The Solid State
Band Theory of Solids
“You do not really understand something unless you can explain it to your
grandmother.”—A. Einstein
Typo in Syllabus—News Flash—Typo in Syllabus
428 points gets you a C
429 points gets you a B
I already generously rounded the cutoff percentages down!
(429 is not quite 79.5%)
I have added two typo points to your grades.
The grades spreadsheet shows the grade you will get if you do
nothing for the rest of the semester.
Please make sure all earned bonus points are recorded!
10.6 Band Theory of Solids
What happens in crystalline solids when we bring atoms so
close together that their valence electrons constitute a single
system of electrons?
Band structure of diamond. http://home.att.net/~mopacmanual/node372.html
The energy levels of the
overlapping electron shells
are all slightly altered.
The energy differences are
very small, but enough so
that a large number of
electrons can be in close
proximity and still satisfy the
Pauli exclusion principle.
The result is the formation
of energy bands,
consisting of many states
close together but slightly
split in energy.
The energy levels are so close together that for all practical
purposes we can consider bands as a continuum of states,
rather than discrete energy levels as we have in isolated atoms
(and in the core electrons of atoms of metals).
A detailed analysis of energy bands shows that there are
as many separate energy levels in each band as there are
atoms in a crystal.*
Suppose there are N atoms in a crystal. Two electrons
can occupy each energy level (spin), so there are 2N
possible quantum states in each band.
Let’s consider sodium as an example. Sodium has a single
outer 3s electron.
*Kind of. I need to explain.
When you bring two sodium
atoms together, the 3s
energy level splits into two
separate energy levels.
Things to note: 4 quantum
states but only 2 electrons.
You could minimize electron energy by putting both 3s electrons
in the lower energy level, one spin up and the other spin down.
There is an internuclear separation which minimizes electron
energy. If you bring the nuclei closer together, energy
increases.
When you bring five sodium
atoms together, the 3s
energy level splits into five
separate energy levels.
The three new energy
levels fall in between the
two for 2 sodiums.
There are now 5 electrons occupying these energy levels.
I’ve suggested one possible minimum-energy configuration.
Notice how the sodium-sodium internuclear distance must
increase slightly.
When you bring N (some
big number) sodium atoms
together, the 3s energy
level splits into N separate
energy levels.
The result is an energy
band, containing N very
closely-spaced energy
levels.
There are now N electrons occupying this 3s band. They go
into the lowest energy levels they can find.
The shaded area represents available states, not filled states.
At the selected separation, these are the available states.
Now let’s take a closer look at
the energy levels in solid
sodium.
Remember, the 3s is the
outermost occupied level.
When sodium atoms are
brought within about 1 nm of
each other, the 3s levels in the
individual atoms overlap
enough to begin the formation
of the 3s band.
The 3s band broadens as the
separation further decreases.
3s band
begins to
form
Because only half the states in
the 3s band are occupied, the
electron energy decreases as
the sodium-sodium separation
decreases below 1 nm.
At about 0.36 nm, two things
happen: the 3s energy levels
start to go up (remember
particle in box?) and the 2p
levels start to form a band.
Further decrease in interatomic
separation results in a net
increase of energy.
3s electron
energy is
minimized
What about the 3p and 4s
bands shown in the figure?
Don’t worry about them—there
are no electrons available to
occupy them!
Keep in mind, the bands do
exist, whether or not any
electrons are in them.
What about the 1s and 2s
energy levels, which are not
shown in the figure?
The sodium atoms do not get
close enough for them to form
bands—they remain as atomic
states.
Figure 10-20 (the one on the last three slides) shows energy
levels as a function of interatomic separation.
Energy levels for an actual crystal structure also vary with
different directions in space.
http://cmt.dur.ac.uk/sjc/thesis/thesis/node39.html, band structure of silicon
As an aid to visualization, we often represent energy bands like
this (using sodium as an example):
This is highly schematic. Real
bands aren't boxes or lines.
