# Domain walls of ferroelectric BaTiO$_{3}$ within the Ginzburg-Landau-Devonshire phenomenological model 

P. Marton, I. Rychetsky and J. Hlinka 

Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Praha 8, Czech Republic 

(Dated: January 11, 2010) 

Mechanically compatible and electrically neutral domain walls in tetragonal, orthorhombic and rhombohedral ferroelectric phases of BaTiO$_{3}$ are systematically investigated in the framework of the phenomenological Ginzburg-Landau-Devonshire (GLD) model with parameters of Ref. [Hlinka and Marton, Phys. Rev. 74 , 104104 (2006)]. Polarization and strain profiles within domain walls are calculated numerically and within an approximation leading to the quasi-one-dimensional analytic solutions applied previously to the ferroelectric walls of the tetragonal phase [W. Cao and L.E. Cross, Phys. Rev. 44 , 5 (1991)]. Domain wall thicknesses and energy densities are estimated for all mechanically compatible and electrically neutral domain wall species in the entire temperature range of ferroelectric phases. The model suggests that the lowest energy walls in the orthorhombic phase of BaTiO$_{3}$ are the 90-degree and 60-degree walls. In the rhombohedral phase, the lowest energy walls are the 71-degree and 109-degree walls. All these ferroelastic walls have thickness below 1 nm except for the 90-degree wall in the tetragonal phase and the 60-degree S-wall in the orthorhombic phase, for which the larger thickness of the order of 5 nm was found. The antiparallel walls of the rhombohedral phase have the largest energy and thus they are unlikely to occur. The calculation indicates that the lowest energy structure of the 109-degree wall and few other domain walls in the orthorhombic and rhombohedral phases resemble Bloch-like walls known from magnetism. 

Domain structure is an important ingredient in func- tionality of ferroelectric materials. Among others, it has impact on their nonlinear optical properties, dielec- tric permittivity and polarization switching phenomena. Since domain boundaries in ferroelectric perovskite mate- rials can simultaneously play the role of the ferroelectric and ferroelastic walls, such domain walls also strongly influence the electromechanical material properties: they facilitate switching of spontaneous polarization and spon- taneous deformation, thus giving rise to a large extrinsic contribution to e.g. piezoelectric constants, which makes ferroelectric materials extremely attractive for applica- tions. The domain structure also provides additional de- gree of freedom for tuning of material properties. In gen- eral, further development of domain engineering strate- gies requires deeper understanding of the physics of fer- roelectric domain wall itself. 

The Ginzburg-Landau-Devonshire (GLD) theory pro- vides a feasible tool for such a purpose. Landau- Devonshire model describes phase-transition properties of single-domain crystal using a limited number of pa- rameters, which are determined experimentally (or re- cently also using ab-initio methods). Introduction of Ginzburg gradient term to the free energy functional enables addressing nonhomogeneous multi-domain fer- roelectric state. The GLD model was previously used for computation of domain wall properties in ferroelec- tric materials (e.g. Refs. 1–5) and in phase-field com- puter modeling of domain formation and evolution. 6–9 The GLD model can be regarded as a bridge model cov- ering length-scales inaccessible by ab-initio and micro- 

PACS numbers: 77.80.-e, 77.80.Dj, 77.84.Dy 

## I. INTRODUCTION 

mechanical models. 

BaTiO$_{3}$ represents a typical ferroelectric material which undergoes a sequence of phase transitions from high-temperature paraelectric cubic Pm 3 m ( O$_{h}$ ) to the ferroelectric tetragonal P 4 mm ( C$_{4}$$_{v}$ ), orthorhombic Amm 2 ( C$_{2}$$_{v}$ ) and rhombohedral R 3 m ( C$_{3}$$_{v}$ ) phase. En- ergetically equivalent directions of spontaneous polar- ization vector, identifying possible ferroelectric domain states in a particular ferroelectric phase, are displayed in Fig. 1. Domain boundaries separating two domain states are characterized by the rotational angle needed to match spontaneous polarizations on both sides of the bound- ary. For example, the boundary separating domains with mutually perpendicular spontaneous polarization is com- monly called 90 ◦ wall, while the one between antiparal- lel spontaneous polarization regions is called 180 ◦ wall. Other angles are possible in orthorhombic and rhombohe- dral phases of BaTiO$_{3}$, where the spontaneous polariza- tion is oriented along the cubic face diagonals and body diagonals, respectively. 

The aim of this paper is to calculate basic characteris- tics of all electrically neutral and mechanically compat- ible domain walls in all ferroelectric phases of BaTiO$_{3}$. For a better comparison of domain wall properties like their thickness or energy density, we employ an Ising- like approximation leading to previously proposed an- alytically solvable one-dimensional solutions. 1 The pa- per is organized as follows. In Section II. we give an overview of the different kinds of mechanically compati- ble domain walls in the three ferroelectric phases. It fol- lows from general theory about macroscopic mechanical compatibility of adjacent domain states. 10,11 The GLD parameters used for calculation of the domain wall prop- 

