In this video, we will discuss drift current. Just a quick recap, upon application of an external electric field, electrons in holes will be drifted, will be accelerated by the electric field. At the same time, they collide with lattice vibrations and also impurities and defects that may be present in the material. So that leads to energy loss and that limits that collision process limit, the velocity, that is being added by the electric field. So that interplay between the acceleration by the electric field and the energy loss due to collision process leads to a finite, steady state velocity added by the electric field. And that velocity's called the drift velocity. And that is the velocity that leads to the current that is produced by the electric field. So to be quantitative, we start with the Newton's second law, which says that the applied force is simply equal to the rate of change of the momentum. Now, the momentum, in our case, is simply the mass times the velocity. And again, this is the drift velocity that we're talking about here. So that is the velocity added by the electric field in addition to the thermal velocity, which is quite high. So that's the momentum, change of momentum, and then the time is the average time in between two successive collisions. So that will be the mean scattering time, tau sub c. And we add it to subscript n here to indicate that we're talking about conduction of an electron here. So tau sub c n is the scattering time for electrons. And the force, of course, is the charge times the electric field. So from this equation we can solve for the drift velocity. And that leads to this equation here. And we lump all the constants in front of the electric field, and we call them mu sub n, and this is the mobility of the electron. So the mobility is the proportionality constant between the drift velocity and the electric field. So it characterizes how fast electron is moving for a given applied electric field. Similarly, we can derive an the expression for hole mobility. And the only difference is that there is a sign change, because hole is a positively charged carrier, and then the mean scattering time for holes and, of course, the hole effective mass. So using these mobilities, we can now write down the current, the drift current, current induced by applied electric field. The current density for electron is given by this equation, so q times n, the electron concentration, times mu, the mobility, times the applied electric field. Similarly, the whole drift current is given by q times p times mu sub p times E. Now, there can be and there are in general multiple scattering mechanism that lead to energy loss of carriers. We already talked about two scattering mechanism, lattice vibrations and impurities. And there are in general multiple vibrational modes, multiple phonon modes that scatter with carriers at different rate. And also, there are in general different types of defects and impurities that may be simultaneously present in your semiconductor. So all of those things will scatter carriers at different rates. And in the presence of these multiple scattering mechanism, the total rate of scattering is simple given by the sum of the scattering rates for individual mechanism. So in terms of probability, you can say that the total probability of scattering is simply the sum of the probabilities of individual scattering mechanism. In terms of scattering time, tau sub c, what that means is that the total scattering time is given by the harmonic mean of the scattering time for individual mechanism. And this result shows that the shortest scattering time will dominate. In other words, whichever mechanism that scatters the carriers the most will dominate, and will primarily determine the total scattering time. So because the scattering time is given by the harmonic mean, the mobility is also, total mobility, is given by the harmonic mean of the mobilities, separately defined for individual scattering mechanism. So here are some graphs of electron mobility and hole mobility as a function of temperature. At high temperature, including room temperature and high, the mobility generally decrease with increasing temperature. And this is because in this temperature range, phonon scattering dominates, and the phonon scattering increases with increasing temperature. This is easily understood, because phonons' lattice vibrations are due to thermal energy primarily. In high temperature, more thermal energy, therefore more active vibrations. And therefore, more scattering by these lattice vibrations of the carriers. Now, as you decrease the temperature, phonon scattering becomes less and less efficient. And at some point, the phonon scattering rate becomes smaller than the impurity scattering rates. And the impurity scattering has an opposite temperature dependence. In other words, it decreases, the scattering rate decreases, with increasing temperature. So that's why you see this change of this maximum here. And at very low temperature, the mobility has the opposite temperature dependence, it increases with increasing temperature here. And that's because at very low temperatures, impurity scattering dominates, and you're seeing the effect of the impurity scattering in the temperature dependence. On the figure on the right here, it shows the mobility as a function of carrier concentration at a fixed temperature. And at low doping densities the impurity scattering is small compared to the lattice scattering or phonon scattering. So the mobility is basically independent of the doping density. However, as the doping density increases, impurity scatting rate increases, eventually exceeds that of the phonon scattering. And therefore it now shows a distinct dependence on the doping density and purity density. And this is where the impurity scattering dominates over the lattice scattering. Now, lastly, we can define conductivity. Re-write the Ohm's law, as shown here, in terms of the current density. You may be familiar with the Ohm's law saying that current is proportional to voltage. And if you rewrite it in terms of current density, that same law says that the current density is proportional to the electric field. And the proportionality constant in this case is called the conductivity. And we can easily find the expression for conductivity from the drift current equation that we derived before in terms of mobility. So q electronic charge times n times mu sub n, carrier concentration, times its mobility. If there are two carrier species equally contributing, then you need to include both the electron term and the hole term here. But for doped semiconductors, extrinsic semiconductors, one carrier species dominates. The majority carrier concentration tends to become many, many orders of magnitude greater than the minority carrier concentration. So in that case, you only need to include only one of these two terms for the majority carriers. And here in the graph shows the doping density and the resistivity, which is the inverse of the conductivity. So you can see that as you increase your doping density, your resistivity goes down, which means that your conductivity goes up. And they are generally linearly related, as shown by the expression for the conductivity, which has this carrier concentration term shown here. Now the last comment before we move on to diffusion current is velocity saturation. So far we assumed that the electric field strength is not so high that the scattering time, tau sub c, is independent of the applied field. And this is a good approximation as long as the drift velocity remains small compared to the thermal velocity. However, if you keep on increasing your electric field, at some point drift velocity becomes comparable to the thermal velocity. And in that case, the average time in between collisions will actually reduce, due to the electric field. And these types of carriers are called the hot carriers. And one of the effects of the hot carriers is that the scattering time gets reduced as you increase your electric field. And therefore, as the higher electric field adds more momentum, or increases more velocity, adds more velocity to the carrier, the increased scattering time takes energy more efficiently as well. And therefore, at some point, even as you keep on increasing your electric field, velocity doesn't increase anymore. And this phenomenon is called velocity saturation, and it is a universal phenomenon. It occurs in all materials, not only semiconductors, but in all materials. And in this graph you show that the drift velocity eventually saturates for electrons here, and holds here. And what does that mean? That means the current doesn't increase anymore, even as you increase your electric field. And in terms of mobility, that means your mobility decreases, mobility degrades. And this is an important effect that we must consider, for example, in the modern field effect transistors, which tends to have a very small gate length, and therefore a very high electric field. We will revisit this issue when we discuss the metal oxide semiconductor field effect transistor, MOSFET.
