If we only had to worry about simple conductors, life would not
be very complicated, but on the other hand we wouldn't be able
to 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 actually
work, we really have to resort to quantum mechanics. Electrons
in a solid are very tiny objects, and it turns out that when
things get small enough, they no longer exactly following the
classical "Newtonian" laws of physics that we are all familiar
with from everyday experience. It is not the purpose of this
course to teach you quantum mechanics, so what we are going to
do instead is describe the results which come from looking at
the behavior of electrons in a solid from a quantum mechanical
point of view.
Solids (at least the ones we will be talking about, and
especially semiconductors) are crystalline materials, which
means 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. Silicon
crystallizes in a structure called the diamond
crystal lattice. This is shown in Figure 1.
Each silicon atom has four covalent bonds, arranged in a
tetrahedral formation about the atom center.
In two dimensions, we can schematically represent a piece of
single-crystal silicon as shown in
Figure 2. Each
silicon atom shares its four valence electrons with valence
electrons from four nearest neighbors, filling the shell to 8
electrons, and forming a stable, periodic structure. Once the
atoms have been arranged like this, the outer valence electrons
are no longer strongly bound to the host atom. The outer shells
of 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 conductivity
as we discussed earlier.
This is not the complete story however, for it turns out that
due to quantum mechanical effects, there is not just one band
which holds electrons, but several of them. What will follow is
a very qualitative picture of how the electrons are distributed
when they are in a periodic solid, and there are necessarily
some details which we will be forced to gloss over. On the
other hand this will give you a pretty good picture of what is
going on, and may enable you to have some understanding of how a
semiconductor really works. Electrons are not only distributed
throughout the solid crystal spatially, but they also have a distribution in energy
as well. The potential energy function within the solid is
periodic in nature. This potential function comes from the
positively charged atomic nuclei which are arranged in
the crystal in a regular array. A detailed analysis
of how electron
wave functions, the mathematical
abstraction which one must use to describe how small quantum
mechanical objects behave when they are in a periodic potential,
gives rise to an energy distribution somewhat like that shown in
Figure 3.
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 configurations
called
energy bands. There are no allowed levels
at zero energy and for some distance above that. Moving up from
zero energy, we then encounter the first energy band. At the
bottom of the band there are very few allowed states, but as we
move up in energy, the number of allowed states first increases,
and then falls off again. We then come to a region with no
allowed 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
Figure 3, where the small cups represent allowed
energy levels, and the vertical axis represents electron energy.
It turns out that each band has exactly
2N
2
N
allowed states in it, where N
N is the total number of atoms in the particular crystal
sample we are talking about. (Since there are 10 cups in each
band in the figure, it must represent a crystal with just 5
atoms in it. Not a very big crystal at all!) Into these bands
we must now distribute all of the valence electrons associated
with the atoms, with the restriction that we can only
put one electron into each allowed state. (This is
the result of something called the Pauli exclusion
principle.) Since in the case of silicon there are 4
valence electrons per atom, we would just
fill up the first two bands, and the next would be empty. (If
we make the logical assumption that the electrons will fill in
the levels with the lowest energy first, and only go into higher
lying levels if the ones below are already filled.) This
situation is shown in Figure 4.
Here, we have represented electrons as small black balls with a
"-" sign on them. Indeed, the first two bands are completely
full, and the next is empty. What will happen if we apply an
electric field to the sample of silicon? Remember the diagram
we have at hand right now is an energy
based one, we are showing how the electrons are distributed in
energy, not how they are arranged spatially. On this diagram we
can not show how they will move about, but only how they will
change their energy as a result of the applied field. The
electric field will exert a force on the electrons and attempt
to accelerate them. If the electrons are accelerated, then they
must increase their kinetic energy. Unfortunately, there are no
empty allowed states in either of the filled bands. An electron
would have to jump all the way up into the next (empty) band in
order to take on more energy. In silicon, the gap between the
top of the highest most occupied band and the lowest unoccupied
band is 1.1 eV.
(One eV is the potential energy gained by an electron moving
across an electrical potential of one volt.)
The mean free path or distance
over which an electron would normally move before it suffers a
collision is only a few hundred angstroms (
≈300×10-8
≈
300
10
-8
cm) and so you would need a very large electric field
(several hundred thousand
voltscm
volts
cm
) in order for the electron to pick up enough energy to
"jump the gap". This makes it appear that silicon would be a
very bad conductor of electricity, and in fact, very pure
silicon is very poor electrical conductor.
A metal is an element with an
odd number of
valence electrons so that a metal ends up with an upper band
which is just half full of electrons. This is illustrated in
Figure 5. Here we see that one band is full, and
the next is just half full. This would be the situation for the
Group III element aluminum for instance. If we apply an
electric field to these carriers, those near the top of the
distribution can indeed move into higher energy levels by
acquiring some kinetic energy of motion, and easily move from
one place to the next. In reality, the whole situation is a bit
more complex than we have shown here, but this is not too far
from how it actually works.
So, back to our silicon sample. If there are no places for
electrons to "move" into, then how does silicon work as a
"semiconductor"? Well, in the first place, it turns out that
not all of the electrons are in the bottom two bands. In
silicon, unlike say quartz or diamond, the band gap between the
top-most full band, the next empty one is not so large. As we
mentioned above it is only about 1.1 eV. So long as the silicon
is not at absolute zero temperature, some electrons near the top
of the full band can acquire enough thermal energy that they can
"hop" the gap, and end up in the upper band, called the
conduction band. This situation is shown in
Figure 6.
In silicon at room temperature, roughly
1010
10
10
electrons per cubic centimeter are thermally excited
across the band-gap at any one time. It should be noted that
the excitation process is a continuous one. Electrons are being
excited across the band, but then they fall back down into empty
spots in the lower band. On average however, the
1010
10
10
in each
cm3
cm
3
of silicon is what you will find at any given instant.
Now 10 billion electrons per cubic centimeter
seems like a lot of electrons, but lets do
a simple calculation. The mobility of electrons in silicon is
about 1000
cm2volt-sec
cm
2
volt-sec
. Remember, mobility times electric field yields the
average velocity of the carriers. Electric field has units of
voltscm
volts
cm
, so with these units we get velocity in
cmsec
cm
sec
as we should.) The charge on an electron is
1.6×10-19
1.6
10
-19
coulombs. Thus from
this equation:
σ=nqμ=10101.6×10-191000=1.6×10-6mhoscm
σ
n
q
μ
10
10
1.6
10
-19
1000
1.6
10
-6
mhos
cm
(1)
If we have a sample of silicon 1 cm long by
1mm1mm
1
mm
1
mm
square, it would have a resistance of
R=LσA=11.6×10-60.12=62.5MΩ
R
L
σ
A
1
1.6
10
-6
0.1
2
62.5
MΩ
(2)
which does not make it much of a "conductor". In fact, if this
were all there was to the silicon story, we could pack up and
move on, because at
any reasonable
temperature, silicon would conduct electricity very poorly.
"This book is available print on demand by going to the course page and selecting "Order a printed copy"."