Thermodynamics of organic electrochemical transistors

Despite their increasing usefulness in a wide variety of applications, organic electrochemical transistors still lack a comprehensive and unifying physical framework able to describe the current-voltage characteristics and the polymer/electrolyte interactions simultaneously. Building upon thermodynamic axioms, we present a quantitative analysis of the operation of organic electrochemical transistors. We reveal that the entropy of mixing is the main driving force behind the redox mechanism that rules the transfer properties of such devices in electrolytic environments. In the light of these findings, we show that traditional models used for organic electrochemical transistors, based on the theory of field-effect transistors, fall short as they treat the active material as a simple capacitor while ignoring the material properties and energetic interactions. Finally, by analyzing a large spectrum of solvents and device regimes, we quantify the entropic and enthalpic contributions and put forward an approach for targeted material design and device applications.

The circles indicate the charged species in play, namely P SS − , holes and sodium cations, and the arrows indicate the Coulomb interaction, either attraction (green) or repulsion (red). Despite the complex chemical process involving multiple species, the energetics of the system does not change dramatically when PEDOT goes from fully oxidized to fully reduced. This explains why an OECT must be described in terms of entropy, and why enthalpy play a secondary role, marking a strong contrast with capacitive systems such as FETs.

Supplementary Note 1: Gibbs free energy of the electrolyte
The Gibbs free energy of the electrolyte:OMIEC system is given by the sum of their free energy G e and Go, respectively. Before the OMIEC is immersed in the electrolyte, the electrolyte has a free energy G e,1 . When the OMIEC is immersed, some ions migrate from the bulk electrolyte into the polymer, effectively diluting the electrolyte and bringing the electrolyte to a free energy G e,2 . Upon dilution, the difference in energy is given by where c 1 and c 2 are the concentrations before and after the OMIEC has been immersed. The enthalpy of the solution depends on the ion concentration (although it is a good approximation to neglect it because of the low ion concentration 100 mM i.e., ideal solution). When the OMIEC is immersed, roughly 10% of its charged moieties are counterbalanced by ions from the electrolyte ( Figure 2 in the main text). Let's assume PEDOT:PSS has a PSS density (hence a maximum charge carrier density) of 1x10 20 cm −3 (upper limit of a highly doped semiconductor, PEDOT:PSS is probably much less). By using a film of size 200 µm x 50 µm x 100 nm, one finds that 1x10 5 charges move from the electrolyte to the organic film.
For these experiments, the OMIECs were immersed in a beaker (V≈ 1 l) containing the electrolyte. Such volume contains 6.2x10 22 ions. Often, OECT measurements are performed by casting a drop of electrolyte on top of gate and channel. Considering a volume of 1 ml, the calculation yields 6.2x10 19 ions. Therefore, after subtracting the above-calculated number of ions that diffuses in the film, one finds that c 1 ≈ c 2 . Thus ∆G e = 0. Therefore, the thermodynamics of the electrolyte is not altered by the presence of such a miniaturized thin film. We conclude that the Gibbs free energy of the electrolyte should be counted, but it adds up to the free energy of the polymeric system as a constant. As such, when measuring variation of energy (∆G), or when we do the derivative to calculate the chemical potential µ, such a constant is lost. It is therefore physically meaningful and mathematically correct to carry out the thermodynamics analysis on the thin film only.

Supplementary Note 2: calculation of the drain current
In order to find the number of charges accumulated in the channel by the gate voltage, Ohm's law (Eq. 7) must be integrated over the channel length from 0 to L (or equivalently from V ds to 0) as done typically for thin-film transistors.
after considering a linearly dropping potential V ch along the channel length (direction ℓ) and its differential: In doing so we are assuming i) that the gradual channel approximation is valid i.e., the field across the channel/electrolyte interface is much higher than the field between source and drain across the channel, and ii) a linearly dropping potential across the channel. The former holds true thanks to the exponential decay of the field in the electrolyte. The second is not necessarily true and strongly depends on the gate voltage. However, the result of the integral is not expected to vary much depending on the charge distribution. To solve Eq. Supplementary Eq. 3, we need ϕ(µ), that can be found by inverting µ(ϕ) in Eq. 8: ). The application of a gate potential has the effect of shifting the system from its chemical equilibrium. By introducing the electrochemical potentialμ, µ becomes a function of the potentials applied as e(V gs − V ch ) as explained in the main text.
Combining Supplementary Eq. 3 and Supplementary Eq. 6 β ln( Ze βV ds + e βVgs Z + e βVgs ) (Supplementary Eq. 7) with β = e k B T (elementary charge divided by thermal energy). Thus, the current that flows in the channel of an OECT, assuming no enthalpy of mixing, depends on V ds , V gs , and T as follows: and has the typical (and peculiar) bell-shaped form of g m (V gs ) of OECTs. For the transfer measurements in Fig. 4 with V ds = 5mV , we exploit the fact that V gs − V ch ≈ V gs in order to rewrite Supplementary Eq. 3 as (Supplementary Eq. 10) from which the material-specific property ϕ(µ) can be extracted.

Enhancement/accumulation mode OECTs
For an enhancement mode OECT, the FET model would foresee the following I-V relationship Eq. 7 becomes and Eq. 10 becomes Ze −βVgs + 1 )).

Gate-channel coupling
A final note on the gate voltage: in order to fully capture the current dependency on the applied gate voltage V gs one must take into consideration that the gate voltage applied differs from the effective gate voltage V g,ef f "felt" by the channel that drives the device at the channel/electrolyte interface due to the drop at the gate/electrolyte interface and across the solution. In this work, since we focused on the modeling of the material parameters, we simplified it as V gs,eff = αV gs (Supplementary Eq. 14) where 0 < α < 1 is the remaining percentage of potential applied after dropping across the electrolyte and at the gate/electrolyte interface. Although the voltage drop from gate/electrolyte interface to electrolyte/PSS interface can have a more complex, nonlinear, voltage-dependent and concentration-dependent, we use a linear approximation for the sake of simplicity, and may be the root of the unmatching fit in Fig. 3 Fig. 4. The experimental I-V transfer curves are transformed into a µ − ϕ as explained in as explained in Supplementary Note 2. This is possible only because the curves are measured at vanishing drain voltage. By integrating the chemical potential, the Gibbs free energy is obtained, which is the curve on which the fit is carried out.