 | Intuitively,
we expect that "normal" electrons, not feeling any diffraction, pretty much obey the relation for the total energy E as before:
|
| |
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
edge, i.e. for a kBZ electron. Instead of E(k)
= ( k)2/2me we obtain. |
| |
E(k BZ) | = | ( kBZ)2 2m |
± |
DE |
| |
|  |
In words: Electrons at the BZ edge 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 electron – there is now an energy gap in
the E = E(k) relation for all k vectors ending on
a Brillouin zone wall. |
 | The time-honored way to
visualize this energy gap is to look at a one-dimensional crystal – i.e. a chain of atoms , periodically spaced with the distance a. |
|  |
Since in this case, we have for the electron wave meeting the Bragg condition
... |
| |
|
|  | ...
the electron wave will be reflected back on itself. Therefore, 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
2L | ö ÷ ø |
1/2 | · | æ è | exp
(ikBZx) + exp (–ik
BZx) | ö ø |
y – | =
| æ ç è |
1 2L |
ö ÷ ø | 1/2 | · | æ
è | exp (ikBZx) –
exp (–ikBZx) |
ö ø | | |
|
 |
For the first Brillouin zone, we have g1 = 2
p/a = k – k' = 2kBZ, and so k
BZ = p/a. Since this can be easily generalized
for higher Brillouin zones, the same consequences will occur also there. To understand the
basics, however, it is sufficient to consider just the edge of the first Brillouin zone. |
 |
Both solutions are no longer propagating plane waves with y · y* = const. throughout the crystal
but standing waves ... |
|
| |y +|2 | = |
2 L | · cos2
| æ ç è |
px a |
ö ÷ ø |
|y
–|2 | =
| 2 L | ·
sin2 | æ ç
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 now a potential energy is involved, but only because
we now implicitly 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 directly 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 zone edges; the amount of splitting, however, may
differ . |
|  | A
general relation yields for the energies of the kBZ electron waves |
| | E(k
BZ) | = | ( kBZ)2 2m | ± | |
U(g)| | | |
|  |
With U(g) = Fourier
component of the periodic potential for the reciprocal lattice vector g
relevant for the kBZ vector considered. |
|
| |
| Representation of Energy Gaps and Band Structures |
| |
 |
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 on the energy axis and with a different
width. Nevertheless, this is already the first step to understand the electronic
band structure of crystalline solids. |
|  | 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 (Semiconductors - Script)