In this video, we will discuss Schottky contact under bias. First, the biasing convention or especially the polarity of the bias. So, we assume that the semiconductor is grounded and we apply a voltage onto the metal side. So, the positive voltage or forward bias means the positive voltage is applied on the metal side. In other words, the metal is placed at a higher voltage, a higher potential than the semiconductor and the reverse bias or negative voltage, negative bias, is the opposite. Further, we assume that when a bias voltage is applied, there is no voltage drop in metal, metal is considered a perfect conductor. Also, we further assume that there is no voltage drop in the semiconductor quasi-neutral region. Again, in reality, there is always a finite resistance. However, even in that case, even in the case where you have to specifically explicitly account for this resistance, you can always add a series resistance at the end of your analysis. So, this is always a good approximation that simplifies our analysis. So, with this assumption, the entire voltage that you apply is dropped across the depletion region. So, this is part of the depletion approximation and in this case, the potential drop across the junction across the contact is simply given by V sub i, the built-in potential which is the potential difference at equilibrium minus your applied voltage and the minus sign here is due to the definition, the biasing convention that I explained at the beginning here. So, all we really need to do is to replace V sub i in the equilibrium equations that we derived with V sub i minus V_a and that basically explains all the energy band structure of the Schottky contact under bias. So, under forward bias, you lower the potential drop across the junction by the applied bias amount here. So, you're band bending decreases and therefore, there are more migration, more electrons migrating from semiconductor to metal. Now, if you recall how the equilibrium is establishing Schottky contact, initially, immediately after forming the junction, electrons in semiconductor, migrate over to metal because metal side have lower energy. As they migrate over, they leave ionized donors behind and these ionized donors produce electric field that opposes that migration when those two balancing each other you'd have equilibrium. Now, you have lowered the energy barrier. What does that mean? That means you have reduced the electric field built within the depletion region. Therefore, the tendency of electron migration from semiconductor to metal prevails. So, you now have electron migration from semiconductor to metal and this results in a large current. On the other hand, if you apply reverse bias, then you are increasing the band bending, you are increasing the potential drop across the junction and that also means that you are increasing the electric field built within the depletion region. So, there are less electrons migrating from semiconductor to metal, but what about the other side? The migration of electrons from metal to semiconductor, that does not really depend on your voltage, the energy barrier for the metal electrons to overcome to go to semiconductor side is determined by this barrier height which is the conduction band, tip of the conduction band here of the semiconductor minus the Fermi level. So, this barrier height is the electron affinity of semiconductor and the difference between the electron affinity of the semiconductor and the work function of metal both of which are material parameters and do not depend on the bias. So, you do not increase the current the other side. So, on the reverse bias, you really have no change in current and that current is generally very small and that's how you get a rectifying behavior in Schottky contact. So, under reverse bias, there is very little DC current flowing. So, the response of your Schottky contact to applied bias on the reverse biasing condition is really capacitive. So, they change the built-in charge across the contact and that behavior increase and decrease of charge in accordance with the applied bias is characterized by capacitance. So, the small signal capacitance is defined as dQ, dV_a, the change in charge in response to change in voltage and we use the space charge equation that we derived in the previous class by replacing V sub i minus V_a and we take a derivative with respect to the a then you get this equation here. That again using the definition, using the expression for the depletion region width, this whole equation is reduced to a very simple form of permittivity of semiconductor divided by the depletion region width. So, this equation tells you that one over C squared is proportional to the applied voltage. So, if you plot one over C squared versus applied voltage, you get a straight line. The slope of this straight line is inversely related to your doping density on the semiconductor side and the x-intercept is simply the built-in potential as shown here right there. So, if you recall, that this expression is exactly the same as the one sided pn junction once again. So, a lot of the electrostatic behavior of the Schottky contact is oftentimes identical to that of the one sided pn junction. You can generalize this expression to a non-uniform doping profile and if your doping density is not uniform for whatever reason, then these capacitance voltage characteristic gives you a doping density at the position corresponding to the depletion region width at that DC reverse bias. So, if you recall how you do these AC small signal capacitance measurements, you apply a DC voltage reverse bias and that establish these depletion region width. If you increase your reverse bias, depletion region width will increase and if you reduce the DC reverse bias, then this X sub d will decrease here. So, you establish this X sub d with your DC and then you apply a very small measuring AC voltage which modulates these depletion region width about this steady-state DC value. So that modulation of the depletion region width, it occurred due to the small AC measuring signal is dX and that dX produces this change of charge dQ_s. So that dQ_s of course is determined by the doping density at that position. So, by measuring CV characteristic, capacitance voltage characteristics you can actually prove the doping profile even in the non-uniform doping case.
