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.

Spintronic elements based on spin transfer torque have emerged with potential for on-chip memory, but they suffer from large energy dissipation due to the large current densities required. In contrast, an electric-field-driven magneto-electric storage element can operate with capacitive displacement charge and potentially reach 1-10 µJ cm^-2 switching operation. Here, magneto-electric switching of a magnetoresistive element is shown, operating at or below 200 mV, with a pathway to get down to 100 mV. A combination of phase detuning is utilized via isovalent La substitution and thickness scaling in multiferroic BiFeO₃ to scale the switching energy density to ≈10 µJ cm^-2 . This work provides a template to achieve attojoule-class nonvolatile memories.