Atomic-resolution differential phase contrast STEM on ferroelectric materials: A mean-field approach

The ultimate challenge in the investigation of ferroelectric properties lies in the quantitative measurements of their polarization at the unit cell scale. Such investigations are commonly performed using an indirect approach, by measuring the atomic displacements from atomic resolution images. Differential phase-contrast (DPC) scanning transmission electron microscopy (STEM) allows mapping the electric field with atomic resolution. This unique capability offers a direct way to study the polar properties in ferroelectrics. However, the effects of ferroelectric polarization on the contrast of high-resolution DPC-STEM imaging have not been addressed so far. In this work, we perform a theoretical study on the origin of the DPC-STEM contrast in ferroelectric materials and propose a modified multislice algorithm for STEM image simulations. Our results demonstrate that the mesoscopic polarization induces asymmetries in the detected electric fields, which are in line with our previous experimental observations. Moreover, we discuss the dependence of the DPC-STEM sensitivity on the polar field amplitude, specimen thickness, and defocus, and provide a route to discriminate between mesoscopic polarization and specimen misorientation.

Figure 1

(a) Sketch of a ferroelectric crystal slab when no external field is applied. The small black arrows give the microscopic dipoles of the $\mathrm{BiFe}{\mathrm{O}}_3$ pseudocubic unit cells, while the large grey arrow depicts the mesoscopic field due to whole the crystal polarization. One possible electron trajectory is given by the black dashed line. (b) Effect of a uniform ferroelectric polarization on the potential slices for multislice simulations. An additive linear term induces disruption of the periodicity of the potential slices. (c) Modified multislice approach including the interaction of the electrons with a transverse electric field in the free-space propagation. After interacting with the potential slice $V_i$, the electron wavefunction is propagated in the free-space until the next potential slice $V_{i+1}$, applying a fractional beam tilt corresponding to the effect induced by the ferroelectric polarization of one slice.

Figure 2

Crystal structure of $\mathrm{BiFe}{\mathrm{O}}_3$ viewed along (a) the ${\left[001\right]}_P$ (top view) and (b) the ${\left[010\right]}_P$ (side view) directions. The lateral size of the crystal is 15.874 Å. Along the $z$ direction, the crystal is 7.937-Å thick. Two slices (SL1, SL2) are cut for generating the projected potential. (c) Total projected electrostatic potential $(V_{\mathrm{SL}1}+V_{\mathrm{SL}2})$ as obtained from the independent atom model. (d) Line profiles of the projected electrostatic potential—as obtained from IAM and DFT calculations—along the lines marked by the white arrows in (c).

Figure 3

(a) Schematic representation of the simulated DPC-STEM measurement on a ferroelectric crystal. A convergent electron probe is scanned over the specimen. The ferroelectric polarization induces a beam tilt—detectable using a segmented annular detector. A dynamical treatment of the beam propagation is considered. (b) Evolution of an electron trajectory as it propagates through a ferroelectric crystal (neglecting the channeling effects and multiple scattering). The polarization-induced beam tilt (colored lines) is shown for different polarization values and specimen thicknesses. The white dashed lines serve as a guide for the eye as they depict the trajectories at constant angles spaced 0.3 mrad between each other.

Figure 4

Thickness and defocus series of the $\mathrm{BiFe}{\mathrm{O}}_3$ imaged in the ${\left[001\right]}_P$ projection. (a) HAADF signal, (b) DPC amplitude for the nonpolar case $(P=0\mu \mathrm{C}\mathrm{c}{\mathrm{m}}^{-2})$, and (c) DPC amplitude with polar field $(P=100\mu \mathrm{C}\mathrm{c}{\mathrm{m}}^{-2})$. [(d)−(f)] DPC amplitude for the specimen tilt angles of 0.6, 1.8, and 3.0 mrad. The thickness and defocus ranges are the same for all the subplots. A different focus and thickness dependence is observed for HAADF and DPC.

Figure 5

Sketches of the crystal structures in the ${\left[001\right]}_P$ zone axis for the three cases of (a) nonpolar, (b) polar and (c) tilted specimens. The yellow arrow gives the direction of the polarization. The magnified views of the simulated DPC amplitude for the cases $P=0$, $P=100\mu \mathrm{C}\mathrm{c}{\mathrm{m}}^{-2}$, and $t=1.8\mathrm{mrad}$ (corresponding to the areas marked by the white boxes in Fig. 4, i.e., $z=20.6\mathrm{nm}$ and $\mathrm{\Delta }f=-174\AA$) are shown below the sketches. (d) Line profiles obtained integrating the DPC amplitude with mesoscopic polarization over an area like the one marked by the white dashed box shown in (a). (e) Similar line profiles obtained integrating the DPC amplitude for different values of crystal tilt angle. (f) Simulated CBED pattern for the nonpolar specimen ($z=20.6\mathrm{nm}$ and $\mathrm{\Delta }f=-174\AA$). Difference of CBED patterns of the (g) polar $(P=100\mu \mathrm{C}\mathrm{c}{\mathrm{m}}^{-2})$ and (h) tilted $(t=1.8\mathrm{mrad})$ specimens with respect to the reference one. Comparison of line profiles for the different patterns for the (i) polar and (j) tilted cases, taken along the black arrows.

Figure 6

Thickness and defocus dependent maps of the average shift of the CoM for (a) different values of mesoscopic polarization and (b) crystal tilt angles. [(c) and (d)] Line profiles of the bright field disks obtained for a thickness $z=20.6\mathrm{nm}$ and a defocus $\mathrm{\Delta }f=-174.3\AA$, with mesoscopic polarization and crystal tilt, respectively. The analyzed bright field disk corresponds to the value marked by the white box in (a) and (b). The magnified views on the right show the relative positions of the peaks for the different polarization and tilt values.

Figure 7

Line profiles of the CoM shifts for the different polarization and tilt values. (a) Defocus-dependent line profiles of the CoM shifts for different polarization values. The line profiles are taken at the constant specimen thickness $z=20.6\mathrm{nm}$ [dotted rectangles in Fig. 5]. (b) Thickness-dependent line profiles of the CoM shifts for the different polarization values, taken at constant defocus $\mathrm{\Delta }f=-174.3\AA$ [dashed rectangles in Fig. 5]. The inset in (b) shows the comparison of the curves of $P=80$ (red) and the rigid-shift model (black). [(c) and (d)] Similarly obtained line profiles for the various specimen tilt angles.