Sodium has a single 3s
electron, so the 3s energy
band contains twice as many
states as there are electrons.
The band is half full.
3s
2p
2s
1s
At T=0 the band is filled exactly halfway up, and the Fermi
level, εF, is right in the middle of the band.
Sodium is a metal because an applied field can easily give
energy to and accelerate an electron.
εF
Magnesium has two 3s electrons.
You expect the 3s band to be full, 3p
the 3p band to be empty, with a
forbidden gap in between.
Magnesium should be an insulator.
(Why?)
3s
But magnesium is a metal
(actually, a “semimetal”).
The 3p and 3s bands overlap.
3p
There are many empty states
nearby into which electrons can be
3s
accelerated.
Materials which have bands either
completely full or completely
empty are insulators (unless band
overlap occurs, as was the case for
magnesium).
3p
3s
In a carbon atom, the 2p shell contains 2 electrons. There
are 6 available states, so one would expect the 2p band to be
1/3 full* and carbon to be a conductor.
But carbon is an **insulator. Why?
*2N 2p electrons in a crystal with N atoms. 6N 2p states, when you include spin.
**If the diamond in your diamond ring conducts electricity, it’s time to take it back!
Figure 10.23 shows energy
bands in carbon (and
silicon) as a function of
interatomic separation.
At large separation, there is
a filled 2s band and a 1/3
filled 2p band.
But electron energy can be lowered if the carbon-carbon
separation is reduced.
There is a range of carbon-carbon separations for which the 2s
and 2p bands overlap and form a hybrid band containing 8N
states (Beiser calls them “levels”).
But the minimum total
electron energy occurs at
this carbon carbon
separation.
At this separation there is a
valence band containing 4N
quantum states and
occupied by 4N electrons.
The filled band is separated by a large gap from the empty
conduction band. The gap is 6 eV—remember, kT is about
0.025 eV at room temperature. The gap is too large for
ordinary electric fields to move an electron into the conduction
band. Carbon is an insulator.
Silicon has a similar band
structure. The forbidden
gap is about 1 eV.
The probability of a single
electron being excited
across the gap is small,
proportional to
exp(-Egap/kT).
However, there are enough 3s+3p electrons in silicon that some
of them might make it into the conduction band. We need to
consider such a special case.
On page 355 Beiser says there are as many levels in a band (N) as there are atoms
in a crystal. In figure 10.23 the caption implies there are 2N levels in a 2s band.
This is inconsistent. Let’s call a “level” an energy level and a “state” a quantum
state, including spin. Then figure 10.23 should talk about 2N states in a 2s band.
A semiconductor is a material which has a filled (at T=0)
valence band separated by a small gap from an empty (at
T=0) conduction band.
A semiconductor at
T= 0 K.
conduction
band
valence
band
Because the gap is small, at room temperature there will be a
few electrons in the conduction band. These electrons can be
accelerated by an applied electric field.
A semiconductor at room
temperature.
CB
Although e(-Egap/kT) is small,
Ne(-Egap/kT) can be significant.
The smaller the gap, the more
charge carriers in the
conduction band.
F
VB
Thus, at very low temperatures, silicon is an insulator, but at
room temperature, it is a weak conductor (intermediate
between conductor and insulator, hence semiconductor).
A rough rule of thumb: a band gap of less than 3 eV gives rise
to a semiconductor.
It is possible to “engineer” properties of semiconductors to make
them more suitable for use in devices.
We do this by doping them with impurities.
Here’s an example: arsenic impurities in silicon.
As has an outer electron configuration of 4s2 4p3. After it
shares 4 of its outer electrons with neighboring silicons, the
remaining electron is very loosely bound.
Homework problems 10.21 and 10.22
(assigned although listed under section 10.8)
show that kT at room temperature is more
than enough energy to ionize the fifth outer
electron.
+
The electrons from the As+ ions are then free to move
throughout the crystal.
