Adding nuclei to the Crystal
Now we add a very crude representation of positively charged nuclei to our little 20 angstrom 1-dimensional world. Each nucleus will be represented by a central region of constant lower potential, with a rise in potential between each nucleus. In the above example we had a zero potential bounded at each end (at x=0 &x=20) by an infinite potential. The universe was only those 20 Angstroms; there was no space outside. We will set the zero potential as a point away from our positively charged nuclei, and actually allow space to continue to plus and minus infinity. Since we have a potential well, we can keep the electron inside, if the energy is not too high. Figure 2-8 shows the potential levels of the 20 angstrom crystal, with 7 “nuclei”. The first 3 wave and probability functions, ψ2i , are also shown. Note that the electron concentration tends to build up around the lower potential of the “nuclei”.
FIgure 2-8
Even in this simple example we can see the origin of the electron band gap so essential to the workings of a semiconductor. The gap between n=7 and n=8 , in table 1, is due the "energy band gap". When the wave length of the energy is about the same as the distance between the “nuclei”, the band gap naturally forms. Intuitively this makes sense, because there are two energy levels with the same wave lengths.
Table 1.
n=1 | n=2 | n=3 | n=4 | n=5 | n=6 | n=7 | n=8 | n=9 | n=10 |
-8.59 | -8.55 | -8.49 | -8.41 | -8.32 | -8.25 | -8.21 | -5.66 | -5.47 | -5.17 |
The lower energy level has the probability nestled around the low potential “nuclei”, where as the next higher energy level, with the same wave length, has the electron concentration between the “nuclei”, giving it a higher energy level. This is not important for our discussion, but does show the power of even simple calculations.
Figure 2-9 shows the electron concentration of the lower 7 energy levels added together. Note that, although higher around the “nuclei", the electron concentration evens out over the 7 nuclei, as intuitively it should. The electron probability around each nuclei has contributions from every free electron in the crystal.
Figure 2-9
A more Realistic Nuclear Potential
Now we give our “nuclei” a little more realistic potential, as seen in figure 2-10, along with the n=2 wave function. The potential used here is an effort to show how the inner, core, electrons in the metal tend to repel the free electrons. If there where no “central core” of electrons, the free electrons would feel the full force of the very large nuclear charge (46 for palladium) all of the time. The core electrons shield and actually repel the free electrons to some extent. These complex interactions will be glossed over here and our potential will be used as shown. We have seen how subtle changes in the potential do not change the wave function drastically, our only justification for this sloppiness.
Figure 2-10
The wave functions for this potential look very much like the previous example. The wave function for n=2, figure 2-10, is very similar to the n=2 wave function in the crude potential. The other energy levels will also look similar. The energy levels for the corresponding “n”, of the two different potentials, will of course be different, but the shape of the wave functions will remain very similar.