Zigzag charged domain walls in ferroelectric ${\mathrm{PbTiO}}_{3}$

Zigzag charged domain walls in ferroelectric ${PbTiO}_{3}$

Pavel Marton, Mauro A. P. Gonçalves, Marek Paściak, Sabine Körbel, Věnceslav Chumchal, Martin Plešinger, Antonín Klíč, and Jirka Hlinka

Phys. Rev. B 107, 094102 – Published 6 March 2023

Abstract

We report a theoretical investigation of a charged $180^{\circ}$ domain wall in ferroelectric ${PbTiO}_{3}$ , compensated by randomly distributed immobile charge defects. For this we utilize atomistic shell-model simulations and continuous phase-field simulations in the framework of the Ginzburg-Landau-Devonshire model. We predict that domain walls form a zigzag pattern and we discuss their properties in a broad interval of compensation-region widths, ranging from a couple to over 100 nm. The zigzag is accompanied by a local polarization rotation which we explain to provide an efficient mechanism for charge compensation.

Received 10 November 2022
Revised 7 February 2023
Accepted 16 February 2023

DOI:https://doi.org/10.1103/PhysRevB.107.094102

Physics Subject Headings (PhySH)

Defects Domain walls Domains Ferroelectricity

Ferroelectrics

Landau-Ginzburg-Devonshire theory Phase-field modeling

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Pavel Marton ^1,2,*, Mauro A. P. Gonçalves ¹, Marek Paściak ¹, Sabine Körbel ^1,3, Věnceslav Chumchal ⁴, Martin Plešinger ⁴, Antonín Klíč ¹, and Jirka Hlinka ¹

¹Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 182 00 Praha 8, Czech Republic
²Institute of Mechatronics and Computer Engineering, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic
³Institute of Condensed Matter Theory and Solid State Optics, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany
⁴Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic

^*marton@fzu.cz

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Vol. 107, Iss. 9 — 1 March 2023

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Figure 1
Sketch of distribution of charged-wall-compensation charges in the $x − y$ cross section of the simulation box. Positive charges (red) are distributed randomly in a layer of thickness $L$ (region of interest in simulations); negative charges (blue) are located on a plane. The charged regions are separated by neutral regions. The electrostatics dictates that two domain walls will develop: A tail-to-tail domain wall (DW1) in the positively charged region, and a head-to-head wall (DW2) in the negatively charged plane. The properties and location of the DW1 are not known a priori and are the subject of this study, while the DW2 is collocated with the negatively charged plane.
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Figure 2
A zigzag domain wall as obtained from shell-model simulations. Arrows stand for the polarization of individual unit cells projected to the $x y$ plane, and color represents the $P x$ , $P y$ , and $P z$ components of polarization in individual panels. The domain wall in the figure was obtained for the $L = 48$ and $W = 45$ unit cells.
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Figure 3
A zigzag domain wall obtained from phase-field simulations. Three types of defect-charge distributions were considered, all of them leading to the same average charge density in the wall. (a) Homogeneous charge, (b) charges as in shell-model calculations, and (c) charges equal to the elementary charge. Color represents $P_{x}$ and $P_{y}$ in the first and second panel for each case [and $P_{z}$ in the third panel of (c)]. d) $P_{y} (y)$ evaluated for $x = 24$ shows linear dependence inside the triangular domain for all three studied cases, and a steep decrease within the domain wall.
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Figure 4
Polarization-originated bound charges, stemming from the $P_{x}$ (upper), $P_{y}$ (middle) components of polarization, and their sum (bottom). For better visibility of the DW1 region we artificially reduce the intensity of charge in the DW2 by a factor of 10.
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Figure 5
Top: Schematic picture of the zigzag pattern, which is used in the derivation of $W_{natural} (L)$ . 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 as in Fig. 1. Bottom: Dependence of $W_{natural} (L)$ . 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, as in Fig. 3. Solid line: Simplified model. The dotted line indicates dependence $W = L$ , i.e., a triangle with identical width and height.
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Figure 6
Zigzag domain wall as obtained from mutually independent phase-field simulations for $L = 200 Δ = 80$ nm and $W \in {0.80, 1.20, 1.60, 1.76, 2.00} \times W_{natural}$ . The color represents $P_{x}$ . White regions are areas where the polarization is oriented along the $y$ axis.
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Physical Review B

covering condensed matter and materials physics

Zigzag charged domain walls in ferroelectric ${PbTiO}_{3}$

Pavel Marton, Mauro A. P. Gonçalves, Marek Paściak, Sabine Körbel, Věnceslav Chumchal, Martin Plešinger, Antonín Klíč, and Jirka Hlinka

Phys. Rev. B 107, 094102 – Published 6 March 2023

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Physical Review B

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Zigzag charged domain walls in ferroelectric PbTiO3

Pavel Marton, Mauro A. P. Gonçalves, Marek Paściak, Sabine Körbel, Věnceslav Chumchal, Martin Plešinger, Antonín Klíč, and Jirka Hlinka

Phys. Rev. B 107, 094102 – Published 6 March 2023

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Zigzag charged domain walls in ferroelectric ${PbTiO}_{3}$