NOVELTY - Transition layer(s) having lower surface, upper surface, thickness (h), and lattice spacing (at(z)) which varies between the lower and upper surfaces, is formed on upper surface (14) of a substrate, and desired film is formed on the upper surface of the transition layer. The lattice spacing (at(0)) at the lower surface and lattice spacing (as) of substrate satisfy preset relation, with lattice mismatch (m1) of 7%, and lattice spacing (at(h)) at the upper surface and lattice spacing (af) of desired film satisfy preset relation, with lattice mismatch (m2) of less than 7%.    USE - Epitaxial growth of desired film e.g. final film (claimed) on crystalline substrate used for forming electronic and opto-electronic device e.g. high-power transistor, high electron-mobility transistors, laser diode, solar cell and detector.    ADVANTAGE - The method enables economical epitaxial growth of desired film on crystalline substrate, without carrying out thermal processing, with reduced dislocation defects.    DETAILED DESCRIPTION - Transition layer(s) having lower surface, upper surface, thickness (h), and lattice spacing (at(z)) which varies between the lower and upper surfaces, is formed on upper surface of a substrate, and desired film is formed on the upper surface of the transition layer. The lattice spacing (at(0)) at the lower surface and lattice spacing (as) of substrate satisfy the relation: m.at(0)=n.as to within lattice mismatch (m1) of 7%, and lattice spacing (at(h)) at the upper surface and lattice spacing (af) of desired film satisfy the relation: i.at(h)=j.af to within lattice mismatch (m2) of less than 7%, where n, m, I and j are integers. INDEPENDENT CLAIMS are included for the following:    (1) formation of template substrate for growing desired film; and    (2) epitaxial growth of final film.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic view explaining formation of transition layer.    Substrate (10)    Component (11)    Upper surface (14)    Layers (42,42L)    Laser beam (LB)