Hello, I hope you've been learning from the lectures so far. In the previous lectures, you saw how one calculates free carrier densities in a semiconductor, both holes and electrons at thermodynamic equilibrium. Let us now consider this topic in more detail by working through an example question. The question is stated as follows; calculate volume concentration of free electrons and holes in a crystalline silicon wafer at room temperature, and with a phosphorus impurity concentration of five times 10 to the 17 per cubic centimeter. Furthermore, the question asks, what is the position of the Fermi level energy relative to the conduction and valence band for this situation? 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. Before we begin to consider this question, let's take inventory of what we know. The wafer is at thermodynamic equilibrium, which implies no light or other radiation, and no charge extraction or injection. For us today, it means a single Fermi level that defines the concentration of both electrons and holes. At thermodynamic equilibrium, the following equation holds true. The intrinsic carrier concentration, n_i, is equal to the product of the electron concentration n, and the hole concentration p. This is only true at thermodynamic equilibrium, and is a consequence of the law of mass action. As well, since we know that it's a crystalline silicon wafer, we know the value of n_i at room temperature. We also know a few more things about crystalline silicon. For one, phosphorus is an n-type donor dopant, and we can assume that all dopant atoms are ionized, and therefore active. This means that the concentration of free electrons will be equal to the concentration of phosphorus atoms. We also know that the band gap of silicon at room temperature is about 1.1 eV. Finally, we can look up the band edge effective densities of states at the conduction and valence bands, N_C, and N_V. Again, these are material properties of crystalline silicon. Now that we've established what we know, let's solve the problem. To calculate the hole concentration, the easiest way is to use the mass action law. We know the intrinsic carrier concentration, and the free electron concentration. So, obtaining the free hole concentration is straightforward. Reorganizing the equation and plugging in values, we obtain an answer of 450 per cubic centimeter. As a sanity check here, we confirm that in this n-doped wafer, there are far more electrons than holes. So, far so good. Onto the position of the Fermi level. The concentration of free electrons in the conduction band is exponentially related to the distance between the Fermi level and the conduction band through the following equation. Note, that this equation is only true as long as the Fermi level is at least a few kT away from either energy band edge, and we should confirm this assumption at the end. This equation can be reorganized to tell us the distance between the Fermi level and the conduction band edge. Plug-in numbers, we obtain an answer of 0.108 electron volts. On an energy band diagram, this would look as follows. Again, as a sanity check, we confirm that E_C minus E_F is a positive value, and so the Fermi level lies within the band gap. As well, since the material is n-doped, the Fermi level is indeed closer to the conduction band edge. As one last point, we confirm that the Fermi level lies more than a few kT away from the conduction band edge, and so all our approximations are appropriate. Now, we can do a cross check, and see what value of hole concentration we obtain using the analogous distance to the valence band edge of 0.99 eV. Using the equation to calculate hole concentration and plugging in the known values, we obtain a free hole concentration of 523 per cubic centimeter. Considering that we're calculating values that range over 14 orders of magnitude, this is pretty close, and it's simply due to rounding errors. The fact that we obtain the same value using this other strategy reminds us that in fact, these material properties are interrelated. If we start with the mass action law, and we plug in the carrier concentration equations, the Fermi level is cancelled out as a variable, and we obtain the following expression for the intrinsic carrier concentration. Note, that this shows that we can obtain the value of n_i if we know N_C, N_V, and the band gap. That's why we got about the same answer when solving for the hole concentration in two different ways. Through this works problem, I hope you've gained some insight into how to calculate carrier concentration in the semi-conductor in thermal equilibrium. Thank you for your attention, and see you again soon.
