 |
Electron waves with
wave vectors on or near a BZ are diffracted; all others are not. |
|
 |
This means simply that electrons with wave
vectors near or at a BZ - let's call them kBZ
electrons - feel the periodic potential of the crystal while the others
do not (in a first approximation). |
|
|
In other words, kBZ
electrons interact with the crystal, and
this must express itself in their energies. |
|
Contrariwise, we expect that
"normal" electrons not feeling any diffraction, still pretty much
obey the relation for the total energy
E |
|
|
|
|
|
| E |
= |
Ekin |
= |
p2
2m |
= |
( k)2
2m |
|
|
|
|
|
|
|
For kBZ electrons,
however, we must expect major modifications. |
|
What we will get in the most general
terms is a splitting of the energy
value if a given k ends exactly at the Brillouin zone,
i.e. for a kBZ electron. Instead of
E(k) = (k)2/2me
we obtain. |
|
|
|
|
|
| E(kBZ) |
= |
(k)2
2m |
± |
DE |
|
|
|
|
|
|
|
In words: Electrons at the BZ can have
two energies for the same wave vector and
thus state. One value is somewhat lower than the free electron gas value, the
other one is somewhat higher. |
|
Energies between these values are
unobtainable for any electrons - there is
now an energy gap in the E =
E(k) relation for all k vectors ending on a
Brillouin zone. |
|
The time-honored way to visualize this energy gap
is to look at a one-dimension crystal - i.e. a chain
of atoms. |
|
|
In this case the electron wave meeting the Bragg condition
will be reflected back on itself. We have |
|
|
|
|
|
|
|
|
|
|
|
in this case - and the effect will be a
standing wave. |
|
|
The solutions of the Schrödinger equation will be
described by the possible superpositions of the two waves and there are
two possibilities to do that: |
|
|
|
|
|
| y+ |
= |
æ
ç
è |
1
L |
ö
÷
ø |
3/2 |
· |
æ
è |
exp (ikr) + exp
–(ikr) |
ö
ø |
| y–
|
= |
æ
ç
è |
1
L |
ö
÷
ø |
3/2 |
· |
æ
è |
exp (ikr) – exp
–(ikr) |
ö
ø |
|
|
|
|
|
|
Both solutions are no longer propagating plane
waves with y · y*
= const. throughout the crystal, but standing
waves. |
|
|
y · y*, i.e. the probability
density of finding the electron, is no longer the same everywhere in
the crystal, but follows a relation given by |
|
|
|
|
|
| y · y* |
= |
const. |
· cos2 |
æ
è |
px
a |
ö
ø |
|
|
|
|
|
|
|
With the maxima being at
the coordinates of the atoms for the y– solution and between the atoms for the y+
solution. |
|
|
In the first case the potential energy of the electrons is
lowered, in the second case it is raised - there is an energy gap! |
|
Note that we now have a potential energy, but only
because we now implicitely assumed that the potential is no longer
constant. |
|
While this is a relatively painless
way to envision the occurrence of an energy gap, the three dimensional case
needs a few more considerations. |
|
|
Waves with k
» kBZ, while diffracted,
do not have to run back in themselves - after some more reflections, however,
they will. |
|
|
This leads to a splitting of the energy for
all positions on the Brillouin zones;
the amount of splitting, however, may
differ. |
|
|
A general relation yields for the energies of the
kBZ electron waves |
|
|
|
|
|
| E(kBZ) |
= |
(k)2
2m |
± |
|U(g)| |
|
|
|
|
|
|
|
With U(g) =
Fourier component of the
periodic potential for the reciprocal lattice vector g
considered. |
|
|
|
|
Bearing this in
mind, we now can construct the E(k) diagram in a principal
way. |
|
|
|
|
|
|
|
|
|
|
In different directions we still would have an energy gap, but
at different positions and with a different width. |
|
That is about as far as the free electron gas
model with diffraction added (and therefore
by necessity some unspecified periodic potential) will get us. |
|
|
For more insights we will actually have to solve the
Schrödinger equation for some kind of periodic potential. This is
difficult, even for very simple (unrealistic) periodic potentials. cf. the
link. |
|
|
For this we first need a halfway realistic potential - e.g.
for a Si or a GaAs crystal - which we then use in the
Schrödinger equation. The solutions will depend on the precise kind of
potential and, as we must expect, they will not be easy to find (or even to
express in closed form). |
|
|
However, since the potential is periodic,
which means it doesn't matter if we look at it at r or at
r + R with R = any translation vector of the lattice - it always looks
the same - we may confidently expect that the solutions mirror somehow this
property. After all, it should not matter much either, at which
crystallographically equivalent crystal positions we look at the
electrons. |
|
This is exactly what the celebrated Bloch
theorem asserts: No matter what kind of periodic potential is
plucked into the Schrödinger equation, the solutions must have certain
properties which can be specified in a very general way. |
|
|
We will deal with this in the next subchapter. |
|
|
|
© H. Föll (Semiconductor - Script)
