Subsections

• 2.3.1 Theory
• 2.3.2 Low Electric Field Regime
  • 2.3.2.1 Results and Discussion
  
  
• 2.3.3 High Electric Field Regime
  • 2.3.3.1 Results and Discussion

2.3 Temperature and Electric Field Dependence of the Mobility

VRH theory has been applied successfully to describe the temperature dependence of conductivity in organic materials [17,43,59]. However, it is more difficult to obtain the experimentally observed electric field dependence. In this section, we extend the VRH theory to get a temperature and electric field dependent conductivity model.

For a disordered organic semiconductor we assumed that localized states are randomly distributed in both energy and space coordinates, and that they form a discrete array of sites. The presented theoretical calculations are applied to explain recent experiment. A good agreement between theory and experiment is observed.

2.3.1 Theory

When an electric field $F$ exists, the transition rate of a carrier hopping from site $i$ to site $j$ is described as [60]

$$\omega_{ij}=\nu_0\left\{\begin{array}{r@{\quad:\quad}l}\exp\left[... ...left(-2\alpha R_{ij}\right) & E_j-E_i\leq qFR_{ij}\cos\theta \end{array}\right.$$ (2.10)

where $\theta$ is the angle between $E$ and $R_{ij}$. Assuming no correlation between the occupation probabilities of different localized states, the current between the two sites is given by

$$I_{ij}=\nu_0\exp\left[-2\alpha R_{ij}-\frac{\mid E_j-E_i+qF\cos\t... ...{E_i-\mu_i}{2k_BT}\right)\cosh\left(\frac{E_j-\mu_j}{2k_BT}\right)\right]^{-1},$$

where $\mu_i$ and $\mu_j$ are the chemical potentials of sites $i$ and $j$, respectively [64].

2.3.2 Low Electric Field Regime

To determine the conductivity of an organic system, one can use percolation theory, regarding the system as a random resistor network [61,62]. In the case of low electric field, the resulting voltage drop over a single hopping distance $\left(\Delta\mu\ll k_BT\right)$ is small. The conductance between sites $i$ and $j$ can be simplified from (2.11) to the form

$$\sigma_{ij}=\sigma_0\exp\left(- 2\alpha R_{ij}+\frac{\mid E_i-E_F\mid+\mid E_j-E_F\mid+\mid E_j-E_i+qF\cos\theta R_{ij}\mid}{2k_BT}\right).$$ (2.11)

Using the same derivation discussed in the previous section, we obtain as a result the percolation criterion for an organic system as

$$B_c\approx N_t\frac{\pi k_BT}{2qF}\left(\frac{T_0}{T}\right)^3\exp\left(\frac{E_F+s_ck_BT}{k_BT_0}\right)\zeta\left(F,T\right),$$ (2.12)

with

$$\zeta=\left(2\alpha-\frac{qF}{k_BT}\right)^{-2}-\left(2\alpha+\frac{qF}{k_BT}\right)^{-2}$$

This yields the expression for the conductivity as

$$\sigma=\sigma_0\left\{\frac{\pi k_BT\delta N_t}{2qFB_c}\left(\fra... ...3\frac{1}{\Gamma(1-T/T_0)\Gamma(1+T/T_0)}\zeta\left(F,T\right)\right\}^{T_0/T}.$$ (2.13)

Equation (2.13) is obtained assuming

• that the site positions are random,
• the energy barrier for the critical hop is large compared to $k_BT$,
• and the charge carrier concentration is very low.

To describe the mobility, we use the mobility definition given by
[63]

$$\mu=\sigma(\delta,T)\frac{T_0}{T}\frac{1}{q\delta N_t}.$$ (2.14)

Figure 2.6: Plot of $\textrm { log }\sigma$ versus $T^{-1/4}$ at the electric field $100 V/cm$.

Figure 2.7: Conductivity and mobility versus temperature for ZnPc as obtained from the model (2.13) and (2.14) in comparison with experimental data (symbols).

2.3.2.1 Results and Discussion

Using expression (2.13), the conductivity has been calculated as a function of T at an electric field of $100$V/cm, as shown in Fig 2.6. One can see the linear dependence of conductivity on $T^{-1/4}$ (the dashed line is a guide to the eye). We also use the presented model to calculate the temperature and electric field dependences of the conductivity and mobility of ZnPc (Zinc phthalocyanine). In Fig 2.7, the results are obtained from (2.13) using $\sigma_0=12.5\times 10^5S/m$, $T_0=485K$ and $\alpha^{-1}=0.3$$\AA$. The experimental data is from [63].

