Revisiting Einstein's diffusion-mobility relation for universal quantum materials: A generalized paradigm

We have revisited the Einstein diffusion-mobility ratio, on the basis of the generalized Einstein relation, which quantifies both the quantum and classical levels of understanding on electron-hole dynamics in ordered and disordered materials. While the Einstein relation (diffusion-mobility ratio, D/${\mu}$ ) works only for classical systems under equilibrium cases, our generalized paradigm is quite general and works for systems ranging from classical to quantum systems in any dimensions. The proposed D/${\mu}$ formalism relies on the chemical potential which provides the cooperative nature between electronic and temperature counterparts on fundamental transport. Our generalized paradigm is applicable to both degenerate and non-degenerate cases, and explains the transition from linear to nonlinear relationship between the charge density and chemical potential. The symmetrical nature of electron-hole transport in highly degenerate two-dimensional quantum materials gives linear dispersion, preserving the time-reversal invariance property, while the symmetry is found to be broken in the nonlinear regime. Our generalized diffusion-mobility relation explains both the relativistic (Dirac particles) and non-relativistic regimes (Schrödinger particles) from low temperature to high temperature, and reproduces the classical Einstein relation in an appropriate limit