Modeling of charge transport in polymers with embedded crystallites

Modern organic semiconductors frequently adopt a semicrystalline morphology which makes the analysis of charge transport challenging. In this paper, we employ Monte Carlo (MC) simulation to study the charge-carrier mobility in an amorphous layer—typically 100 nm thick—that contains well-ordered nanosized domains (crystallites). The case of a low carrier concentration, typical for applications in photovoltaics and LEDs, is considered. We study the dependence of mobility on temperature, on the amount $\left(V\right)$ of the crystallites, on the energy offset between crystalline and amorphous regions $\left(E_t\right)$, and we compare the results with those for a system with solitary traps. We find that, in a system with solitary traps, the mobility can exceed that of the neat phase if the trap depth is comparable with the standard deviation of the density of states (DOS) of the amorphous phase. Controlled by the electronic overlap in the amorphous phase and the crystallites, the mobility in a system with crystallites can increase by up to two orders of magnitude upon increasing V. It features a pronounced maximum when $E_t$ is close to the standard deviation of the DOS of the amorphous phase, while for large values of $E_t$, the mobility is practically independent of $E_t$. The results can be rationalized in terms of an interplay between percolation at large $E_t$ and hopping transport within the cumulative densities near the transport of the system. We developed a generalized analytic multiple-trapping-release model to rationalize the results of the MC simulations.

Figure 1

Schematic view of (a) a polymer layer with amorphous and aggregated regions and (b) the energetic distribution of electronic states in this polymer.

Figure 2

Illustration of how a two-phase structure is generated. (a) Generating of the points where the crystallites grow. (b) Defining the orientations of chains in the crystallites. (c) The structure after 5 cycles of growth. (d) The final structure.

Figure 3

Illustration of the contribution of diffusion to the mobility as defined from Eq. (7) in an amorphous material with (a) $V=0$ and (b) $V=0.25$; in a two-phase material with (c) $V=0.25$ and (d) $V=0.39$. $T=298$ K, ${\sigma }_1=50\mathrm{meV}$, ${\sigma }_2={\sigma }_1/3$, $E_t=-2{\sigma }_1$, $2{\gamma }_{2}a=5$, and $2{\gamma }_{1}a=10;{\mu }_C=\left(e/kT\right){\nu }_0{a}^2\exp \left(-2\gamma a\right),{\mu }_0$ is the mobility value at $V=0$.

Figure 4

The drift mobility, parametric in the mean energy depth of the lower Gaussian $E_t$. The other parameters: $T=298$ K, $\langle l\rangle =8$ nm, $eFa=kT$, ${\sigma }_1=50\mathrm{meV}$, ${\sigma }_2={\sigma }_1/3$; $2{\gamma }_{2}a=5$, and $2{\gamma }_{1}a=10$ for the hops involving and not involving crystallites, respectively. (a) Sample with crystallites, i.e., the ${\mathrm{M}}_1$ model; (b) sample with traps, i.e., the ${\mathrm{M}}_0$ model.

Figure 5

Dependence of the bulk mobility in an amorphous material with trap sites (${\mathrm{M}}_0$ model) for two values of temperature on the average energy of the G2 states $E_t$ and for various trap fractions V. The other parameters are the same as in Fig. 4. The horizontal black line shows the mobility value at $V=0$. (a) Data for 298 K. The horizontal red line shows the analytic estimate for $E_t\to -\infty$, $V=0.39$, according to Eq. (9) below. (b) Data for 200 K.

Figure 6

Dependence of the mobility in a material with crystallites (${\mathrm{M}}_1$ model) on the mean energy of states in crystallites $E_t$. Here, ${\mu }_{\left(100\right)}$ is the mobility calculated for the sample thickness 100 nm, and ${\mu }_{\mathrm{bulk}}$ is the mobility for the infinite sample. The data are shown for different values of the fraction of crystallites V for room temperature. The other parameters are the same as in Fig. 4.

Figure 7

Temperature dependences of the mobility in material of the ${\mathrm{M}}_1$ model at various values of the mean energies of states in crystalline phase $E_t$; $V=39\%$, for (a) a 100-nm film and (b) a bulk film. The solid black curves show analytic estimates from (1) the Gaussian disorder model (GDM) model ($V=0$, 2γa $=5$), (2) Eq. (16), and (3) Eq. (17), further below. The other parameters are the same as in Fig. 3. The mobility at $V=0$ and $T=298$ K, 2γa $=10$ is denoted as ${\mu }_0$.

Figure 8

Comparison of Monte Carlo (MC) results with analytic results of multiple trapping and release (MTR), see Eq. (9), and revised MTR, see Eq. (3), normalized to the mobility ${\mu }_0$ of a trap-free amorphous material ($V=0$).

Figure 9

Dependences of the average energy, as calculated from Eq. (15), on the fraction of G2-states, V (a) and temperature (b). Solid and dashed vertical arrows show the values of $E_t$ at which the maximum mobility is achieved in material with crystallites (${\mathrm{M}}_1$ model, value taken from Fig. 6) or with singular traps (${\mathrm{M}}_0$ model, value taken from Fig. 5), respectively. The dashed-dotted diagonal line indicates the values for $E_{\mathrm{av}}=E_t$. Here, ${\sigma }_1$ is 50 meV. The horizontal dashed lines denote to the average equilibrium energies of occupied G1 states $-{\sigma }_1^2/kT$.