Ferromagnetic Heusler alloys can be used in combination with semiconductors to create spintronic devices. The materials have cubic crystal structures, making it possible to grow lattice-matched heterojunctions by molecular beam epitaxy. However, the development of devices is limited by the difficulty of growing epitaxial semiconductors over metallic surfaces while preventing chemical reactions, a requirement to obtain abrupt interfaces and achieve efficient spin-injection by tunneling. We used a solid-phase epitaxy approach to grow crystalline thin film stacks on GaAs(001) substrates, while preventing interfacial reactions. The crystallized Ge layer forms superlattice regions, which are caused by the migration of Fe and Si atoms into the film. X-ray diffraction and transmission electron microscopy indicate that the trilayers are fully crystalline, lattice-matched, and have ideal interface quality over extended areas.