In this video, we will discuss Schottky barrier lowering. So, the barrier height for a Schottky contact is defined as the difference, if you recall, between the semiconductor electron affinity and the metal work function. Semiconductor electron affinity, vacuum energy level minus the conduction band, and the metal work function, vacuum energy minus the fermi level of metal, they both are materials specific parameters, and they don't depend on the applied voltage. So the barrier height to a first approximation is assumed to be a quantity that is independent of your biasing condition. However, in reality, you will find that the barrier height does change a little bit as a function of your applied voltage, and this effect is called the Schottky barrier lowering. This effect is due to a phenomenon called the Schottky barrier lowering, and the Schottky barrier lowering is due to the fact that the presence of metal or good conductor modifies the electric field near the surface of metal. You can see in this diagram here, in this in this figure on the right, if you have a positive charge in a free space, in a free homogeneous dielectric medium, this positive charge is going to produce electric field in the radial direction and the amplitude or the magnitude of your electric field will fall with the distance according to the Coulomb's law. However, when you place a metal nearby, then the electric field near the metal surface is going to bend, and also the magnitude of the electric field will change so that all of these electric field will be perpendicular to the metal surface. The absolute value of the electric field magnitude also changes due to the presence of this metal. This phenomenon can be modeled, electric field in this situation can be modeled very well by a technique called the immediate charged technique, and that is, you assume that you have an equal and opposite charge at the same distance away from the metal dielectric interface. So, if your positive charge is located at a distance x away from the metal surface, then you presume a negative charge at the same distance x in the opposite side into the metal, and you assume that the entire space is the same dielectric, and the electric field produced by this pair of positive and negative charge in a uniform dielectric medium matches the electric field on the dielectric side in the presence of metal surface, okay. So, it's a very simple technique that allows you to calculate the electric field modification due to metal surface. So, you can calculate the Coulomb force between the positive and negative charges here separated by the distance 2x, x from the metal surface to positive and x from the metal surface to the negative charge, so, the total distance is 2x. So, this is the force due to Coulomb's law, and if you integrate the force over the distance, then that will give you the total potential energy due to this negative charge or the presence of a metal. Now, adding that potential here, this Coulomb potential due to energy discharge or due to your metal surface, this additional potential energy to the original potential energy due to the donor impurities that we calculated from the Poisson's equation before, then that gives you the total potential. So, the total potential energy which is the charge times the electric potential, is the sum of the two terms. Here is the Coulomb potential due to energy charge, and this here is the original potential energy. Here, we're assuming that the electric field is constant at the maximum value. So, in reality, your electric field, there is linearly inside the depletion region but if we are considering a region very close to the metal surface, we can ignore this linear dependence and just consider that the electric field is constant at its maximum value, and that gives you a linearly varying potential. So, that linearly varying potential is this term, this graph here, and then the first term, the Coulomb potential due to image charge is this guy here, and if you add those two, then you get the total potential which is the solid line here. So, the maximum potential, the maximum barrier height is not simply the electron affinity minus the metal fermi level, and that's q phi b note. Both the actual barrier height is lowered by this amount here, two times delta phi, and the real barrier height use of phi B, is determined by the combination of your original potential and the Coulomb potential due to energy charge. You can calculate this maximum point using this equation by simply taking the derivative and setting equal to zero, and if you do that, then you will find that the potential drop, the decrease of the barrier height is proportional to the square root of your E max, the maximum electric field at the junction, and this E max is a quantity that depends on your bias. So, the actual barrier height is actually dependent, has some dependence on the applied voltage. It is a small dependence, relatively small dependence but nevertheless, it is there. So, the barrier height in reality is a voltage dependent quantity and that causes some deviation from the ideal diode behavior for an actual real Schottky diodes.
