A transition crystal structure is disclosed for providing a good lattice and thermal match between a layer of single crystal silicon carbide and a layer of single crystal gallium nitride. The transition structure comprises a buffer formed of a first layer of gallium nitride and aluminum nitride, and a second layer of gallium nitride and aluminum nitride adjacent to the first layer. The mole percentage of aluminum nitride in the second layer is substantially different from the mole percentage of aluminum nitride in the first layer. A layer of single crystal gallium nitride is formed upon the second layer of gallium nitride. In preferred embodiments, the buffer further comprises an epitaxial layer of aluminum nitride upon a silicon carbide substrate.