Modeling of the transient mobility in disordered organic semiconductors with a Gaussian density of states

The charge-carrier mobility in organic semiconductors is often studied using non-steady-state experiments. However, energetic disorder can severely hamper the analysis due to the occurrence of a strong time dependence of the mobility caused by carrier relaxation. The multiple-trapping model is known to provide an accurate description of this effect. However, the value of the conduction level energy and the hopping attempt rate, which enter the model as free parameters, are not a priori known for a given material. We show how for the case of a Gaussian density of states both parameters can be deduced from the parameter values used to describe the measured dc current-voltage characteristics within the framework of the extended Gaussian disorder model. The approach is validated using three-dimensional Monte Carlo modeling. In the analysis, the charge-density dependence of the time-dependent mobility is included. The model is shown to successfully predict the low-frequency differential capacitance of sandwich-type devices based on a polyfluorene copolymer.

Figure 1

(a) Schematic structure of PF-TAA. (b) Experimental $C\left(V\right)/C_{V=0}$ curves (symbols) for a 97-nm device at $T=295$ K for various frequencies[33] and calculated 100-Hz curve without relaxation.

Figure 2

Schematic current density response in (a) a spatially uniform system to (b) a small sudden increase of the carrier density, and (c) a schematic representation of the time-dependent population of the conduction level and the Gaussian DOS. The dashed lines indicate the initial and final positions of the top of the density of occupied states.

Figure 3

Calculated time dependence of the zero-field mobility relative to ${\mu }_0$ (the dc mobility in the Boltzmann limit, BL) for PF-TAA ($\widehat{\sigma }=5.1$, including results for various concentrations) and for systems with $\widehat{\sigma }=4$ and 3 (BL). Results from the MT model (solid spheres) with $E_{\mathrm{c}}=-0.75\sigma$, $-0.65\sigma$, and $-0.60\sigma$, respectively, are compared with the results from MC calculations (solid curves, smoothened). The dashed curves give the MT results for $\widehat{\sigma }=5.1$ (BL) with $E_{\mathrm{c}}=-0.5\sigma$ (I) and $-1.0\sigma$ (II). For PF-TAA, the time range corresponds to 0.08 ns to 80 s.

Figure 4

Comparison of theoretical and experimental $C\left(V\right)$ curves at 295 K using $E_{\mathrm{c}}=-0.75\sigma$, for (a,b) 97-nm and (c,d) 121-nm PF-TAA devices. In (a), the dashed curves show MT results at 100 Hz for $E_{\mathrm{c}}=-0.5\sigma$ and $-1.0\sigma$.

Figure 5

(a) Calculated concentration dependence of the real and imaginary components of ${\mu }_{\mathrm{ac}}/{\mu }_0$ for PF-TAA. (b) Local relaxation contribution to the capacitance (normalized, solid curve) and carrier concentration (dashed) for a 97-nm device at 4 V and 250 Hz.