4.3 Doping Characteristics

Figure 4.1: Temperature dependence of the conductivity in a disordered hopping system at different doping concentrations.

Figure 4.2: Temperature dependence of the conductivity in an organic semiconductor plotted as $\textrm { log }\sigma$ versus $T^{-2}$. The dashed line is to guide the eye.

Figure 4.3: Conductivity of doped ZnPc at various doping ratios as a function of temperature. The lines represent the analytical model, experiments (symbols) are from [63].

Figure 4.4: Conductivity as a function of the dopant concentration with temperature as a parameter.

Figure 4.5: Conductivity of PPEEB films versus the dopant concentration. The line represents the analytical model. Experiments (symbols) are from [139].

Fig 4.1 illustrates the temperature dependence of the carrier conductivity for different doping concentrations. Parameters are $\alpha^{-1}=0.37$Å, $E_d=0.5$eV, $T_0=800$K and $T_1=400$K. An Arrhenius-like temperature dependence

$$\log\sigma\propto -E_A/k_BT$$

can be observed clearly in Fig 4.1. In Fig 4.2, we plot $\log\sigma$ versus $T^{-2}$, which is observed to deviate slightly from a straight line (dashed in Fig 4.2). This is because at higher temperatures almost all the carriers occupy the intrinsic states such that the dopants do not change the trap-free hopping relation $\log\sigma\propto T^{-2}$ [95]. The doping process is quite efficient for ZnPc with dopant F4-TCNQ [63]. In Fig 4.3, we compare the measured conductivity at room temperature and the theoretical model (4.7). The agreement is quite satisfactory. The fit parameters are the same as those used in Fig 4.1, and have been chosen according to [63]. From Fig 4.1 and Fig 4.3 we can see that the conductivity increases considerably with the dopant concentration, especially in the lower temperature regime.

The superlinear dependence of conductivity on the doping concentration has been investigated extensively by several groups [81,91,96,97], where the empirical formula

$$\sigma\propto N_d^\gamma$$

is used to describe this dependence. Using our model, such superlinear increase of the conductivity upon doping can be predicted successfully. We show this in Fig 4.4, where the parameters are the same as in Fig 4.1. Our model gives $\gamma=4.9$ for $T=250$K, and $\gamma=3.9$ for $T=200$K. Note that these choices are consistent with those in [81], where the $\gamma$ is chosen in the range $[3,5]$. In Fig 4.5, we compare the predictions of our model with the experimental data of doped PPEEB [91]. The parameters are $\alpha^{-1}=6$Å, $E_d=0.6$eV, $T_0=1000$K and $T_1=500$K. The predictions fit the experimental data very well.

In Fig 4.6 we plot the relation between the conductivity and the doping ratio, defined as

$$\frac{N_d}{N_t+N_d},$$

for different temperatures with parameters $T_0=1000$K, $T_1=500$K, $E_d=0.5$eV and $\sigma_0=1\times 10^7$S/cm. We can see that the conductivity increases with both temperature and doping ratio. More specifically, there is a transition in the increase of the conductivity of an organic semiconductor upon doping, which is manifested by a change in the slope of the curve as shown in Fig 4.7. The conductivity increases linearly for low doping levels, and superlinerly for high doping levels. This transition has been interpreted in [92] in terms of the broadening of the transport manifold due to the enhanced disorder from the dopant.

Figure 4.6: Conductivity as a function of the doping ratio with temperature as a parameter.

Figure 4.7: Conductivity at T=200K as a function of the doping ratio. The dashed line is to guide the eye.

Figure 4.8: Activation energy ($E_A$) as a function of the doping ratio.

Assuming a simple Arrhenius law

$$\sigma\propto \exp\left(\frac{-E_A}{k_BT}\right),$$

we can obtain the relation between activation energy $E_A$ and doping ratio, as shown in Fig 4.8. $E_A$ decreases with the doping ratio, indicating that less and less energy will be required for a carrier activated jump to neighboring sites when the doping ratio increases. Similar to Fig 4.7, we can also observe a transition between the two doping regimes visible as a change in the slope.