Theory and simulations
Zigzag charged domain walls in ferroelectric PbTiO3
We report a theoretical investigation of a charged 180-degree domain wall in ferroelectric PbTiO3, compensated by randomly distributed immobile charge defects. We predict that domain walls form a zigzag pattern and we discuss their properties in a broad interval of compensation-region widths. The zigzag is accompanied by a local polarization rotation which we explain to provide an efficient mechanism for charge compensation [Phys. Rev. B 107, 094102 (2023)].
Our study delved into a theoretical exploration of charged domain walls in ferroelectric PbTiO3, which were compensated by randomly distributed immobile charge defects located within a relatively broad slab. To achieve this, we employed a combination of atomistic shell-model simulations and continuous phase-field simulations based on the Ginzburg-Landau-Devonshire model. Our findings showed that domain walls form a zigzag pattern, and we examined their properties across a broad range of compensation-region widths, ranging from a few nanometers to over 100 nm, focusing in particular on understanding the zigzag modulation lengths in terms of material properties of PbTiO3. The zigzag formation is accompanied by a local ferroelectric-polarization rotation, which we proposed as an efficient mechanism for local charge compensation. Our study provides a new understanding of the behavior of charged domain walls in ferroelectric materials and highlights the significance of the considered compensation charges in their formation. The insights gained from our study may contribute to the development of advanced ferroelectric materials with applications in the field of smart-materials for electronics.
Figure:
Top panel: Schematic picture of the zigzag pattern, which is used in the derivation of natural width of triangles.
Red and blue colors and arrows correspond here to positively and negatively oriented ferroelectric domains
(with respect to the x axis). The color of the defects is red for positive and blue for negative point defects.
Bottom panel: Dependence of natural width of the zigzag triangles. Bullets: Phase-field simulations
with homogeneous compensation charge; the dashed line is just a connection of these.
Crosses: Phase-field simulations with randomly distributed defect charges.
Solid line: Simplified analytical model. Numerical and analytical approaches show
an excellent corespondence for large thicknesses of the compensation-region L.
[1] P. Marton, M. A. P. Gonçalves, M. Paściak, S. Körbel, V. Chumchal, M. Plešinger, A. Klíč, and J. Hlinka, Zigzag charged domain walls in ferroelectric PbTiO3, Phys. Rev. B 107, 094102 (2023).
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Multidomain ordered metal–ferroelectric superlattices
By combination of advanced experimental techniques and phase-field simulations, we found that electric dipoles in superlattices, composed of layers of a ferroelectric material separated by thin metallic spacers, form an unusual pattern of nanoscale domains that order in three dimensions. These ferroelectric multidomain ordered superlattices exhibit an outstanding dielectric response and their engineered modulated structural and electronic properties can be controlled using electric field [Nat. Mater. 20, 495 (2021)].
Figure:
Two-dimensional base motif of the ferroelectrically ordered PbTiO3–SrRuO3 superlattices as seen by
(a-d) phase-field simulations and (e,f) transmission electron microscopy:
(a) electric polarization showing ferroelectric domain structure in two PbTiO3 layers separated by SrRuO3 spacers
(b) gradient energy density coming from domain walls and boundaries between layers,
(c-f) in-plane (exx) and out-of-plane (ezz) strain components demonstrating correlations between ferroelectric PbTiO3 layers.
The arrows in (a,b) panels show direction of electric polarization forming characteristic flux-closure patterns.
[1] M. Hadjimichael, Y. Li, E. Zatterin, G. A. Chahine, M. Conroy, K. Moore, E. N. O’ Connell, P. Ondrejkovic, P. Marton, J. Hlinka, U. Bangert, S. Leake, and P. Zubko, Metal–ferroelectric supercrystals with periodically curved metallic layers, Nat. Mater. 20, 495 (2021).
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Polarization of domain boundaries in SrTiO3 studied by layer group and order-parameter symmetry
Based on a recently developed combination of layer group analysis with order-parameter symmetry, we study the polarity of antiphase domain boundaries (APBs) and ferroelastic twin boundaries (TBs) in SrTiO3 [Phys. Rev. B 102, 184101 (2020)].
In addition to the celebrated layer group analysis of domain twins, the present method allows us to investigate tensor properties of domain walls also for the case where order-parameter variables other than the spontaneous ones are active [1].
