Calculate n(E) for an infinitely deep well of width a, and compare it with the results for free three-dimensional electrons (Figure 4.7(c)). There is a slight difficulty in comparing these two functions, because the step like density of states for the well is a density per unit area, whereas the E 112 function for free electrons is a density per unit volume. We can allow for this by multiplying the three-dimensional density of states by the width a of the well, which turns it into a density of states per unit area that can be directly compared with the two-dimensional result (or, of course, both can be converted to three-dimensional units). Plot on the same axes the density of states for free electrons in 3D and the density of states for a GaAs well 10 nm wide and for a well 20 nm wide. Show that the top of each step just touches the parabola. What happens as the well gets wider?
It is desired to put as many electrons as possible in the lowest subband without occupying the second subband. Does it help to use a material with a different effective mass? Would the conclusion be different if a finite well were used?