Hi everyone, I hope you've been learning from the lecture so far. In the previous lectures you saw how the different properties of semiconductors such as their workfunction and band gap can affect the way that a junction is formed. Let us now consider this topic in more detail, by working through an example question. The question is stated as follows, draw the band diagram of a stack of materials with the following sets of properties. First, a p-doped material with a work function WF of 6.3 electron volts, and electron affinity chi of 4.1 electron volts, and an energy band gap E sub G of 2.3 electron volts. Secondly, an n-doped material with a work function of 5.6 eV, an electron affinity of 5.5 eV, and a band gap of 1.1 eV. Thirdly, a metal with a work function of 5.5 eV. Finally, we're told to assume that each layer is much thicker than the space charge region that forms. If you wish at this point you may stop the video and try to solve the problem on your own. If not, let's continue. To draw this band diagram, we must work in two steps. In the first step, we draw the three materials as they would appear if not in contact, that is with a flat vacuum level. This step will tell us the offsets between the bands as the work functions will tell us the band alignment when the materials are joined. Secondly, we will draw the band diagram at thermal equilibrium with a single flat Fermi level. In this case as we will see, the vacuum energy level will be continuous but not flat. The conduction and valence bands will follow its curvature. Let's see what this all looks like in practice. We start with the vacuum energy level, which before the materials are in contact and can exchange charges gives us an absolute reference level. Now, we draw the first material noting the work function of 6.3 eV, the electron affinity of 4.1, and the band gap of 2.3 eV. Even if we were not told we could tell that this is the p-doped layer as the Fermi level is closer to the valence band than the conduction band. Second, we draw the n-doped material. Work function of 5.6, chi of 5.5, and gap of 1.1. Again, the position of the Fermi level shows us that this is an n-doped material. At this point, we can see the band offsets between these materials. These offsets must be maintained even when the materials are in contact. In this example there's a conduction band offset of 1.4 eV and a valence band offset of 0.2 eV. In both cases lower for the n type material. Both the sign and magnitude of these offsets will be maintained in the final device. Now we draw the metal. For a metal, the work function and the electron affinity can be considered as synonymous. This leads to a conduction band offset of 0 eV. Between the metal and the n-type material. We will soon see what consequences this has and the final stack. Now let's draw the stack at thermal equilibrium. All of the layers will share the same Fermi level, as free charges will have moved around to establish equilibrium. We first draw the p-doped layer as before. No surprises yet. Next up is the n-doped layer. Note now that the vacuum levels are not aligned. We know that it must be continuous but not flat and any slope in the vacuum level indicates an electric field. Without worrying too much about the details, we can draw the curved line connecting the two vacuum levels. In reality this line will contain information about the relative doping levels of the two materials and you could think about this in more detail when time permits. For now, it's just a continuous curved line. The curvature of the conduction bands will follow the curvature of the vacuum level but then must also include the same band offset that we calculated before, in this case 1.4 eV. The same applies to the valence band, follows the curvature and keeps the same band offset 0.2 eV. Now, we add the metal to the diagram. Again, the work function and the electron affinity are equivalent for this metal as far as we're concerned. We again connect the vacuum levels. Typically in a metal the charge carrier density is so high that we consider all of the space charge region to occur in the semiconductor. This is not so obvious here but you'll see it elsewhere. Following the same rule, we draw the curvature of the conduction and valence bands. Since there was no offset in the conduction band originally there isn't one now either. We now have the band diagram for this complete structure. We can now look at the constructed band diagram without all the clutter. At the surface of the metal, there is a small but existing barrier for holes to get to the metal. Remember that this is a electron potential energy picture so holes flow uphill. We only worry about the barriers in this case because normally a metal is an excellent recombination surface. So, we can consider there to be no holes at the metal surface. On the other hand, there's no barrier for holes to get to the p-doped material. In fact, the space charge region there will sweep them away into the p-region. Also, electrons cannot get to the p-region. They're blocked by a barrier created by the band offset. Electrons will therefore defuse everywhere and are much more likely to be collected at the metal surface. Through this worked problem, I hope you've gained some insight into how to construct a band diagram knowing the material properties of each layer involved. I highly suggest trying this exercise with a randomly chosen set of material properties, to see if a solar cell results. In the meantime, thanks for your attention, and see you again soon.
