Piezoresponse force microscopy is used to study the velocity of the polarization domain wall in ultrathin ferroelectric barium titanate (BTO) films grown on strontium titanate (STO) substrates by molecular beam epitaxy. The electric field due to the cone of the atomic force microscope tip is demonstrated as the dominant electric field for domain expansion in thin films at lateral distances greater than about one tip diameter away from the tip. The velocity of the domain wall under the applied electric field by the tip in BTO for thin films (less than 40 nm) followed an expanding process given by Merz’s law. The material constants in a fit of the data to Merz’s law for very thin films are reported as about 4.2 KV/cm for the activation field,
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