In this video, we will discuss Schottky contact at equilibrium. Let's first start with a few definitions. So, work function, as before, is defined as the vacuum energy level, the energy of free electron minus the Fermi level. So, this here is the metal work function, and this represents the minimum energy required to extract an electron out of metal. You can define the work function for a semiconductor in the same way, vacuum energy level minus the Fermi level, but in semiconductor, Fermi level is located inside the bandgap. So, there are no electrons at the energy equaling the Fermi level. So, more realizable, more meaningful physical quantity is electron affinity, which is the vacuum energy minus the bottom of the conduction band. As the most of the electrons inside the semiconductor are congregated near the bottom of the conduction band, electron affinity represents the energy required to extract an electron outside of n-type semiconductor. The barrier height is defined to be the difference between the metal work function, and the semiconductor is electron affinity and pictorially, that's shown here. This is the conduction band of the electron minus the metal work function. This represents the energy barrier for metal electrons to overcome in order to move into the semiconductor. Now, let's consider the Schottky contact, which is a kind of a metal semiconductor contact that behaves a rectifying current voltage characteristics. So, to understand the formation of Schottky contact, we first consider the energy band structure or energy band diagram for a metal and semiconductor before junction. Those are isolated metal and the isolated semiconductor. The metal energy band diagram is very simple. By definition, metal has a one contiguous energy band that is partially filled. So, the energy electrons are filled up to the Fermi level. Once again, the difference between the vacuum and the Fermi level is the work function. The semiconductor band structure is basically a conduction band, valence band, and the Fermi level somewhere in the middle. Here we consider an n-type semiconductor. So, Fermi level is closer to the conduction band. Now, immediately after formation of the junction or the contact, then the band structure basically will remain the same, metal Fermi level here, the semiconductor Fermi level is here, conduction band and valence band of the semiconductor. Now, the contact is assumed to be intimate, meaning that it allows free exchange of carriers. So, electrons can move from semiconductor to metal and metal to semiconductor freely. This time, the electrons movement are dictated by the energy level. Even though metal has much more electrons than semiconductor, there is an energy difference as shown here. So, semiconductor, the majority carriers of the semiconductor electrons here wants to move to metal to lower their energy. So, electrons will move initially from semiconductor to metal. As they move, they leave ionized donors behind, and these ionized donors are positively charged, produces electric field pointing from semiconductor to metal, and that electric field resist or opposes the electron migration from semiconductor to metal. So, when that electric field is strong enough to stop the electrons migration, then you have reached the equilibrium. So, at equilibrium, as in any system, Fermi level is constant throughout the system. So, the metal Fermi level should line up with the semiconductor Fermi level. In addition, the vacuum energy level must be continuous. You can't really have a jump in the free electron energy. Fermi level must be continuous, as I mentioned. The semiconductor region far away from the contact must retain the properties, that is, the work function and the electron affinity, of the isolated material. So, what this means really is that the E_c minus E_F, the conduction band and the Fermi level difference, must be the same as the isolated semiconductor. That is, it is determined solely by the doping density. So, here, once again, is the band diagram immediately after forming the contact. In order to produce or draw equilibrium energy band diagram, you need to bring this Fermi level of metal to line up to the Fermi level of the semiconductor. In order to do that, you need to produce a band bending on the side of the semiconductor, and you get a band diagram looking like this. So, Fermi level here is continuous throughout the system metal, semiconductor Fermi level line up, and it requires a band bending of the conduction band and valence band of the semiconductor and also the vacuum energy level on the semiconductor side. So, the total band bending is characterized by the built-in potential here, and far away from the junction or the contact, E_c minus E_F, or the semiconductor work function minus the semiconductor electron affinity. That remains the same as that of the isolated semiconductor, which is related to the doping density. Here, the discontinuity of the conduction band at the junction is the barrier height. So, we can do the electrostatic analysis to find out exactly how much band bending there is, and how large is the depletion region width is near the contact, and how much charge is building up or how much electric field is building up near the contact. We can do all that. We can find all those information by solving the Poisson's equation, which is the second derivative of potential is equal to the negative charge density divided by the permittivity. Now, we've doped the depletion approximation here, just as we did in the p-n junction case. The depletion approximation says that the semiconductor region near the junction or near the contact can be divided into two regions, depletion region and quasi-neutral region. Depletion region is where the mobile carriers are completely depleted, whereas the quasi-neutral region is the region where you retain exactly the same characteristics as the neutral, isolated semiconductor, and there is an abrupt boundary between these two region. So, that's the depletion approximation, and that simplifies our analysis. We can write down the Poisson's equation for the depletion region using this charge density. Inside the depletion region, because there are no mobile carriers, the charge is entirely due to your ionized dopants, in this case, donor. Outside the depletion region, because the semiconductor is neutral, number of electron is exactly the same as the number of donors, so your net charge density is zero. So, this is the essence of the depletion approximation, and that allows us to write down the Poisson's equation inside the depletion region as shown here. If you integrate this equation once, you get the electric field because the doping is assumed to be uniform and d, the donor density, is independent of position. Therefore, if you integrate it once, you get a linear function for electric field. Electric field varies linearly within the depletion region, reaching maximum at the junction, at the contact. Then, if you integrate the electric field one more time, you get the electrostatic potential. The electrostatic potential is a quadratic function of position. Outside the depletion region, your E field is zero, because there is no charge, and therefore, your electrostatic potential is constant. So, the potential difference across the junction or across the Schottky contact is given by integrating the electric field across the depletion region. This is the region where your E field is non-zero. So, if you do that, then you get an expression for a built-in potential, a potential built across the Schottky contact, which is linearly proportional to the doping density and quadratically dependent upon the width of the depletion region. Solving for depletion region width, you get this equation here, depletion region width is proportional to the square root of the built-in potential. The total charge inside the depletion region is the charge density times the width of the depletion region, and that is again a quantity that's proportional to the square root of the built-in potential. So, all these equation here, if you recall, look very similar to those derived for p-n junction, and they're identical if the p-n junction is one-sided. In other words, the doping density on one side is much, much greater than the other side, then all these equation for built-in potential depletion region width and space charge for the one sided p-n junction are exactly the same as these equations that we derived for Schottky contact.
