Subsections

• 1.3.1 Gaussian Disorder Model (GDM)
• 1.3.2 Percolation Theory
• 1.3.3 Transport Energy
• 1.3.4 Multiple Trapping Theory

1.3 Theoretical Concepts

The incoherent dynamics of carriers as well as excitons can be described by a master equation,

$$\frac{df_i\left(t\right)}{dt}=f_i\left(t\right)\left(1-f_j\left(t... ...t\right)\left(1-f_i\left(t\right)\right)\omega_{ji}+\lambda_if_i\left(t\right).$$ (1.2)

with $f_i$ denoting the occupation probability of state $i$ and $\omega_{ij}$ the electron or hole transition rate of the hopping process between the occupied state $i$ to empty state $j$. Defining $\mu_i$ as the chemical potential at the position of state $i$ and $E_i$ as the energy of state $i$, the occupation probability is given by the Fermi-Dirac distribution function

$$f_i=\frac{1}{1+\exp\left(\frac{E_i-\mu_i}{k_BT}\right)}.$$ (1.3)

Assuming no correlations between the occupation probability of different localized states, the steady-state current between these two sites is given by

$$I_{ij}=q\left[f_i\left(t\right)\left(1-f_j\left(t\right)\right)\omega_{ij}-f_j\left(t\right)\left(1-f_i\left(t\right)\right)\omega_{ji}\right].$$ (1.4)

Substituting the Miller-Abrahams rate (1.1) in (1.4) the current becomes

$$I_{ij}=\frac{q\nu_0\exp\left(-2\alpha R_{ij}-\frac{\mid E_i-E_j\m... ...\left[\frac{E_i-\mu_i}{2k_BT}\right]\cosh\left[\frac{E_j-\mu_j}{2k_BT}\right]}.$$ (1.5)

In the case of low electric field, resulting in a small voltage drop over a single hopping distance ( $\Delta\mu=\mu_j-\mu_i\ll k_BT$), the following conductance is obtained

$$\sigma_{ij}=\frac{I_{ij}}{\Delta\mu}\propto\exp\left[-2\alpha\mid... ...j}\mid-\frac{\mid E_i-\mu\mid+\mid E_j-\mu\mid+\mid E_i-E_j\mid}{2k_BT}\right].$$ (1.6)

Here $\mu_i\approx\mu_j\approx \mu$. This expression was introduced in 1960 by Miller and Abrahams [7] and is often referred as the Miller-Abrahams conductance. Equation (1.6) has an important implication. Even if the energies are moderately distributed, the exponential dependence of $\sigma_{ij}$ on these energies makes them enormously broadly distributed. This can be used to reduce the computations of the effective properties of the network, since the broadness of the distribution of $\sigma_{ij}$ implies that there are many small conductances that can be removed from the network. This resulting network is called the reduced network [13].

Miller and Abrahams [7] were the first to calculate the hopping conductivity $G$ of semiconductors using reduced networks. They assumed that the statistical distribution of the resistances depends only on $R_{ij}$ and not on the site energies. This was justified because the experimental data for some semiconductors indicated that the impurity conduction exhibits a well-defined activation energy. But Mott [16] pointed out that the exponential dependence of the resistances on the site energies can not be ignored in most cases. When a carrier close to the Fermi energy hops away over a distance $R$ with an energy $\Delta E$, it has $\frac{4}{3}\pi R^3\rho\Delta E$ sites to choose from, where $\rho$ is the site density function. In general, the carrier will jump to a site for which $\Delta E$ is as small as possible. The constraint to find a site within a range $\left( R, \Delta E\right)$ is given by $\frac{4}{3}\pi R^3\rho\Delta E\approx 1$. Substituting this relation into (1.6) yields

$$G\propto \exp\left[-2\alpha R-\frac{1}{k_BT\left(4/3\right)\pi R^3\rho}\right].$$ (1.7)

The optimum conductance is obtained by maximizing $G$ with respect to the hopping distance $R$ as

$$G\propto\exp\left[-\left(\frac{T_1}{T}\right)^{1/4}\right],$$ (1.8)

with $k_BT_1\propto\gamma^3/\rho$.

