A semiconductor device comprising a semiconductor crystalline substrate having projections each thereof having an area of 0.01 .mu.m.sup.2 to 4 .mu.m.sup.2 or stripe projections each thereof having a width of 0.01 .mu.m to 1 .mu.m and semiconductor crystalline layers formed on the projections, each of the layers having lattice constants different from those of the semiconductor crystalline substrate preferably by 0.5% or more. The semiconductor device is free of dislocations and thermally stable. The semiconductor device can be fabricated by performing such processes as forming projections on the substrate and forming semiconductor crystalline layers on the projections by molecular beam epitaxy.