Figure 2.8: Logarithm of the mobility versus $T^{-1}$. The electric field is $10^5 V/$cm, $\sigma _0=1.1\times 10^9 S/cm$, $T_0=340$K, $\alpha ^{-1}=0.5$ $\AA$

Figure 2.9: The same data as in Fig 2.8 plotted versus $T^{-2}$.

Fig 2.8 and Fig 2.9 show the mobility plotted semilogarithmically versus $T^{-1}$ and $T^{-2}$, respectively. Symbols are TOF (time of flight) experimental data for ZnPc from [65] and the solid lines are the results of the analytical model. The dashed line is to guide the eye. In both presentations a good fit is observed. But when plotted as $\log\mu$ versus $T^{-2}$, the slope is reduced when temperature is lower than the transition temperature $T_c\approx 210$K. This transition has also been observed by Monte-Carlo simulation [48].

Figure 2.10: Plot of $\textrm { log }\sigma$ versus $F^{1/2}$ at $T=204$ K.

Figure 2.11: Electric field dependence of the mobility at 290K. Symbols represent Monte Carlo results [49], the line represents our work with parameter $T_0$=$852$K.

The field dependence of the conductivity is presented in Fig 2.10. The conductivity is approximately constant for very low fields, and increases as we increase the field. This is the result of the fact that the field can decrease the activation energy for forward jumps, enabling the motion of carriers. In Fig 2.11 we also compare the mobility (2.14) to the Monte-Carlo result reported in [49].

2.3.3 High Electric Field Regime

With increasing electric field, the voltage drop over a single hopping distance increases. If this voltage drop is of the order of $k_BT$ or larger, the approximate expression (2.13) for conductivity does not longer hold. The current between the two sites depends on the chemical potential of the sites, which in turn depends on the strength and direction of the electric field. Therefore, a percolation model is usually adopted, assuming site-to-site hopping currents instead of conductance [64].

However, in this case, a conductivity model for the high electric field regime can only be obtained after some approximations. According to percolation theory, the critical percolation cluster of sites would comprise a current carrying backbone with at least one site-to-site current equal to the threshold value. Since a steady-state situation would prescribe a constant current throughout the whole current carrying backbone, the charge will redistribute itself along the path, thus changing the chemical potentials of sites. Hapert omitted this rearrangement by optimization of the current with tunneling [64]. Potentially, the redistribution of charge would change the tunneling current, but this effect seems negligible compared to large spread $I_{ij}$. As a result, the conductivity between two sites is given by

$$\sigma_{ij}\approx\exp\left(-s_{ij}\right),$$ (2.15)

with

$$s_{ij}=2\alpha R_{ij}+\ln\left(\frac{qF}{2k_BT}R_{ij}\right).$$ (2.16)

Combining (2.2), (2.8) and (2.16), the following expression for the percolation criterion is obtained.

$$B_c\approx \frac{N_t}{2}\left[1+\frac{\delta}{\Gamma\left(1-T/T_0... ...mma\left(1+T/T_0\right)}\right]\int{d\bf {R_{ij}}}\theta\left(s_c-s_{ij}\right)$$ (2.17)

This gives the conductance as

$$\ln\left(\sigma/\sigma_0\right)=-2\alpha\eta-\ln\left(\frac{qF\eta}{2k_BT}\right)$$ (2.18)

where

$$\eta=-\frac{2k_BT}{qF}\ln\left[1-\left(\frac{qF}{2k_BT}\right)^3\... ...(1+\delta/{\Gamma\left(1-T/T_0\right)\Gamma\left(1+T/T_0\right)}\right)}\right]$$ (2.19)

2.3.3.1 Results and Discussion

In Fig 2.12, the conductivity is presented logarimically as a function of $F^{1/2}$ for high electric field . In this case, a field-saturated drift velocity, i.e. $\sigma\propto F^{-1}$, is observed in accordance with the simulation work [66] and experiment [67]. At very high fields the effective disorder seen by a migrating carrier vanishes and backward transitions are excluded [9]. The temperature dependence of conductivity at the electric field of $1\times 10^5$ V/cm is presented in Fig 2.13. An Arrhennius-like temperature dependence $\ln\sigma\propto -E_a/(k_BT)$ is also observed at low temperature.

Figure 2.12: Field dependence of the conductivity at different temperatures.

Figure 2.13: Temperature dependence of the conductivity at different electric fields.