Methods are provided for producing SiGe-on-insulator structures and for forming strain-relaxed SiGe layers on silicon while minimizing defects. Amorphous SiGe layers are deposited by CVD from trisilane and GeH4. The amorphous SiGe layers are recrystallized over silicon by melt or solid phase epitaxy (SPE) processes. The melt processes preferably also cause diffusion of germanium to dilute the overall germanium content and essentially consume the silicon overlying the insulator. The SPE process can be conducted with or without diffusion of germanium into the underlying silicon, and so is applicable to SOI as well as conventional semiconductor substrates.