Transient Behavior of Organic Electrochemical Transistors Based on a Fractional Extension of the Bernards–Malliaras Model

We present a fractional-order generalization of the Bernards–Malliaras model for organic electrochemical transistors (OECTs), extending its applicability to devices with solid or polymer-supported electrolytes. The framework employs a fractional law to describe ionic transport dynamics, naturally capturing memory effects and anomalous diffusion arising from ion trapping, percolation, and polymer–electrolyte interactions. The ionic current is assumed proportional to a fractional time derivative of the voltage across an effective fractional capacitance, and analytical expressions for the channel charge and transient drain current are derived in closed form using the Mittag–Leffler formalism. The fractional order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $</tex-math> </inline-formula> quantifies deviations from normal diffusion. The model reproduces experimentally observed long-tailed transient responses, allowing to predict the small-signal transconductance in the frequency domain as well as the switching behavior of the OECTs. This fractional Bernards–Malliaras framework provides a physically consistent and compact description of coupled ionic–electronic transport in OECTs, enabling improved interpretation of transient dynamics and guidance for the design of next-generation bioelectronic and neuromorphic circuits.