Influence of magnetic quantization on the Einstein relation in non-linear optical, optoelectronic and related materials: Simplified theory, relative comparison and suggestion for experimental determination

In this paper, we have investigated the Einstein relation for the diffusivity-to-mobility ratio (DMR) under magnetic quantization in non-linear optical materials on the basis of a newly formulated electron dispersion law by considering the crystal field constant, the anisotropies of the momentum-matrix element and the spin–orbit splitting constant, respectively, within the frame work of k·p formalism. The corresponding result for the three-band model of Kane (the conduction electrons of III–V, ternary and quaternary compounds obey this model) forms a special case of our generalized analysis. The DMR under magnetic quantization has also been investigated for II–VI (on the basis of Hopfield model), bismuth (using the models of McClure and Choi, Cohen, Lax and parabolic ellipsoidal, respectively), and stressed materials (on the basis of model of Seiler et al.) by formulating the respective electron statistics under magnetic quantization incorporating the respective energy band constants. It has been found, taking n-CdGeAs2, n-Hg1−xCdxTe, p-CdS, and stressed n-InSb as examples of the aforementioned compounds, that the DMR exhibits oscillatory dependence with the inverse quantizing magnetic field due to Subhnikov de Haas (SdH) effect with different numerical values. The DMR also increases with increasing carrier degeneracy and the nature of oscillations are totally dependent on their respective band structures in various cases. The classical expression of the DMR has been obtained as a special case from the results of all the materials as considered here under certain limiting conditions, and this compatibility is the indirect test of our generalized formalism. In addition, we have suggested an experimental method of determining the DMR for degenerate materials under magnetic quantization having arbitrary dispersion laws. The three applications of our results in the presence of magneto-transport have further been suggested.