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In the most simplified version of the
free electron gas, the true three-dimensional potential was ignored and
approximated with a constant potential (see the
quantum mechanics script
as well) conveniently put at 0 eV. |
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The true potential, however, e.g. for a Na
crystal, is periodic and looks more like this (including some electronic
states): |
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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. However, this
is much easier said than done: There are several ways to do this (for real
potentials always numerically), but they are all mathematically rather involved
– and, thus, beyond the scope of this lecture. |
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Luckily, as stated before, it can be shown that all solutions must have certain general properties.
These properties can be used to make calculations easier – as well as to
obtain a general understanding of the effects of a periodic potential on
the behavior of electron waves. |
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The starting point is a potential V(r)
determined by the crystal lattice that has the periodicity of the lattice, i.e.
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With T = any translation vector of the lattice
under consideration. |
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We then will obtain some wavefunctions y(r) which are solutions of the
Schrödinger equation for V(r).
In addition, these wavefunctions have to fulfill the boundary conditions,
since we are still dealing with a kind of "particle in a box"
problem – the electrons are confined inside the crystal. |
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The Bloch
theorem in essence formulates a condition that all solutions y(r), for any periodic potential V(r)
whatsoever, have to meet. In one version it ascertains |
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| y(r) |
= |
u(r) · exp (i
· k · r) |
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With k = any allowed wave vector for the
electron that is obtained for a constant
potential, and u(r) = some functions
with the periodicity of the lattice, i.e.
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Any wavefunction meeting this requirement we will
henceforth call a Bloch wave. |
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As before, we choose periodic
boundary conditions; this ensures that no restiction on the translation
vectors T needs to be considered. |
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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. |
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We notice that, in contrast to
the case of the constant potential, so far, k is just a wave
vector in the plane wave part of the solution. Due to the periodic potential,
however, its role as an index to the wave function is
not the same as before – as we will
shortly see.
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Bloch's theorem is a proven theorem with perfectly general validity. We
will first give some ideas about the proof of this theorem and then discuss
what it means for real crystals. As always with hindsight, Bloch's theorem can
be proved in many ways; the links give some examples. Here we only look at
general outlines of how to prove the theorem: |
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It follows rather directly from applying group theory to crystals. In this case one looks
at symmetry properties that are invariant under translation. |
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It can easily be proved by working with operator algebra in the context of formal quantum
theory mathematics (see the
quantum mechanics script
again). |
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It can be directly proved in
simple ways – but then only for special
cases or with not quite kosher "tricks". |
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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. |
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Bloch's theorem can also be
rewritten in a somewhat different form, giving us a second version: |
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| y(r +
T) |
= |
y(r) ·
exp(ikT) |
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This means that any
function y(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. |
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This implies immediately that the probability of finding an
electron is the same at any equivalent position in the
lattice, exactly as we
expected, because |
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| |y(r +
T)|2 |
= |
|y(r)|2 |
· |
|exp(ikT)|2 |
= |
|y(r)|2 |
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This is so because |exp(ikT)|2 = 1
for all k and T. |
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That this second
version of Bloch's theorem is equivalent to the first one may be
seen as follows: |
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If we write the wave function in the first form y(r) = u(r) ·
exp(ikr) and consider its value at an equivalent lattice position
r + T we obtain |
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| y(r +
T) |
= |
u(r + T) |
· exp [ik · (r + T)]
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= |
u(r) · exp
(ikr) |
· exp (ikT) |
= |
y(r) ·
exp(ikT) |
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= u(r ) |
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= y(r) |
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q.e.d. |
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Bloch's theorem has many more forms and does not
only apply to electrons in periodic potentials, but for all kinds of waves in
periodic structures, e.g. phonons. However, we will now consider the theorem
to be proven and only discuss some of its implications. |
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Most importantly, we will now clarify what index to apply to
the wave function y(r) and, thus, also
to its lattice-periodic part u(r). |
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Implications of the Bloch Theorem |
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One difference to the constant potential case
is most crucial: If we know the wavefunction for one particular k value, we also know
the wavefunctions for infinitely may other k values, too. |
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This follows from the fact that in the "second version" of the Bloch
theorem, y(r + T) = y(r) · exp(ikT), the vector
k is not unique, since for any reciprocal lattice vector
g it holds that exp(igT) = 1. Hence, when
k is replaced by k' = k + g, the
Bloch theorem is also fulfilled: |
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| y(r + T) |
= |
y(r) ·
exp(ik'T) |
= |
y(r) ·
exp[i(k + g)T] |
= |
y(r) ·
exp(ikT) · exp(igT) |
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= |
y(r) · exp(ikT) |
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Obviously it is not clear which reciprocal space vector,
k or k', is to be associated with this Bloch wave;
this is why the index k was omitted so far. (That this
non-uniqueness also holds for u(r) can be seen from its Fourier
series representation; cf. the relevant proof of the Bloch theorem.) |
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The deeper reason behind this property is that, similar to
E being an eigenvalue of the Hamilton operator
H in the equation
Hy = Ey, the exponential expressions exp(ikT)
are the eigenvalues of the translation operator
T in the equation
Ty = exp(ikT)y (cf. the relevant
proof
of the Bloch theorem), and these eigenvalues do not change when k
is replaced by k' = k + g. |
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Therefore, in principle not a
single vector k is to be attributed as an index to a specific
wavefunction, but all k' =
k + g. |
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All these points in k-space are equivalent,
because for any reciprocal lattice vector g it holds that
yk + g(r) = yk(r). |
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On the other hand, this permits to restrict the
k vector to the first Brillouin zone, because any k'
not in the first Brillouin zone can always be written as k' = k
+ g, with k now being in the first Brillouin zone.
