Now, we will describe the electrical properties of possibly doping semiconductors. We will determine carrier densities, for example, the electrons in the conduction band. Therefore, we'll take into account the intrinsic or the extrinsic ionizations, due to the presence of impurities in the semiconductor. If a phosphorus atom is ionized, it has lost an electron, creating a positive charge of density, N_D plus, which is related to the donor effect. Likewise, the acceptor effect will induce the presence of band negative ions of density, N_A minus. And therefore, the electrical neutrality requires that the number of carriers in the conduction band plus the number of band ions is equal to the carriers in the valence band plus the phosphorous ions. We must correctly interpret this equation. Small n, correspond to free electrons, which can be transported with an electric field from the silicon to an external low circuit. However, N_A minus corresponds to band ions which are fixed in the lattice. The same remark applies to p and N_D plus. It can be shown that the occupation probabilities of a donor by an electron or an acceptor by a hole is given by the following equations. The factor half will be a little bit difficult to demonstrate. As an explanation, it can be simply argued that two generated states cannot be occupied simultaneously due to the Coulomb repulsion. Then n, p and u can obtained from these equations. Generally, solve numerical methods. As an example, we'll treat the case of the semiconductor only with donors. This is called the n doped semiconductor. Moreover, we will neglect the presence of acceptors. It means that only phosphorus atoms have been substituted to the silicon atoms. So, N_A will be neglected leading to a simplification of equations. So at zero degree k, the donor levels are occupied. Indeed, because of the lack of thermal oxidation, donor ionization cannot be obtained. As you can see, the chemical potential (Fermi level) is located between the donor level, E_d, and the minimum of the conduction band, E_c. I remind you that a chemical potential corresponds to the maximum energy of occupied states at zero k. So, if the donor is not ionized, it means that its energy state is occupied. Therefore, the Fermi level is located above the donor level. The is the situation at zero degree k. Then, when the temperature increases the donor levels will be ionized and the Fermi level will decrease below E_d, leading to an ionization probability of the presence of free electrons in the conduction band. At this step, electrons are the majority carriers as the donor level is close to the conduction band and holes as the minority carriers. Then, as the temperature increases all the donors will be ionized. At that time, small n becomes constant since the temperature is not sufficient to induce intrinsic ionization, that is to say, the transition between valence and conduction bands. I remind you that the difference between E_c and E_d is very small as compared to the band gap. All donor levels being filled, the number of carrier is constant. This is the so-called saturation regime which corresponds to the case of silicon at room temperature. Because of this order of magnitude, the electrons are the majority carriers. When the temperature increases, at higher temperature transition between valence and conduction band will become significant. Therefore, the Fermi level will naturally move towards the middle of the band gap which corresponds to the intrinsic regime. So the above calculation establishes this relationships. So as a function of temperature, different regime can take place. Let us show now, the variation of log, small n versus the inverse of temperature, T. At very low temperature, so 1/T is very high. This is called extrinsic ionization regime. At higher temperatures, the donor are all ionized leading to a plateau since the temperature is too low to induce transition between valence and conduction bands. When the temperature increases, we obtain the intrinsic regime. This regime is characterized by an increase of thermal transition with temperature. The number of electrons in the conduction band becomes larger and the doping effects become negligible. It corresponds to the intrinsic behavior. The figure below displays the variation of the chemical potential with temperature, p-chart. At zero k, Mu is above level donor since donor are not ionized. Then Mu, decreases as a function of T relative to doping effect. In contrast, at very high temperatures when the transition between valence and conduction bands become dominant, the intrinsic regime takes place. The transition between extrinsic and intrinsic regimes depend on the doping level and the value of the band gap. Thermal conduction are more probable in the case of germanium as compared to silicon. In other words, the importance of doping is always related to weak impurity in combustion is found to decrease as a function of temperature. At very high temperature, the semiconductor behaves as an intrinsic semiconductor whatever the initial doping. The shift of Fermi level towards the middle of the band gap will lead to account for the decrease of the conversion efficiency of solar cells like silicon with temperature. So, we can summarize the values, behaviors previously described. The top figure corresponds to intrinsic semiconductors. The electrons in conduction bands are produced by transition from the valence band to the conduction band. So the creation of an electron is connected to simultaneous creation of hole, the chemical potential being located in the middle of the band gap. And doping, middle, corresponds to a chemical potential of Fermi level close to the conduction band. The electron density, n, is much larger than p due to the presence of donors. The electrical neutrality implies that the number of electrons in the conduction band is compensated by the positive ions phosphorus ions, which were produced by ionization. The P in the p case display a similar behavior. In this last case, p is much bigger than n and electrical neutrality is satisfied, because of the presence of negative ions. Thank you. I invite you to refer to the appendix 3 which provides the order of magnitude of the acceptor on donor levels in the band gap.
