Unified Entropy-Ruled Einstein’s Relation for Bulk and Low-Dimensional Molecular-Material Systems: A Hopping-to-Band Shift Paradigm

We present a unified paradigm on entropy-ruled Einstein’s diffusion-mobility relation (μ/ D ratio) for 1D, 2D, and 3D free-electron solid state systems. The localization transport in the extended molecules is well approximated by the continuum time-delayed hopping factor within our unified entropy-ruled transport method of noninteracting quantum systems. Moreover, we generalize an entropy-dependent diffusion relation for 1D, 2D, and 3D systems as defined by Dd,heff=Dd,heff=0⁡exp((d−1)heffd+2), where h eff and d are the effective entropy and dimension ( d = 1, 2, 3), respectively. This generalized relation is valid for both equilibrium and nonequilibrium transport systems since the parameter h eff is closely connected with the nonequilibrium fluctuation theorem-based entropy production rule. Importantly, we herein revisit the Boltzmann approach using an entropy-ruled method for mobility calculation for the universal quantum materials that is expressed as μd=[(dd+2)qdheffdη]vF2τ2, where vF2τ2 is the diffusion constant for band transport systems and η is the chemical potential. According to our entropy-ruled μ/ D relation, the Navamani–Shockley diode equation is transformed.