Figure: Representation of two different ferroelastic twins with normal n=[110] at position p=(0,0,0) in tetragonal SrTiO3, together with the symmetry elements of the corresponding layer groups. (a) Pure orientational twin (11|21). The corresponding twin wall is an easy (HT) wall. The twin in (b) is of mixed type, i.e., (11|22), where the corresponding twin wall is a hard (HH) one. The arrows at the centers show polarization predicted by the theory.
[1] W. Schranz, C. Schuster, A. Tröster, and I. Rychetský, Polarization of domain boundaries in SrTiO3 studied by layer group and order-parameter symmetry, Phys. Rev. B 102, 184101 (2020). (show less)
Curie-Weiss susceptibility in strongly correlated electron systems
We succeeded in identifying a microscopic mechanism combining adequately quantum and thermal fluctuations in metals with strong electron correlations that lead to the genesis of local magnetic moments and the Curie-Weiss susceptibility [Phys. Rev. B 102, 205120 (2020)].
The magnetism of solids is macroscopic evidence of microscopic quantum dynamics of spins of valence electrons. Magnetic susceptibility of metals with delocalized conduction electrons should theoretically be of Pauli character. However, the transition metals react on the magnetic field via the Curie-Weiss law due to local magnetic moments. We identified the microscopic mechanism combining adequately quantum and thermal fluctuations in metals with strong electron correlations that lead to the genesis of local magnetic moments and the Curie-Weiss susceptibility [1].
Figure: Temperature dependence of the product of temperature T and magnetic susceptibility χ, the constant value of which reflects the Curie-Weiss law, determined from our analytic theory, left panel, and from Monte-Carlo simulations, right panel, for the single-impurity Anderson model. Here Δ is the width of the band of the conduction electrons, and U is the strength of electron correlations. Both results prove the existence of the Curie-Weiss susceptibility for sufficiently strong electron correlations in a temperature interval above the Kondo temperature below which the susceptibility goes over to the Pauli one, and the plotted curves must fall to zero. The analytic theory predicts qualitatively correctly the universal character of the Curie-Weiss susceptibility but misses the nonuniversal numerical value of the Curie constant.
[1] V. Janiš, A. Klíč, J. Yan, and V. Pokorný, Curie-Weiss susceptibility in strongly correlated electron systems, Phys. Rev. B 102, 205120 (2020). (show less)
Local properties and phase transitions in Sn doped antiferroelectric PbHfO3 single crystal
Pb(Hf0.77Sn0.23)03 crystals were characterized using x-ray diffraction and 119Sn Mossbauer spectroscopy in a wide temperature range. The nature of two intermediate phases, situated between antiferroelectric ground-state and high temperature paraelectric phase, has been unveiled [J. Phys.: Condens. Matter 32, 435402 (2020)].
The lower-temperature one is characterized by incommensurate modulations while the higher-T intermediate phase is defined by anti-phase rotations of oxygen octahedra. Two kinds of quadrupole splitting indicate that the environment of Sn is locally non-centrosymmetric even in the cubic phase [1].
Figure: Diffraction signatures of four phases observed for Pb(Hf0.77Sn0.23)03. Diffuse streaks emerging from Bragg reflections at high temperatures are a signature of disorder (locally correlated displacements of Pb ions). They transform into broad maxima at 463 K, incommensurate peaks at 440 K and finally commensurate Bragg reflections at 385 K in the ground-state phase of this material.
[1] I. Jankowska-Sumara, M. Paściak, M. Kądziołka-Gaweł, M. Podgórna, A. Majchrowski, and K. Roleder, Local properties and phase transitions in Sn doped antiferroelectric PbHfO3 single crystal, J. Phys.: Condens. Matter 32, 435402 (2020). (show less)
Spatio-temporal distribution of relative Ti-O6 displacements in cubic BaTiO3
BaTiO3 is often considered a model ferroelectric material in which the dielectric properties are defined by the displacements of Ti ions with respect to surrounding oxygen atoms. However, despite the decades of a dedicated research, certain controversies have remained as to the description of collective movements of the Ti ions. We approached this problem using nonoscale-oriented X-ray scattering methods and large-scale atomistic simulations [1]. Together these allowed us to show that the Ti dynamics can be exhaustively explained by phonons excited on a timescale of picoseconds.
The three-dimensional distribution of the x-ray diffuse scattering intensity of BaTiO3 has been recorded in a synchrotron experiment and simultaneously computed using molecular dynamics simulations of a shell model. Together, these have allowed the details of the disorder in paraelectric BaTiO3 to be clarified. The narrow sheets of diffuse scattering, related to the famous anisotropic longitudinal correlations of Ti ions, are shown to be caused by the overdamped anharmonic soft phonon branch. This finding demonstrates that the occurrence of narrow sheets of diffuse scattering agrees with a displacive picture of the cubic phase of this textbook ferroelectric material. The presented methodology allows one to go beyond the harmonic approximation in the analysis of phonons and phonon-related scattering.
