1.2 Organic Semiconductor Physics

In inorganic semiconductor crystals such as silicon or germanium, the strong coupling between the constituting atoms and the long-range order lead to the delocalization of the electronic states and the formation of allowed valence and conduction bands, separated by a forbidden gap. By thermal activation or photo-excitation, free carriers are generated in the conduction band, leaving behind positively charged holes in the valence band. The transport of these free carriers is described in quantum mechanical terms by Bloch functions, wave vector-space and dispersion relations.

However, because of structural or chemical defects in organic semiconductors, the motion of carriers is typically described by hopping transport, which is a phonon-assisted tunneling mechanism from site to site (Fig 1.2). Many hopping models are based on the Miller-Abrahams equation [7]. In this model hopping from a localized state $i$ to a state $j$ takes place at frequency $\nu_0$, corrected for a tunneling probability and the probability to absorb a phonon for hops upwards in energy:

$$\omega_{ij}=\nu_0\left\{\begin{array}{r@{\quad:\quad}l}\exp\left(... ...-E_i\ge 0 \exp\left(-2\alpha R_{ij}\right) & E_j-E_i\leq 0\end{array}\right.$$ (1.1)

Here $\alpha$ is the inverse localization length, $R_{ij}$ the distance between the localized states, $E_i$ the energy at the state $i$, and $\nu_0$ the attempt-to-escape frequency.

Figure 1.2: Charge transport mechanism in solids. The left image describes the band transport. In a perfect crystal, depicted as a straight line, free carriers are delocalized. There are always lattice vibrations that disrupt the crystal symmetry. Carriers are scattered at these phonons, which limit the carriers mobility. The image on the right describes hopping transport. If a carrier is localized due to defects, disorder or selflocalization, the lattice vibrations are essential for a carrier to move from one site to another. The figure is from [8].

Since the hopping probability depends on both the spatial and 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.

In organic solids, interactions are mainly covalent, but intermolecular interactions are due to much weaker van der Waals and London forces. These organic semiconductors typically have narrow energy bands, the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), which can be easily disrupted by disorder. Thus, even in molecule crystals, the concepts of allowed energy band is of limited validity and excitations and interactions localized on individual molecules play a predominant role. The charge transport sites have a Gaussian distribution of energies and are localized [8]. The shape of the density of states (DOS) is suggested to be Gaussian based on the observed Gaussian shape of the optical spectra [9].

Transport energy [10] is a useful concept for the analysis of hopping transport in organic semiconductors. Importance of the transport energy stems from the fact that this is the energy that maximizes the probability for a carrier to hop upward in energy. It does not depended on the carrier initial energy, thus serving as an analog to the mobility edge.

For polycrystalline organic semiconductor layers, the temperature dependent transport data is often interpreted in terms of a multiple trapping and release model [11,12]. In this model the organic semiconductor film consists of crystallites which are separated from each other by amorphous grain boundaries. In the crystallites the carriers are delocalized, while the carriers in the grain boundaries become trapped in localized states. The transport description in terms of trapped carriers that can be thermally activated to transport level, is very similar to hopping transport as discussed above.