-
The arsenic impurity creates a donor impurity level. Because it
takes only a few meV to ionize the As and place the resulting
electron in the conduction band, the donor impurity level sits just
below the conduction band.
A few donor impurities can
produce many electrons in the
conduction band and give rise
to significant conductivity.
The Fermi level lies somewhere
between the top of the valence
band and the highest-energy
electron in the conduction
band.
CB
F
VB
These semiconductors are n-type, because conduction is by
n egative electrons.
Here’s another example: gallium impurities in silicon.
Ga has an outer electron configuration of 4s2 4p1. It “wants” to
borrow 3 electrons from its 4 silicon nearest neighbors.
After a Ga “borrows” 3 electrons, it is
easy for the 4th silicon to “convince”
the Ga to “borrow” one more
electron, creating a Ga- ion.
The gallium impurity creates an
acceptor impurity level.
-
+
The lattice depicted here is
highly schematic, not
realistic, and should be used
as a visual aid only!
Because the acceptor ionization energy is very small, the
acceptor levels sit just above the valence band (a small fraction
of an eV).
Electrons from the valence band
can easily get to the acceptor
bound states. That leaves holes
in the valence band.
The holes represent states into
which other electrons in the
valence band can move. Thus,
electrons can easily move around
in the presence of an applied
field.
CB
F
VB
This is called a p-type
semiconductor because
“p”ositive holes are the charge
carriers.
Alternatively, we can look at the holes, say they move around,
and say that conduction is due to holes.
Again, it turns out that there don't have to be very many
acceptor atoms around to result in a significant number of
holes in the valence band.
What's all this fuss about holes. It seems like a hole is really
just an electron missing from the valence band and sitting in an
acceptor state.
Why don't we just talk about conduction by these electrons
moving from state to state, instead of worrying about holes?
Answer: holes really are more than just missing electrons. For
one thing, electrons in this case are "stuck" on acceptor atoms,
but the holes are free to move about in the valence band.
For another thing, we can dope semiconductors so that there
are excesses of holes and electrons. A "hole" really is more
than just a missing electron which is somewhere else.
Optical properties of solids are closely related to band
structures.
Visible photons have energies ranging from about 1 eV to
about 3 eV.
Metals can absorb visible photons
because there are many empty states
for electrons to move to.
In diamond, the valence band is full and the
conduction band is empty.
A 3 eV photon cannot excite an electron across
the 6 eV band gap. Diamond cannot absorb
visible photons. Diamonds are transparent.
“Shouldn’t all insulators be transparent?”
Yes. They are opaque because they contain impurities and
irregularities which scatter visible light.
Insulators are transparent to high-energy UV (ultraviolet)
photons.
Now, skipping ahead briefly…
Section 10.8 is a lengthy section which attempts to
give a better justification of energy bands.
The section is not required, but does contain a few key
points:
The periodicity of a crystal lattice leads naturally
(through the solution of Schrödinger’s equation)
to allowed and forbidden energy bands for
electrons.
Electrons cannot exist in forbidden energy states
because there are no solutions to Schrödinger’s
equation there.
Electrons do not (in general) scatter from a
perfect lattice, but they do interact with the
lattice.
Sometimes the lattice “holds” the “free” electrons
in place. These electrons act like they have an
unusually high mass (meffective > melectron).
Sometimes the electrons act like they have an
unusually small mass (meffective < melectron).
Sometimes the electrons even have a negative
effective mass.
Huh?
Sure! Push on an electron in one direction. If
the push gives it enough energy to Bragg scatter
off the lattice (the exception to the “no
scattering” rule), the electron goes in the
opposite direction. Negative mass!
Ceramics majors would benefit from studying section
10.8. The only material you are required to know (for
exam or quiz purposes) is that given above.
One more thing… the band structure I showed earlier:
Notice the gap that extends in all directions through
the crystal.
Let’s move on to “lecture 34” and see how far we get…