Fig. 1: One component of the relative Ti-O6 displacement is mapped within one layer of the material using different time averages (a-c) which clearly shows that chain correlations exist on a timescale of picoseconds. The displacements averaged over the simulation time and all unit cells of the crystal have a cuboidal distribution with shallow minima along diagonal directions.
[1] M. Paściak, T. R. Welberry, J. Kulda, S. Leoni, and J. Hlinka, Dynamic Displacement Disorder of Cubic BaTiO3, Phys. Rev. Lett. 120, 167601 (2018).
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Domain wall contribution to lattice dynamics and permittivity of BiFeO3
Ferroelectric materials are known for their exceptionally high dielectric permittivity. It turns out, that important part of it originates from a material's complicated microstructure and in particular from interfaces between ferroelectric domains.
In this study we investigate different types of interfaces in a multiferroic BiFeO3. By means of atomistic modelling we show that some configurations of interfaces can greatly enhance the permittivity by hosting terahertz-range collective polar fluctuations. The key result is that the permittivity is modified in the THz range. This fundamental finding and its understanding can be used in a design of new materials [ J. Hlinka, M. Pasciak, S. Körbel, and P. Márton, Phys. Rev. Lett. 119, 057604 (2017) ].
Figure: Change of the phonon spectrum due to presence of dense domain-interface structure (right panel) in comparison with the single-domain case (left panel). Change of the (polar-active) phonon spectrum due to presence of dense domain-interface structure of BiFeO3 (right panel) in comparison with the single-domain case (left panel). Additional low-frequency Γ-point mode (labelled A) is responsible for 25x enhancement of permittivity in the domain engineered material.
First-principles-based Landau-Devonshire potential for BiFeO3
We describe a first-principles-based computational strategy for determination of the Landau-Devonshire potential.
It exploits the configuration space attached to the eigenvectors of the modes frozen in the ground state. This allows to probe the energy surface in the vicinity of the ground state, which is most relevant for the properties of the ordered phase. We apply this procedure to BiFeO3 in order to determine potential energy associated with strain, polarization, and oxygen octahedra tilt. [ P. Marton, A. Klíč, M. Paściak, and J. Hlinka, Phys. Rev. B 96, 174110 (2017) ].
Figure: Energy profiles along selected paths in order parameter space of BiFeO3, connecting the cubic reference state (REF) and the ground state (Ps , As , es). Point symbols are outcomes of the first-principles calculations, lines are evaluated from the present Landau-Devonshire potential.
Existence conditions for ferroaxial materials
All 212 species of structural phase transitions with a macroscopic symmetry breaking were inspected with respect to the simultaneous occurrence of the ferroelastic, ferroelectric, and ferroaxial properties.
For each species, a matrix of representative equilibrium property tensors in both high-symmetry and low-symmetry phases, showing emergence of spontaneous components, were explicitly worked out [http://palata.fzu.cz/species/13x13axial]. Results can serve as a useful tool in search of fundamentally new material properties [J. Hlinka et al., Phys. Rev. Lett. 116, 177602 (2016)].
Figure: Dichromatic matrix of material property tensors for a symmetry-breaking structural phase transition from tetragonal to monoclinic phases, showing relations between tensor components of both phases and emergence of spontaneous components (red points).
Ising lines: Natural topological defects within ferroelectric Bloch walls
Phase-field simulations demonstrate that the polarization order-parameter field in the Ginzburg-Landau-Devonshire model of rhombohedral ferroelectric BaTiO3 allows for an interesting linear defect, stable under simple periodic boundary conditions. This linear defect, here called the Ising line, can be described as an about 2-nm-thick intrinsic paraelectric nanorod acting as a highly mobile borderline between finite portions of Bloch-like domain walls of opposite helicity.
These Ising lines play the role of domain boundaries associated with the Ising-to-Bloch domain-wall phase transition. Therefore, the electric field parallel to the Ising line can be used to induce the Ising line motion, which mediates the switching of both polarization and chirality of the Bloch wall without changing the bulk domain polarization. Detailed description of this study can be found in the paper [ V. Stepkova, P. Marton, and J. Hlinka, Phys. Rev. B 92, 094106 (2015)]
Figure: Ising line defect in rhombohedral BaTiO3. When circumventing the core of this defect following a closed path (dashed line) around its axis t, the polarization on this path rotates around the [-211] axis, which is perpendicular to t.
