Index of refraction changes in semiconductors with population inversion

The effect of population inversion on the index of refraction in semiconductors is calculated. A simple model of the absorption edge is modified to include the effects of population inversion. By correlating the density of states in the conduction band with the absoption edge, a modification of the absorption edge occurs because of the filling of states in the conduction band. The effective density of hole states in the valence band is neglected because the density of states in the valence band is much larger than the density of states in the conduction band. When the exponential tail in the conduction band is filled to an energy E
f
, a propagating electromagnetic wave does not attenuate for energies below E
f
defined as the quasi-Fermi level for electrons in the inverted region. No attenuation occurs because electrons in the valence band can not be excited to the filled states of the conduction band. For a wave with an energy above E
f
, band-to-band transistions are allowed because of vacant states in the conduction band. Therefore, for waves with energies
E = h\omega > E_{f}
, the absorption process is the same as the absorption process for a material without population inversion. Free carrier effects are neglected. The change in the index of refraction is calculated for various slopes of the band tails and for various values of Fermi levels below the nominal-band edge. The results show that at low temperatures there is a positive contribution to the index for energies slightly below the Fermi level. At energies much less than the Fermi level, the contribution to the index of refraction is negative. At absolute zero, the crossover between a negative contribution and a positive contribution to the index is
E_{t} = E_{f} - 0.3725E_{0}
where E
0
is the slope of the conduction-band tail. As the temperature increases, the value of E
t
increases. The magnitude of the index of refraction change increases as the slope E
0
increases.