In this video, we'll discuss currents in semiconductor and also introduce drift current, one type of current that is possible in semiconductor. So, in general, current density is given as a product of charge density times the velocity of those charge. So, in a semiconductor, there are two types of charge carriers species: electrons in the connection band, holes in the valence band. So, the charge density is expressed by the electronic charge, -q, and the electron concentration in the conduction band, n. And that's the charge density. And if you multiply to that the velocity at which these electrons move, then you get the current density due to the electrons in the conduction band. So, that's given here in this first equation, -q, electronic charge, times the electron concentration, n, and their velocity, v_sub_n. Likewise, the current density due to hole is given by the product of q, +q, a hole's charge, times the hole concentration in the valence band, p, and the velocity of holes, v_sub_p. So, you only need to know two things, the carrier concentration in each band and their velocity, because q is a constant. And we already answered the first question, how many carriers are there in each band? We have derived the expressions for n and p, electron concentration in the conduction band, hole concentration in the valence band for various types of semiconductor: intrinsic, extrinsic and different doping conditions. So now, we're ready to answer the second question, and that is how fast are they moving? What is their velocity? Now, to answer this, we need to consider two cases, because there are two possible mechanisms that can drive carrier motion that can induce non-zero velocity that could contribute to the current density. One is electric field. When you apply electric field, it pushes the charge carriers one way or the other depending on the sign of their charge, and that causes movement and that leads to current. And that type of current is called a drift current. And the quantity or the parameter that describes how fast electrons and holes are moving for a given electric field is mobility. So, we're going to look at that. And then another mechanism that causes carrier motion is carrier concentration gradient. If the carrier concentration is not uniform, then the carriers will diffuse from high concentration region to the low concentration region, and this motion is characterized by a quantity called diffusion coefficients. So, we're going to look at those two in that order. Now, before that though, we would like to discuss the thermal motion of carriers. Now, carriers are not at rest in general. They move due to their thermal energy. At a finite temperature, for example room temperature, there is a finite thermal energy that these carriers possess, and because of that, they move around. And these motions are generally random and therefore, they do not add up to any particular current. So, when you have a uniform semiconductor with no applied electric field, then naturally you don't get any current. However, that doesn't mean that the carriers are not moving. They are moving, but they are moving randomly. So, on average, when you add them all up, they add up to zero. Now, in quantum mechanics, carriers move around inside the crystal without any collisions or scattering that leads to energy loss. However, in real crystals, there are always defects and impurities, and these impurities cause scattering, and this scattering leads to energy loss. So, that's one type of carrier scattering that goes on in semiconductor, and that is impurities scattering. And another type of collision is through the vibrating lattice. When the lattice itself, the atoms that make up the semiconductor vibrate due to their own thermal energy, then at any given time, the atom is not at the exact crystallographic lattice position, they're slightly deviated from it, and you can consider that as defects as breaking the crystal symmetry, perfect symmetry. So, vibrating lattice, when the atoms are vibrating, then they also collide with carriers and that leads to energy loss. So, this type of collision is called a phonon collision, or phonon scattering. Phonon is the quantized lattice vibration. Now, both of these collision, I already mentioned that these collisions are inelastic scattering event, that means there is energy loss. When the electrons collide with either phonon or impurities, the electron lose its energy. And the average time, this is a random scattering event, so it's a random event, but you can always talk about average time in-between collision, and this quantity is very important. The time it takes for carriers to travel in-between two successive collision is called a mean collision time or mean scattering time. And we're going to call it Tau_sub_c. Now, let's consider parabolic energy band. And as I mentioned before in previous videos, this is a good approximation for bandage near the bottom of the conduction band than near the top of the valence band. And almost all important phenomena take place in this region. So, this is a good model. And in a parabolic energy band model, the energy of an electron is expressed by a band of the conduction band, E_sub_c, plus this term, that is quadratic