cell. Thus, the assembly of cells \({{\mathcal {B}}}_{c}\) with reverse periodic boundary condition does not require any macroscopic electric field to stabilize it. Figure 2 illustrates the effect of the standard and reverse boundary conditions. The reverse periodic boundary condition encourages the formation of additional \(180^{\circ }\) domain walls and hence more complex domain patterns.

### 3.3 Side conditions for strained states

Next, we introduce mechanical side conditions to apply four different representative strained states. We use these representative strained states to explore the general trends in temporal evolution of polarization patterns and to explore the complexities of 3D-domain arrangements.

*c*axis aligned to \(\mathbf{{e}}_{1}\), \(\mathbf{{e}}_{2}\) and \(\mathbf{{e}}_{3}\), respectively. The cubic lattice constant \(a_{cub}\) is replaced by tetragonal lattice constants \(c_{tet}\) and \(a_{tet}\), respectively, giving remanent strain components

*c*axis aligns with the polarization \(\mathbf{{P}}\) in the minimum energy state. For \(\hbox {BaTiO}_{3}\), set \(a_{cub}=4.000\,\mathrm{\AA }\), \(c_{tet}=4,0328\,\mathrm{\AA }\), and \(a_{tet}=3.9892\,\mathrm{\AA }\), based on experimental data and theoretical results presented in [48]. Then, \(\varepsilon _{c}=0.0082\) and \(\varepsilon _{a}=-0.0027\) giving \(2\,\varepsilon _{a}+\varepsilon _{c}=0.0028\), which indicates that the unit cell undergoes a non-isochoric process if unconstrained. Thus, it is desirable for the simulations to allow an overall straining of \({{\mathcal {B}}}_{c}\), corresponding to displacement of the boundaries \(\partial {{\mathcal {B}}}_{c}\).

*c*axis aligned to \(\mathbf{{e}}_{1}\). By contrast, setting all \(f_{i}=0\), \(i=1\dots 3\), produces a state free of average strain, while setting all \(f_{i}=1/3\) produces an equiaxed volume dilation that matches the dilation of the cubic–tetragonal phase transformation.

**I**sochoric

**F**erroelectric

**P**hase

**T**ransition (IFPT). Thus, the volume of \({{\mathcal {B}}}_{c}\) is held constant during the simulation. A negative spherical stress is likely to arise to restrain \(\bar{{\varvec{\varepsilon }}}_{r}\) if the material evolves polarized domains such that volume fractions \(\lambda _{i}>0\).

**A**daptive

**S**pherical

**S**tress

**F**ree (ASSF). Note that ASSF may generally yield unequal strains along the three axes, but tracks any volume dilation accompanying the transformation, so that typically \(\mathrm{tr}[{\bar{{\varvec{\varepsilon }}}}]=(2\,\varepsilon _{a}+\varepsilon _{c})=0.0028\) at equilibrium.

**I**sotropic

**S**pherical

**S**tress

**F**ree (ISSF) case. This side condition tends to eliminate any average spherical stress.

**T**ransversely

**I**sotropic

**D**eformation (TID). Note that all four cases allow for locally inhomogeneous strain within \({{\mathcal {B}}}_{c}\). Further, the volume fractions \(\lambda _{i}\) need not generally coincide with imposed strain parameters \(f_{i}\), except for the case of ASSF, where there is continuous tracking to match these parameters.

The strained states might bear similarities to experimentally imposed mechanical strains. However, the spirit of the current work is to indicate the general trends for complexity and types of patterns that form in bulk ferroelectrics.

## 4 Results and discussion

Domain topologies resulting from the use of periodic boundary conditions in 3-dimensions, combined with each of the four strain conditions (IFPT, ASSF, ISSF, TID), are presented in this section. It is instructive first to consider the evolution of domains within a single simulation, then to explore the effect of the periodic cell size in the range 9-41nm on the topologies that may form. Finally, a cell size large enough to generate complex domain patterns (41nm) is selected and the domain topologies resulting from each set of boundary conditions are described.

### 4.1 Evolution of a 2-dimensional domain topology

Figure 3 shows a series of states of the simulation as it evolves towards equilibrium; each image shows four copies of the simulated volume \({{\mathcal {B}}}_{c}\) to illustrate the periodic continuation in the \(\mathbf{{e}}_{1}\) and \(\mathbf{{e}}_{3}\) directions. The reverse periodic boundary condition encourages (but does not enforce) the formation of a \(180^{\circ }\) domain wall normal to the \(\mathbf{{e}}_{3}\) direction.

