Periodically-arrayed ferroelectric nanostructures induced by dislocation structures in strontium titanate

Kairi Masuda*a, Le Van Lichb, Takahiro Shimadaa and Takayuki Kitamuraa
aDepartment of Mechanical Engineering and Science, Kyoto University, Nishikyo-ku, Kyoto 615-8540, Japan. E-mail:
bSchool of Materials Science and Engineering, Hanoi University of Science and Technology, No 1, Dai Co Viet Street, Hanoi, Vietnam

Received 25th July 2019 , Accepted 23rd September 2019

First published on 24th September 2019

A dislocation induces ferroelectricity around it in incipient ferroelectric SrTiO3 due to some reasons such as electro-mechanical coupling and it being a one-dimensional ferroelectric nanostructure. Furthermore, this microstructure is arrayed periodically in the material and dislocation structures such as a dislocation wall are formed. Due to these facts, periodically-arrayed ferroelectric nanostructures, which show various intriguing polarization configurations and functionalities depending on the internal periodic structure, may be fabricated by dislocations. The phase-field simulation exhibits that a ferroelectric nano-region induced by the strain concentration and incidental electric field around a dislocation connects with each other in a dislocation wall. As a result, a periodic ferroelectric nano-region, which is a periodically-arrayed ferroelectric nanostructure embedded in paraelectric matrices, is formed. Our findings provide a new pathway for the fabrication of novel functional nanodevices in ferroelectric systems.

1 Introduction

In nanoscale ferroelectrics, polarization tends to align along the material surface because of the depolarization field.1 Due to this geometric confinement, specific polarization configurations, which cannot be stabilized at the macroscale, are formed in ferroelectric nanostructures and intriguing phenomena are realized.2–4 For example, polarization vortices, which are stabilized in nanodots,5–7 exhibit a new order parameter, or toroidal moment. In particular, periodically-arrayed ferroelectric nanostructures, which have periodic internal structures such as nanoporous ferroelectrics, exhibit novel polarization configuration such as micro-vortices in a macro-vortex,8–10 which is difficult for isolated counterparts, because of periodicity besides the geometric confinements. More importantly, several unprecedented functionalities are realized including unusual piezoelectricity,11 multilevel hysteresis loops,12 and the coexistence of ferroelectricity and ferrotoroidicity.8 The use of rationally designed ferroelectric nanostructures provides a means for achieving new functionalities and novel future nanodevices.

In recent years, novel synthesis methods for ferroelectric nanostructures have been developed by engineering lattice defects and their microstructures. Nanoscale ferroelectricity is experimentally observed around a dislocation in SrTiO3 despite this material being in the paraelectric state in the absence of dislocations.13 The region of dislocation-induced ferroelectricity is considered as a one-dimensional ferroelectric nanostructure surrounded by a paraelectric material. Moreover, since dislocations sometimes are arrayed periodically in the material, e.g. a dislocation wall,14,15 even in SrTiO3,16 the distribution of dislocation provides a new degree of freedom to tailor the ferroelectric nanostructures emerging in SrTiO3. On the other hand, due to electro-coupling, flexoelectricity and nonstoichiometry as mentioned later, polarization configurations that appeared in SrTiO3 with multiple dislocations may possess distinguished characteristics, which cannot be mimicked by artificial ferroelectric nanostructures with the control of outer shapes.

However, the previous studies mainly focus on ferroelectricity around a single dislocation both in the experiment13 and numerical simulation,17 and those of collective dislocations are not clarified. Furthermore, the formation mechanism of ferroelectricity around dislocations in SrTiO3 has not been fully understood. It is well-known that ferroelectricity can be triggered in SrTiO3 by a large enough strain due to the electro-mechanical coupling and polarization due to displacements of cations (Sr2+, Ti4+) and anions (O2−) appearing along the strain direction.18–20 Besides this coupling, polarization can also be induced by flexoelectricity21 and nonstoichiometry such as Ti-antisites,22 and these effects appear simultaneously near dislocations because of strain concentration, large strain gradients, and atomic disorders.13

In this study, we investigate the ferroelectricity around a single dislocation, dislocation dipoles and a dislocation wall in SrTiO3 by using phase-field calculations, for simplicity, excluding flexoelectricity and nonstoichiometry. Through these calculations, we clarify the ferroelectricity by an assembly of dislocations and its formation mechanism.