Phase transition in ferroelectric domain walls of BaTiO3
The seminal paper by Zhirnov (1958 Zh. Eksp. Teor. Fiz. 35 1175–80) explained why the structure of domain walls in ferroelectrics and ferromagnets is so different. We have recently realized that the antiparallel ferroelectric walls in rhombohedral ferroelectric BaTiO3 can be switched between the Ising-like state (typical for ferroelectrics) and a Bloch-like state (unusual for ferroelectric walls but typical for magnetic ones) [see Figure 1] by a compressive epitaxial stress.
Phase-field simulations using a Ginzburg–Landau–Devonshire model allows to explore this strain-induced phase transition within the domain wall in detail [Figure 2]. Strain-tunable chiral properties of ferroelectric Bloch walls promise a range of novel phenomena in epitaxial ferroelectric thin films. Results are published in JPCM [ V. Stepkova et al., J. Phys.: Condens. Matter 24, 212201 (2012)].
Figure 1: Variation of polarization P within Bloch- and Ising-type domain walls. The Bloch-type structure (a,b) conserves the magnitude of the local vectorial order parameter. This is typically encountered in ferromagnetism. In contrast, ferroelectric materials usually have large anisotropy and the magnitude of the polarization can be more easily adjusted. Therefore, antiparallel walls in ferroelectrics are known to correspond to the achiral Ising solution (c). We show here that in rhombohedral BaTiO3, both types of 180° domain walls can be stabilized.
Figure 2: Chiral order parameter. The extremal value of the Pt polarization component in the core of the domain wall is plotted as a function of the epitaxial stress. This quantity can be taken as an order parameter of this chiral phase transition within the domain wall. For the wall under study, the phase transition is continuous and it can be associated with the Landau free-energy sketched in the figure.
Dielectric response of a phase transition in domain wall
Computer simulations based on Ginzburg-Landau-Devonshire theory was used to study details of a phase-transition between Ising and Bloch type of a domain wall.
Such transition was predicted to appear for 180 degree domain wall in the rhombohedral ferroelectric phase of BaTiO3 close to 2GPa compressive stress [ V. Stepkova et al., J. Phys.: Condens. Matter 24, 212201 (2012)]. We confirm the hypothesis that the phase transition is of the first-order and predict corresponding divergent increase of the dielectric permittivity in the vicinity of the transition. In case of dense domain-wall pattern it can represent a substantial part of the dielectric response of the material [ P. Marton et al., Phase Transitions 86, 103 (2012)].
Figure 1: Dielectric permittivity in dependence on mechanical pressure in the vicinity of the transition from Ising (larger pressure) to Bloch-type of the domain wall.
Figure 2: Bloch-to-Ising phase transition within a rhombohedral 180° domain wall.
Piezoelectric response of nanotwinned ferroelectric perovskites
Computer simulations based on Ginzburg-Landau-Devonshire theory has benn used to investigate piezoelectric properties of tetragonal BaTiO3 crystals. We have shown that piezoelectric response of twinned BaTiO3 increases with increasing density of 90-degree domain walls.
A considerable enhancement of piezoelectric coefficients is predicted below 50 nm. We have also shown that main contribution to the longitudinal piezoelectric coefficient comes from volume of domain, rather than from the domain wall region. Nevertheless, the experimentally reported impact of domain wall density is even stronger than in our simulations. Our most recent theoretical investigations [P. Marton et al, Physical Review B 81, 144125 (2010)] suggest that ferroelectric perovskites such as BaTiO3 most likely support also more exotic domain walls, resembling Bloch walls known from ferromagnetism. This kind of domain walls might account for additional enhancement of the electromechanical material response.
Figure 1: Dependence of the effective longitudinal piezoelectric coefficient on domain size calculated from the response of the ideal 2D laminar domain structure to weak uniaxial stresses applied along the [111] crystallographic direction. The inset shows the geometry of the initial 2D laminate domain structure assumed in the simulations.
Figure 2: Set of mechanically compatible and electrically neutral domain walls in the three ferroelectric phases o f BaTiO3. Letter denominating the phase (T, O and R stay for tetragonal, orthorhombic and rhombohedral phase, resp.) is followed by an angle between the polarization in adjacent domains and in some cases also by the domain-wall normal.