In our previous lecture, we have [INAUDIBLE] Drude's model when charged transport is limited by collisions. This led us to the concept of mobility. Mobility is a proportionality factor between the mean velocity of the charge carriers and the electric field. However, the parameter that measured the ability of a solid to conduct electricity is the electrical conductivity, defined by the factor of proportionality between the current density and the electric field. The electrical conductivity is proportional to the mobility, but it also depends on the density of charge carriers, n. The conductivity of solids varies over a very wide range, from as low as 10 to the power of minus 23 Siemens per meter for insulators, to up to 10 to the 8th for metals. The conductivity of semiconductors lies between these two extremes. However, its main characteristics is to be extremely variable, because the density of charge carriers may be greatly varied. As stated in the aid of previous lecture, organic semiconductors have low mobility. Today we will see that the main reason is that their charge carriers are localized. Localization of charge carriers may occurs through various mechanisms. In this lecture, we deal with localization through polarization. The next lecture, we'll describe localization by disorder. To evaluate the effect of polarization on charge transport, we have to estimate the time scale of each state of the process. The residence time is the time charge carriers stay on the given molecule before moving to the next one. The polarization time is the time it takes for the polarization cloud to set up. To estimate these time scales, we use a basic principle of quantum mechanics known as Heisenberg's uncertainty principle. The first principle says that the position and momentum of a particle can not be exactly known simultaneously. The second principle state the same for energy and time. The second principle mathematically expresses by writing that the energy scale delta E multiplied by the time scale tau has the same magnitude as the reduced Planck constant. So the second uncertainty principle gives us simple way to evaluate time scales. Now the question is what energy to associate to each time scale. We first associate the residence time to the width of the HOMO and LUMO bands. The major difference between organic and inorganic semiconductors is that the binding energy between the constitutive elements that is weak in organic solids and strong in inorganic solids. As a consequence, the valence and conduction band are much larger in inorganic than in organic semiconductors. That is ten electron volt for the former, and only a few tenths of electron volt for the later. So the residence time is 100 times shorter in inorganic semiconductors than in organic semiconductors. Now we turn to the polarization time. This process can be associated to the energy gap, which is roughly identical in inorganic and organic semiconductors with an order of magnitude of one electron volt. We now see that for an inorganic semiconductor, the residence time is around ten times lower than the polarization time so that polarization has not enough time to set up. By contrast, the residence time in organic semiconductors is ten times longer than the polarization time so polarization can take place. Molecular polarization refers to the movement of the nuclei of the molecule. This only concerns organic semiconductors. The characteristic energy is at the vibration of the molecule. That is around 0.1 eV, as estimated by infrared absorption. This scale compares with that of the residence time. So the event of molecular polarization will vary on each particular case. In summary, in inorganic semiconductor, the residence time is so short that the polarization cloud has no time to form. In organic semiconductors, the residence time is longer, electronic polarization is always present. By contrast, the residence time and the molecular polarization time have the same order of magnitude. So the presence of molecular polarization depends on the precise nature of the organic solid. Polarization tends to push the HOMO and LUMO levels towards each other by an amount that corresponds to the polarization energy. Occasionally, an additional shift is induced by molecular polarization. By shifting the HOMO and LUMO levels, polarization induces self-localization of the electron, as shown here for a negative charge. The self-localized charge is also called the polaron. Now charge transfer does not intervene through delocalized level. Instead, it takes place by hopping between adjacent localized states. Because charge transfer only takes place between states at the same energy, the transfer will happen when, under the effect of thermal agitation, an adjacent neutral molecule adopts a configuration of a charged molecule. After the transfer, the first molecule relaxes to its neutral configuration. In summary, polarization of charge carrier in organic semiconductors is a direct consequence of the weak intermolecular bonding in organic solids. The charge carrier, surrounded by its polarization cloud, is called a polaron. Because of the sheet of the HOMO and LUMO levels, the polaron is self-localized. Charge transport takes place through hopping between self-localized polarons. Charge transfer occurs when an adjacent neutral molecule adopts the configuration of a charged molecule under the effect of thermal energy. Delocalized and localized charge carrier transport can be differentiated by the temperature dependence of the mobility. For Drude's and bond model, the mobility follows a power law with negative exponent so that the mobility increases when temperature decreases. In a hopping model, the mobility is thermally activated so that it's decreased when temperature decreases. In the next lecture we will address localization induced by disorder. Thank you for your attention.
