In this video, we will discuss Heterogeneous materials. So, so far, we have discussed homogeneous materials. So, material itself is the same everywhere and also the doping density is the same everywhere. That's what we assume so far. Now, let's generalize the discussion to consider a non-uniform materials. So, let's first consider the case of non-uniform doping. So when the doping is not uniform, you can imagine this process. So initially, the carrier concentration should be equal to the doping density, because the donors will donate, produce electronic conduction band. Acceptors will produce holes in the valence band. At the very beginning, the electrons and holes are generated at the location of the donor and acceptor where they're originally from. So for an anti-semiconductor, your n, the majority carrier concentration profile, should be equal to the profile of your doping. Then, when carrier concentration is not uniform what happens? Well, the diffusion take place. Carriers are free, unlike the donors and acceptors, which are stacked at the Lattice position, crystal positions. These carriers are free to move, so they diffuse. Electrons will move from high concentration region to the low concentration region, and holes will do the same. So diffusion current will take place. Until when? Until the diffusion would like to continue until the concentration is uniform everywhere. However, in this particular case of semi-conductor, as the electron moves away from the donor it came from, it leaves a positively charge donor behind. Donor is ionized, so donor is positively charged. And the positively charged entity produces electric field. And the electric field direction, is in such a way that it wants to bring electrons back. So the electric field builds up in the direction that opposes diffusion. And when the diffusion tendency produced by this non-uniform carrier concentration is balanced by the electric field building up as a result of this diffusion process, then you have an equilibrium. And statistical physics says, that when we reach an equilibrium, the Fermi level, E_sub_F, should be constant everywhere. Now, let's consider a parabolic energy band. Parabolic energy band equation looks like this. And the second term here, that's proportional to k squared, can be written in terms of momentum. And this is the familiar equation for a kinetic energy, so that if the total energy is kinetic energy plus this, you can imagine that this is essentially the potential energy. So the E_sub_C, the bottom of the conduction band, represents the potential energy for electron. But how do you measure E_sub_C? What is your zero? What is your reference point with which you measure the position of your bottom of the conduction band? When you need some convenient reference point, and an ideal reference point should be the same everywhere in your material. And we just talked about something that is constant everywhere whenever your semi-conductor is at equilibrium. And that's the Fermi level. So your Fermi level is a natural reference point from which you can measure the position of your E_sub_C or E_sub_E for a valence band. So, define potential energy as E_sub_C minus E_sub_F. And if you define the potential energy of electrons this way, then you can define electrostatic potential which is basically potential energy divided by the charge. This is the electric potential. Now recall, that potential energy reference can be shifted by any arbitrary amount. In other words, the reference point for your potential energy, can be arbitrarily chosen. The only physically meaningful quantities, is the potential difference. So you can shift the reference point of your potential energy by half the band gap and use E_sub_I, the intrinsic Fermi level, which is always in the middle of the band gap, roughly, instead of E_sub_C. So E_sub_F minus E_sub_I, could be another definition of your electrostatic potential or potential energy. Once you know your electrostatic potential, the electric field is simply the negative gradient of your electrostatic potential. And because E_sub_F is independent of position, this gradient of potential is simply the gradient of your E_sub_I. Now, at equilibrium, both the electron current and hole current should be zero by definition. So, your electron current, which is the sum of drift current and the diffusion current is zero, from which you can solve for electric field and get an equation like this. And the E-field, by definition, is the negative gradient of the potential again, and that should be equal to this quantity derived here. And from this, you can see that the carrier concentration at two different positions, this subscript two and three represent two different location. So the ratio of the carrier concentration for two different positions, is related to the electrostatic potential of the two different position logarithmically. And this definition is- you can see that it's consistent with this equation that we derive for carrier concentration. For a non-degenerate semiconductor, the quantity here, is simply related, EF minus EI is related to potential. And so, you can write down the carrier concentration in the non-degenerate semiconductor as this, and this relationship is perfectly in agreement with the equation that we derive just above. Now, the most general equation that relates the electrostatic potential with charge density is the Poisson equation. The