in k, and therefore if you plot E versus k, then you get a parabola and hence the name parabolic energy band. The h_bar times k that is in the numerator of the second term is momentum through the de Broglie's matter wave hypothesis. So, the second term here in this parabolic energy band model is just the kinetic energy. So, p_square over 2m. So, E, the total energy of the electron, minus the bottom of the conduction E_sub_c leaves you the kinetic energy. So, any excess energy that your electron has above the bottom of the conduction band is considered kinetic energy, and likewise any energy, excess energy, that a hole possesses below the valence band but on top of the valence band is also considered the kinetic energy of a hole. So, kinetic energy of a hole is expressed as E_sub_v, top of the valence band, minus the actual hole energy. That difference is consider the kinetic energy of the hole. Now, if the electrons and holes follow the Maxwell-Boltzmann distribution, now this is an assumption, the same assumption that we made for non-degenerate semiconductor, so, rigorously speaking, electrons and holes follow Fermi-Dirac distribution. However, if the doping density is mild and your Fermi level is relatively far away from the band edges, then Maxwell-Boltzmann distribution is a good approximation to use instead of the Fermi-Dirac function. And in this case, we can take that approximation again and express the kinetic energy of the electron, E minus E_sub_c, which is basically one half mv_square, that kinetic energy is equal to three halves times kT. This is the kinetic energy of ideal gas molecules that follow Maxwell-Boltzmann distribution. So, if the electron follows the Maxwell-Boltzmann distribution, and that is the case for non-degenerate semiconductor once again, then we can use the ideal gas results and equate the kinetic energy to three-halves kT. From this equation, we can calculate the thermal velocity of electrons, and that turns out to be about a 140 miles per second at room temperature. This is a very, very high speed. So you can see that electrons actually move at very, very high speed. However, they move at random directions, and therefore, all of those motion add up to zero current as it should. Now, when you apply electric field however, then this electric field, non-zero electric field, adds another extra velocity component that pushes the electron. If this is an electron, then the electron will move in the direction opposite to the electric field. So, there is an added motion of electrons along the direction opposite to the applied electric field. So, it does go through these random collisions still, but as it goes through random collision, it drifts along the direction opposite to the electric field. This extra velocity added by the presence of the applied electric field is called the drift velocity. And if the electric field strength is not very high, then this added velocity component will remain small compared to the thermal velocity which is very high as we just have seen. In that case, you can approximate that this time in between two successive collisions will remain the same because the actual velocity of an electron at any given moment is dominated by the thermal velocity still. So, the average time between collision still dictated more or less by the thermal velocity alone. So, the scattering time, mean scattering time, tau_sub_c, remains the same, but as they go through the collision at the same rate, the electron drifts along one direction. In the energy picture, this process, this collision process, can be viewed as shown here in the energy band diagram. So, here is the conduction band, and here is the valence band. And your Fermi level is somewhere here in the upper half of the band gap, indicating that this is an N-type semiconductor, electron being the majority carriers. And the electron is accelerated by the electric field. And as the electron gets accelerated, It moves away from the band edge. Remember, the kinetic energy of an electron is specified by the difference E minus E_sub_c. So, electron acquires kinetic energy as it gets accelerated by the applied electric field. This is the added velocity component due to electric field. But, when they collide at some point, they lose their energy and they go back down, they go back down to the bottom of the conduction band. And then, from then on, they get accelerated again until they collide again and lose all the energy. And this process keeps on going. So, if there was no collision, the electron will get accelerated indefinitely and the velocity of the electron will increase indefinitely, reaching eventually infinity. However, there is this constant collision process that limits the velocity of the electron. So, electron reaches a finite velocity that is proportional to the applied electric field because of this interplay between the acceleration in between collision and the energy loss upon collision. So, this steady velocity that the electron reaches in the presence of applied electric field is called the drift velocity. And the drift velocity is then used in the current density equation that I've shown you in the very first line, and that gives us the expression for a drift current density.
