Hello. Electronic devices operate through electrical currents generated by electrical voltages. An electrical current occurs when charge carriers move. So today's lecture is about charge carrier transport in semiconductors. The movement of electrons is easy to conceive when they are in vacuum, like in the vacuum tube. Here electrons emitted by the heated cathode are accelerated by the electric field between the negatively polarized cathode and the positively polarized anode. The movement is ballistic because electrons do not experience any collision on their way from the cathode to the anode. If you now turn to solids, we have seen that a crystal is generated by an elemental unit cell characterized by the three elemental vectors a, b, and c which repeat themselves along the three directional space. According to Bloch's theorem the wave function of electrons in a crystal has the same translational periodicity as the crystal itself. More precisely, wave function is defined by the product of a plane wave represented here by the imaginary exponential function, times a function u_k that has a periodicity of the crystal. Here, k is a wave vector. Its dimension is that of a reverse distance. According to quantum mechanics, the probability density of finding of an electron at a position r is given by the squared modulus of the wave function. Because the imaginary exponential has a modulus of one, the probability density reduced to the square modulus of the periodic function u. In other words, the probability of finding an electron somewhere in the elemental cell is identical in every elemental cell of the whole crystal. These de-localized states are called Bloch's states. They exist in all kinds of crystal, whether it is a metal, an insulator, or a semiconductor. Thus quantum mechanics leads to this counter intuitive statement that electrons in the conduction band and holes in the valence band nearly behave as electrons in vacuum. Hence, their name of nearly free charge carriers. However, at variance with electrons in vacuum, charge carriers in solids are submitted to collision at various scattering centers. A good way to describe the movement of charge carriers in a solid is by Drude's model. When electrons and holes are regarded as non interrupting gases. In the absence of any electric field, electrons and holes move at a constant velocity between two scattering events. Collisions make all directions of motion equally probable, so that, statistically, the mean position of a charge carrier remains constant, so the charges do not carry any electrical current. When an electrical field F is applied, the charge carriers feel the Coulomb force, proportional to the field. So between two collisions they are accelerated according to Newton's law of dynamics and there is a net displacements of the charge. The acceleration of the charge can be written as the mean velocity divided by tau where tau is the mean time between two collision. Combining with Newton's law the mean velocity reveals proportional to the magnitude of the electric field. The coefficient of proportionality is called the mobility. In Drude's model the mobility equals the mean scattering time multiplied by the elemental charge and divided by the electron mass. Another parameter of importance is the mean free path, that is, the mean distance covered by an electron or a hole between two collisions. The mean free path is given by the mean collision time multiply by the thermal velocity. It can also be written in terms of mobility from the definition given earlier, that is, lambda equals the mobility times m times the thermal velocity divided by the elemental charge. The validity of Drude's model is conditioned to the fact that the mean free path is significantly higher than the mean distance between two scattering centers. In the case of silicon, typical values of the parameters are the elemental charge, the electron mass, a mobility around 1,000 square centimeters per volt and per second, and the thermal velocity at room temperature of 10 to the fifth meter per second, which results in a typical mean free path of 10 nanometers, which is indeed more than 10 times larger than the mean atomic distance. Drude's model dominates in inorganic semiconductors like silicon. The associated charge-carrier transport mechanism is called band transport because it is chiefly based on the fact that charge carriers are delocalized as stated by Bloch's theorem. One of its remarkable feature is that mobility increases when temperature decreases. However in most organic semiconductors, Drude's model is not valid because the mobility is too low. The reason for low mobility is the localization of the charge carriers. Accordingly, we have to invoke other charge transport mechanisms. Before closing this lecture I would like to introduce the concept of density of states or DOS. The density of states g(e) per unit volume and per unit energy is defined so that g(e) times dE is the number of the available states for energies between E and E + dE. For a perfectly crystalline, three-dimensional semiconductor, the density of states near the band edges has this shape of a lying parabola. There is a parabola pointing to the right for the conduction band, and a parabola pointing to the left for the valence band. The empty space between the two bands correspond to the energy gap, where the density of state is strictly equal to zero. Our next two lectures will deal with charge transfer mechanisms in organic semiconductors. I thank you for your attention.
