Subsections

• 2.4.1 Theory
• 2.4.2 Results and Discussion

2.4 Unified Mobility Model

Most existing models are only valid for either very low [43,68,69] or very high electric fields [60,73]. Another common simplification stems from the transport energy concept [68,69,70], where exothermic, i.e. non-activated jumps, and jumps against the field are neglected.

The following is based on Apsley's work [19]. We will derive a formula for both the conductivity's temperature as well as its field dependence in the case of a Gaussian density of states mirroring the molecular disorder. We assume that (i) the localized states are distributed randomly in both space and energy, (ii) the states are occupied according to the Fermi-Dirac statistics, (iii) both hops upwards and hops downwards are regarded, (iv) the state energies are uncorrelated, and (v) the electric field may assume any value. Finally, the mobility's concentration dependence is discussed.

2.4.1 Theory

The amorphous structure of organic semiconductors is mirrored in localized states, which are distributed randomly in space and energy. The carrier transport between them is described as hopping, i.e. as a series of incoherent, thermally activated tunneling events. We can define the hopping range $R$ as

$$P_{ij}=\exp(R).$$ (2.20)

where $P_{ij}$ is the Miller Abrahams rate. In the presence of an electric field $F$, the actual energy differences will be modified from $E_j-E_i$ to $(E_j-E_i)-qER_{ij}\cos\theta$, where $\theta$ is the angle enclosed by the jump and the field direction. Using the reduced coordinates $R_{ij} '=2R_{ij}\alpha$ and $\epsilon=E/{k_BT}$, the hopping range may be re-written to

$$R=\left\{\begin{array}{r@{\quad:\quad}l} (1+\beta\cos\theta)R_{ij... ... \epsilon_j>\epsilon_i, R '_{ij} & \epsilon_j<\epsilon_i. \end{array}\right.$$ (2.21)

where $\beta=Fq\alpha/(2 k_BT)$. Since the hopping probability depends on both the spacial and the energetic difference between the hopping sites, it is natural to describe the hopping processes in a four-dimensional hopping space, which is spanned by three spacial and one energy coordinate. The hopping range $R$, as given by (2.21), defines a metric on this space.

In various disordered systems, a Gaussian density of states has been used to describe the hopping transport in band tails.

$$g\left(\epsilon\right)=\frac{N_t}{\sqrt{2\pi}a}\exp\left(-\left(\frac{\epsilon-\epsilon_0}{\sqrt{2}a}\right)^2\right),$$ (2.22)

where $E_0$ is the center of Gaussian function and $a=\sigma_0/{kT}$.

Let $F\left(E,\xi\right)$ be the normalized Fermi-Dirac distribution function. Then the carrier concentration can be written as

$$n\left(\xi\right)=\int_{-\infty}^{\infty}g\left(\epsilon\right)F\left(\epsilon,\xi\right)d\epsilon.$$ (2.23)

with $\xi$ denoting the normalized chemical potential. The conductivity can be written as

$$\sigma\left(T, \beta\right)=-\frac{e\nu_0}{F}\int_{-\infty}^{\infty}dE '_ig\left(E '_i\right)F\left(E '_i\right)X_f\exp\left(-{R_{nn}}\right).$$ (2.24)

Here $X_f$ is the forward hopping distance in the direction of the electric field and, $R_{nn}$ stands for the nearest neighbor hopping range in the hopping space. To calculate the conductivity, we need to calculate $R_{nn}$. First, the number of unoccupied states $N$ within a radius $R$ in the hopping space is calculated [19].

$$N\left(T,\beta,R,\epsilon_i\right)=\int_0^{\pi}\int_0^{R}\int_{-\... ...ight)\right]\frac{1}{8\alpha^3} 2\pi R^{ '2}\sin\theta d\epsilon_jdR 'd\theta$$

Here $K_f={R+\epsilon_i-R '\left(1+\beta\cos\theta\right)}$. The factor $kT\alpha^3/8$ arises from the reduced coordinate system.

According to Mott, $R_{nn}$ will be the value of the radius in the hopping space for which only one available vacant site is enclosed [72]. In other words, $R_{nn}$ can be obtained by solving the equation

$$N\left(R,\epsilon_i, T, \epsilon\right)\mid_{R=R_{nn}}=1.$$ (2.25)

Similarly, the expression for $X_f$ can be calculated as [19]

$$X_f=\frac{I_1+I_2}{I_3+I_4},$$ (2.26)

where

$$I_1=\int_0^{\pi}\sin\theta\cos\theta d\theta\int_{\epsilon_j-R_{n... ...\frac{R_{nn}-\epsilon_i+\epsilon_j}{1+\beta\cos\theta}\right]^3d\epsilon_i\quad$$

$$I_2=\int_0^{\pi}\sin\theta\cos\theta d\theta\int_{-\infty}^{\epsi... ...3 \left[1-F\left(\epsilon_i,\xi\right)\right]d\epsilon_i\qquad\qquad\qquad\quad$$

$$I_3=\int_0^{\pi}\sin\theta d\theta\int_{\epsilon_j-R_{nn}\beta\c... ...n}-\epsilon_i+\epsilon_j}{1+\beta\cos\theta}\right]^2d\epsilon_i\quad\quad\quad$$

$$I_4=\int_0^{\pi}\sin\theta d\theta\int_{-\infty}^{\epsilon_j-R_{n... ...n_i,\xi\right)\right] R_{nn}^2d\epsilon_i\qquad\qquad\quad\qquad\quad\quad\quad$$

$R_{nn}$ depends only weakly on $\epsilon_j$ [19]. Therefore, for $\epsilon_j=0$, we can obtain the value for $R_{nn}$ by solving (2.25) numerically. Then the mobility for electrons at energy $\epsilon_i$ amounts to

$$\mu\left(\epsilon_i,T,\beta\right)=\frac{\nu_0}{F}X_f\exp\left(-R_{nn}\right).$$ (2.27)

Finally, the total conductivity for the organic semiconductor can be calculated numerically according to (2.24). The mobility can be determined from

$$\mu=\frac{\sigma}{nq}.$$ (2.28)

2.4.2 Results and Discussion

Figure 2.14: The calculated mobility (symbols) as a function of $\sigma /{kT}$.

