A fractional calculus approach to analyze physicochemical relaxation processes of charge trapping in organic based nano-photodiodes

Fractional calculus deals with differentiation and integration of arbitrary real or complex orders. Its application to physicochemical processes in organic-based nano-photodiodes offers a distinct advantage by enabling the explicit characterization of memory effects, phenomena that are often hided in conventional calculus. We present a novel application of fractional calculus to quantify metastable effects in organic nano-photodiodes, revealing temperature-dependent trap states characterized via Mittag-Leffler kinetics. Therefore, the primary analyses focuses on the metastable capacitance, ΔC, polarisation, P, and conductance, G. Although these metastable effects have been reported previously for organic-based transistors. It is remarkable that these effects are also commonly observed in organic-based nano-photodiodes. The model structure consists of a poly(3-hexylthiophene) (P3HT):[6,6]-phenyl-C₆₁-butyric acid methyl ester (PCBM) active layer deposited on a glass substrate coated with indium tin oxide (ITO) and a poly(3,4-ethylenedioxythiophene):polystyrene sulfonate (PEDOT:PSS) buffer layer, with aluminum (Al) serving as the top electrode. Measurements were carried out from 122.0 K to 335.0 K. The trap activation temperature was determined from the dielectric loss (G/ω) measured at a frequency of 1 kHz (f = 2πω), which is well below the cutoff frequency of 650 kHz. The system dynamics were modeled by deriving a fractional differential equation within the Caputo paradigm, with transient solutions expressed as Mittag-Leffler functions ML(α,−t/τ), where α is the fractional order and τ the relaxation time constant. Main findings are: (1) A unique fractional calculus approach was applied to analyze metastability in organic nano-photodiodes, leading to a first-principles derivation of a fractional equation for transient capacitance and polarisation. (2) The unresponsive polarisation between 191.3 and 253.9 K reflects balanced hole–electron recombination rates. (3) Temperature-dependent trap regimes below 191.3 K and above 253.9 K provides a roadmap for optimizing organic photodiodes by tailoring active-layer compositions to suppress recombination in critical temperature ranges.