Thickness scaling of ferroelectricity in BiFeO₃ by tomographic atomic force microscopy

Nanometer-scale 3D imaging of materials properties is critical for understanding equilibrium states in electronic materials, as well as for optimization of device performance and reliability, even though such capabilities remain a substantial experimental challenge. Tomographic atomic force microscopy (TAFM) is presented as a subtractive scanning probe technique for high-resolution, 3D ferroelectric property measurements. Volumetric property resolution below 315 nm³, as well as unit-cell-scale vertical material removal, are demonstrated. Specifically, TAFM is applied to investigate the size dependence of ferroelectricity in the room-temperature multiferroic BiFeO₃ across two decades of thickness to below 1 nm. TAFM enables volumetric imaging of ferroelectric domains in BiFeO₃ with a significant improvement in spatial resolution compared with existing domain tomography techniques. We additionally employ TAFM for direct, thickness-dependent measurements of the local spontaneous polarization and ferroelectric coercive field in BiFeO₃ The thickness-resolved ferroelectric properties strongly correlate with cross-sectional transmission electron microscopy (TEM), Landau-Ginzburg-Devonshire phenomenological theory, and the semiempirical Kay-Dunn scaling law for ferroelectric coercive fields. These results provide an unambiguous determination of a stable and switchable polar state in BiFeO₃ to thicknesses below 5 nm. The accuracy and utility of these findings on finite size effects in ferroelectric and multiferroic materials more broadly exemplifies the potential for novel insight into nanoscale 3D property measurements via other variations of TAFM.