Band Gaps of Semiconductor Alloys

It is shown, on the basis of perturbation theory and a simple assumption about the matrix elements of the random scattering potential, that the effect of alloy disorder is inevitably to reduce the band gap of a semiconductor alloy below its virtual-crystal value. The conclusion, which was previously suggested by Van Vechten and Bergstresser (VB) on the basis of a more intuitive argument, is in agreement with existing theory and calculation. In contrast to VB, however, the present treatment predicts that the magnitude of the disorder-induced shift will vary from one direct gap to another in the same alloy, and that the uppermost gaps may be larger than a straight-line interpolation of the gaps of the constituents. This prediction appears to be corroborated by the electroreflectance studies of Thompson et al. on $\mathrm{Ga}\left({\mathrm{As}}_x{\mathrm{P}}_{1-x}\right)$.