In this video, we will continue our discussion on extrinsic semiconductors. So, in order to calculate the number of electrons at the donor energy level not ionizing to conduction boundaries, and the number of holes in the acceptor energy level not ionizing to the valence band again. We basically use the same approach, calculate the probability of finding an electron at the donor energy level, multiply that with the donor density, then that will give you total electrons, number of electrons at the donor energy level. Likewise, calculate the probability of finding an empty state at the acceptor energy level, multiply that with the acceptor density. This will give you the total density of holes in the acceptor energy level, and they are shown here. So, I will not show the derivation of this probability function. Its uses the statistical physics technique using partition functions. But I will just use these expressions. And if you multiply to this ND the density of donors or NA to this probability, then they will give you the electron density of the donor energy level, which we call N sub capital D, and the whole density at the acceptor energy level piece of capital A. Now, for shallow impurities where the donor energy levels they're close to conduction band, and the acceptor energy level they're close to the valence band. Then, we should expect in general that the donor energy is far from the fermi level. And acceptor energy level also far from the fermi level. Now, if this is the case then from the equations in the previous slide, we should expect the electrons at the donor energy level and the whole concentration at the acceptor energy level should be zero. What that means is, that all of these impurities are fully ionized producing free electrons in the conduction band and holes in the valence band. Now, when the doping density, the density of donors, and sub D, and density of acceptors and survey is much greater than the carrier concentration, then we should expect the carrier concentration will then be determined simply by the density of your dopers, donors or acceptors. If you have the donors and acceptors simultaneously present, in the same semiconductor, then the net doping density will be determined by the difference between the two. In the case when ND, donor density is greater than the acceptor density then the electron density is given by ND minus NA. When the opposite is true, then the whole density is determined by NA minus ND. Now, for deep impurities where these inequalities don't apply, then there will be a substantial number of electrons and holes still stuck at the donor level and the acceptor level. And these carriers don't move. They are trapped at the impurity site, and therefore these electrons and holes do not contribute to conductivity. We call these impurities carrier traps, because they're trapping carriers. Now, in order to calculate the electron densities and whole densities in an extrinsic semiconductor, electron densities in the conduction band and hole densities in the valence band that is, we invoke charge neutrality condition meaning that, all of the negative charge in your material should equal to all of the positive charge in the material. So that overall, your material is charge neutral. This condition allows us to determine the fermi level in general. So, assume shallow impurities and non-degeneracy. So, what that means inequality this first inequality is the shallow impurity conditions. And also non-degeneracy condition, because this implicitly assume that EF is far away from the vendanges as well. So, in these cases as we mentioned already, donors and acceptors are fully ionized. We can ignore any electron density at the donor level, and hole densities at the acceptor level. These are zero. So, this simplifies the charging neutrality conditions. So, N sub capital D and P sub capital E goes away. And it simply is this. So what this equation says is that, the total number of electrons in a conduction band plus total number of ionized acceptors. Now, ionized acceptor they have lost hole. So, they are negatively charged. So, this contributes to the negative charge. So, these two terms represent the total negative charge in your material. Likewise the ionized donors have lost electron, so they are positively charged. And then the total number of holes in the valence band, they make up the total positive charge in their material. So, sum of these total positive charge should equal to the sum of these two terms which represents the total negative charge. Now, without loss of generality, we just assume that ND is greater than NA. If the opposite is true, then you can just reverse this inequality and go through the same procedure, you get a similar answer. So, we assume that the donor density is greater than the acceptor density and defined delta N being the difference between the electron concentration in the conduction band, and the whole concentration in the valence band. Now, from the charging neutrality condition, this should equal to the difference in doping densities. Now, we invoke the law of mass action which says, nP product should be equal to a ni squared ni squared being the intrinsic care of concentration. Then, the charging neutrality conditions becomes a simple quadratic equation, and the solution to that simple quadratic equation is given by the quadratic formula shown here. If we spell out the definition of delta N using this equation up here, then the electron concentration is given by this equation. Now, there are two solutions because we have a quadratic equation, we will choose plus sign for this, because negative sign will give you a negative