 |
In the most simplified version of the
free electron gas, the true three-dimensional potential was ignored and
approximated with a constant potential conveniently put at 0 eV (see quantum
mechanics script as well) |
|
 |
The true potential, however, e.g. for a Na
crystal including some energy states, is periodic and looks more like
this: |
|
|
|
 |
Semiconducting properties will not emerge without
some consideration of the periodic potential - we therefore have to solve the
Schrödinger equation for a suitable periodic potential. There are several
(for real potentials always numerical) ways to do this, but as stated before,
it can be shown that all solutions must
have certain general properties. These properties can be used to make
calculations easier and to obtain a general understanding of the the effects of
a periodic potential on the behavior of electron waves. |
|
 |
The starting point is a potential V(r)
determined by the crystal lattice that has the periodicity of the lattice, i.e.
|
|
|
|
|
 |
With T = any translation vector of the lattice
under consideration. |
|
 |
We then will obtain some wavefunctions yk(r) which are solutions of the
Schrödinger equation for V(r).
As before, we use a quantum number
"k" (three numbers, actually) as an index to
distinguish the various solutions. |
 |
The Bloch
theorem in essence formulates a condition that all solutions yk(r), for any periodic potential V(r)
whatsoever have to meet. In one version it ascertains |
|
|
yk(r) |
= |
uk(r) · exp (i
· k · r) |
|
|
|
 |
With k = any allowed wave vector for the
electron that is obtained for a constant
potential, and uk(r) = arbitrary functions
(distinguished by the index k that marks the particular solution
we are after), but always with the periodicity of the
lattice, i.e. |
|
|
|
 |
Any wavefunction meeting this requirement we will
henceforth call a Bloch wave. |
 |
The Bloch theorem is quite remarkable, because, as
said before, it imposes very special conditions on any solution of the Schrödinger equation, no
matter what the form of the periodic potential might be. |
|
 |
We notice that exactly as in the
case of the constant potential ,
the wave vector k has a twofold
role: It is still a wave vector in the plane wave part of the
solution, but also an index to yk(r) and
uk(r) because it contains all the quantum numbers,
which ennumerate the individual solutions. |
 |
Blochs theorem is a proven theorem with perfectly general validity. We
will first give some ideas about the prove of this theorem, and then discuss
what it means for real crystals. As always with hindsight, Blochs theorem can
be proved in many ways; the links give some examples. Here we only look
ageneral outlines of how to prove the theorem: |
|
 |
It follows rather directly from applying group theory to crystals. In this case one looks
at symmetry properties that are invariant under translation. |
|
 |
It can easily be proved by working with operator algebra in the context of formal quantum
theory mathematics. |
|
 |
It can be directly proved in
simple
ways - but then only for special cases or with not quite kosher
"tricks". |
|
 |
It can be proved (and used for further calculations), by
expanding V(r) and y(r) into a Fourier
series and then rewriting the Schrödinger equation. This is a
particularly useful way because it can also be used for obtaining specific
results for the periodic potential. This proof is demonstrated in detail in the
link, or in the book of
Ibach and
Lüth. |
 |
Blochs theorem can also be
rewritten in a somewhat different form, giving us a second version: |
|
|
yk(r +
T) |
= |
yk(r) ·
exp(ikT) |
|
|
|
 |
This means that any
function yk(r) that is a
solution to the Schrödinger equation of the problem, differs only by a
phase factor exp(ikT) between equivalent
positions in the lattice. |
|
 |
This implies immediately that the probability of finding an
electron is the same at any equivalent position in the
lattice since, exactly as we
expected, because |
|
|
[yk(r +
T)]2 |
= |
[yk(r)]2 |
· |
[exp(ikT)]2 |
= |
[yk(r)]2 |
|
|
|
 |
Since [exp(ikT)]2 = 1 for all
k and T. |
 |
That this second
version of Blochs theorem is equivalent to the first one may be seen
as follows. |
|
 |
If we write the wave function in the first form yk(r) = uk(r) ·
exp(ikr) and consider its value at an equivalent lattice position
r + T we obtain |
|
|
yk(r +
T) |
= |
uk(r + T) |
· exp [ik · (r + T)]
|
= |
uk(r) · exp
(ikr) |
· exp (ikT) |
= |
yk(r) ·
exp(ikT) |
|
|
|
|
|
|
|
|
|
|
|
= uk(r ) |
|
|
= yk(r) |
|
|
q.e.d |
|
|
 |
Blochs theorem has many more forms and does not
only apply for electrons in periodic potentials, but for all kinds of waves,
