The size scaling of the threshold voltage required for the amorphous-to-crystalline transition in phase-change memory (PCM) is investigated using planar devices incorporating individual GeTe and Sb2Te3 nanowires. We show that the scaling law governing threshold switching changes from constant field to constant voltage scaling as the amorphous domain length falls below 10 nm. This crossover is a consequence of the energetic requirement for carrier multiplication through inelastic scattering processes and indicates that the size of PCM bits can be miniaturized to the true nanometer scale.