Vogel-Tammann-Fulcher model for charging dynamics in an organic
electrochemical transistor: electronic supplementary information
Parisa Shiri, Earl Jon S. Dacanay, Brennan Hagen, Loren G. Kaake
Simon Fraser University, Department of Chemistry
8888 University Drive, Burnaby, BC V5A 1S6 Canada
Mathematical model
The results shown in fig. 2 demonstrate that two processes are primarily responsible for charging dynamics in organic
electrochemical transistors. The two processes are ion currents in the dielectric and mixed ion-carrier diffusion in the organic
semiconductor. These processes are coupled, ion-carrier pairs cannot diffuse into the organic semiconductor prior to the polarization of
ions in the dielectric. We begin by specifying the problem in terms of two limiting cases. In the first case, the rate of ion-carrier diffusion is
orders of magnitude faster than the movement of ions in the dielectric. In this case, the system can be modeled by considering only the
behavior of the dielectric. It is common to use a stretched exponential function to model currents in a polymer electrolyte,
1, 2
we therefore
model the RC charging behavior using the following function:
() =
0
(1 − [−(
)
]) (1)
In eq. 1,
0
represents the charge carrier concentration at steady state,
= ()
−1
is the time constant of device charging from a simple
RC circuit model and is a constant between 0 and 1 which is indicative of static and dynamic disorder in the ion conducting material.
In the second limiting case, the rate of ion polarization in the dielectric is orders of magnitude faster than the diffusion of ion-
carrier pairs in the semiconductor. In this case, we model the problem as a simple case of 1-D diffusion with a single diffusivity, , that is
constant both in time and in position. To solve the differential equation, we assume a constant boundary condition at the semiconductor-
dielectric interface and a no-flux boundary condition at the opposing side of the film. In our experiments, the opposing side of the film is in
contact with a silicon substrate that is not being electrically addressed. We assume that it is impermeable to the ions. The solution to this
problem is well known and is given in terms of a Fourier series.
3
(, ) =
1
(1 −
4
∑
∞
[
] [−
2
2
]) (2)
In eq. 2,
1
represents the steady state solution, a spatially uniform distribution of that value. Terms in the Fourier series are distinguished
by
= (2 − 1)/2 which is used in the calculation of the coefficients,
along with the semiconductor film thickness, .
=
2
∫ (ℎ(, 0) −
1
) [
]
0
(3)
In equation 3, ℎ(, 0) is the initial condition, the ion-carrier pair concentration at =0.
Combining these two limiting cases into a single solution requires that the boundary condition at the dielectric-semiconductor
interface changes with time as described by eq. 1. This stipulation makes the boundary conditions for the diffusion equation
inhomogeneous. We chose to develop a solution numerically. The calculation begins by assuming that there are no ion-carrier pairs
throughout the semiconductor at =0. It is likely that trapped charges linger from a previous charge/discharge cycle, but in our data
analysis, we subtract out any signal from trapped charge, causing ℎ(, 0) = 0 for the experiment as well as the simulation.
In the first time step, which we will call
1
, we set
1
= (
1
) and solve for (, Δ
1
) where Δ
=
−
−1
is the time variable
that is input into equation 3 during each step of the numerical solution. During the calculation for the next time step, we set
1
= (
2
)
and calculate the Fourier coefficients using the previous solution as our initial condition. In symbols, we set ℎ(,
2
) = (, Δ
1
) during our
calculation of
for the second step of the time series. This process is repeated, always using the previous solution as the initial conditions
for the next time step. Because of the iterative nature of the solution, convergence is not guaranteed. However, choosing sufficiently fine
time samples at short timescales aids in obtaining convergence for rapidly varying components and choosing sufficiently fine spatial
sampling aids in obtaining convergence for slowly varying components.
Electronic Supplementary Material (ESI) for Journal of Materials Chemistry C.
This journal is © The Royal Society of Chemistry 2019