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 D d , h e f f = D d , h e f f = 0 ⁡ exp ( ( d - 1 ) h e f f d + 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 μ d = [ ( d d + 2 ) q d h e f f d η ] v F 2 τ 2 , where v F 2 τ 2 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.