introduction to semiconductors, mainly looking at the behavior of electrons in a solid from a quantum mechanical point of view.
If we only had to worry about simple conductors, life would not
be very complicated, but on the other hand we wouldn't be ableto make computers, CD players, cell phones, i-Pods and a lot of other
things which we have found to be useful. We will now move on,and talk about another class of conductors called
semiconductors.
In order to understand semiconductors and in fact to get a more
accurate picture of how metals, or normal conductors actuallywork, we really have to resort to quantum mechanics. Electrons
in a solid are very tiny objects, and it turns out that whenthings get small enough, they no longer exactly following the
classical "Newtonian" laws of physics that we are all familiarwith from everyday experience. It is not the purpose of this
course to teach you quantum mechanics, so what we are going todo instead is describe the results which come from looking at
the behavior of electrons in a solid from a quantum mechanicalpoint of view.
Solids (at least the ones we will be talking about, and
especially semiconductors) are crystalline materials, whichmeans that they have their atoms arranged in a ordered
fashion. We can take silicon (the most important semiconductor)as an example. Silicon is a group IV element, which means it
has four electrons in its outer or valence shell. Siliconcrystallizes in a structure called the
diamond crystal lattice. This is shown in
.
Each silicon atom has four covalent bonds, arranged in atetrahedral formation about the atom center.
Crystal structure of silicon In two dimensions, we can schematically represent a piece of
single-crystal silicon as shown in
. Each
silicon atom shares its four valence electrons with valenceelectrons from four nearest neighbors, filling the shell to 8
electrons, and forming a stable, periodic structure. Once theatoms have been arranged like this, the outer valence electrons
are no longer strongly bound to the host atom. The outer shellsof all of the atoms blend together and form what is called a
band . The electrons are now free to move about
within this band, and this can lead to electrical conductivityas we discussed earlier.
A 2-D representation of a silicon crystal This is not the complete story however, for it turns out that
due to quantum mechanical effects, there is not just one bandwhich holds electrons, but several of them. What will follow is
a very qualitative picture of how the electrons are distributedwhen they are in a periodic solid, and there are necessarily
some details which we will be forced to gloss over. On theother hand this will give you a pretty good picture of what is
going on, and may enable you to have some understanding of how asemiconductor really works. Electrons are not only distributed
throughout the solid crystal spatially, but they also have a distribution in energyas well. The potential energy function within the solid is
periodic in nature. This potential function comes from thepositively charged atomic nuclei which are arranged in
the crystal in a regular array. A detailed analysisof how electron
wave functions , the mathematical
abstraction which one must use to describe how small quantummechanical objects behave when they are in a periodic potential,
gives rise to an energy distribution somewhat like that shown in
.
Schematic of the first two bands in a periodic solid showing
energy levels and bands Firstly, unlike the case for free electrons, in a periodic solid,
electrons are not free to take on any energy value they wish.They are forced into specific energy levels called
allowed
states which are represented by the cups in the figure. The allowed states are not distributed uniformly
in energy either. They are grouped into specific configurationscalled
energy bands . There are no allowed levels
at zero energy and for some distance above that. Moving up fromzero energy, we then encounter the first energy band. At the
bottom of the band there are very few allowed states, but as wemove up in energy, the number of allowed states first increases,
and then falls off again. We then come to a region with noallowed states, called an energy
band gap . Above the band gap,
another band of allowed states exists. This goes on and on,with any given material having many such bands and band gaps.
This situation is shown schematically in
, where the small cups represent allowed
energy levels, and the vertical axis represents electron energy.
