Ab-initio calculations of structural,phonon and thermal properties of $$hbox {Al}_x hbox {Ga}_{1-x} hbox {As}$$ alloy

We provide the energy spectrum of an electron in a degenerately doped semiconductor of parabolic band. Knowing the energy spectrum, the density-of-states (DOS) functions are obtained, considering the Gaussian distribution of the potential energy of the impurity states, showing a band tail in them e.g., energy spectrum and density-of-states. Therefore, Fermi integrals (FIs) of DOS functions, having band tail, are developed by the exact theoretical calculations of the same. It is noticed that with heavy dopings in semiconductors, the total FI demonstrates complex functions, containing both real and imaginary terms of different FI functions. Their moduli possess an oscillatory function of (eta ) (reduced (hbox {Fermi energy} = E_{mathrm{f}}/k_{mathrm{B}}T), (k_{mathrm{B}}) is the Boltzmann constant and T is the absolute temperature) and (eta _{e}) (impurity screening potential), having a series solutions of confluent hypergeometric functions, (Phi (a, b; z)), superimposed with natural cosine functions of angle (theta ). The variation of (theta ) with respect to (eta ) indicated a resonance at (eta =1.5). The oscillatory behaviour of FIs show the existence of ‘band-gaps’, both in the real as well as in the forbidden bands as new band gaps in the semiconductor.