e.g. phonons. However, we will now consider the theorem to be proven and only
discuss some of its implications. |
|
|
|
Implications of the Bloch Theorem |
|
|
|
One way of looking at the Bloch theorem is to
interprete the periodic function uk(r) as a kind of
correction factor that is used to generate
solutions for periodic potentials from the simple solutions for constant
potentials. |
|
 |
We then have good
reasons to assume that uk(r) for k
vectors not close to a Brillouin zone will
only be a minor correction, i.e. uk(r) should be close
to 1. |
 |
But
in any case, the quantity k, while still being the wave vector of
the plane wave that is part of the wave function (and which may be seen as the
"backbone" of the Bloch functions), has lost its simple meaning: It
can no longer be taken as a direct
representation of the momentum p of the wave via p =
k, or of
its wavelength l = 2p/k, since: |
|
 |
The momentum of the electron moving in a periodic potential is
no longer constant (as we will see shortly); for the standing waves resulting
from (multiple) reflections at the Brillouin zones it is actually zero (because the velocity is zero), while
k is not. |
|
 |
There is no unique wavelength to a plane wave modulated with
some arbitrary (if periodic) function. Its Fourier decomposition can have any
spectra of wavelengths, so which one is the one to associate with
k? |
 |
To make this clear, sometimes the vector
k for Bloch waves is called the "quasi wave vector". |
 |
Instead of associating
k with the momentum of the
electron, we may identify the quantity
k,
which is obviously still a constant, with
the so-called crystal momentum
P, something like
the combined momentum of crystal and
electron. |
|
 |
Whatever its name,
k is a
constant of motion related to the particular wave yk(r) with the index
k. Only if V = 0, i.e. there is no periodic
potential, is the electron momentum equal to the crystal momentum; i.e. the
part of the crystal is zero. |
|
 |
The crystal momentum P, while not a
"true" momentum which should be expressible as the product of a
distinct mass and a velocity, still has many properties of momentums, in
particular it is
conserved during all
kinds of processes. |
|
 |
This is a major feature for the
understanding of semiconductors, as we will see soon enough! |
 |
One more difference to the constant potential case
is crucial: If we know the wavefunction for one particular k-value, we also know
the wavefunctions for infinitely may other k-values, too. |
|
 |
This follows from yet another formulation of Bloch's theorem:
|
|
 |
If yk(r) =
uk(r) · exp(ikr) is a particular Bloch wave
solving the Schrödinger equation of the problem, then the following
function is also a solution. |
|
|
yk + g(r)
|
= |
uk + g(r) · exp i[k +
g]r |
|
|
|
 |
With g = arbitrary reciprocal lattice vector as
always.
|
 |
This is rather easy to show and you should
attempt it yourself. It has a far reaching consequence: |
|
 |
If yk(r)
is a solution of the Schrödinger equation for the system, it will always
be associated with a specific energy E(k) which is a constant of
the system for the particular sets of quantum numbers embodied by
k. Since yk(r) is identical to yk + g(r), its specific energy
E(k + g) must be identical to
E(k), or |
|
|
|
 |
This is a major insight. However, there is also a
difficulty: |
|
 |
The equation does not mean
that two electrons with wave vectors k and k + g
have the same energy (see below), but that
any reciprocal lattice point can serve as
the origin of the E(k) function. |
|
 |
Lets visualize this for the case of an infinitesimaly small
periodic potential - we have the periodicity, but not a real potential. The
E(k) function than is practically the same as in the case
of free electrons, but starting at every
point in reciprocal space: |
|
|
|
|
 |
Indeed, we do have E(k +
g) = E(k), but for
dispersion curves that have a different origin. |
|
 |
We have even more, we now have also many energy values for one given k, and in particular all possible energy values are contained within the first
Brilluoin zone (between -1/2g1 and
+1/2g1 in the picture). |
 |
It thus is sufficient to consider
only the first Brillouin zone in graphical representations - it contains all
the information available about the system.This is called a
reduced representation of the band
diagram, which may look like this: |
|
|
|
|
 |
The branches outside the 1. BZ have been "folded
back" into the 1. BZ, i.e. translated by the approbriate reciprocal
lattice vector g. To make band diagrams like this one as
comprehensive as possible, the symmetric branch on the left side is omitted;
