In the previous lecture, we have described the geometry of the organic diode. And its operating mode. Today we will analyze the diode in more details, and in particular explain what is the difference between the weak and strong injection regimes. Here we recall the geometry of the diode with the two electrodes, anode and cathode, and the definition of the diffusion potential, as the difference between the work function of the anode and that of the cathode. When the diode is biased, an electric field is induced. In reverse bias, the field goes against injection of both electron and holes, and no current is flowing. In principle, the current should only flow when the electric field reverses, that is, when the forward bias is in excess of the division potential. This actually corresponds to the strong injection regime. However, when looking at the semi-logarithmic plot of the current voltage curve, we see that the forward current starts as soon as a positive voltage is applied, and steeply increase in the so-called weak injection regime. So we are facing a paradox, how can a current flow against the electric field? To resolve the paradox let us recall that an electrical current may actually have two origins, drift and diffusion. First, we distinguish between electron and hole currents. Drift current is driven by an electrical field; it is proportional to the density of charge-carriers and to the mobility. This is actually how the electrical conductivity sigma is defined. Diffusion current results of a gradient of charge-carrier density. It is proportional to the gradient through another fundamental parameter called diffusion coefficient. Like for mobility, the diffusion coefficient differs for electrons and holes. The differences between drift and diffusion currents are summarized in this table. First point, diffusion may occur in the absence of any electric field. Next, drift current is proportional to the density of charge carriers, diffusion current to the gradient of the charge carriers, so no diffusion current if the density is uniform. Drift current is along the electric field, diffusion along the charge carrier gradient. And finally, drift current obeys Ohm's law, diffusion current does not. Most theoretical models for electronic devices relies on two basic equations: Drift diffusion, here written for holes, that gives the electrical current, and Poisson's equations that connects electrical potential, hole density, and electric field. In spite of their apparent simplicity, in most cases, these equations can only be solved numerically. Now we show the numerical resolution at different regimes. The drift current is in red, the diffusion current in blue and the total current in dashed green. The anode is on the left hand side, the cathode is on the right hand side. In the weak injection regime, the drift current is negative, meaning that it goes against the total current, which is consistent with the direction of the electric field. However, the diffusion current is higher than the drift current. So the total current goes against the direction of the electric field. When the applied voltage equals the division potential the electric field inside the organic layer vanishes. Accordingly, the drift current goes to 0. The total current corresponds to the diffusion current. In the strong injection regime, the drift current is now in the same direction as the total current, and as expected it largely dominates the diffusion current. The last point we want to address is the particular regime that occurs in the strong injection regime, known as space-charge limited current, or Mott-Gurney law. Unlike Ohm's law, Mott-Gurney law predicts a current proportional to the square of the voltage. A simple derivation of the space-charge limited current can be made when we consider the organic semiconductors as a dielectric, so the diode behaves like a capacitor. The capacitance per unit area is the permittivity epsilon divided by the thickness of the semiconductor. Applying a voltage V between the electrodes induces a charge Q that equals the capacitance times the applied voltage. When the charge moves, its transit time is equal to the distance divided by the mean velocity, which in turn equals the mobility times the electric field, the latter being equal to the voltage divided by the distance. Now the current corresponds to the charge, divided by the transit time. All in all, we find that the current is proportional to the square of the voltage, and inversely proportional to the cube of the distance between the electrodes. This very simple gives a value that only differs by a factor of 9/8 from the exact Mott-Gurney law. This lecture completes the theoretical part of my course. With the next lecture, we will start the study of real organic electric devices, light emitting diodes, solar cells and field-effect transistors. So thank you for your patience.