In the early stages of the simulation, the local values of \(|\mathbf{{P}}|\) are much less than \(P_{0}\), but the energetic driving force for the formation of polarized regions rapidly produces domains. These initially nucleate in a random, high energy pattern, and gradually resolve sharply defined \(90^{\circ }\) and \(180^{\circ }\) domain walls. As can be seen from Fig. 3c–e, the domain pattern is complex as it evolves towards equilibrium, but eventually reaches a simpler, lower energy state.

The final equilibrium state shows a herringbone-type pattern that was also identified by Tsou et al. [5], using a sharp interface model, as a member of the family \(\{1314\}\) of compatible rank-2 domain patterns. In this case, the final pattern is essentially 2-dimensional, with the polarization vector always lying in the \(\mathbf{{e}}_{1}-\mathbf{{e}}_{3}\) plane; this was not forced, but rather was a natural outcome of the relaxation towards a minimum energy state. The same pattern was noted in earlier work using a phase field model, but in a 2-dimensional simulation with a 40nm periodic cell size [14]. However, in that study a fixed average strain was imposed to stabilize the pattern. Here, the adaptive strain condition allows the topology to form and adopt an average strain state that minimizes energy.

### 4.2 Cell size variation with standard periodic boundary conditions

In this section, the effect of periodic cell size by varying the cell size \(L_{c}\) from 9 to 41 nm is under investigation. Below about 9 nm size, the cell is too small to support multiple domains (domain wall widths in the model are in the region of 2–3 nm. Beyond 41 nm the computation becomes excessively cumbersome due to the scaling of the number of degrees of freedom in the problem. From our previous work on scale effects in polarization patterns [13], we expect that increasing the cell size will enable polarization patterns with multiple domain walls to form. However, in the current work we stop at 41nm cell size because it is sufficient for complex domain patterns to form within practical computation time.

As shown in Fig. 4, simple \(90^{\circ }\) domain stripes arise in most of the simulations. However, the TID constraint, which imposes a strain condition close to the free strain of the single domain, strongly encourages a monodomain to form. Generally, smaller cell sizes result in simpler domain patterns. Only the 41 nm cell is large enough that more complex domain patterns can form if this reduces the energy. Thus, for the subsequent study of domain topologies, the 41 nm cell size is employed.

To understand the meaning of \({\varvec{\alpha }}\) and \({\varvec{\beta }}\) in terms of domain topologies, note that for a simple monodomain \({\varvec{\alpha }}={\varvec{\beta }}=[100,0,0]\). Alternating \(180^{\circ }\) domains with equal volume fractions give \({\varvec{\alpha }}=[0,0,0]\), \({\varvec{\beta }}=[100,0,0]\), while alternating \(90^{\circ }\) domains with equal volume fractions give \({\varvec{\alpha }}=[50,50,0]\), \({\varvec{\beta }}=[50,50,0]\). Differences between \({\varvec{\alpha }}\) and \({\varvec{\beta }}\) indicate the formation of oppositely polarized regions. Values in \({\varvec{\alpha }}\) close to zero indicate that there are directions with little or no net polarization while a value in \({\varvec{\beta }}\) close to zero would indicate the presence of an axis to which few or no domains have their *c* axis aligned. Both measures are invariant to rotations or reflections of the periodic unit cell in the symmetry group of the cube. Table 1 shows the normalized Helmholtz free energy along with \({\varvec{\alpha }}\) and \({\varvec{\beta }}\) as percentage values for the equilibrium states, computed with 41 nm cubes.

The normalized overall free Helmholtz energy \({\overline{\Psi }}=\int _{{{\mathcal {B}}}_{c}}\Psi /L_{c}^{3}\,\mathrm{d}V\) is always negative, as the zero datum is the high energy, cubic, state. However, the ratio \({\overline{\Psi }}/\Psi _{0}\), where \(\Psi _{0}=-0.5345\,E_{0}\,P_{0}\) corresponds to the energy of a stress-free single domain, is a positive number less than 1, see Table 1. Imposing a strained state (IFPT, ISSF, TID) often does not lead to the single domain solution as can be seen from the \({\varvec{\alpha }}\), \({\varvec{\beta }}\) values; even when a monodomain does result, the energy is higher than that of the stress-free state. Allowing the periodic cell to adapt its strain as domains develop (ASSF) results in the lowest energy solutions, which may be a simple monodomain, but can develop a domain pattern if the random starting conditions nucleate one. The isochoric conditions (IFPT) result in the highest vales of free energy, as the periodic cell is rigidly constrained to match that of the cubic state.

