Charge transport in disordered materials is assumed to be due to hopping between the localised states. However, charge transport in polycrystalline organic semiconductors is significantly different due to the two distinct regions; the grains and the grain boundaries. The charge transport is dependent on the grain size but is said to be limited by the grain boundaries due the existence of trap states. The crystalline-like grains are much more conductive such that the Fermi level is able to move somewhat freely compared to the grain boundaries. The barrier, in the grain boundary, is believed to be controlled by the density of states in the disordered grain boundary irrespective of the position of the potential barrier within the material. Therefore, Fermi level pinning is expected in disordered grain boundaries. A small change in the external field at the grain boundary can cause a significant change in the energy levels in the grains. Gaussian shaped
distribution of states (DOS) is expected in organic semiconductors with disorder [76, 77]. This can be precisely approximated to an exponential function for DOS when the carrier concentrations are expected to be small in the band tail of the Gaussian distribution, Fig.2.4 [20, 78-80]. A polycrystalline semiconductor can also be expected to follow the Gaussian DOS as the grain boundaries limit the carrier flow due to discontinuities in the bonding. In this case, the equations already developed for a disorder model seem to be applicable to polycrystalline organic materials. Furthermore, the value of the Meyer Neldel Energy associated with disorder in the system is obtained in polycrystalline organic materials even though the basic concept relates only to truly disordered semiconductors [35]. As a result, the charge transport in a polycrystalline material can be approximated as of disordered nature.
The section that follows defines the Universal Mobility Law in terms of disordered organic materials to demonstrate the significance of effective mobility in polycrystalline organic semiconductors.
I. Universal Mobility Law in Disordered Organic Semiconductors and its Significance in Polycrystalline Materials.
A single mobility term can be used to describe carrier mobility in organic crystalline semiconductors. However, as disordered materials follow exponential DOS, the mobility term is not constant and described in terms of effective mobility [81].
The transport of carriers in an exponential DOS is governed by the thermally activated hopping between the states. According to Variable range hopping mechanism, the movement of carriers is dependent on the distance between the states in DOS and the energetic distribution of states. In an exponential DOS, the analytical model discussed can be developed without taking into account the precise mechanism by which the carriers move [82].
Increasing the number of states per unit area causes an immense increase in the hopping between the states hence leading to an increase in carrier density which is proportional to mobility. For disordered materials, the effective mobility and the carrier density can be written in terms the characteristic temperature of the exponential DOS, where and is valid for small carrier concentrations. Brown et al. were one of the first to investigate the UML with measurements carried out on field effect transistors (FETs) doped with poly (B’-dodecyloxy- α, α’, -α’,
α’’terthienyl) (polyDOT3) [81]. The relation can also be applied on organic disordered materials
such as P3HT and PTAA.
2.17
2.18
K is the mobility prefactor, is the mobility prefactor, is the effective mobility of the
material, n is the carrier concentration and m is the exponent dependent on and the material. The carriers in a disordered material do not possess a constant mobility as they are distributed in energy so that Mott-Gurney law for space charge limited currents does not apply (Section 3.3.2). The carriers occupy a range of energy levels and behave differently than carriers in conventional semiconductors. This means that a more generalized expression with dependence of mobility on the carrier concentration has to be considered.
The carrier concentration (taken equivalent to the dopant concentration), obtained for these materials was extracted from the reverse current characteristics using the quarter power law by applying the Mott-Schottky relationship [83]. The extracted dopant concentrations in reverse bias assume that the dopant concentration in the depletion region is uniform and that the dopant ions are stationary. The assumption of the carrier concentration equivalent to the dopant concentration is made as the electrical conductivity in the organic material is dominated by these dopant related carriers. This is except when the carrier concentration n is enhanced by field effect such that in a thin film transistor or by optical excitation. It is therefore extremely important that the existing concepts developed for inorganic disordered semiconductors are used with caution when defining conduction in organic disordered materials.
In disordered organic semiconductors, the carrier occupancy of the states is dependent on the position of the Fermi level which lies in the band tail of the Gaussian DOS. At energies below the Fermi level, the states are almost full with the location of localised states further apart from each other such that they can be considered immobile with the occupancy probability close to one (Fig.2.4). For energies above the Fermi level, the density of states is considerably more with many unoccupied states so it is easier for the charge carriers to hop around from state to state. The carriers above and below the Fermi level can thus be evaluated with relative ease. Carriers above the Fermi level being free where as the carriers below being trapped and not contributing to
current, under external bias. Below the Fermi level, the energy levels are Donor-like whereas the energy levels above the Fermi level are Acceptor-like. Donor-like traps are positive if unoccupied or neutral if occupied whereas Acceptor-like traps are either negative if occupied or neutral if unoccupied [14].
The integration from the Fermi level to infinity is chosen to include all the charge carriers contributing to current flow. The combined effect of decreasing charge occupancy and increasing energy states indicates that the charge transport is dominated by the hopping events close to the Fermi level. At the peak of the DOS, the carrier concentration is insignificant such that integrating to infinity does not add any excess charge which is not taking part in conduction. For n-type disordered semiconductors, this is given by (Appendix A)
2.19
where E is the energy of the localised states, TC is the characteristic temperature of density of states (DOS) and N(0) is the number of states per unit volume at E=0. In this case, only hopping of carriers, observed above the Fermi level, contribute to current conduction. However, this limits the application of the model to low carrier concentrations and temperatures so that the charge transport can be dominated by the band tail of the exponential DOS.
Ignoring diffusion currents and assuming that drift currents are most dominant in disordered materials leads to 2.20
A direct comparison to the ordered crystalline grains is made to understand , the effective
mobility. Comparison can be made at the edge of the grain where the disordered boundary meets the crystalline grain for which the equation was initially developed. Ignoring the band bending at the interface as well as in the absence of current flow, the flux to and fro the grain boundary and the grain must be equal in thermal equilibrium (Fig.2.9). This causes an effective transport level to be formed in the grain boundary much like the ET level in the grain such that it appears to be a
single energy level. Existence of an effective transport level must apply for all currents including the current at zero bias. At the edge of the grain, the carrier concentration for an ordered material is given by
Disordered Ordered Disordered
ET
EF
Etr Etr
e e e e
Figure 2.9 Ordered organic semiconducting material in contact with a disordered organic material to explain universal mobility law in organic polycrystalline semiconductors. The transport level in the grains ET is assumed equivalent to the
transport energy level in the grain boundaries Etr.
2.21 2.22
where NT is the hypothetical carrier density at the transport level at the grain edge, and is the energy corresponding to the transport energy level in the grain. It should be noted that ET in the grain and Etr at the edge of the boundary are assumed to be similar. Combining Eq. (2.21) and Eq. (2.22) and assuming that the drift currents dominate gives
2.23
The effective mobility can then be written in terms of carrier concentration as in Eq. (2.17) and refers to as the universal mobility law (UML). µ0 is the mobility prefactor explained in terms of
2.24
The use of a single mobility prefactor for the exponential DOS in a disordered material is not typical. Although µ0 can be defined experimentally, the physical significance of the mobility prefactor is hard to understand. A higher transport energy level means an increase in mobility value because of the increase in hopping probability. Although the interpretation of disordered organic materials in terms of standard inorganic semiconductors is unconventional, it is a useful tool to understand motion of carriers in disordered materials (Appendix B). The exponent in Eq. (2.17) is seen to be dependent on the characteristic temperature of the DOS, TC as seen in Eq. (2.18) [84].