In this video, we will discuss the short channel Mosfet theory, and the next effect that we want to discuss is the mobility degradation. Now the electrons and holes in the inversion layer undergo many collisions with impurities and interfaces and other carriers. These collision process ultimately limit the drift velocity that these carriers can achieve and therefore that limit the mobility. If in a short channel device, the field is very high and therefore there will be more and more collisions and that tends to degrade the mobility. This is the mobility degradation effect. This mobility degradation effect is typically described empirically. So, set up your effective field using these charge densities in the depletion region and the inversion layer and you describe your mobility degradation using these empirical equation. So, these are E naught, mu naught and nu here are all fitting parameters. Typical impurity degradation behavior looks like this. This is an empiric mobility, I'm sorry, mobility degradation behavior looks like this, Y-axis is mobility, X-axis is usually effective field. This equation typically works well for room temperature where phonon scattering is dominant, but if you go to a very low temperature or extremely heavily doped semiconductor where impurity scattering becomes dominant, then it begins to break down. Now, in the ultimate case where your mobility degrades all the way down to zero, then you have velocity saturation. So, in the simple description of your drift current, your drift velocity of your carrier is given by mobility times the field. So in this case, your drift velocity will increase linearly with your field. However, when your field becomes too high, then the drift velocity becomes so high. It becomes comparable to the thermal velocity of the carriers in the semiconductor. Then as you increase your field, your drift velocity is so fast. It reduces the collision time so much and your actual drift velocity does not increase anymore because of so many collisions. That's the velocity saturation. The drift velocity as a function of electric field is shown here. As you can see that at low field, there is a certain slope and this slope is your mobility as shown here, but as you increase your field, the slope decreases. That is the mobility degradation. Eventually the slope becomes zero. That's when your mobility goes all the way to zero. That's the velocity saturation. This is a universal phenomenon. It happens in all materials when your applied field is very high and it's not unique to Mosfet but surely it happens in Mosfet as well. It is an effect only shown in high field situation. If you have a short channel device, then the field inside the device tends to be very high and therefore the short channel devices more susceptible to the velocity saturation effect. Here is the typical values of saturated velocities for electrons and holes in the silicon based Mosfet device, six to 10 times, 10 tilde six centimeters per second or four tilde eight times 10 raised to six centimeters per second for electrons and holes respectively. The drift velocity is typically expressed by this type of equation, with this mu effective and an E saturation as a parameter value that fits the experimentally observed velocities. The saturation electric field can be calculated by setting your drift velocity equal to the saturated velocity. So, your E saturation is given by this equation here. Then you rewrite the drain current equation for a long-channel device using this. So, instead of for the velocity here, this is where your drift velocity comes in. But instead of the simple drift velocity, given by mobility times the field, you use this equation, which describes the saturated velocity saturation effect. Then you integrate, this is a function of Y, so you integrate along the channel as we did before in the long-channel Mosfet. Then you can derive this equation for the drain current with the velocity saturation effect. When the drift velocity saturates, your drift current will also saturate. So, this equation, drain current equation shows these drain current saturation effect. You can see that when the saturation field is much greater than VD over L, which is the actual field inside the channel, then your field inside the channel is much weaker than the saturation field. So, velocity saturation effect is not important. In that case, this equation simply reduces back to the long-channel Mosfet equation. There are several approximations that we made. These may not be the most reliable accurate approximations and you will have to sort of consider this when using these equations in particular cases. Nevertheless, this equation describes very well the experimentally observed behavior in most Mosfets. Now in order to really have a quantitative description of the drain current with the velocity saturation effect in short channel device, we write down the inversion layer carrier density using this. So, the drain current saturation occurs at VD saturation. You plug that in and the inversion layer carrier density at the drain at the end of the channel will be given by this, and this charge will be moving at the saturated velocity if you are saturated. If you are in the velocity saturation regime, the drift velocity will simply be equal to the V saturation. So, multiply V saturation to Q sub nD will give you the saturated drain current. Now plug in the expression for V saturation, the saturated velocity into this then you can get an equation for the saturation drain current. Once again note that VD saturation simply converges into the over-dry voltage when E saturation is very large and therefore velocity saturation is not important as it should. Now, important parameter that we use to characterize the performance of Mosfet is a trans-conductance. The trans-conductance by definition is the differentiation of the derivative of the drain current with respect to the gate voltage. So, if you use a saturation drain current then that will be the saturation trans-conductance. In the long-channel device, we already know the equation for ID saturation and which is a quadratic function of VG. So, if you take a derivative with respect to VG, you get a linear function here. You can rewrite it as this. But the point is that your trans-conductance is linearly dependent on your gate voltage V sub G. Now, if you include the velocity saturation effect in a short channel device, then we must use equation two in the previous slide to calculate the trans-conductance. That's states, in order to calculate this derivative term, we use equation three, the VD saturation equation and you get this. Now, if you look at this, then numerator is of the order of VG square, denominator is also of the order of VG square. So, you can see that the trans-conductance in the long-channel device goes linearly as VG whereas trans-conductance in short channel device is zeroth-order in VG or independent of VG. What that means is that the ID saturation is a quadratic function of VG in the case of long-channel device, but in the case of short channel device, your ID saturation is a linear function of VG. So, ID increases linearly with VG in the short channel device. What that means is that, as you decrease your channel length, if you bring your channel lengths to zero, then VD saturation tends to zero. In the case of the long-channel device that makes your drain current goes to infinity, blows up but in the short channel device your ID is limited to a finite value. That will be this value here. So, if you plot your drain current saturation drain current as a function of channel length in the long-channel device, in the long-channel theory, predict these unphysical results that your ID saturation blows up in the limit of zero channel length. However, in the short channel device, because of the velocity saturation effect, it approaches a finite value. It says the maximum value. What that means is that the drain current in general, in a short channel device for a given a channel length, your actual short channel current is always smaller than what's predicted by the long-channel theory. Therefore, the improvement or advantage that you should expect by reducing your channel is generally overestimated by the long-channel theory and the short channel theory gives you a realistic expectation of the benefits that you can achieve and you can enjoy by going to a short channel device.
