In this video, we will discuss diffusion current and finish with a summary of both drift and diffusion current in semiconductor. So, the diffusion is a universal phenomenon. The thermal motion of carrier species, or molecules, or whatever particle that you're interested in, will lead to motion of those particles or carriers from high concentration region to low concentration region. So, for example, if you have a perfume bottle and if you open it, and the perfume molecules will diffuse out of the bottle, inside the bottle, perfume molecules concentration is very high. Outside, concentration is very low. So, they will diffuse out and that process will continue until the entire room is filled uniformly with those molecules when there is no difference in the density or concentration. So, likewise, carriers inside a semiconductor, if for whatever reason, you initially have carriers very highly concentrated on one side and very low density on the other side, then because of this difference in carrier concentration, these carriers will diffuse from high concentration region to the low concentration region. This process is called diffusion, and this process takes place until the entire material is filled with uniform concentration. And the carrier movement induced by this carrier concentration difference, will lead to current, and this current is called the diffusion current. Now, to derive a quantitative expression for diffusion current, we consider a simple one dimensional case. So, it's a one-dimensional solid, electrons can move only in along the one-dimension, positive x direction or negative x direction. And let's assume some random profile of carrier concentration. Now, this X=0 is the position that we're interested in, it's an arbitrary position obviously, and the electrons here in this one-dimensional solid is subject to thermal motion. And what does that mean? That means they move at the average thermal velocity determined by the temperature, and in doing so, they collide with lattice, lattice vibration or impurities. So, that collision process is characterized by this mean scattering time tau_sub_cn. If you multiply the thermal velocity to the mean scattering time, you get the average distance that the electron travels in between collisions, and that quantity we call the lambda, is called the mean free path. So, at any given moment, at any given position, half of these electrons will be moving to the right and half of these electrons will be moving to the left. And on average, they will travel the distance of lambda, mean free path, in either direction, and then they will collide. And at that point, half of those electrons will be moving to the left, and half of them moving to the right, same thing here happening on this side. So, if you look at the electron flux at this position X=0, then the electron flux, the movement, flow of electrons at this position is not determined by the concentration at this position, because the n(0), the carriers here, half of them will be moving to the right and half of them will be moving to the left, and therefore they cancel each other out producing zero flux. And likewise, half of these electrons located at X=λ, mean free path away from this position of interest, half of these guys will be moving to the left, and this left moving flux from X=λ position, coming into X=0 will contribute to the electron flux at X=0, and likewise, half of these electrons at X=-λ moving will be scattered to the right, so they will be moving to the right and coming to X=0, and this guy will contribute to the electron flux at X=0. So, there are two electron flux contributing to the total electron flux at X=0, and that is the half of the electrons at X=-λ, moving to the right, and that at the velocity of thermal velocity of course, and then half of the electrons at X=λ moving to the left at the velocity of equaling thermal velocity, those two will add up to produce, to determine the actual flux at X=0. So, mathematically, you can write that down here. So, there is a negative sign here because those two flux are moving in the opposite direction and therefore, you need to take the difference, and if lambda is very small and in general, lambda is actually small, then you can take Taylor expansion at about X=0, and retain only the linear term, first solder term here on both case, and then, if you do that then the zero solder term will cancel each other, cancel out. So, this and this cancel, and you're left with these two adding up. So, the flux is given by the thermal velocity, times lambda mean free path, times the carrier concentration gradient at X=0. Now, this is the flux, number of electrons moving per unit time. So, the current density is simply given by multiplying electronic charge to the flux, -q to the flux. So, the diffusion current of the electron is given by this. Now, we need to then eliminate the thermal velocity and the mean free path using the usual Maxwell-Boltzmann distribution. So, the thermal velocity of an electron is related to the Maxwell-Boltzmann distribution. In this case, we're talking about one-dimensional case, so the total thermal energy is one-half K_B_T. In the three-dimensional case, this will be three halves K_B_T. So, using this and also recall the fact that lambda is related to the thermal velocity through the mean collision time, lambda divided by tau_sub_ c, is equal to the thermal velocity. So, using those two, you eliminate V_th and lambda, and rewrite it in terms of K_T, and also the collision time. Then, you can derive this equation here, diffusing current of electron is proportional to the electronic charge, and proportional to the carrier concentration gradient, and the remaining constant is lumped into this quantity here called the Ds of n, and we call that the diffusion coefficient. And if you compare the equation of the Ds of n, diffusion coefficient, you will find that it is related to mobility. And and through these K_T divide by q, coefficient, and this relationship is called the Einstein's relationship. And you may wonder that why diffusion coefficient is related to mobility, because mobility describes a drift phenomenon, a carrier or current induced by electric field. Diffusion is diffusion coefficient, this describes the current due to the diffusion process. Those two seem to be two fundamentally different processes induced by two fundamentally different mechanisms, and why are they related like this? And that's because mobility, actually not only accounts for the effect of acceleration by electric field, but also the collision process. The collision process with the electron, with the lattice vibrations, and the impurities, which is a temperature dependent process. Also diffusion is driven by this collision process. Collision with lattice vibration and impurities, actually drive diffusion. So, these collision process is at the heart of both the drift current and the diffusion current, and that's why these diffusion coefficient and the mobility are fundamentally interrelated. So, so far we have discussed only the electrons, but you can do the same argument, make the same argument, same procedure for holes and derive a similar equation. So, the hole diffusion current is again proportional to the gradient of the whole concentration. And there is a negative sign here because a hole has a different opposite sign charge than the electron. So, we can summarize now. There are two types of carrier species in semiconductors that can carry current, electrons in the conduction band, holes in the valence band. There are two types of currents: drift and diffusion. Drift current is induced by applied electric field, diffusion current is induced by non-uniform carrier concentrations. Drift current is proportional to the applied electric field, and it's proportional to the carrier concentration here, n and p, and also the mobility is a material parameter that specifies how fast carriers are moving for a given applied electric field. So, high mobility will generally mean high current, for a given bandage or a given applied electric field, and low mobility of course means low drift current. And the diffusion current is given by this equation here, it's proportional to the carrier concentration gradient, not the carrier concentration itself, and the proportionally constant is the diffusion coefficient, and the diffusion coefficient characterizes how fast that these carriers are moving for a given concentration gradient. So, the total current is then given by some of the drift and diffusion. So, for a one-dimensional case, you can write it like this for electrons, and for holes likewise, and you can readily extend this to three-dimensional case by simply vectorizing these quantities for the drift current, you can just use a vector field, and for the diffusion current, the derivative is replaced with the gradient operator. So, this is the general expression for a three-dimensional current, current density for electrons, and this is a three-dimensional current density for holes in semico-
