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

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

The charge and energy fluctuations in molecular solids are crucial factors for a better understanding of charge transport (CT) in organic semiconductors. The energetic disorder-coupled molecular charge transport is still not well-established. Moreover, the conventional Einstein's diffusion (D)-mobility (μ) relation fails to explain the quantum features of organic semiconductors, including nonequilibrium and degenerate transport systems, where kB is the Boltzmann constant, T is the temperature and q is the electric charge. To overcome this issue, a unified version of the entropy-ruled D/μ relation was proposed by Navamani (J. Phys. Chem. Lett., 2024, 15, 2519-2528) for hopping and band transport systems as where d, η and heff are the dimension (d = 1, 2, 3), chemical potential and effective entropy, respectively. Within this context, we investigate the CT properties of 2,5-bis(4-methoxyphenyl)thiazolo[5,4-d]thiazole (MOP-TZTZ) and 2,5-bis(2,4,5 trifluorophenyl)-thiazolo[5,4-d]thiazole (TFP-TZTZ) molecular solids using electronic structure calculations and the entropy-ruled method. The CT key parameters such as charge transfer integral and site energy are computed by matrix elements of the Kohn-Sham Hamiltonian. Using Marcus theory, the charge transfer rate is numerically calculated for MOP-TZTZ and TFP-TZTZ molecular crystals under different site energy disorder (ΔEij(E⃑)) situations. Using our entropy-ruled method, the exact diffusion-mobility (D/μ) and other transport quantities such as thermodynamic density of states, conductivity, and current density are calculated for these derivatives at different applied electric field values via the site energy disorder. The theoretical results show that the molecule TFP-TZTZ has good hole mobility (∼0.012 cm2 V-1 s-1) at a site energy disorder value of 90 meV. The obtained ideality factor from the Navamani-Shockley diode current density equation categorizes the typical transport as either the Langevin-type or Shockley-Read-Hall mechanism in the studied molecular solids. Our analysis clearly shows that both the electron and hole transport in these MOP-TZTZ and TFP-TZTZ molecules follow the trap-free Langevin mechanism, which is indeed ideal for designing charge-transporting molecular devices.

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 heff 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 heff 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.

Quantum-Classical Transition Analogy of the Diffusion-Mobility Relation for Hopping and Band Transport Systems: Molecules to Materials

We propose a quantum-classical transition analogy for Einstein's diffusion-mobility (D/μ) relation to reveal electron-hole dynamics in both the degenerate and nondegenerate molecular and material systems. Here, one-to-one variation between differential entropy and chemical potential (Δη/Δh_s) 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.

Generalization on Entropy-Ruled Charge and Energy Transport for Organic Solids and Biomolecular Aggregates

Herein, a generalized version of the entropy-ruled charge and energy transport mechanism for organic solids and biomolecular aggregates is presented. The effects of thermal disorder and electric field on electronic transport in molecular solids have been quantified by entropy, which eventually varies with respect to the typical disorder (static or dynamic). Based on our previous differential entropy (h s )-driven charge transport method, we explore the nonsteady carrier energy flux principle for soft matter systems from small organic solids to macrobiomolecular aggregates. Through this principle, the synergic nature of charge and energy transport in different organic systems is addressed. In this work, entropy is the key parameter to classify whether the carrier dynamics is in a nonsteady or steady state. Besides that, we also propose the formulation for unifying the hopping and band transport, which provides the relaxation time-hopping rate relation and the relaxation time-effective mass ratio. The calculated disorder drift time (or entropy-weighted carrier drift time) for hole transport in an alkyl-substituted triphenylamine (TPA) molecular device is 9.3 × 10-7 s, which illustrates nuclear dynamics-coupled charge transfer kinetics. The existence of nonequilibrium transport is anticipated while the carrier dynamics is in the nonsteady state, which is further examined from the rate of traversing potential in octupolar molecules. Our entropy-ruled Einstein model connects the adiabatic band and nonadiabatic hopping transport mechanisms. The logarithmic current density at different electric field-assisted site energy differences provides information about the typical transport (whether trap-free diffusion or trap-assisted recombination) in molecular devices, which reflects in the Navamani-Shockley diode equation.