Subsections

• 5.2.1 Theory
• 5.2.2 Results and Discussion

5.2 Diffusion Controlled Injection Model for OLEDS

Due to the low mobility in organic semiconductors $\left(\mu\ll 10^{-3} cm^2/Vs\right)$, the diffusion transport is important for the charge injection process. Therefore, the aim of this section is to develop an analytical, diffusion-controlled charge injection model particularly suited for organic light-emitting diodes (OLED). This model is based on drift-diffusion and multiple trapping theory. The latter can be used to describe hopping transport in organic semiconductors [114]. The presented model can explain the dependence of the injection current on the temperature, the electric field and the energy barrier height. The theoretical predictions agree well with experimental data.

5.2.1 Theory

The potential barrier $q\varphi\left(x\right)$ formed at the metal semiconductor interface is a superposition of an external electric field and a Coulomb field binding the carrier on the electrode [115,116],

$$q\varphi\left(x\right)=\Delta-\frac{q^2}{16\pi\epsilon_0\epsilon_r x}-qFx.$$ (5.1)

Here, $x$ is the distance to the metal-organic layer interface. Since the rapid variation of the potential 5.1 takes place within $x_d$ (about 50Å     [117]) in front of the cathode, the field $F$ can be regarded as being nearly constant.

Using the drift-diffusion theory, the hole current $J$ can be written as

$$J=-k_BT\mu\left[\frac{q}{k_BT}p_e\left(x\right)\frac{d\varphi\left(x\right)}{dx}+\frac{dp_e\left(x\right)}{dx}\right],$$ (5.2)

where $\mu$ is the mobility. On taking $J$ and $\mu$ as constant, and solving for $n$, we obtain

$$p_e\left(x\right)=\left[N-\frac{J}{k_BT\mu}\int_0^x \exp\left(\fr... ...t)}{k_BT}\right)dx'\right]\exp\left(-\frac{q\varphi\left(x\right)}{k_BT}\right)$$ (5.3)

where $N$ is the hole concentration at $x=0$. In multiple trapping theory [118], the total carrier concentration is given by a sum of the carrier concentrations in the extended states $p_e\left(x\right)$ and the localized states,

$$p\left(x\right)=p_e\left(x\right)+\int_0^{\infty} g\left(E,x\right)f\left(E,E_F\right)dE.$$ (5.4)

Here, $g\left(E\right)$ is the density of the localized states, $f\left(E, E_F\right)$ is the Fermi Dirac distribution, and the quasi-Fermi energy $E_F$ can be written as [118]

$$E_F\left(x\right)=k_BT\ln\left[\frac{\nu_0\tau_0N_t}{p_e\left(x\right)}\right],$$

where $N_t$ is the total concentration of localized states, $\tau_0$ is the lifetime of carriers, and $\nu_0$ is the attempt-to-escape frequency.

In the injection regime, very close to the contact all the traps are filled. Moreover, the carrier concentration in the extended states is much higher than that in the trapped states. At large distance from the injection contact, the main contribution to the total carrier concentration comes from the occupied localized states [112]. So we propose here the concept of a critical distance $x_d$, where the carrier concentration in the extended states equals the carrier concentration in localized states, i.e.,

$$p_e\left(x_d\right)=\int_0^{\infty} g\left(E,x_d\right)f\left(E,E_F\right)dE.$$ (5.5)

Substituting (5.1),(5.4) and (5.5) into the Poisson equation,

$$\frac{d^2\left(q\varphi\right)}{dx^2}=-\frac{q}{\epsilon_0\epsilon}p\left(x\right),$$ (5.6)

then the critical distance $x_d$ can be calculated as

$$1=\int_0^{\infty}\frac{16\pi x_d^3 g\left(E-q\varphi\left(x_d\right)\right)}{1+16\pi x_d^3\nu_0\tau_0N_t\exp\left(-E/{k_BT}\right)}.$$ (5.7)

Solving (5.7) with a Gaussian DOS numerically, we can obtain the critical distance $x_d$. The free carrier concentration at $x_d$ is calculated by (5.5). Finally, the injection current can be calculated as

$$J=k_BT\mu\frac{\left[N-p_e\left(x_d\right)\exp\left(\frac{q\varph... ...t)\right]} {\int_0^{x_d}\exp\left(\frac{\varphi\left(x\right)}{k_BT}\right)dx}.$$ (5.8)

5.2.2 Results and Discussion

The barrier height $\Delta$ plays an important role for the injection efficiency. We calculate the relation between the injection current and the electric field for different $\Delta$, as shown in Fig 5.1. The parameters are $N_t=1\times 10^{18}$cm$^{-3}$, $\sigma=0.1656$eV, $\nu_0=10^{11}$s$^{-1}$, $\tau_0=10^{-11}$s, $T=300$K and $\mu=1\times 10^{-9}$cm$^2$/Vs. The injection current increases with the electric field, and the lower the $\Delta$, the higher the injection current as intuitively expected. But the slope of $\log J$ versus $\log F$ is not constant.

Figure 5.1: Dependence of the injection current on the barrier height.

Fig 5.2 shows the temperature dependence of the injection current for $\Delta=0.3eV$, where the other parameters are the same as in Fig 5.1. The temperature coefficient decreases strongly with increasing electric field. The coefficient reverses sign at high electric field, which has also been observed in [115] theoretically.

Figure 5.2: Temperature dependencies of the injection current.

A comparison between the model and experimental data [112] is shown in Fig 5.3. The fitting parameters are $N_t=1\times 10^{17}$cm$^{-3}$, $\mu=2.56\times 10^{-11}$ cm$^2$/Vs for PPV-ether and $2.51\times 10^{-9}$ cm$^2$/Vs for PPV-imine, respectively. The other parameters are the same as in Fig 5.1.

Figure 5.3: Comparison between the model and experimental data.

The mobility in organic materials depends on the local electric field $F$ as [119]

$$\mu\left(F\right)=\mu_0\exp\left(\gamma\sqrt{F}\right).$$ (5.9)

Here $\mu_0$ denotes the mobility of carriers at zero field and $\gamma$ is the parameter describing the field dependence. We first substitute (5.9) into (5.2) to obtain the carrier concentration,

$$p_e\left(x\right)=\left[N-\frac{J}{k_BT\mu_0\exp\left(\gamma\sqrt... ...)}{k_BT}\right)dx'\right]\exp\left(-\frac{q\varphi\left(x\right)}{k_BT}\right).$$ (5.10)

Then, by combining (5.7), (5.10), Gaussian DOS and (5.8), we obtain the injection current with the field-dependent mobility. Fig 5.4 illustrates the relation between injection current and electric field with field dependent mobility. Parameters are $\mu_0=7.3\times 10^{-6}$cm$^{2}$/Vs, $\gamma=1\times 10^{-4}$ (m/V)$^{1/2}$ and $\Delta=0.3eV$. For comparison, the injection current with constant mobility is plotted as well.

Figure 5.4: Comparison between injection currents for field dependent mobility and constant mobility.