instead the band diagram in a different direction in reciprocal space is
shown. |
 |
Again, this looks like a specific
electron could now have many energies all at once - this is, of course,
not the case. |
|
 |
Different energies, formerly distinguished by different
k - vectors, are still different energies, but now the branches
in the 1st Brillouin zone coming from larger k - vectors
belong to different bands. Every energy branch in principle should carry an index denoting
the band; this is, however, often omitted. |
 |
The identical construction, but now for the energy
functions of a periodic potential as
given before, now looks like
this |
|
|
|
|
 |
We now have band gaps - regions
with unattainable energies - in all directions of the reciprocal lattice. |
|
 |
A numerical example for the Kroning Penney Model is shown in
this link. |
 |
What does this mean for a particular electron, say
one on the lowest branch of the blue diagram with the wave vector
k1? It has a definite energy E
associated with it. |
|
 |
But it also could have larger energies: all the values
obtained for the same k but in higher branches of the band
diagram. |
|
 |
For a transition to the next higher branch the energy DE1 is needed. It has to be supplied from the outside world. |
|
 |
After the transition the electron has now a
higher energy, but the wave vector is the same. But
wait, in the reduced band diagram, we simply omitted a reciprocal
wave vector, so its wave vector is actually k1 +
g. If we index the situation after the transition with
"2", before with "1", we have the following
equations. |
|
|
E2 |
= |
E1 + DE |
|
|
|
k2 |
= |
k1 + g |
|
|
|
|k1| |
¹ |
|k2| |
|
|
 |
This is simply
Braggs law, but now for inelastic scattering, where the magnitude of
k may change - but only by a specified amount tied to a
reciprocal lattice vector. |
 |
Since we interpreted
k as
crystal momentum, we may consider
Braggs law to be the expression for the conservation
of momentum in crystals. |
|
 |
The reduced band diagram representations thus allow a very
simple graphical representation of allowed transitions of electrons from one
state represented by to another state (E2 ,
k2): the states must be on a vertical line through the
diagram, i.e. straight up or down. |
|
 |
An alternative way of describing the states in the spirit of
the reduced diagram is to use the same wave vector k1
for all states and a band index for the energy. The transition then goes from
(En , k) to
(Em, k) with n,
m = number of the energy band involved. |
 |
The possibility of working in a reduced band
diagram, however, does not mean that wave vectors larger than all possible
vectors contained in the 1. BZ are not meaningful or do not exist: |
|
 |
Consider an electron "shot" into the crystal with a
high energy and thus a large k - e.g. in an electron microscope.
If you reduce its wave vector by subtracting a suitable large g
vector without regard to its energy and band number, you may also reduce its
energy - you move it. e.g., from a band with a high band number to a lower one.
While this may happen physically, it will only happen via many transitions from
one band to the next lower one - and this takes time! |
|
 |
Most of the time in normal applications the electron will
contain its energy and its original wave vector. And it is this wave vector you
must take for considering diffraction effects! An Ewald (or Brillouin)
construction for diffraction will give totally wrong results for reduced wave
vectors - think about it! |
 |
If you feel slightly (or muchly) confused at this
point, that is as it should be. Blochs theorem, while relatively
straightforward mathematically, is not easy to grasp in its implications to
real electrons. The representation of the energy - wave vector relationship
(the dispersion curves) in extended or reduced schemata, the somewhat unclear
role of the wave vector itself, the relation to diffraction via Braggs law, the
connection to electrons introduced from the outside, e.g. by an electron
microscope (think about it for minute!), and so on, are difficult concepts not
easily understood at the "gut level". |
|
 |
While it never hurts to think about these questions, it is
sufficient for our purpose to just accept the reduced band structure scheme and
its implications as something useful in dealing with semiconductors - never
mind the small print associated with it. |
|
 |
However, if you want to dig deeper: These problem are to some
extent rooted in the formal quantum mechanics behind Blochs theorem. It has to
do with Eigenvectors and Eigenvalues of Operators; a
glimpse of these issues
can be found in an advanced module. |
© H. Föll (Semiconductor - Script)