Light, Einstein Relation, Quantization and the Heavily Doped Opto-Electronic Materials

In this paper we study the Einstein relation (ER) in heavily doped (HD) quantum confined opto-electronic materials on the basis of newly formulated electron dispersion law within the framework of k . p method in the presence of intense light waves which changes the dispersion relation fundamentally from real to complex one due to the existence of the poles in the finite complex plane of the corresponding electron energy spectrum in accordance with the three band model of Kane in the absence of band tails. Taking heavily doped n-InAs and n-InSb as examples of III-V compounds, n-Hg1-xCdxTe and n-In1-xGaxAsyP1-y lattice matched to InP as examples of HD ternary and quaternary materials we observe that under magnetic quantization the ER oscillates with inverse quantizing magnetic field due to Shubnikov-de Haas effect. The ER also decreases with increasing light intensity and wavelengths in different manners which are band structure dependent. Under cross fields configuration, the ER increases with increasing electric field and in quantum wells ER increases with increasing concentration, decreasing intensity and wavelengths respectively. The numerical values are different in different cases because of the signature of the quantization of the wave vector space due to quantizing magnetic field, cross field configuration and size confinements respectively. The ER decreases with increasing alloy composition in all the cases and we have suggested an experimental method of determining the ER for HD materials. Besides we have investigated the effective electron mass (EEM) and sub-band energy as a collateral study. The EEM exists in the forbidden zone, which is impossible without the effect of band tailing. In the absence of band tails, the effective mass in the band gap of semiconductors is infinity. Under certain limiting conditions all the results get transformed into the well-known formula of ER as derived for the first time by Landsberg [P. T. Landsberg, Eur. J. Phys. 2, 213 (1981)] and thus confirming the compatibility test.