An adaptive phase field model is used to study the morphological instability and the facet formation during a directional solidification of SiGe alloys. Using highly anisotropic surface free energy and cusp kinetic functions, as well as a proper temperature gradient, the simulated critical growth velocity for various alloy concentrations is in good agreement with classical theories and experimental observations. The simulated facet formation and kinetics are also consistent with the experiments as well (C) 2013 Elsevier B.V. All rights reserved.