number which doesn't make sense, which is non-physical solution obviously. Now, here we have a solution for electron concentration in the case when donor impurity is greater than donor impunity density is greater than the acceptor density. This type of material is called the end type material because this type of material naturally has more electrons and holes. And so, electrons are the predominant or dominant charge carriers, and this type of material is called the anti-material. In an anti-material, electrons, the dominant carriers are called the majority carriers. Holes, the non-dominant carriers are called the minority carrier. So, this equation gives you a majority carrier concentration. And how do you calculate the minority carrier concentration? Go back to law of mass action. And law of mass action gives you the minority carrier concentration. And this equation is varied for very wide range of concentration, and it's a very powerful and useful equation. Now, we can consider two extreme cases that would be quite illuminating. So, in the low doping density limit, that is the Nd minus Na, the net doping density is much smaller than the intrinsic carrier concentration. In that case, you can ignore, you can approximate these full quadratic formula and to simplify that into this. So, in low doping limit, your carrier concentration remain close to intrinsic carrier concentration, and is slightly lifted and slightly reduced by the net doping density. So, if the majority carrier concentration is increased by the doping density, minority carrier concentration is decrease by half of the net doping density. In the high doping density limit, the net doping density is much much greater than intrinsic carrier concentration, then the equation in the previous slide simply gets reduced to this. The majority carrier concentration simply equals to the net doping density, and the minority care concentration is given by the law of mass action. So, if you plop the carrier concentration in an extrinsic semi-conductor as a function of temperature, so here is the extrinsic limit where we have just considered. So, in this case, when the temperature is in such a way that it fully analyzes all of your donors and acceptors, and yet the intrinsic carrier concentration remains much smaller than your doping density, then you're in the extrinsic limit. And in this in extrinsic limit, your electron density, majority carrier concentration is independent of temperature because it is determined entirely by the doping density. Now, if you decrease your temperature a lot, this axis is one over t. So, this is low temperature region, then at some point your Fermi Energy is no longer large enough to ionize your donors and acceptors. So, some of the carriers that would have been ionized are now trapped at the donor levels and acceptor levels, and that leads to decrease of carrier concentration here. This region is called the freezer region. If on the other hand if you go to a higher temperature region, then your intrinsic carrier concentration increases exponentially with temperature if you recall. And therefore, at some point your intrinsic carrier concentration begins to exceed your doping density. In that case, your doping density becomes smaller compared to the intrinsic carrier concentration. Carrier concentration is predominantly determined by thermal excitation, and the carrier concentration simply equals intrinsic semi-conductor. So, even though you have impurities in your semiconductor at this very high temperature region, your doping density is negligible compared to the intrinsic carrier concentration. So, for all practical purpose, your material is an intrinsic semiconductor. Now, the last task is to calculate Fermi Level in an extrinsic semiconductor. And that's now easy. So, we know the carrier concentration, majority carrier concentration and minority carrier concentration. Majority carrier concentration is equal to the doping density, the minority carrier concentration is given by the law of mass action. Now, plug that into our favorite formula of this carrier concentration equation that we derive for the non-degenerate semiconductor. And so for Ef here. For a degenerate semiconductor, of course, you can't use the simple formula, you have to use this full integration. So, you can't really solve an expression for Ef analytically, but you can always do numerical calculations. So, it looks like this. So, depending on the doping density as a function of doping density, if you plop the Fermi Level at very low doping density, they are in the middle of the bad gap. This is the position of the intrinsic Fermi Level. And as you increase your doping density, if this doping is donor, then you're turning your material into an n-type semiconductor with electrons as a majority carrier, then the Fermi Level will go up as shown here. Now, if this doping density represents the density of accepter, then you're turning your material into p-type semiconductor with holes being the majority carrier. In that case your Fermi Level will go down towards the variant site. Now, the slope of this Fermi Level as a function of doping density, will be different for different temperature. So, at very high temperature, your intrinsic carrier concentration is very high. Therefore, your doping density has to be very high even greater than that large intrinsic carrier concentration in order to turn your material into an extrinsic semiconductor. While at low temperature, your intrinsic carrier of concentration is very small. Therefore, even very small density of donors and acceptors will push the Fermi Level down below or higher up very quickly.
