The Einstein relation in nanostructured materials: simplified theory and suggestion for experimental determination

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 tetragonal compounds on the basis of a newly formulated electron energy spectrum taking into account the combined influences of the anisotropies in effective electron mass, the spin–orbit splitting, and the presence of crystal field splitting, respectively. The results for quantum-confined III–V compounds form a special case of our generalized analysis. The DMR has also been studied for QWs and QWWs of II–VI and IV–VI materials. Taking QWs and QWWs of CdGeAs₂, InAs, CdS and PbSe as examples, it was found that the DMR increases with increasing carrier statistics and decreasing film thickness respectively in various oscillatory manners emphasizing the influence of dimensional quantizations and the energy band constants in different cases. An experimental method of determining the DMR in nanostructures with arbitrary dispersion laws has also been suggested and the present simplified analysis is in agreement with the suggested relationship. The well-known results for nanostructures with parabolic energy bands have also been obtained as special cases from this generalized analysis under certain limiting conditions.