Quantum-Classical Transition Analogy of Diffusion-Mobility Relation for Organic Semiconductors

Abstract

We propose the quantum-classical transition analogy for Einstein’s diffusion-mobility (D/μ) relation to reveal the electron hole dynamics in both the degenerate and nondegenerate molecular solids. Here, one to one variation between differential entropy and chemical potential (Δη/Δhs) is the proposed analogy, which unifies quantum and classical transport. The degeneracy stabilization energy on D/μ decides whether the transport is quantum or classical; accordingly, the transformation occurs in the Navamani-Shockley diode equation.

Entropy-ruled nonequilibrium charge transport in thiazolothiazole-based molecular crystals: a quantum chemical study

“…The device performance is generally characterized by the diode ideality factor (N id ). According to our earlier reports, 27,60 . Here, the device (e.g., diode) performance in terms of J-V characteristics is closely associated with the enhancement factor (g), which is here derived from the ratio between entropy-ruled D/m and Einstein's D/m factor.…”

“…8, the applied electric fielddriven degeneracy strength increases the chemical potential (refer eqn (15)), which shows the deviation in the Einstein relation from its original value of k B T/q (see Table S7, ESI †). The value of deviation in Einstein relation is normally quantified by the enhancement factor, g. 60 In our analysis, the chemical potential increases in a quasi-linear manner with respect to the differential entropy. In this case, the DE ij ( -E)-driven carrier energy flux (along the hopping sites) causes the production of degeneracy, which is characterized by the differential entropy, h S .…”

“…To overcome this limitation of the Einstein relation, the promising entropy-ruled transport method was developed for both quantum and classical systems, and it works for the entire range from equilibrium to nonequilibrium. 27,28,60 In this work, we have used the entropy-ruled diffusion-mobility relation (see eqn (14)) to calculate the exact value of the diffusion-mobility relation for these molecules. According to this theoretical procedure, we have estimated the differential entropy and chemical potential at different sets of electric field-assisted site energy values (DE ij ( - E)) at finite temperature, which provides the precise diffusion-mobility value for these (MOP-TZTZ and TFP-TZTZ) molecular solids.…”

“…The enhancement factor (g) is a dimensionless parameter that quantifies the deviation of the D/m value from the classical Einstein relation (D/m = k B T/q). 60 According to our entropy-ruled method, the enhancement factor depends on the DZ/Dh s value, which also acts as the quantum-classical transition analogy. 60 The enhancement factor (g) for MOP-TZTZ and TFP-TZTZ is 3.103 and 3.135 (for hole transport), and 3.239 and 3.205 (for electron transport) respectively.…”

“…60 According to our entropy-ruled method, the enhancement factor depends on the DZ/Dh s value, which also acts as the quantum-classical transition analogy. 60 The enhancement factor (g) for MOP-TZTZ and TFP-TZTZ is 3.103 and 3.135 (for hole transport), and 3.239 and 3.205 (for electron transport) respectively.…”