2 Methods

2.1 The strain field induced by dislocations

The eigen-strain of a dislocation loop is described as
image file: c9cp04147h-t1.tif(1)
where bi, d, and ni are the Burgers vector, the interplanar distance of the slip plane and a unit vector normal to the dislocation loop, respectively. δ(Sr) is the Dirac delta function with S being the surface surrounded by the dislocation loop and r being the spatial vector. In the Fourier space, the relationship between the eigen-strain and the elastic strain due to the dislocation loop without homogeneous strain [small epsilon, Greek, macron]ij is described as23
image file: c9cp04147h-t2.tif(2)
image file: c9cp04147h-t3.tif(3)
where I is an imaginary number and q is a unit vector in the reciprocal space. [small epsilon, Greek, tilde]d,eigenmn(q) and [small epsilon, Greek, tilde]d,elasmn(q) are the Fourier transform of εd,eigenmn(r) and εd,elasmn(r). Gim(q) is the Green tensor whose inverse tensor is defined as Gim−1(q) = cijmlqjql with cijml being the elastic stiffness tensor. By the inverse Fourier transformation of [small epsilon, Greek, tilde]d,elasmn(q), the dislocation-induced elastic field in the real space εd,elasij(r) is obtained. The eigen-strain of multiple dislocations is represented by a superposition of each dislocation loop, that is, image file: c9cp04147h-t4.tif.24 Thus, elastic strain fields induced by arbitrarily distributed dislocations are obtained by eigen-strain of dislocation loops.

2.2 The phase-field modeling with dislocations

The phase-field modeling successfully reproduces the above-mentioned phase transitions in SrTiO3 caused by the electro-mechanical coupling25–27 and interactions between microstructures and ferroelectricity.24,28–30 Taking the polarization p = (p1, p2, p3) as the order parameter, the total free energy F can be described as
image file: c9cp04147h-t5.tif(4)
where fbulk, felas, fgrad, and felec are the Landau energy density, elastic energy density, gradient energy density, and electrostatic energy density, respectively. Detailed expressions of energy densities are presented in previous studies25–27,31 and in the ESI.

Note that, due to the superposition of elastic strain εelasij(r), the elastic energy density cijklεelasij(r)εelaskl(r)/2 with dislocations is described as28

image file: c9cp04147h-t6.tif(5)
where εp,elasij(r) is elastic strain induced by polarization. Thus, by calculating an elastic field of dislocations from the eigen-strain and incorporating it into the elastic energy density, the effect of dislocations on polarization is realized in the phase-field modeling.

The temporal evolution of polarization is calculated by the time-dependent Ginzburg–Landau (TDGL) equations,

image file: c9cp04147h-t7.tif(6)
where t and Lp represent time and the kinetic coefficients related to the domain mobility for polarization, respectively. δFpi(r,t) denotes the thermodynamic driving force for polarization evolution. It should be noted that, in the phase-field modeling of this study, the effects of flexoelectricity, nonstoichiometry, and antiferrodistortive are not included although they are observed around a dislocation in the experiments13 because our concern in this study is polarization induced by dislocations through the electro-mechanical coupling.

Larger dislocation structures can be formed32–34 as shown in Fig. 1 and ferroelectricity might be induced around them. To systematically investigate the ferroelectricity around dislocations, we firstly consider the polarization distribution around an edge dislocation (Fig. 1(a)). This dislocation type is an atomic disorder with the extra half-plane at the upper side as shown in Fig. 1, and widely seen in SrTiO3 and other perovskite structures.35–37 Based on the experimental observations,38–40 the Burgers vector and slip plane are [100] and (010).

image file: c9cp04147h-f1.tif
Fig. 1 (a) The schematic illustration of assemblies of dislocations and ferroelectricity expected to be induced around them represented by a colored region. (a) A single dislocation, (b) dislocation dipole: (i) dipole A and (ii) dipole B, and (c) dislocation wall.

Secondly, we employ the simplest dislocation structure, or dislocation dipoles (Fig. 1(b)) to understand the emergence of ferroelectricity around dislocation structures. Due to the lack of the experimental observation of SrTiO3, we refer to the observation of Ba0.3Sr0.7TiO3,41 that is, the burgers vectors and slip planes are [±100](010), and the distance between two dislocation cores and angle are 14 nm and 45 degrees, respectively. Here, we calculate two dislocation dipoles, that is, dipole A in which the extra half-planes face apart, and dipole B, in which the extra half-planes face each other. This is because how the extra-half planes face affects the elastic field around dipoles and may affect ferroelectricity.

