In the previous lecture, we have addressed localization through polarization. In practice, most organic solids used in plastic electronics devices are disordered. So the main origin of charge carrier localization is disorder. In the perfect solid, the electron potential exactly reproduces at each site of the crystal, and charge-carriers are delocalized. In Anderson's localisation model, disorder in the solid manifests itself through a variation of the electron potential from one side to the other, thus leading to localization of the charge carriers. Delta V measures the mean value of the variation of the potential. Because disorder leads to a loss of the translational symmetry of the crystal, the electron wave function is no longer described by Bloch's states. Instead, the wave function is given by an exponential function centered on the particular site at position r_o. Here, alpha is the localization length that is the spatial extension of the wave function. The localization strength can simulated by comparing the magnitude of the disorder factor, delta V, to the width, W, of the valance or conduction band. Delta V over W lower than 3 corresponds to weak localization, while delta V over W larger than 3 means strong localization. W is of the order of 10 eV in inorganic solids, while it is only 0.1 eV in organic semiconductors. As a consequence, disorder leads to weak localization in say, amorphous silicon, while strong localization is found in most organic solids. In single-crystal silicon, the density of state (DOS) has a parabolic shape with a parabola pointing to the left for the valance band and a parabola pointing to the right for the conduction band. Because of weak localization, amorphous silicon still has extended states; however, we now have a density of localized states that extends within the energy gap between the valence and the conduction bands. Because of these localized states, an energy gap cannot be precisely defined as in the case of single crystal silicon; instead, one defined mobility edges that separates localized and delocalized states. This is based on the fact that the mobility in the localized states is considerably reduced as compared to that in the extended states. The model of choice for charge transfers in weakly disordered semiconductors is the so-called Multiple Trapping and Release or MTR model. The MTR model assumes that charge carriers move in the extended states with the mobility mu_not. During the transfer, charge carrier are trapped by localized states. So the effective mobility is given by the mobility in the extended states times the ratio between the time elapsed in the extended states to that in the localized states. Because the time while trapped is much longer than the time in the extended states, the mobility is largely reduced. Because the release of charge carriers from the traps is a thermally assisted process, the mobility is now thermally activated with an activation energy equal to the difference between the mobility edge, E_c and the mean energy of the localized states, E_t. The HOMO and LUMO bands in organic semiconductors are very narrow, 0.1 eV as compared to about 10 eV in silicon. Because of the random distribution of energy states, the density of states in disorder organic semiconductors adopts a Gaussian shape. The general form of DOS is given by the equation at right hand side. Here, sigma correspond to the width of the distribution and N_o is the molecular density. A strong difference with amorphous silicon is that now, all states are localized. When all states are localized, charge carrier transport occurs via hopping. The transport is described by the hopping, rate nu_ij between an initial state i and define a state j. In Miller and Abrahams' model, the hopping rates is a product of two terms; a distance dependent term that represents the probability of the charge carrier to tunnel between the two states and a term that depends on the energy difference between the two sites, which is one when arrival state is at an energy lower than initial states. In the reverse situation, the term is thermally activated. So hopping to states with a lower energy are much more likely than hopping to states with a higher energy. Because of that, a charge-carrier starting from the dense states near the maximum of the DOS tends to go down to sites with lower energy. However, when reaching sites of energy far from the center of the Gaussian, the density of site decrease a lot, so the mean distance between the sites becomes very high and the first exponential in the hopping rate drastically decreases. So now, there is a tradeoff between hopping distance and energy difference, and hopping to a site closer to the center of the DOS become more likely. After that, the transport consists of an alternation of jumps downwards and upwards. In practice, all charge carriers localized in a deep level will sooner or later jump to a site at an energy close to some universal value, traditionally referred to as the transport energy localized slightly lower to the maximum of the DOS. The transport energy plays a similar role as the mobility edge in the MTR model. Hopping transport in a Gaussian DOS present several unique features. First, mobility is not constant with charge carrier density. It increases when increasing charge carrier concentration. The temperature dependence of the mobility also changes when changing the density of charge carriers. At high charge carrier density, the mobility follows an Arrhenius behavior like with the multiple trapping and release model and the polaron model. This means that mobility decreases when temperature decreases. At low charge carrier density, the dependent is different. This is shown by the so-called Gaussian Disorder Model or GDM. The Gaussian Disorder Model was first developed by Heinz Bässler. It describes charge transport through Miller-Abrahams' hopping rate in a Gaussian distribution of states. Furthermore, it includes a Gaussian positional disorder. A numerical estimation from Monte Carlo simulation lead to the following empirical equation. The equation contains two exponential terms. The first one relates to the energy disorder. It gives the temperature dependence of the mobility. Note that this is not a strict Arrhenius dependence because the term inside the exponential is squared. The second term relates to the positional disorder. It shows that the mobility is also electric field dependent. Because the term between the square brackets can be positive or negative, the mobility can increase or decrease with the electric field. But most of the time, the mobility increases with the field. In summary, in most organic semiconductors, charge transport is limited by disorder, which induces a localization of the energy states. Because of the weak nature of the intermolecular bonds, bandwidth is narrow and localization is strong, resulting in a Gaussian density of states. In this context, charge carrier transport occurs through hopping between localized states. Hopping model predict that the charge-carrier mobility depends on both the density of charge carriers and the electric field. The next lecture will deal with optical properties of organic semiconductors. I thank you for your attention.
