Nonequilibrium drift-diffusion model for organic semiconductor devices

Two prevailing formalisms are currently used to model charge transport in organic semiconductor devices. Drift-diffusion calculations, on the one hand, are time effective but assume local thermodynamic equilibrium, which is not always realistic. Kinetic Monte Carlo models, on the other hand, do not require this assumption but are computationally expensive. Here, we present a nonequilibrium drift-diffusion model that bridges this gap by fusing the established multiple trap and release formalism with the drift-diffusion transport equation. For a prototypical photovoltaic system the model is shown to quantitatively describe, with a single set of parameters, experiments probing (1) temperature-dependent steady-state charge transport—space-charge limited currents, and (2) time-resolved charge transport and relaxation of nonequilibrated photocreated charges. Moreover, the outputs of the developed kinetic drift-diffusion model are an order of magnitude, or more, faster to compute and in good agreement with kinetic Monte Carlo calculations.

Figure 1

Schematic band diagram under near-equilibrium conditions, indicating the electron mobility edge $E_c$ and the quasi-Fermi level $E_F$ in the disordered semiconductor that is sandwiched between two metallic contacts (blue areas). The arrows indicate a single release-transport-trapping cycle characterized by rates ${\nu }_{1,0}$ and free mobility μ*. The blue curve indicates the Gaussian DOS $g_0\left(\mathrm{E}\right)$.

Figure 2

Flow diagram of the numerical implementation of the kDD model. Typical values for both the number of particles (NoP) and the number of time steps (NoT) are ${10}^3$. When a particle annihilates by recombination or extraction at one of the contacts within NoT, it is recreated by injection or photogeneration. The weight factor for the new solution when averaging with the old one is typically 0.2–0.05.

Figure 3

(a) Temperature-dependent current-voltage characteristics of a hole-only TQ1:PCBM device with $L=110\mathrm{nm}$. Open symbols are experiments, and connected solid symbols are calculations by MC. (b) Comparison between calculations by MC (connected solid symbols) and kDD (dashed lines). Simulation parameters are ${\sigma }_{\mathrm{DOS}}=0.085\mathrm{eV},{\nu }_0=5.53\times {10}^{10}{\mathrm{s}}^{-1}$, and $a_{\mathrm{NN}}=1.8\mathrm{nm}$, giving ${\mu }_0=1.8\times {10}^{-8}{\mathrm{m}}^2\mathrm{V}{\mathrm{s}}^{-1}$. Injection barriers of 0.2 eV were used for the Ohmic contacts.

Figure 4

Thickness dependence of space-charge limited conduction as calculated by MC (connected solid symbols) and kDD (dashed lines). $T=300\mathrm{K}$, and $L=50$, 110, 200, and 300 nm; otherwise, the same parameters are used as in Fig. 3.

Figure 5

Simulation of combined TREFISH and transient photocurrent experiments. (a) Extracted charge vs time as calculated by MC (connected open symbols) and kDD (solid red lines), using the hole parameters from Fig. 3 and electron parameters ${\sigma }_{\mathrm{DOS}}=0.12\mathrm{eV},{\nu }_0=1\times {10}^{13}{\mathrm{s}}^{-1},L=70$ nm, and $T=300\mathrm{K}$ from Ref. [21]. (b) Measured (heavy black lines; data taken from Ref. [21]) and calculated by kDD (solid red lines) charge extraction for a $\mathrm{TQ}1:\mathrm{P}{\mathrm{C}}_{71}\mathrm{BM}$ organic photovoltaic device. Parameters are obtained by a least-squares fitting procedure and are ${\sigma }_{\mathrm{DOS}}=0.125\mathrm{eV}$ and ${\nu }_0=1.84\times {10}^{13}{\mathrm{s}}^{-1}$ for electrons and ${\sigma }_{\mathrm{DOS}}=0.113\mathrm{eV}$ and ${\nu }_0=4.93\times {10}^{10}{\mathrm{s}}^{-1}$ for holes. The thin dashed lines are an MC calculation using the parameters from Ref. [21]: ${\sigma }_{\mathrm{DOS}}=0.10\mathrm{eV}$ and ${\nu }_0=1\times {10}^{10}{\mathrm{s}}^{-1}$ for holes and ${\sigma }_{\mathrm{DOS}}=0.12\mathrm{eV}$ and ${\nu }_0=1\times {10}^{13}{\mathrm{s}}^{-1}$ for electrons.

Figure 6

Time-dependent relaxation of photocreated electrons and holes as calculated by MC (symbols) and kDD (lines). Same parameters as Fig. 5.