Finally, we investigate the polarization distribution around a dislocation wall (Fig. 1(c)). In SrTiO3,16 dislocation walls are experimentally observed although they are composed of [10−1](101) dislocations. For simplicity, however, we consider the dislocation wall composed of [±100](010) dislocations, using the concept of the Taylor–Nabarro lattice,42–44 which is the idealization of the dislocation wall. Here, we employ the same distance and angle as dislocation dipoles to compare the results of dipoles and walls.

The spontaneous formation of polarization around the dislocation structures is carefully calculated by phase-field modeling using the semi-implicit Fourier-spectral method.45,46 Details of the simulation models and the simulation procedures are found in the ESI.

3 Results and discussion

3.1 Ferroelectricity around an edge dislocation

To understand the ferroelectricity around dislocations, we start from the strain field around an edge dislocation. Fig. 2(a) shows the strain distribution obtained from the eigen-strain. A compressive strain appears on the side of the slip plane while a tensile strain is on the other side symmetrically. In thin film SrTiO3, ferroelectricity is induced by a tensile strain as large as about ε11 = 0.01.25 Thus, in this study, we regard high strain intensity region |ε11| ≥ 0.01 as the strain concentration region. That region is formed in about 7 nm × 7 nm area around the dislocation. On the other hand, a ferroelectric nano-region composed of two vortices with opposite rotations, i.e., clockwise and counter-clockwise, is formed around the dislocation (Fig. 2(b)) although SrTiO3 does not exhibit any polarization without dislocations.
image file: c9cp04147h-f2.tif
Fig. 2 (a) The elastic strain ε11 distribution around an edge dislocation in bulk SrTiO3. The contour indicates the elastic strain intensity. White lines indicate |ε11| = 0.01, 0.02, and 0.03. (b) The polarization distribution around the dislocation. The colored contour region shows where polarization emerges. The contour color indicates the angle between the polarization vector (black arrow), and the x1 direction. The gray zone indicates a paraelectric region. (c) The polarization distribution where the contour color and the length of arrows indicate the magnitude of polarization. (d) The electric field distribution. The colored contour region shows where electric fields emerge. The contour color indicates the magnitude of the electric field. The arrow indicates the direction of the electric field.

To elucidate the underlying formation mechanism of this ferroelectric nano-region, we investigate the correspondence between the elastic strain field and the ferroelectricity. Fig. 2(c) also shows the polarization distribution around the dislocation while the contour color and the length of the arrow indicate the magnitude of polarization. In the compressive strain region (upper side of the dislocation), the magnitude of polarization is small regardless of the high strain intensity. It is attributed that the local compressive strain does not contribute to the appearance of polarization in bulk SrTiO3 due to plane strain states. On the other hand, the magnitude of polarization exhibits the maximum value of 0.23 C m−2 in the −x1 direction right under the dislocation where the strain exhibits the highest tensile intensity. Thus, the ferroelectricity corresponds with the tensile strain around the dislocation. In other words, the local ferroelectricity is induced by the strain concentration of the dislocation due to the electro-mechanical property of SrTiO3 while low strain regions remain the paraelectric phase. This suggests that other types of dislocations such as screw dislocations16,47 may exhibit different ferroelectric nano-regions from edge dislocations because they form different strain fields.

On the other hand, the size of the ferroelectric nano-region, i.e., 12 nm × 12 nm is larger than the size of the strain concentration region. Therefore, ferroelectricity should also be induced by other factors. Since the SrTiO3 material intrinsically includes long-range electrostatic interaction, it is expected that this interaction may play an important role in the emergence of the ferroelectricity. To make this point clear, the distribution of the electric field is shown in Fig. 2(d). As shown in the figure, the electric field appears locally around the dislocation although SrTiO3 does not have internal electric fields without a dislocation. In this region, the electric field is along the x1 direction, which is opposite to the polarization right under the dislocation. In addition, the magnitude of the electric field exhibits the maximum value of 2.4 × 107 V m−1 right under the dislocation where the polarization exhibits the maximum value. These facts indicate that this electric field is the depolarization field incidentally induced by the strain-induced polarization. More specifically, due to the polarization induced locally at the high strain region (Fig. 3(a)), uncompensated charges at the para-ferroelectric interface (Fig. 3(b)) form the depolarization field in the material (Fig. 3(c)) and, as a result, two polarization vortices appear. Here, the size of the electric field is about 11 nm × 11 nm and approximately corresponds with the ferroelectric nano-region. This indicates that the ferroelectricity around the dislocation is also formed by the incidental depolarization field.

image file: c9cp04147h-f3.tif
Fig. 3 Schematic illustration of the appearance of the ferroelectric nano region. (a) Strain concentration is formed around a dislocation. (b) Polarization P is induced by strain concentration due to the electro-mechanical coupling of SrTiO3, then interface charges are accumulated between paraelectric and ferroelectric regions. (c) Electrical field E is induced by the interface charges.