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Keeping this in mind, we can now simply write
yk(r); further
implications are discussed below. |
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In general, a Bloch wave yk(r) = uk(r) ·
exp(ikr) can be understood as a lattice-peridocally modulated plane
wave. |
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One way of looking at this "first version" of the Bloch theorem
is to interprete the lattice-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. |
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We then have good
reasons to assume that uk(r) for k
vectors not close to a Brillouin zone edge
will only be a minor correction, i.e. uk(r) should be
close to 1, then. |
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But in any case, the quantity k, while still being the wave vector of
the plane wave that is part of the Bloch wave function (and which may be seen
as the "backbone" of the Bloch functions), has lost its simple
meaning: Besides not being unique, there are explicit reasons why it can no
longer be taken as a direct representation
of the momentum p of the electron via p =
 k, or of
its wavelength l = 2p/k: |
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The momentum of the electron moving in a periodic potential,
having the wave function
yk(r) = uk(r)
· exp(ikr), can be calculated from –i yk(r), thus it is not only
related to k but also to uk(r),
which represents the influence of the lattice. As an example, for the standing waves
resulting from (multiple) reflections at the Brillouin zone edges the momentum
of the electron is actually zero (because
the velocity is zero), while k is not. |
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There is no unique wavelength to a plane wave modulated with
some arbitrary periodic function. Its Fourier decomposition can have any
spectrum of wavelengths, so which is the one to be relevant for being
associated with k? |
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To make this clear, sometimes the vector
k for Bloch waves is called the "quasi wave vector". |
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Nevertheless, k is
a quantum number related to the translational symmetry
of the lattice. Thus, instead of associating it with the momentum of
the electron which is related to the translational symmetry of free
space, we may identify the quantity k with the so-called
crystal momentum
P, being relevant
for the movement of the electron in the periodic lattice.
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The crystal momentum P is something like the
combined momentum of crystal and electron.
While not being a "true" momentum (which should be expressible as the
product of a distinct mass and a velocity), it still has many properties of
momenta, in particular it is
conserved during all
kinds of processes (as we will see later on). |
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This is a major feature for the
understanding of semiconductors, as we will see soon enough! |
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Only if V = 0, i.e. if there is no periodic
potential, then the crystal momentum is equal to the bare electron momentum;
i.e. then the part of the crystal is zero. |
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Reduced- and Repeated-Zone Schemes |
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We now consider the far-reaching consequences of
the fact that yk + g(r) = yk(r) for the energy value(s)
associated with that wavefunction; with g = arbitrary reciprocal
lattice vector. |
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Writing this as y(k + g,
r) = y(k, r) and taking
k as a continuous variable, this means that
y(k, r) is periodic in
k-space. |
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As a solution of the Schrödinger equation for the system,
associated with y(k, r)
there is a specific energy E(k) which necessarily is also periodic: |
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It thus is sufficient to consider
only the first Brillouin zone in graphical representations of the
E(k) function – it contains all the information
available about the system. |
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This is called a
reduced representation of the band
diagram (or reduced-zone scheme), which may look like this: |
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The branches outside the 1. BZ have been shifted
("folded back") into the 1. BZ, i.e. translated by the
appropriate reciprocal lattice vector g. |
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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. |
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Altogether, this now looks like
a specific electron (i.e., with a specific k) could have many
energies all at once – which is, of course,
not the case. |
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Different energies, formerly distinguished by different
k-vectors, are still different energies, but in the first Brillouin
zone they are distinguished by considering them to form different
bands. |
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Every energy branch coming from larger k-vectors
carries an index n denoting the band;
this index specifies the original Brillouin zone that this branch originated from.
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Formally, in the first Brillouin zone we have the function
En(k) associated with the Bloch wave
yn,k(r).
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The identical construction, but now for the energy
functions of a periodic potential as
given before, now looks like
this: |
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We now have band gaps –
regions with unattainable energies – in all directions of the reciprocal
lattice. |
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A numerical example for the Kroning–Penney Model is
shown in this link. |
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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. |
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But it also could have larger energies – all the values
obtained for the same k but in higher branches of the band
diagram. |
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For a transition to the next higher branch the energy DE1 is needed. It has to be supplied from the outside world. |
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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: |
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| E2 |
= |
E1 + DE |
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| k2 |
= |
k1 + g |
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| |k1| |
¹ |
|k2| |
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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. |
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Since we interpreted
k as
crystal momentum, we may consider
Braggs law to be the expression for the conservation
of momentum in crystals. |
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The reduced band diagram representation thus provides a
very simple graphical representation of allowed transitions of electrons from
one state, represented by (E1,
k1), to another state (E2,
k2): the states must be on a vertical line through the
diagram, i.e. straight up or down. |
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An alternative way of describing the states in the spirit of
the reduced diagram is to use the same wave vector k 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. |
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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: |
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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! |
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Most of the time in normal applications the electron will keep
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!
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If you feel slightly (or muchly) confused at this
point, that is as it should be. Bloch's 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 Bragg's 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". |
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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. |
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However, if you want to dig deeper: All these effects that are
difficult to grasp are to some extent rooted in the formal quantum mechanics
behind Bloch's theorem. It has to do with eigenvectors, eigenvalues and
commutation of operators; so, if you know about that, all these effects come
rather naturally. |
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© H. Föll (Semiconductors - Script)
