A ZnO thin film is fabricated on the c-surface of a sapphire substrate through use of a laser molecular beam epitaxy (MBE) method-which is effective for epitaxial growth of an oxide thin film through control at an atomic level. The thus-formed ZnO thin film has a considerably high crystallinity; the half width of an X-ray rocking curve was 0.06.degree.. The thin film is of an n-type and has a carrier density of 4.times.10.sup.17 /cm.sup.3. The thin film fabricated in a state in which oxygen partial pressure is held constant at 10.sup.-6 Torr has a structure in which hexagon-shaped nanocrystals of uniform size are close-packed, reflecting the crystal behavior of a wurtzite type. Since in each nanocrystal there is observed a spiral structure formed by steps of a unit cell height (0.5 nm), the nanocrystals are considered to grow in a thermodynamically equilibrated state. The lateral size of the nanocrystal can be controlled within the range of approximately 50 to 250 nm. A II-oxide optical semiconductor element utilizes a zinc oxide thin film containing magnesium or cadmium in a solid-solution state. Through addition of magnesium or cadmium, the band gap of zinc oxide can be controlled within the range of 3 to 4 eV.