In this video, we will discuss the basic operation of MOS MOSFET. In this basic theory, we will assume the source and the body are at the same potential. This is a typical operation, but we will relax this assumption later for more general case. As I explained in the previous video, in order for MOSFET to operate, you apply a voltage on the gate terminal in order to bias the MOS device forming the channel region into inversion. That way, the inversion layer then provides the conducting path between the source and drain region. Now, if you bias your MOS device into strong inversion, the carrier concentration will be very high, and therefore, the conductivity is high and the primary conduction mechanism will be drift. So, in order to describe the current in your MOSFET, you write down drift current equation as shown here in this equation. Now, in order to understand this equation, I like to quickly sketch the device structure here source, drain, and then there is the oxide, and then there is the gate electrode, and you apply a voltage to create a channel inversion layer here that connect source and drain. Now, the coordinate is set up so that the vertical direction is X and the horizontal direction along the channel is Y. Then the out-of-plane direction is the G direction. So, the Q here, the inversion layer charge density is the charge against two-dimensional charged density in the YZ plane, and W here is the width of the channel along the G direction. So, W times Q sub N gives you a charge density per unit length along the Y direction. Then you multiply that with the drift velocity, that will give you the total current drift current along the channel. Now, for the small drain voltage, the drain voltage is the one that produces the Y direction of electric field, which drives a current between source and drain. When the drain voltage is small, then the drift velocity is simply given by the mobility times the electric field. So, if you recall, when field is very, very large, then the velocity may saturate. So, we're saying that that is not where far from it, and we're in the normal conduction regime where drift velocity is linearly proportional to the applied electric field, and the proportionality constant is the mobility. Now, plug this into the drift current equation before, and use the fact that the electric field is simply the negative derivative of your potential. Then, this is the equation for your drain current. Now, we make a few assumptions saying that my mobility is constant, this is not dependent on Y coordinate, and also the threshold voltage is not a function of Y. So, my MOS device is uniform across the channel and the threshold voltage for my MOS device doesn't change as a function of Y position along the channel. Also, the inversion layer and the bulk charge, the depletion charge is controlled only by the X directional field. The X directional field is the field perpendicular to the semiconductor-oxide interface, and that field is controlled by your gate voltage only. So, this is called the gradual channel approximation. It's a powerful approximation very useful because it allows us to use 1D simple theory in order to calculate the drain current, which we will do here. So the inversion layer charge as a function of Y now can be written simply by this. This is the V_G minus V_ T minus V here is the charging voltage of your MOS device, and multiply by the C oxide will give you the inversion layer charge. This is something that we derived in the previous module on MOS device. The only difference is that now we have this full Y dependent voltage that's produced by the drain voltage. Now, you plug this equation two into equation one, drift current equation in the previous slide, and then you integrate that equation from source to drain. So for y-variable, you integrate from zero to L. For the voltage, you integrate from zero, at the source to V sub D, the drain voltage at the drain. When you do that, it's a simply integration will give you this equation I_D which is quadratic in V_D. This is called the long channel MOSFET equation. This describes the current that you may have in your MOSFET, as a function of two control voltages is here, V_G, gate voltage and V_T, the drain voltage. Now, for a given V sub G and D sub G has to be greater than V sub T, the threshold voltage. Otherwise you don't have inversion layer. So, your devices in cut off. There is no current irrespective of how much voltage that you apply on the drain terminal. But if you're V_G is greater than V_T, then you are in strong inversion and the channel is turned on, so now you can have a current as you apply a voltage on the drain. Now, the previous equation, the long channel MOSFET equation shows that your I_D, the drain current is a quadratic function of V sub D. It's actually an upside down parabola. So, it increases quadratically and reaches a maximum as you increase your V sub D. The slope actually decreases as you increase your V sub D because it's an upside down parabola. The slope becomes zero when it reaches the top maximum. That's what happens when V sub D is equal to V_G minus V_T. This is sometimes called the overdrive voltage. If you increase the voltage beyond that, then you will see that your I_D will decrease because once again this is an upside down parabola. So, parabola goes down. Which is clearly unphysical result. So, what happens when your drain voltage V_D is greater than V_G minus V_T? In order to see that, you have to go back to equation two, which is the charge density equation. So, at Y is equal to L, this voltage here is equal to V_D. When V_D is greater than V_G minus V_T, what does that make? That makes your Q sub N positive, which means that you no longer have inversion. You no longer have an inversion layer in that case. So, if you don't have strong inversion, then you cannot use the drift current equation that's so, and therefore the whole formalism that we have followed breaks down. That's why it leads to this non-physical result. Therefore, the long channel MOSFET equation that we derived is valid only for the drain voltage less than the overdrive voltage. Then, what is the correct description for the drain voltage greater than the overdrive voltage? Now to look at this, you have to look at the actual charge distribution. So, this is the case when there is no drain voltage. Then your channel is uniform. The charge density in the channel region is given by the inversion layer charge carrier concentration. As you increase your V_D, then you are reducing your charge density in the inversion layer is given by this equation two. So, because of this last term, your charge density is small in on the drain side, large on the source side. When your V_D is greater than V_G minus V_T, then there is a point in the middle of channel where your carrier concentration goes to zero, inversion layer carrier concentration goes to zero. This is called the pinch off. What happens there then? Well, your carriers are drifted up to this pinch-off point. At that point, they enter the depletion region. So, from that point on, it's no longer drift current but your carriers are swept away by the large built-in electric field inside the depletion region here. So, how do you describe this situation? Well, it depends on the length of this L minus L prime. This region where the channel is pinched off. This region here, is this region large or small compared to this entire channel? If this region is very small, then we can just ignore this small shift, small depleted region at the end of the channel, and say that my charge density profile remains the same, and therefore my current remains the same. That will be the first approximation, and that will be a good description when your channel length is very very long. So, for long channel device, which is what we are describing now. When your V_D is equal to the overdrive voltage, V_G minus V_T, then the channel pinches off and your drain current remains constant. It does not depend on the V_D. As you increase your V_D further, the drain current still remains constant. This region where the drain current is independent of your drain voltage is called saturation region of operation for your MOSFET. So, if you combine the long channel MOSFET equation that we derived first, and then the saturation drain current equation that we just we just derived, you can put them together, you get the typical I-V characteristics of your MOSFET as shown here. Drain current versus drain voltage is plotted into two regions. One is the quadratic region where the long channel MOSFET equation is valid. That's when your drain voltage V_D is less than V_G minus V_T. Then the saturation region where your V_D is greater than V_G minus V_T. In that case, your current is constant independent of your V_D. Therefore, the current depends only on the gate voltage. So, these represent several different values of gate voltage.