### 4.3 Domain patterns simulated with standard periodic boundary conditions

Helmholtz free energy and volume fractions (\({\varvec{\alpha }}\), \({\varvec{\beta }}\)) for standard and reverse periodic boundary conditions (PBC) and strain controls in the \(L_{c}=41\) nm cubes. The parameters \(f_{i}\) define the imposed strain controls and \(\lambda _{i}\) tracks volume fractions of domains in the evolving simulation for the adapted spherical stress-free case (ASSF)

Standard PBC | Reverse PBC | ||
---|---|---|---|

IFPT | \({\overline{\Psi }}/\Psi _{0}\) | 0.520 | 0.472 |

\(f_{i}=0\) | \({\varvec{\alpha }}\) | 39, 30, 14 | 36, 0, 0 |

\({\varvec{\beta }}\) | 42, 31, 27 | 39, 33, 28 | |

ISSF | \({\overline{\Psi }}/\Psi _{0}\) | 0.642 | 0.588 |

\(f_{i}=1/3\) | \({\varvec{\alpha }}\) | 40, 39, 11 | 37, 0, 0 |

\({\varvec{\beta }}\) | 44, 40, 16 | 40, 31, 29 | |

TID | \({\overline{\Psi }}/\Psi _{0}\) | 0.602 | 0.565 |

\(f_{1}=1/2\) | \({\varvec{\alpha }}\) | 100, 0, 0 | 0, 0, 0 |

\(f_{2}=f_{3}=1/4\) | \({\varvec{\beta }}\) | 100, 0, 0 | 100, 0, 0 |

ASSF | \({\overline{\Psi }}/\Psi _{0}\) | 0.870 | 0.752 |

\(f_{i}=\lambda _{i}\) | \({\varvec{\alpha }}\) | 48, 48, 0 | 59, 0, 0 |

\({\varvec{\beta }}\) | 50, 50, 0 | 59, 39, 2 |

Close examination reveals that the laminate does not have perfectly planar interfaces between domains, but instead the intersections of domains form complicated angular shapes that result in vortex-like regions at the triple junctions where domains polarized along the \(\pm \,\mathbf{{e}}_{3}\) and \(-\,\mathbf{{e}}_{2}\) directions meet. There has been great interest in such vortex centres [11, 38, 49] because of their potential use in applications such as memory storage, and their interesting topological properties as skyrmions. The simulations here suggest that it may be possible to generate arrays of vortices by suitably constraining crystals during the cubic–tetragonal phase transition.

In summary, the model tends to form more complex domain patterns, including some of the vortex structures and ribbon-like domains shown in Figs. 5 and 6, if the mechanical side condition of periodic boundary conditions constrains the ferroelectric phase transition. The reason for this is that the constraint forces the material to produce a mixture of domains to match the imposed strain. In experimental work, the constraint can be imposed by the use of epitaxial growth with lattice mismatch. This method has been used in strontium titanate, for example, to provide biaxial tensile stresses that induce incipient ferroelectricity [52]. By contrast, our work suggests that lattice mismatch or stress gradients in the sample could be used in thin films of ordinary ferroelectrics to promote complex domain structures, or to encourage specific patterning of domains [53]. For example, a stress gradient generated by an atomic force microscopy tip was used to mechanically write polarization patterns on a barium titanate thin film [54].

### 4.4 Domain patterns simulated with the reverse periodic boundary condition

*c*axis aligned parallel to the \(\mathbf{{e}}_{1}\), \(\mathbf{{e}}_{2}\) and \(\mathbf{{e}}_{3}\) directions, as indicated by \({\varvec{\beta }}=[39,33,28]\). Vortex centres are evident at the junctions of the \(\mathbf{{e}}_{3}\) and \(\pm \,\mathbf{{e}}_{2}\) domains in Fig. 9d.

