Charge-carrier relaxation in disordered organic semiconductors studied by dark injection: Experiment and modeling

Understanding of stationary charge transport in disordered organic semiconductors has matured during recent years. However, charge-carrier relaxation in nonstationary situations is still poorly understood. Such relaxation can be studied in dark injection experiments, in which the bias applied over an unilluminated organic semiconductor device is abruptly increased. The resulting transient current reveals both charge-carrier transport and relaxation characteristics. We performed such experiments on hole-only devices of a polyfluorene-based organic semiconductor. Modeling the dark injection by solving a one-dimensional master equation using the equilibrium carrier mobility leads to a too-slow current transient, since this approach does not account for carrier relaxation. Modeling by solving a three-dimensional time-dependent master equation does take into account all carrier transport and relaxation effects. With this modeling, the time scale of the current transient is found to be in agreement with experiment. With a disorder strength somewhat smaller than extracted from the temperature-dependent stationary current-voltage characteristics, also the shape of the experimental transients is well described.

Figure 1

(a) Measurement setup for the dark-injection (DI) experiments. (b) Applied voltage pulse sequence. An offset voltage of 1.5 V is used. (c) DI transients for a device with a 122-nm-thick polymer layer and a pulse amplitude of 8 V. Solid lines, measurements; blue dotted lines, results of the one-dimensional master equation (1D ME) approach; green dashed and red dash-dotted lines, results of the three-dimensional master equation (3D ME) approach for two different values of the disorder strength $\sigma$. The arrows indicate the peak position in the transients, and the circle a shoulder in the 3D ME transient for $\sigma =0.12$ eV. The error bars of all presented results are negligible. The inset in (c) shows the chemical structure of the used light-emitting polymer.

Figure 2

DI transients for different pulse amplitudes for devices with LEP-layer thicknesses of (a) $L=67$, (b) $L=98$, and (c) $L=122$ nm. The parameters used in the 1D and 3D ME modeling are given in Table . The transient presented in (c) for a pulse amplitude of 8 V is the same as in Fig. 1c.

Figure 3

(a–c) Peak times [see arrows in Fig. 1c] as a function of pulse amplitude of the DI transients for devices with different LEP-layer thicknesses. Circles, experiments; gray lines, Many-Rakavy theory;[22] squares, 1D ME modeling results with $\sigma =0.12$ eV; triangles, 3D ME modeling results with $\sigma =0.08$ eV. The 3D ME transients with $\sigma =0.12$ eV do not show a peak [see Figs. 1c and 2]. In addition to the transients shown in Fig. 2 also the peak times of transients at other pulse amplitudes were included.

Figure 4

Current density versus voltage, $J$-$V$, characteristics of the three devices with different LEP-layer thicknesses. Fits to 1D and 3D ME modeling results lead to the parameters given in Table .

Figure 5

Comparison of the DI transient obtained with the 3D ME approach with that obtained with the 3D Monte Carlo (MC) approach, for the device with $L=67$ nm, a pulse amplitude of 5 V, and $\sigma =0.12$ eV. The load resistance is not included. The error bars in the 3D MC results are shown. In the 3D MC simulations explicit effects of Coulomb interactions between the charge carriers are included, but not in the 3D ME calculations. No shoulder is in this case visible in the transients.