Statistical mechanics for organic mixed conductors: phase transitions in a lattice gas

TL;DR

The problem addressed is how to describe high-density, strongly interacting charge carriers in organic mixed conductors beyond conventional semiconductor theory. The authors propose a minimal lattice-gas model in the grand canonical ensemble, bridging statistical mechanics with device physics. They show a first-order vapor–liquid-like transition with vapor and liquid carrier-density phases and metastability, explaining mesoscale domains and hysteresis in OECTs. By linking gate bias to an effective chemical potential and deriving a mean-field transfer relation, the work connects thermodynamics to drain-current behavior and highlights universality near criticality.

Abstract

Organic mixed conductors (OMCs) represent a promising class of materials for applications in bioelectronics, physical computing, and thermoelectrics. Rather unparalleled, OMCs feature dynamics spanning multiple length and time scales, involving an intricate coupling between electronic, ionic, and mass transport. These characteristics set them notably apart from traditional semiconductors and hinder the description by conventional semiconductor theory. In this work, we approach the charge carrier modulation of OMCs using statistical mechanics. We discuss OMCs from a first-principles perspective and contrast them with established semiconductor materials, highlighting key differences in their collective charge carrier dynamics. This motivates our toy model describing OMCs as a lattice gas, which we analyze within the grand canonical ensemble. The model exhibits a first-order phase transition analogous to a classical vapor$\unicode{x2013}$liquid transition, governed by temperature and chemical potential. In doing so, it explains the formation of distinct low- and high-density carrier phases $\unicode{x2013}$ a mesoscale phenomenon recently observed experimentally. It also demonstrates how metastability near the phase boundary can give rise to history-dependent characteristics in device operation, a similarly well-reported effect in OMC transistors. This work is intended as a simple motivation for studying OMCs through the lens of statistical mechanics, offering a more natural description than traditional semiconductor models developed for materials of fundamentally distinct character.

Figures (4)

• Figure 1: Phase diagram of a lattice gas model, mapping equilibrium charge carrier density $\rho$ against temperature $T$ and chemical potential $\mu$ ($J=1.0$). At high temperatures, density varies smoothly (supercritical fluid). At low temperatures, a sharp, first-order phase transition occurs at $\mu_c = -J$, dividing a low-density "vapor" phase (free carriers) from a high-density "liquid" phase (correlated carriers). Insets show simulation snapshots for each phase, corresponding to the markers in the phase diagram.
• Figure 2: Dynamical phase diagram showing the average carrier density $\langle \rho \rangle$ as a function of Monte Carlo time $t$ and chemical potential $\mu$ at fixed temperature $T = 0.8$ ($J = 1.0$, $\rho_0 = 0$, averaged over $10^3$ realizations). The dashed line marks $\langle \rho \rangle = 0.5$, indicating the dynamical phase boundary. Inset: the $\langle \rho \rangle = 0.5$ contour for temperatures $T = 0.2$ (blue) to $T = 1.0$ (yellow), showing that metastable states become increasingly long-lived at lower temperatures.
• Figure 3: Interplay of probability and grand potential in the mean-field model ($J=1.0$). (Top) Probability density $p[\rho]$ as a function of carrier density $\rho$. (Bottom) The corresponding grand potential landscape, $\phi_{T,\mu}(\rho)$ (shifted vertically for clarity). The peaks in probability correspond directly to the minima in the grand potential. As the chemical potential $\mu$ increases, the most probable state (global minimum) shifts from a low-density "vapor" phase (blue) to a high-density "liquid" phase (green), passing through a coexistence point where both phases are equally probable (orange). Insets show representative simulation snapshots for each regime.
• Figure 4: Predicted transfer characteristics from the lattice-gas model: drain current $I_D$ versus gate voltage $V_G$ at $V_D = 0.1\,\mathrm{V}$ for several reduced temperatures $T$. Above the critical temperature, the response is smooth; below, the first-order transition produces abrupt switching. Dashed segments indicate where the local chemical potential crosses the coexistence region, corresponding to hysteretic behavior in experiment. Parameters: $J=1.0$, $\mu_{\mathrm{reservoir}}=-1.0$, $\gamma=1.0$, $W=50\,µm$, $L=100\,µm$, $t=100\,\mathrm{nm}$, $n_{\max}=1\times 10^{21}\,\mathrm{cm}^{-3}$, $\mu_{\mathrm{tr}}=1\,\mathrm{cm}^2\mathrm{V}^{-1}\mathrm{s}^{-1}$.