Continuum time-delayed electron hopping in the extended dynamical molecules and entropy-ruled Einstein relation for organic semiconductors

Charge transport (CT) in dynamically disordered molecular systems is still unclear; though it is fundamentally important to understand the semiconducting properties of molecular devices. In this regard, we explore vibronically coupled polaron hopping transport in the extended hopping systems (N + 1 sites) of thiazolothiazole (TZTZ) based molecules. The molecular vibrations correlated charge transfer integral and site energy fluctuation effects on polaron transport are analyzed by kinetic Monte-Carlo simulations. In order to quantify the CT properties more precisely, we have proposed the continuum time delayed CT mechanism, which takes account of typical disordered (static or dynamic) effect via dispersion on each CT quantity (like charge transfer rate, diffusion coefficient, mobility, current density and etc) at each hopping. The charge compressibility analysis further addresses the electronic level understanding of all CT quantities, which originally relates the thermodynamic density of states with CT. Using differential entropy-dependent charge density and diffusion expressions, the drift-diffusion transport has been elucidated for different extended systems of TZTZ derivatives. Besides, we have mainly developed entropy-ruled diffusion-mobility relation for both degenerate and nondegenerate materials to study the validity and limitations of original Einstein relation, which directly pertain to the device performance. Here, the traversing chemical potential along the hopping sites is the deterministic parameter of diffusion-mobility ratio. Using our continuum time delayed model, we can categorize the typical disordered transport in the molecular semiconductors; whether is dynamic or static or intermediate disordered transport. (paper)