It should be noted that the polarization configuration observed in this study, i.e., vortices is different from the experimental observation from Gao et al., i.e., a radial configuration like hedgehogs although we consider the same type edge dislocation.13 This difference can be attributed to the absence of flexoelectricity in our study because the radical configuration is assumed to be induced by flexoelectricity due to the large strain gradient. This indicates that we may alter the topology of polarization drastically by changing the balance of the electro-mechanical coupling and flexoelectricity. Actually, in the experiment, flexoelectricity is strengthened because the distance between dislocations is close to each other, i.e., 3 nm, and the strain gradient is larger than that of an isolated dislocation due to a superposition of tensile and compressive strain regions of dislocation. In addition, if strain gradients are weakened in some ways such as a superposition of a tensile strain region, vortices may appear instead of hedgehogs. Furthermore, we can utilize the nonstoichiometry to control polarization around dislocations. Therefore, ferroelectricity around dislocations can be engineered, and novel polarization configuration may appear by considering the circumstances around dislocations. This remains for future studies.

3.2 Ferroelectricity around dislocation dipoles

To understand ferroelectricity due to multiple dislocations, we investigate the simplest dislocation structure, i.e., dislocation dipoles. Fig. 4(a1) and (a2) shows the polarization distributions around the dislocation dipole A and dipole B in SrTiO3. For dislocation dipole A, double vortices also emerge and surround each dislocation. Interestingly, these vortices connect with each other to form a ferroelectric nano-region that covers two dislocations (Fig. 4(a1)). In this region, a polarization configuration like “step” appears, which is not observed in an isolated edge dislocation. For dislocation dipole B, on the other hand, two isolated ferroelectric nano-regions appear around the dislocations (Fig. 4(a2)). Therefore, how they face each other affects the ferroelectric nano-regions around the dislocation dipoles. This indicates that the ferroelectricity around the dislocation dipole is not a simple superposition of two dislocations and, when we discuss ferroelectricity around dislocation structures, it is necessary to regard dislocation dipoles as different components from a single dislocation.
image file: c9cp04147h-f4.tif
Fig. 4 Polarization distributions, elastic strains, polarization magnitudes and electric fields around (a1–d1) dipole A in which the extra half-planes of two dislocations face apart and (a2–d2) dipole B in which the extra half-planes face each other. The definitions of each figure are the same as Fig. 2 and black solid lines are corresponding physical fields of an isolated dislocation.

Here, we explore the underlying formation mechanism of separated/combined ferroelectric nano-regions. Fig. 4(b1)–(d1) show the elastic strain, polarization magnitudes and electric field distributions around dipole A, respectively. The corresponding physical fields of an isolated dislocation are superimposed. Around dipole A, due to a superposition of the strain fields, the tensile strain concentration region (|ε11| ≥ 0.01) is larger than an isolated dislocation (Fig. 4(b1)). Thus, the intensity of tensile strain increases compared to the isolated dislocation. Due to this high tensile strain, the magnitude of polarization exhibits the maximum value of 0.26 C m−2, which is larger than that of the isolated dislocation (Fig. 4(c1)). This increasing polarization results in the expansion and connection of the incidental depolarization field (Fig. 4(d1)). In the case of dipole B, in contrast, the compressive strain regions confront each other. As a result, the tensile strain intensity, polarization magnitude (maximum: 0.19 C m−2), and electric field decrease (Fig. 4(b2)–(d2)). Therefore, the ferroelectric nano-regions around dislocations in the dipole A enhance each other to make a connection, while they mutually suppress in dipole B to form isolated ferroelectric nano-regions.

Here, as well as SrTiO3, ferromagnetics in NiO48 and multiferroics in PbTiO349 are induced by a single dislocation. As shown above, dislocation arrangements will play an important role in the intensity of ferroelectricity. Thus, in NiO and PbTiO3, their ferroic properties can also be controlled by dislocation arrangements. Therefore, an assembly of dislocations should not be overlooked as the origin of novel ferroic properties.