Further simulations using the isotropic spherical stress-free side condition (ISSF) yield similarly complex patterns, see Fig. 10. These domain patterns have many \(180^{\circ }\) domain walls, along with ribbon-like domains [50] that thread through the structure. In both cases, five of the six polarization variants are present, which can be understood as follows: the strain condition enforces the presence of some domains with *c* axis aligned to each of the three coordinate axes. However, polarization must balance in both the \(\pm \,\mathbf{{e}}_{1}\) and \(\pm \,\mathbf{{e}}_{2}\) directions due to the rotation of the cube during assembly, so all four variants polarized in the \((x_{1}, x_{2})\) plane must be present. Meanwhile, since some domains with *c* axis aligned to the \(\mathbf{{e}}_{3}\) are expected at least five variants are likely to be present overall. From the methods employed by Tsou et al. [5, 6], it is immediately evident that the pattern formed cannot be any laminate of rank less than 3. The patterns shown appear similar to rank-3 laminates, but with imperfect alignment of domains and curved domain walls. The equilibrium domain patterns in Figs. 9 and 10 both exhibit centres of vorticity and ribbon-like domains, with five of the six polarization variants present. Close inspection reveals that they are related variants. This is reflected in their similar \({\varvec{\alpha }}\) and \({\varvec{\beta }}\) values (see Table 1).

The result of applying the adapted spherical stress-free case (ASSF) and reverse boundary condition on the 41 nm cube is the same as that shown in Fig. 3g for the 33 nm cube. Thus, there is no need for an additional figure.

### 4.5 Impact of an outer electric field

In the initial stages, up to \(E=0.08E_{0}\) the smaller domains with polarization in the \(-\,{\mathbf {e}}_{3}\) direction shrink in size and disappear. However, the domains with polarization in \({\mathbf {e}}_{1}\) and \(-\,{\mathbf {e}}_{2}\) direction remain such that the ribbon-like domain pattern is stable up to \(E=0.64E_{0}\), beyond which a striped domain pattern with \({\mathbf {e}}_{1}\) and \({\mathbf {e}}_{3}\) domains forms. On increasing the electric field up to \(E=E_{0}\), the domains with polarization along \({\mathbf {e}}_{1}\) direction remain in the polarization pattern. We interpret this polarization pattern as arising from the competing isochoric mechanical side condition (IFPT) and the applied electric field. At \(E=E_{0}\), the IFPT constraint stabilizes the \({\mathbf {e}}_{1}\)-domain in the presence of opposing external field.

However, at \(E=1.21E_{0}\), the stripe pattern dissolves to form a monodomain with net polarization along \({\mathbf {e}}_{3}\)-axis. The net polarization in the domain pattern decreases during the transition from the stripe pattern to a monodomain. We associate this decrease in net polarization to result from the strong IFPT mechanical constraint. The mechanical strains modelled in the transverse direction to the applied electric field, force the monodomain away from its relaxed domain state.

## 5 Conclusions

In this work, we have applied multiple periodic boundary conditions for the simulation of ferroelectric domain patterns on the nanoscale. For the well-known standard conditions, our solutions tend to evolve patterns with \(90^{\circ }\) domain walls only, if the RVE is allowed to adopt the inherent volumetric growth resulting from the materials phase transition from the cubic to the tetragonal crystal state. This result has been observed for three important variants (ASSF, ISSF, TID) of many more possible mechanical side conditions. However, we have also found a mechanical side condition enforcing \(180^{\circ }\) domain walls. It restricts the RVE concerning its isochoric deformation, which we have denoted the isochoric ferroelectric phase transition (IFPT).

Since many experimental works do not restrict the isochoric deformation of the crystal during its ferroelectric phase transition, and since they have found ferroelectric patterns with \(180^{\circ }\) domain walls, we have proposed the reverse periodic boundary condition, which naturally enforces \(180^{\circ }\) domain walls within the RVE by an enhanced continuation rule of the cells. This extends the capabilities of our simulations and reproduces some domain patterns that are well known from prior theoretical studies and from observations of \(\hbox {BaTiO}_{3}\).

The reverse periodic boundary condition might be of interest for similar simulations in multiphysics; we just discuss one specific application here. But even for pure mechanical problems it widens the solution space of the displacement field. We refer to Fig. 8a, where the exaggerated deformation of the equilibrium state is deliberately shown. Obviously, displacement fields exist such that the assembly of cells requires the superposition of translation and rotation. To the best knowledge of the authors, the reverse periodic boundary condition is novel in the context of RVE and ferroelectric simulation models.

For both kinds of periodic boundary conditions, the simulations also exhibited interesting features such as frustrated domain walls, curved ribbon-like domains, and centres of vorticity. Some of these features can be of use in ferroelectric devices, and the simulations suggest that, by controlling strain during the cubic-ferroelectric transition, it may be possible to generate arrays of domains with these exotic features.

## Notes

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