Twin wall distortions through structural investigation of epitaxial BiFeO₃ thin films

Abstract

In this work, epitaxial (001) BiFeO₃ thin films were deposited on SrTiO₃ and TbScO₃ single-crystal substrates and analyzed with high-resolution x-ray diffraction—reciprocal space mapping. A basic method was developed to extract structural details of the as-grown BiFeO₃ film, including (i) epitaxial strain, (ii) ferroelastic domains, and (iii) lattice rotations. After demonstrating the method, extrinsic distortions at vertical twin walls were determined for specific BiFeO₃ heterostructures. A relatively large distortion (0.20° ± 0.08°) was measured in a multidomain (12) and incoherent film, while a nearly intrinsic distortion (0.04° ± 0.03°) was measured in a two-domain coherent film. This work offers insights into the structure of multiferroic BiFeO₃ thin films with a general approach that is appropriate for low symmetry epitaxial heterostructures.

APPENDIX

The bulk real space structures of the BiFeO₃ film and the substrates were described with individual sets of basis vectors in a single Cartesian coordinate system (see subcells in Fig. 1). After defining the heterostructure in real space, the basis vectors were transformed to reciprocal space using standard Gibb’s relations.31 The real space lattice corresponding to the substrates was fixed. In contrast, the real space lattice of the BiFeO₃ film was matched to the data considering the following parameters: subcell definition (a_p, b_p, c_p, α_p, β_p, and γ_p) and parameters associated with structural variants (R), specifically (r₁, r₂, r₃, or r₄) and lattice rotations ($\hat k$, ω).

The substrate lattice (Fig. 1) transformation matrices were defined for SrTiO₃ with a = 3.905 Å and TbScO₃ with a_m = c_m = ${{\sqrt {a_{\rm{o}}^2 + b_{\rm{o}}^2 } } \mathord{\left/ {\vphantom {{\sqrt {a_{\rm{o}}^2 + b_{\rm{o}}^2 } } 2}} \right. \kern-\nulldelimiterspace} 2}$ = 3.953 Å, b_m = ${{c_{\rm{o}} } \mathord{\left/ {\vphantom {{c_{\rm{o}} } 2}} \right. \kern-\nulldelimiterspace} 2}$ = 3.957 Å, and β_m = $2\tan ^{ - 1} \left( {{{a_{\rm{o}} } \mathord{\left/ {\vphantom {{a_{\rm{o}} } {b_{\rm{o}} }}} \right. \kern-\nulldelimiterspace} {b_{\rm{o}} }}} \right)$ = 87.243°.123233

$${S^{{\rm{STO}}}} = \left[ {\matrix{ a & 0 & 0 \cr 0 & a & 0 \cr 0 & 0 & a \cr } } \right].$$

((1))

((2))

A transformation matrix was also used to define the BiFeO₃ film subcell (F) although in contrast to the substrates, the lattice parameters a_p, b_p, c_p, and intercell angles α_p, β_p, and γ_p were made flexible as to fit the experimental data (general triclinic structure).

((3))

where,

$${\gamma ^/} = {\pi \over 4} - {{{\gamma _{\rm{p}}}} \over 2},$$

$${F^{\left[ {1,3} \right]}} = {F^{\left[ {2,3} \right]}} = 0,$$

The expressions for F were geometrically derived assuming the subcell initially lies flat on the crystallographic growth surface (F^[2,3] = F^[1,3] = 0) and equally spaced from the [100] and [010] axes.

Four ferroelastic variations are in a rhombohedral system which quadruples the number of independent lattices that must be considered. These were accounted for using separate matrices, r_1–4, where each may be visualized as a stretching of the subcell along the <111> diagonals (Fig. 1).28

((4))

Additionally, structural variants (R) were considered to account for the possibility of lattice rotations. This was accomplished considering the ferroelastic variants (r_1–4) and a rotation matrix.31

((5))

Here ω is the magnitude of the rotation about a unit vector $\hat k$ = [k₁, k₂, k₃]. Here discussion was limited to ω_f and ω_d, which are notations used in the text to describe lattice rotation of the entire film due to miscut and individual ferroelastic domains due to twin walls, respectively.

At this point, transformation of all the individual real space lattices of the film and substrate to reciprocal space (H K L) was performed using the standard Gibb’s relations generating respective reciprocal lattices in the same coordinate system (Fig. 1).31

To compare with the measured 2D RSM data, the nearest scattering vector G_HKL of each reciprocal lattice was projected onto the exact plane measured in reciprocal space. The projected G_Proj was calculated with,

$${G_{{\rm<Prefix>oj</Prefix>}}} = P{\left( {{P^/}P} \right)^{ - 1}}{P^/}{G_{HKL}},$$

((6))

where P defines the measured plane in reciprocal space (P^/ is the transpose). For {002} and {103} reflections,

((7))

and for {113} reflections,

((8))

A full list of reflections used in this study is given in Table II with the position equivalent bulk Bragg reflections. The Bragg reflections for single-crystal substrates were used for calibration.

TABLE II.

Position select equivalent Bragg reflections for bulk unit cell and sub-cell