6.3 Results and Discussion

First we solve (6.8) numerically. The position dependence of the quasi Fermi energy is shown in Fig 6.2. The parameters are $N_t=1\times 10^{22}$cm$^{-3}$, $j=0.4$A/cm$^2$, $\mu=1$cm$^2$/Vs, $\epsilon_r=3$, $\sigma/{kT}=4$ and the sample thickness $L=100$nm. It can be seen that the quasi Fermi-energy decreases with position and increases with current density. Near the contact, the quasi Fermi energy decreases very quickly.

Figure 6.2: Spatial distribution of the quasi Fermi energy for different current densities.

Figure 6.3: Spatial distribution of the carrier concentration near the contact for $j=0.4$A/cm$^2$.

We treat the metal electrode as site 0 with Fermi energy as $E_F=-E_m$, where $E_m$ is the metal work function. The Ohmic contact at $x=0$ implies that the field must drop to zero at this coordinate, so that $F\left(0\right)=0$.

Figure 6.4: Spatial distribution of the electric field at different current densities.

Fig 6.3 shows that the carrier concentration decreases from the contact. Fig 6.4 shows the field distribution in OLED with the same parameters as in Fig 6.2. The $j/V$ characteristics are plotted in Fig 6.5 for different $\sigma /{kT}$, where the parameters are the same as Fig 6.2. As we can see, at low voltages and current densities, the current follows a $j\propto V^2$ characteristics, which may suggest either the trap-free case or the shallow-trap case. At higher voltages, the space charge is formed mainly by carriers occupying states above Fermi energy and the current increases with voltage faster than $V^2$. This behavior is also predicted by a SCLC model based on an exponential DOS distribution [124], where

$$j\propto\frac{V^{m+1}}{L^{2m+1}}.$$

The parameter $m=E_t/{kT}$ varies between about 1 and 4, $E_t$ is the characteristic energy of the exponential DOS and $L$ is the layer thickness of LED.

Figure 6.5: Current-voltage characteristics of a sample with Gaussian DOS distribution parametric in temperature.

Figure 6.6: The effect of the field dependent mobility on the space charge limited current.

The available models for SCLC transport assume constant mobility, and include or neglect traps. However, it was found that the mobility in organic semiconductors depends on the local electric field [119].

Integrating (6.5) yields

$$F=\frac{q}{\epsilon_0\epsilon_r}\int_0^x\int_\infty^\infty g\left(E\right)f\left(E\right)dE,$$ (6.8)

(6.6) is rewritten as

$$F\mu_0\exp\left(\gamma\sqrt{F}\right)=\frac{j}{q}\left(\int_{-2\sigma}^\infty g\left(E\right)f\left(E\right)dE\right)^{-1}.$$ (6.9)

Substituting (6.9) into (6.8), we can obtain a new equation for quasi Fermi-energy as

$$\frac{\mu_0 q^2}{\epsilon_0\epsilon_rj}\left(\int_0^x\int_\infty^... ...)= \left(\int_{-2\sigma}^\infty g\left(E\right)f\left(E\right)dE\right)^{-1}.$$

The quasi Fermi energy and electric field distribution can be obtained by solving (6.10) numerically. Fig 6.6 illustrates the effect of electric field dependent mobility on SCLC with $\gamma=1\times 10^{-3}$(m/V)$^{1/2}$ and $\mu_0=1$cm$^2$/Vs. Other parameters are the same as in Fig 6.3. For comparison, SCLC with constant mobility and the standard SCLC model $j\propto V^2$ are also plotted as well. It should be observed that our model departs slightly from the standard one at high current densities.

Figure 6.7: The relation between organic layer thickness and space charge limited current.

For light emitting diodes, it is important to distinguish if the device is controlled by injection at the contact or by currents in the bulk of the organic layer. To determine the dominant mechanism, an understanding of the thickness scaling is required [128,129]. The thickness dependent SCLC in Child's law model is given as [122]

$$j=\frac{9}{8}e\epsilon_0\epsilon_r\mu\frac{V^2}{L^3}.$$ (6.10)

The relation between layer thickness and SCLC in our model is shown in Fig 6.7 assuming the same parameters as in Fig 6.2. The thickness dependence of the current is also of the form $j\propto 1/L^k$ with $k=3.2264$ for the constant mobility case and $k=3.8$ for the field dependent mobility case. In both cases $k$ is slightly bigger than $3$ as in the standard model.