Quantitative lateral and vertical piezoresponse force microscopy on a PbTiO₃ single crystal

Piezoresponse force microscopy (PFM) has emerged as a powerful tool for research in ferroelectric and piezoelectric materials. While the vertical PFM (VPFM) mode is well understood and applied at a quantitative level, the lateral PFM (LPFM) mode is rarely quantified, mainly due to the lack of a practical calibration methodology. Here by PFM imaging on a LiNbO₃ 180° domain wall, we demonstrate a convenient way to achieve simultaneous VPFM and LPFM calibrations. Using these calibrations, we perform a full quantitative VPFM and LPFM measurement on a (001)-cut PbTiO₃ single crystal. The measured effective piezoelectric coefficients d 33 e f f and d 35 e f f together naturally provide more information on a material's local tensorial electromechanical properties. The proposed approach can be applied to a wide variety of ferroelectric and piezoelectric systems.