3.3 Ferroelectricity around a dislocation wall

Fig. 5 shows the polarization distribution around a dislocation wall in SrTiO3. A two-dimensional ferroelectric nano-region is formed along the dislocation wall. Here, this nano-region is a ferroelectric nanostructure embedded in the paraelectric matrix, that is, a two-dimensional ferroelectric thin film with internal periodic zig-zags. Since ferroelectricity is induced by the strain concentration, this result indicates that we can mechanically tailor periodically-arrayed ferroelectric nanostructures by dislocations. Here, the ferroelectric nano-region is induced independently of how dislocations face each other, which is different from dislocation dipoles. This can be attributed to the fact that the dislocation wall consists of repetitions of dipole A and dipole B, and one dislocation in the wall belongs to dipole A and dipole B at the same time. In other words, in the dislocation wall, the shrunk ferroelectric nano-region due to dipole B undergoes expansion by nearby dipole A. As a result, the ferroelectric nano-region connects even in dipole B arrangements. And once ferroelectricity connects, compression between dipole B seems to contribute to enhancing polarization in the x2 direction because the connection of dipole B is larger than that of dipole A (see Supplemental Information 4, ESI). This ferroelectric connection even in dipole B suggests that, although a Taylor–Nabarro lattice consists of multiple dislocation dipoles, the ferroelectricity around this structure is not a simple superposition of its components. Therefore, the shape of ferroelectric nanostructures is not restricted by the components, i.e., a dislocation and dislocation dipoles, and we may fabricate various-shaped ferroelectric nanostructures by controlling the connectivity of the ferroelectric nano-region through the elastic fields.
image file: c9cp04147h-f5.tif
Fig. 5 Polarization distributions around the dislocation wall. The definition is the same as Fig. 1. The white line in the enlarged view is the schematic indication of the shape of the ferroelectric nano-region.

In metals such as copper, not only dislocations and dipoles but also dislocation walls array periodically. As a result, a ladder structure is formed and this large dislocation structure also arrays periodically.32–34 If these larger dislocation structures are observed in SrTiO3, it means that periodically-arrayed ferroelectric nanostructures periodically array and larger periodically-arrayed ferroelectric nanostructures are formed multiple times. Since these multiple hierarchical structures are called fractals, we may fabricate ferroelectric nanofractals by dislocations. Furthermore, dislocations exhibit more various properties such as moving by strain sometimes like an avalanche and reproduction in Frank–Read sources.50 Considering these properties, we may fabricate mobile or reproductive ferroelectric nanostructures which are totally different from artificial media.

It is noted that our results are not limited to ferroics. In another paper,51 we also demonstrate that periodically-arrayed ferroelectric nanostructures can be fabricated by nanoporous SrTiO3 under mechanical loadings due to strain concentration formed around pores. Here, dislocation structures are also formed by material fatigue. Therefore, our studies can be interpreted as that strain concentration and fatigue, which are the cause of fracture and have not been favorable generally, can be a new synthesis method of ferroelectric nanostructures. This will provide a new viewpoint with not only ferroics but also fracture, leading to extending our knowledge of engineering materials.

Before the conclusion, we mention about experimental measurements of these results. Edge dislocations and dipoles are formed in epitaxially grown films41 and they can be detectable at the top surface of the film. Therefore, with regard to a single dislocation and dipoles, ferroelectricity can be measured as piezoresponse by piezoresponse force microscopy (PFM) or displacements of atoms by scanning transmission electron microscopy (STEM). Furthermore, although highly ordered dislocation structures like the dislocation wall may be difficult, dense dislocation arrangements are formed by nano-indentations.52,53 Therefore, ferroelectricity by the assembly of dislocations also can be measured experimentally.

4 Conclusions

In this study, we investigate ferroelectricity around dislocation structures using phase-field modeling. It is shown that, in paraelectric SrTiO3, a ferroelectric nano-region is induced by the strain concentration of an edge dislocation. In addition, this mechanically induced polarization forms an electrical field around the dislocation, and this electrical field induces polarization additionally. Due to these mechanisms, in dislocation structures, ferroelectric nano-regions around each dislocation connect with each other. As a result, a ferroelectric nano-region with periodicity is formed, which is a periodically-arrayed ferroelectric nanostructure embedded in paraelectric matrices. Our findings demonstrate that we can tailor and design novel ferroelectric nanostructures by dislocations and thus provide a new perspective for realizing novel future nanodevices.

Conflicts of interest

There are no conflicts to declare.


This work was supported in part by JSPS KAKENHI Grant No. 18H05241, 18K18806, 17H03145.


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Electronic supplementary information (ESI) available: Details of phase field models; details of simulation models and procedures. See DOI: 10.1039/c9cp04147h

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