1. The Makineni Theoretical Laboratories, Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104–6323, USA
   
2. Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104–6272, USA
   
3. Present address: Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang 790–784, Korea.

Correspondence to: Andrew M. Rappe¹ Correspondence and requests for materials should be addressed to A.M.R. (Email: rappe@sas.upenn.edu).

The motion of domain walls is critical to many applications involving ferroelectric materials, such as fast high-density non-volatile random access memory¹. In memories of this sort, storing a data bit means increasing the size of one polar region at the expense of another, and hence the movement of a domain wall separating these regions. Experimental measurements of domain growth rates in the well-established ferroelectrics PbTiO₃ and BaTiO₃ have been performed, but the development of new materials has been hampered by a lack of microscopic understanding of how domain walls move^{2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. Despite some success in interpreting domain-wall motion in terms of classical nucleation and growth models^{12, 13, 14, 15, 16}, these models were formulated without insight from first-principles-based calculations, and they portray a picture of a large, triangular nucleus that leads to unrealistically large depolarization and nucleation energies⁵. Here we use atomistic molecular dynamics and coarse-grained Monte Carlo simulations to analyse these processes, and demonstrate that the prevailing models are incorrect. Our multi-scale simulations reproduce experimental domain growth rates in PbTiO₃ and reveal small, square critical nuclei with a diffuse interface. A simple analytic model is also proposed, relating bulk polarization and gradient energies to wall nucleation and growth, and thus rationalizing all experimental rate measurements in PbTiO₃ and BaTiO₃.