1.3.1 Gaussian Disorder Model (GDM)

Much theoretical work has been done by investigating the mobilities of organic semiconductors within the framework of GDM [9]. Non-crystalline organic solids, such as molecularly doped crystals, molecular glasses, and conjugated polymers, are characterized by small mean free paths for the carriers, as a result of the high degree of disorder present in the organic system. Therefore, the elementary transport step is the charge transfer between adjacent elements, which can either be molecules participating in transport or segments of a polymer separated by topological defects. These charge transporting elements are identified as sites whose energy are subjected to a Gaussian distribution

$$g\left(E\right)=\frac{N_t}{\sqrt{2\pi}\sigma}\exp\left(-\frac{E^2}{2\sigma^2}\right),$$

where $E$ is the energy measured relative to the center of the density of states and $\sigma$ is the standard deviation of the Gaussian distribution. Within this distribution, all the states are localized. The choice of such distribution was based on the Gaussian profile of the excitonic absorption band, as well as on the recognition that the polarization energy is determined by a large number of internal coordinates, which vary randomly by a small amount, so the central limit theorem of statistics holds.

The Gaussian disorder model has been treated by the Monte Carlo simulation technique based on the Miller Abraham equation [9]. In this simulation charge transport is described as an incoherent random walk. The carriers start their motion from randomly chosen sites at one of the boundaries of the system sample. Their trajectories are specified from the constraint that the probability for a carrier to jump between two transport sites is

$$p_{ij}=\frac{\omega_{ij}}{\sum_{i\neq j}\omega_{ij}}.$$

With this technique, TOF measurements can be simulated [9], in which mobility is derived from the mean arrival time of the carrier at the end of the sample and from their mean displacement. The predictions made concern the temperature and electric field dependence of the mobility.

In the Gaussian disorder model, the strength of electronic coupling among sites is split into separate contributions from the relevant sites, each obtained from a Gaussian probability density. However, the choice for the off-diagonal disorder of a Gaussian distribution is not theoretically sustained unlike in the case of energy disorder, and a more realistic way of representing geometric disorder has been pursued. One of such attempt is described in [14], in which an alternative approach comprising positional and orientation disorder is introduced via fluctuations in the bonds adjoining the various transport sites rather than site fluctuations. This model gets rid of the unnecessary corrections between hops and results in overestimating the contribution of the log hops.

In particular, Gartstein and Conwell's [14] Monte Carlo simulations of hopping with the elementary jump rate described by 1.1, but in which

$$\omega_{ij}=\exp\left(\varsigma_{ij}\right)\exp\left(-2\alpha R_{ij}\right),$$

where $\varsigma_{ij}$ is a uniformly distributed random variable. In this way, the different bonds of a given site with its neighbors are uncorrelated.

Another approach for the description of positional disorder was presented by Hartenstein [15], and was also based on Monte Carlo simulations of transport in a dilute lattice. This treatment employs the GDM, but without the need for defining a distribution function for the electronic coupling between different sites. In this case the hopping sites having the nearest neighbors were grouped in clusters whose size depends on the random intercluster distances, but ignores any contribution from the random orientation of the transporting elements. Nevertheless, the model is adequate for low dopant concentrations for which there are large fluctuations in the intersite distances.

1.3.2 Percolation Theory

Ambegaokar and coworkers argued that an accurate estimate of $G$ is the critical percolation conductance $G_c$ [17], which is the largest value of the conductance such that the subnet of the network with $G_{ij} \ge G_c$ still contains a conducting sample-spanning cluster. They divided the network into three parts. First, a set of isolated clusters of high conductivity where each cluster consists of a group of sites connected together by conductances $G_{ij}\gg G_c$; Second, a small number of resistors with $G_{ij}$ of order $G_c$, which connect together a subset of high conductance clusters to form the sample-spanning cluster, called the critical subnetwork, essentially the same as the static limit of the reduced network discussed above; and third, the remaining resistors with $G_{ij}\ll G_c$. The resistors in the second part dominate the overall conductance of the network. The critical conductance $G_c$ is calculated as follows.

