Important formulas

A Generalized Paradigm on Einstein diffusion-mobility relation for quantum materials/devices


Generalized version of Einstein's diffusion-mobility relation for Schrödinger-type (nonrelativistic electron hole kinetics in 1D, 2D and 3D) quantum materials/Devices is

The governing relation of diffusion-mobility ratio for two-dimensional layered semiconductors is given as,

According to our revisited Einstein relation, the derived thermal conductivity (ĸ)-charge diffusion (D) ratio (i.e., extended version of Wiedemann-Franz law) is expressed as,

The generalized diffusion-mobility relation for Dirac materials (relativistic dynamics of electron and hole) is,

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

Entropy ruled diffusion-mobility relation for organic semiconductors


The proposed entropy-ruled diffusion-mobility relation is described by

According to our entropy-ruled Einstein relation, the modified Shockley diode equation (also referred as, Navamani-Shockley diode current density equation) for molecular quantum devices,

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

Multiscale modeling of charge transport (on the basis of "3-set of analytical procedures") in molecular semiconductors


Mathematical Structures of 3-set of analytical procedure are as follows,


The proposed momentum-energy redistribution expressions to analyze the drift–diffusion property is given as,

Here, Pmom(t) is the carrier momentum during the charge transport, which is non-linearly related with the survival probability P(t)

Pmom,0 is the initial momentum of the particle

ΔU is the Potential energy changes during the charge transport simulation

EK,0 is the Initial kinetic energy of the particle (hole or electron)

(Both ΔU and EK,0 are strongly depends on temperature and chemical potential)


The governed carrier drift energy-current density relation to explore the current density with respect to the applied bias/ applied electric field is expressed as,

The Proposed equations of entropy-dependent charge density and diffusion coefficient to study the validity and limitations of Einstein relation (with Physics of deviation in the molecules/materials) are described as

Here, hS, n(E) and D(E) are the differential entropy, carrier density and diffusion coefficient under the bias or applied electric field condition, respectively.

Using this multiscale model, the material scientists can categorize the applicability of molecules of interest for photovoltaic or light emitting diode applications.

Reference: Effect of site energy fluctuation on charge transport in disordered organic molecules

Density Flux Model on Hopping Conductivity for Disordered Molecular Solids


The proposed density flux model for hopping conductivity formula to calculate electrical transport in disordered molecular solids is expressed as

Here, N is the carrier concentration (or number of particles)

ε is the electric permittivity of medium or molecule

dP/dt is the rate of transition probability, which is equivalent to that of charge transfer rate


Reference: Effect of Structural Fluctuations on Charge Carrier Dynamics in Triazene Based Octupolar Molecules