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We present the exact analytical equation of diffusion-mobility fortwo-dimensional (2D) Schr\"odinger type transport systems, from molecules tomaterials. The density of electronic states in such Schr\"odinger systemspertains to the 2D non-relativistic carrier dynamics. We implement the Gaussianfunction into carrier density derivation; accordingly we develop the electroniccompressibility and diffusion-mobility for both the generic and the degenerateFermi systems. This model is originally developed from generalized Einsteinrelation, along with concern about the thermodynamic effects on many-bodyinteractions. The effect of interactions is included through the imperfectFermi-gas entropy function. Our extended model explains the cooperativebehavior of thermal and electronic counterparts on diffusion-mobility indisordered systems at wide temperature range. Using earlier experimental andtheoretical results, we have shown the validity of our extended Einstein modelfor different 2D degenerate systems. The results validate the original Einsteinequation at certain sets of temperature and chemical potential values fordifferent Gaussian variances. Beyond those combinations, the deviation isobserved. At very low temperature, the diffusion-mobility depends only onchemical potential, which is the extended Einstein equation for ideal quantummaterials.

Extended Einstein diffusion-mobility equation for two-dimensional Schr\"odinger-type quantum materials