The percolation subnetwork consists of conductances with $G_{ij} \ge G_c$. Using (1.6), this condition can be written as

$$\frac{R_{ij}}{R_{max}}+\frac{\mid E_i-\mu\mid+\mid E_j-\mu\mid+\mid E_i-E_j\mid}{2E_{max}}\le 1,$$ (1.9)

with

$$R_{max}=\frac{1}{2\alpha}\ln\left(\frac{G_ck_BT}{e\nu_0}\right),$$ (1.10)

$$E_{max}=k_BT\ln\left(\frac{G_ck_BT}{e\nu_0}\right).$$ (1.11)

$R_{max}$ is the maximum distance between any two sites between which a hop can occur, and $E_{max}$ is the maximum energy that any initial or final state can have. Thus the density of states that can be part of the percolating subnetwork is given by

$$N_s=2\rho E_{max}.$$ (1.12)

Since the sites in the subnetwork are linked only to sites within a range $R_{max}$, this criterion has the form

$$N_sR_{max}^3=\nu_c,$$ (1.13)

with $\nu_c$ being a dimensionless constant. A combination of (1.9) to (1.11) yields Mott's law (1.8), with $k_BT_1=4\nu_c\gamma^3/\rho$.

1.3.3 Transport Energy

According to the Miller Abraham equation (1.1) we can roughly calculate the nearest-neighbor distance for an upward hop from an initial site with energy $E_i$ to a finial site with energy $E_f\ge E_i$ from the equations below [10]

$$\frac{4\pi}{3}R^3\left(E_f\right)\int_{-\infty}^{E_f}g\left(E\right)dE\approx 1.$$ (1.14)

Here $g\left(E\right)$ is the DOS function. The hopping distance can be calculated as

$$R\left(E_f\right)=\left[\frac{4\pi}{3}\int_{-\infty}^{E_f}g\left(E\right)dE\right]^{-1/3}.$$ (1.15)

So the corresponding hopping rate is

$$\nu=\nu_0\exp\left(-2\alpha R\left(E_f\right)-\frac{E_f-E_i}{k_BT}\right).$$ (1.16)

Maximization of (1.16) over energy $E_f$ gives the equation

$$g\left(E_f\right)\left[\int_{\infty}^{E_f}g\left(E\right)dE\right]^{-4/3}=\frac{1}{\gamma k_BT}\left(\frac{9\pi}{2}\right)^{1/3}.$$ (1.17)

The finial energy $E_f$ that maximizes the hopping probability does not depend on the initial energy $E_i$. This particular energy is called transport energy $E_{tr}$ [10].

Arkhipov extended this theory to the effective transport energy [18]. In this theory, the Miller Abrahams equation is rewritten as

$$\nu=\nu_0\exp\left(-\mu\left(R_{ij}, E_i, E_j\right)\right)=\nu_0\exp\left[-2\alpha R_{ij}-\frac{\theta\left(E_j-E_i\right)}{k_BT}\right],$$ (1.18)

with the hopping parameter $\mu$ and the unity step function $\theta$. The average number $n\left(E_i\right)$ of target sites for a starting site with energy $E_i$, whose hopping parameters are not larger than $\mu$ can be calculated as

$$n\left(E_i,\mu\right)=4\pi\int_0^{\mu/{2\alpha}}R_{ij}^2dR_{ij}\int_{-\infty}^{E_i+k_BT\left(\mu-2\gamma R_{ij}\right)}g\left(E_t\right)dE_t.$$ (1.19)

Neglecting the downward jumps and defining

$$E_{tr}=E_i+k_BT\mu$$ (1.20)

transform (1.19) into

$$n\left(E_i,\mu\right)=\frac{\pi}{6}\left(\alpha k_BT\right)^{-3}\int_{E_i}^{E_{tr}}g\left(E_t\right)\left(E_{tr}-E_t\right)^3dE_t.$$ (1.21)

According to variable range hopping theory [19], a hop is possible if there is at least one such hopping neighbor, i.e. $n=1$. This leads to the following equation

$$\int_{E_i}^{E_{tr}}g\left(E_t\right)\left(E_{tr}-E_t\right)^3dE_t=\frac{6}{\pi}\left(\alpha k_BT\right)^3.$$ (1.22)

If the DOS distribution decreases with energy faster than $E^{-4}$ then the integral on the left-hand side of (1.22) depends weakly upon the lower bound of integration for sufficiently deep starting sites, and (1.22) is reduced to

$$\int_{-\infty}^{E_{tr}}g\left(E_t\right)\left(E_{tr}-E_t\right)^3dE_t=\frac{6}{\pi}\left(\alpha k_BT\right)^3,$$ (1.23)

where $E_{tr}$ is the effective transport energy.

