3.3 Results and Discussion

In Fig 3.1 we compare our work with Baranovskii's model for the temperature characteristics of the transport energy. The input parameters are $N_t=1\times 10^{22}$cm$^{-3}$, $E_0=0$eV, $\xi=30kT$, $\alpha^{-1}=1$Å. The two models agree very well when the temperature is high enough, but differ in the low temperature range.

Calculation of transport energy versus the normalized chemical potential $\xi$ is given for different DOS standard deviation in Fig 3.2 with parameters

Figure 3.1: Comparison between the model (3.8) and Baranovskii's model for the temperature characteristics of $E_{tr}$.

Figure 3.2: The transport energy versus the chemical potential for different standard deviations $a$ of the DOS.

Figure 3.3: The transport energy versus the relative carrier concentration for different standard deviations $a$ of the DOS.

Figure 3.4: Dependence of the relaxation time on the chemical potential for different standard deviations $a$ of the DOS .

Figure 3.5: Temperature dependence of the carrier mobility in organic semiconductors. In (a) the data are plotted versus $T^{-1}$, in (b) the same data are plotted versus $T^{-2}$.

a

b

Figure 3.6: Carrier concentration dependence of the mobility in organic semiconductors.

$E_0=0$eV, $N_t=1\times 10^{22}$cm$^{-3}$, $\alpha^{-1}=2$$\AA$ and $\sigma_0=0.05$eV.

As expected, for a very low chemical potential level (very low carrier concentration), the two models agree very well. However, when the chemical potential goes up and thus the concentration increases, the transport energy considering Fermi statistics will increase as well, while in the Baranovskii model the transport energy is independent on the chemical potential. Baranovskii's model of the transport energy can only be used when the carrier concentration is low enough.

The dependence of the transport energy on the relative carrier concentration ($n/N_t$) can be seen in Fig 3.3. The transport energy increases at a relative carrier concentration of about $1\times 10^{-2}$.

For the calculation of the hopping mobility [78], the relaxation time $\tau_{rel}$ is important, which can be calculated as

$$\tau_{rel}=\nu_0^{-1}\exp\left(2\alpha R\left(E_{tr}\right)+\frac{\left(E_{t}-E_\infty\right)}{k_BT}\right).$$ (3.10)

$E_\infty$ is the thermal equilibrium energy of hopping carriers, defined as

$$E_\infty=-\frac{\sigma^2}{k_BT}.$$

We plot the relation between $\tau_{rel}$ and the carrier concentration in Fig 3.4 with parameters $N_t=1\times 10^{22}$cm$^{-3}$ and $\alpha^{-1}=1$Å. We can see that the relaxation time is constant when the chemical potential is low enough, but it increases for $\xi\ge -5$ for our case.

We apply the calculated transport energy to the problem of charge mobility in organic semiconductors. Using the Einstein relation we obtain [78]

$$\mu\propto\left(\frac{q}{k_BT}\right)R\left(E_{tr}\right)^2\langle t\rangle,$$ (3.11)

the average hopping time is determined as

$$\langle t\rangle=\frac{\left(\int_{-\infty}^{E_{tr}} PdE\right)}{\int_{-\infty}^{E_{tr}}g\left(E\right)dE}$$ (3.12)

with

$$P=\nu_0\exp\left(2\alpha R\left(E_{tr}\right)+\frac{\left(E_{tr}-E\right)}{k_BT}\right)g\left(E\right).$$

Fig 3.5 compares the temperature dependence of the carrier mobility as obtained from our model and Baranovskii's model. The input parameters are $N_t=1\times 10^{22}$cm$^{-3}$, $\alpha^{-1}=1$$\AA$ and $\xi=30$ $k_B$T. The graph $\log\mu$ versus $T^{-1}$ and $\log\mu$ versus $T^{-2}$ are plotted in Fig 3.5 (a) and (b). Our model can describe a deviation from straight. In fact, at higher temperature, the mobility is controlled by jumps of carriers that occupy intrinsic sites, so that the occurrence of the traps does not change the linear relation between $\log\mu$ versus $T^{-2}$. At lower temperature, the traps in organic semiconductors play a more important role for charge transport [68].

In Fig 3.6 we plot the relation between the mobility and the carrier concentration. The input parameters are $N_t=1\times 10^{22}$ cm$^{-3}$, $\gamma=1\times 10^{15}$s$^{-1}$ and $\alpha=1 \AA$.

It is illustrated that the mobility remains constant when the carrier concentration is very low. However, it will increase when the carrier concentration is above a critical value. This result coincides with experimental data given in [79] and recent work given in [80].