Figure 2.15: The calculated mobility (symbols) as a function of $\left (\sigma /{kT}\right )^2$.

We investigated the mobility's temperature dependence in a three-dimensional hopping lattice. The crucial system parameters were set to the following values: $\alpha ^{-1}=0.5$Å, $E_0=0$, $N_t=1\times 10^{21}$cm$^{-3}$, $F=1\times 10^{3}$V/cm and $\xi=30kT$. Fig 2.14 depicts the mobility as a function of the lattice temperature $\sigma /{kT}$. A linear dependence is observed between $\sigma/{kT}=$ 3 and 8. Fig 2.15 displays the mobility as a function of $\left (\sigma /{kT}\right )^2$. The range with linear dependence of mobility on $\left (\sigma /{kT}\right )^2$ is not as broad as the one for the dependence of mobility on $\sigma /{kT}$. This can be used to test the validity of Arrhenius law $\mu\propto\exp\left(-E_A/{k_BT}\right)$ and the empirical model $\mu\propto\exp\left(-\left(2\sigma/3\right)^2\right)$, where $E_A$ is the activation energy.

These results are in accordance with the measurements reorted in [79]. As the presented model expresses, only in the regime $\left(\sigma/{kT}\right)^2\leq 50$ an approximately linear relation can be observed.

The mobility versus electric field characteristics predicted by the presented model is shown in Fig 2.16. The parameters are $\alpha^{-1}=1$Å, $E_0=20kT$, $N_t=1\times 10^{21}$cm$^{-3}$ and $\sigma/{kT}=4$. $\beta^{1/2}\le 0.3$, where $\beta={Fe\alpha}/{2k_BT}=\frac{F}{F_0}, F_0\approx 1\times 10^8$V/m, the mobility remains constant. At higher fields, it increases with the field. Therefore, the simple empirical relation between mobility and electric field of the form $\mu\propto F^{1/2}$ is not valid for all electric fields.

Figure 2.16: The conductivity as a function of the scaled electric field, $\beta =F/F_0$.

Figure 2.17: Comparison between our mobility model and analytical expression (2.30) with $p=1/3$ and $C=0.7$.

Figure 2.18: Temperature dependences of parameters $C$ and $p$ extracted from the analytical model.

Figure 2.19: Electric field dependence of parameter $C$ extracted from the analytical model.

In addition to the temperature and electric field dependence, mobility also depends on the carrier concentration. Experiments show that for a hole-only diode and a field effect transistor fabricated from the same $\pi$-conjugated polymer, the mobility can differ up to three orders of magnitude [79]. Empirically, the mobility's dependence on the concentration $N$ of localized states is written in the form

$$\mu\propto\exp\left[-C\left(N\alpha^{-3}\right)^{-p}\right]$$ (2.29)

with constant $C$ and $p=1/3$ [59,69,74,75].

With the parameter $\alpha^{-1}=0.178$Å, we compare the presented mobility model and this empirical formula, as shown in Fig 2.17. The agreement is quite good when we use parameters $C=0.75$ and $p=1/3$. We notice that the value of $C$ is different from $C=2$ given in [69] and $C=3$ given in [76]. Baranovskii [69] has stated that the parameters $p$ and $C$ are temperature dependent. In Fig 2.18, we also show the values of parameters $p$ and $C$ that provide the best fit for the solution of our model with the empirical expression (2.28). The input parameters are $\beta=1\times10^{-5}$, $N_t=1\times 10^{21}$cm$^{-3}$, $E_0=0$ and $\alpha^{-1}=1$Å. As illustrated in Fig 2.18, the parameter value of $p$ is less than $1/3$ for temperatures low enough. The value of $C$ is decreasing with increasing temperature, a result which coincides with [10]. Here $p$ is not constant, since the variable range hopping (VRH) transport mechanism is based on the interplay between the spacial and energy factors in the exponent of transition probability, as given by (2.21). However, assuming nearest neighbor-hopping (NNH) regime, which does not consider the effect of energy dependent terms in (2.21) [69], leads to the values $p=1/3$.

Figure 2.20: Electric field dependence of parameter $p$ extracted from the analytical model.

Figure 2.21: Effect of $\alpha$ on the values of parameters $p$ and $C$ extracted from the analytical model.

Next, we discuss the effect of the electric field on the parameters values $p$ and $C$. The results are shown in Fig 2.19 and Fig 2.20. Input parameters are $\alpha^{-1}=1$Å, $E_0=20$kT, $N_t=1\times 10^{21}$cm$^{-3}$ and $\xi=30$. From these figures we can see that the values $p$ and $C$ are nearly constant in the low electric field regime ( $\beta\leq 1\times 10^{-2}$).

We have shown that, as expected in the variable range hopping picture, (2.24) with $p=1/3$ is only approximately valid for restricted ranges of temperature and electric field strength. So we consider the effect of the material parameter $\alpha$ on the values of $p$ and $C$ in Fig 2.21. The input parameters are $\beta=1\times10^{-5}$, $\sigma/{kT}=2$ and $N_t=1\times 10^{21}$cm$^{-3}$. Remarkably, both parameter values $p$ and $C$ are not constant in the given range of $\alpha$. With increasing $\alpha$, the values of $p$ will decrease and the ones of $C$ will increase.