1.3.4 Multiple Trapping Theory

To investigate charge transport and charge buildup in $SiO_2$ films, an analysis of hole transport has been presented which is predicated on a model involving stochastic hopping transport. This description, based on the work of Scher and Montroll [20], accounts for many of the features of hole conduction in $SiO_2$ and has been termed the continuous-time random walk (CTRW) model. A wide range of experimental observation can be understood on this basis [20,21,22]. However, there has been some reticence to accept the CTRW model completely because of observations that the apparent activation energy for the charge collection process depends on the fraction of charge collected [23]. This observation is at odds with the CTRW model as it has been presented, since that model predicts universality, i.e., charge transport curves obtained at different temperature should superimpose with a simple shift in the time axis [24]. Although charge collection curves curves obtained at different temperature do superimpose approximately, there is some deviation, and this deviation is in the direction predicted by the multiple-trapping model.

The multiple-trapping model for unipolar conduction is defined by the following equations [12]

$$\frac{\partial \rho\left({\bf {x}},t\right)}{\partial t}=g\left({\bf {x}},t\right)-\bigtriangledown\cdot f\left({\bf {x}},t\right),$$ (1.24)

where

$$\rho\left({\bf {x}},t\right)=p\left({\bf {x}},t\right)+\sum_{i}p_i\left({\bf {x}},t\right)$$ (1.25)

and

$$\frac{\partial p_i\left({\bf {x}},t\right)}{\partial t}=p\left({\bf {x}},t\right)\omega_i-p_i\left({\bf {x}},t\right)\gamma_i$$ (1.26)

Here $g\left({\bf {x}},t\right)$ is the local photogeneration rate, $f$ is the flux of mobile charge carriers, the total carrier concentration is $\rho\left({\bf {x}},t\right)$, $p\left({\bf {x}},t\right)$ is concentration of mobile carriers, $p_i\left({\bf {x}},t\right)$ is the carrier concentration temporarily immobilized in the $i$th trap, $\omega_i$ is the capture rate by the $i$th trap and $\gamma_i$ is the release rate.

Later multiple trapping theory was extended for disordered organic semiconductors as:

$$p=p_c+\int\rho\left(E\right)dE.$$ (1.27)

Here $p$ is the total hole concentration, $p_c$ is the hole concentration in extended states and $\rho\left(E\right)$ is the energy distribution of localized (immobile) holes. Since carrier trapping does not change the total carrier concentration $p$, the continuity equation can be written as

$$\frac{\partial p}{\partial t}+\mu_cF\frac{\partial p_c}{\partial x}-D_c\frac{\partial^2p_c}{\partial x^2}=0,$$ (1.28)

with the mobility $\mu_c$ and the diffusion coefficient $D_c$. This equation assumes two simplifications: no carrier recombination and constant electric field (no space charge). Substituting the trapping rate

$$\frac{p_c}{\tau_0}\frac{g\left(E\right)}{N_t}$$ (1.29)

and release rate

$$\nu_0\exp\left(-\frac{E}{k_BT}\right)\rho\left(E\right)$$ (1.30)

gives the following equation

$$\frac{\partial\rho\left(E\right)}{\partial t}=\frac{1}{\tau_0N_t}g\left(E\right)-\nu_0\exp\left(-\frac{E}{k_BT}\right)\rho\left(E\right)$$ (1.31)

In equilibrium the energy distribution of localized carriers is established, and the function $\rho\left(E\right)$ does not depend upon time

$$\frac{\partial\rho\left(E\right)}{\partial t}=0.$$ (1.32)