Unlike inorganic semiconductors, organic semiconductors have very small density of charge carriers when at equilibrium. Charge-carrier injection is therefore a fundamental process in inorganic electronic devices, and this is the topic of today's lecture. First, we address an electrostatic effect that may affect the actual barrier height at the metal semiconductor interface. The effect is known as image-force effect. When a point charge is placed at a distance x from a metal plate, it provokes a rearrangement of the charge in the metal, resulting in a reduced field and potential. It can be shown that the induced field is equivalent to that generated by a mirror point charge of opposite charge located at the same distance. The Coulomb force induced by the mirror charge is proportional to the square of the charge q, and inversely proportional to the square of the distance 2x between the charges. Let's now come back to the energy diagram of the metal vacuum interface. The energy induced by the image force is obtained by integrating the force. There is, now, an additional term, represented by the dashed hyperbolic line. The shape of the potential has changed, but the barrier height E sub b nought remains the same. If now we apply a uniform electric field F, we add an energy that varies linearly with distance. The resulting energy is a sum of this line with the hyperbola. It is shown by the red line. The barrier height is now reduced by an amount proportional to the square root of the electric field. This reduction is called image force lowering. There are three basic mechanisms for charge-carrier injection in organic semiconductors. The first two correspond to a jump over the barrier, so they initiate with a thermal activation of the carriers. The second step can be either a ballistic transfer over the barrier; this mechanism is called thermionic emission, or a diffusion of the carriers. This second mechanism is more likely in materials with low mobility, which is the case for most organic semiconductors. The third mechanism takes place at low temperature or for a thin barrier. In this case, the charge-carriers can tunnel through the barrier, a mechanism specific to quantum mechanics. The equations for the injection current are shown here for the thermionic, diffusion, and tunnel mechanisms. Here, A* is called Richardson's constant. The equations of thermionic emission and diffusion both contains an exponential term that implies the temperature T. Both mechanisms are said to be thermally activated, or alternatively, that they obey an Arrhenius law, which means that the current is strongly temperature-dependent. This is at variance with tunneling, which is temperature independent. In the case of a disordered solid, the states are localized and randomly distributed. Transport now occurs by hopping between localized states. The first step is a jump to a state close to the metal-semiconductor interface. This first state is thermally activated. At this stage, transport occurs against the electric field so the charge carrier can go back and recombine in the metal. Once the charge carrier has passed this first step, it is driven by the electric field, and escapes to the bulk of the organic layer. The injection current in a disordered solid has been theoretically calculated for a Gaussian density of states. Injection strongly depends on the width of the Gaussian DOS. Remarkably, injection current increases when the width of the Gaussian DOS increases. The qualitative explanation for this unexpected feature is that disorder would reduce the injection barrier, by creating localized level in the energy gap. This table summarizes the various charge-carrier injection mechanisms, and gives their likelihood in organic semiconductors. Thermionic emission is not expected because mobility in organic semiconductors is too low, so ballistic transfers is unlikely. Tunnel transfer only become significant at low temperatures of very thin semiconductor layers. The dominant process are diffusion and Gaussian disorder. The simplest electronic device is a diode. An organic diode consists of an organic semiconductor inserted between two metal electrodes. One electrode is made of a high work function metal and is called the anode. The other electrode is made of a low work function metal, and is called cathode. The diffusion potential V sub d is the difference between the work function of the anode and that of the cathode. Why are the electrodes called anode and cathode? Actually this is a reminder of the vacuum diode we have seen in the beginning of this course. In the vacuum diode, the cathode emit electrons that are collected at the anode. However, there is a big difference between the vacuum diode and the organic diode. In the organic diode, the electrical current can be transported either by the negatively charged electrons or positively charged holes. If the electron barrier of the cathode is low, electrons are injected here. Conversely, if the hole barrier of the anodes is low, holes are injected there. So we can now have three kinds of organic diodes. When the injection barrier is low for holes and high for electrons, this is called a hole-only diode. Conversely, when injection is low for electrons and high for holes, we have an electron-only diode. Finally, when both holes and electrons can be easily injected, this is called an ambipolar device. Let's now look at how the diode operates. When we connect the anode and the cathode through an external circuit, their Fermi levels align. This induces a tilting of the HOMO and LUMO levels, and the setting up of a built-in electric field oriented from the cathode to the anode. At this stage, it is a good approximation to assume that the organic semiconductor behaves like an insulator, so the electric field is uniform. We see that the direction of the field goes against the injection of both holes from the anode and electrons from the cathode. If we apply a bias so that the anode is negatively polarized, the tilting of the bands is enhanced and the magnitude of the electric field increases, and no charge-carriers can be injected. This corresponds to the reverse bias. Applying a positive voltage, the bands gradually flatten, up to a point where they are horizontal. This occurs when the applied voltage exactly compensates the diffusion potential. At this point, the electric field is nil. Going beyond will reverse the direction of the electric field. So now both holes from the anode and electrons from the cathode are injected, so that the current flows in the diode. This corresponds to the forward bias. The operation of the diode is well illustrated by the current voltage curve. We clearly see that there is no current in the reverse bias, while the current rapidly increases in the forward bias. The curve is better analyzed when drawn in a semi-log plot. Now we see that there is very small current in the reverse bias. More interestingly, the forward bias current can now be divided in two regimes. One at low bias, when the current increases exponentially; this is called the weak injection regime, and one at higher voltages when the increase is less than exponential, which is called the strong injection regime. In the next lecture, we will analyze in more detail the differences between the two regimes in forward bias. Thank you for your attention.