Poisson equation says that the secondary rate of a potential is equal to the negative charge density divided by the permittivity of the material. And the charge density that we have in our semiconductor is this. There is a positive charge due to holes to holes, negative charge due to electrons, positive charge due to ionized donors and a negative charge due to ionized acceptors. So, you can combine this and express n and p in terms of potential through exponential function. And you get this equation. So, this is the most general equation. If you solve this, then you can calculate potential for a given arbitrary distribution of the donor and acceptor. So, for any arbitrarily, non-uniform, doping profile, you can calculate your Phi, electrostatic potential, once you know your electrostatic potential using the equation in the previous slide, you can calculate the carrier concentrations. Now, this is a nasty equation to solve. You consider an example where ND to doping, donor density, there is two orders of magnitude over several hundred nanometers. This is a relatively severe variations in doping density. And that, if you use this equation, Poisson equation, and estimate the changes in potential, it produces a potential difference of about point one volt. And from that, you can estimate the changes in carrier concentration. And that point one volt potential difference corresponds to the difference in carrier concentration and the actual doping density of less than 10 to the 15th, which is less than 10% of the minimum. So, you can see that we even read a relatively severe variations in doping density. The difference in the majority carrier concentration and the doping density tends to be small. And therefore, we can adopt this approximation called the Quasi-neutrality approximation. And that simply says that even in the non-uniform doping case, your majority carrier concentration is always equal to the doping density. So from that, using that, you can then replace the carrier concentration in the equation in the previous slide shown here, majority carrier concentration, n here, you can replace that with the doping density n_sub_d and your electric field is then given by this. Likewise, if you have a P-type material, then the electric field in the material is related to the gradient of your doping density. So what does this mean? In the non-uniformity dope semi-conductor, despite the fact that your doping density is non-uniform, your majority carrier concentration is still the same as your doping density. Which means that the majority carrier concentration is non-uniform. How come? Why don't they diffuse and equalize and make the carrier concentration the same everywhere? Why? Because there is this electric field. Electric field given by these two equations in the n and p-type material, this electric field opposes the diffusion. This stops the carriers from diffusing away from the original profile. So, within the Quasi-neutrality approximation, majority carrier concentration remain the same as the doping density. And that is held together by a built-in electric field. Built-in electric field is given by these equations here. Now, let's consider another case of heterogeneous material, which is a case where the doping density is uniform but the material itself changes as a function of position. And you can do this by for example, gradually alloying your semi-conductor from one material to another. So, let's consider a hypothetical material where the band gap is continuously varied. Now, if the doping density is the same, doping density is uniform, then the majority carrier concentration should be uniform. So here, we're considering an example of a p-type material, so your whole concentration, majority carrier concentration, should be the same as the doping density. Now, what does that mean? That means that EF which is constant, as long as you're at equilibrium, EF should be constant. And because p here, majority care concentration, a hole concentration is uniform, is the same everywhere, that means that your EF minus EV should be the same everywhere, because whole concentration is related to this quantity, EF minus EV. Which means that your EV is constant, EV is the same everywhere. Which means that your EC should be changing according to the changes in band gap. And as a result, you can see that EC minus EF here changes linearly in this case, in this particular example. And linearly changing EC minus EF results in exponentially varying minority carrier concentration, electron concentration as shown here. So, in this particular case, majority carrier concentration remains constant. It follows the doping density, basically. And because the band gap changes as a function of position, your minority carrier concentration changes, varies exponentially as a function of position. Now, once again, why don't they diffuse? Because there must be an electric field that balances, that stops the minority carriers from the diffusing. So, this here, again, the exponentially varying minor concentration, is a result of a built in electric field. When you have a heterogeneous material, whether the heterogeneity is produced by non-uniform doping or the non-uniform band gap, that non-uniformity produces built-in electric field. And that built-in electric field balances the tendency of diffusion and maintain non-uniform carrier concentration inside the semiconductor material.
