The Einstein Relation in Quantum Wells and Wires of Heavily Doped Nonlinear Optical and Optoelectronic Materials: Simplified Theory and Suggestion for Experimental Determination

In this paper an attempt is made to study the Einstein relation for the diffusivity-mobility ratio (DMR) in quantum wells (QWs) and Quantum well wires (QWWs) of heavily doped non-linear materials forming band tails on the basis of a newly formulated electron dispersion law considering the anisotropies of the effective electron masses, the spin orbit splitting constants and the influence of crystal field splitting within the framework of k.p formalism. The results of the quantum confined III-V optoelectronic compounds form a special case of our generalized analysis. The DMR has also been studied for QWs and QWWs of heavily doped II-VI and IV-VI materials on the basis of newly derived electron energy spectra. It has been found taking QWs and QWWs of heavily doped n-CdGeAs₂, Hg_1−xCd_xTe, In_1−xGa_xAs_yP_1−y lattice matched to InP, CdS and PbSe as examples that the DMR increases with increasing carrier statistics and decreasing film thickness in various oscillatory manners and the nature of oscillations are totally band structure dependent emphasizing the influence of dimensional quantizations and the energy band constants in different cases. A dimension independent experimental method of determining the DMR in heavily doped nanostructures forming band tails having arbitrary dispersion law has been suggested and the theoretical results are in quantitative agreement with the suggested relationship. In addition, the expressions of the DMRs for QWs and QWWs of wide gap un-doped materials have been obtained as special cases of our generalized analysis under certain limiting conditions.