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Real crystals are three-dimensional
and we must consider their band structure in three dimensions, too. |
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Of course, we must consider the reciprocal
lattice, and, as always if we look at electronic properties, use the
Wigner-Seitz cell
(identical to the 1st Brillouin zone) as the unit cell. |
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There is no way to express quantities that change
as a function of three coordinates graphically, so we look at a two dimensional
crystal first (which, incidentally, do exist in semiconductor physics). |
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The qualitative recipe for obtaining
the band structure of a two-dimensional lattice using the slightly adjusted
parabolas of the free electron gas model is simple: |
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Construct the parabolas along major directions of
the reciprocal lattice, interpolate in between, and fold them back into the
first Brillouin zone. How this can be done for the free electron gas
is shown in an illustration
module. |
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An example - taken from
"Harrison" - may
look like this: |
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The lower part (the "cup") is contained
in the 1st Brillouin zone, the upper part (the "top") comes
from the second BZ, but is now folded back into the first one. It thus
would carry a different band index.
This could be continued ad infinitum; but Brillouin zones with energies well
above the Fermi energy are of no real interest. |
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The lower part shows tracings along major
directions. Evidently, they contain most of the relevant information in
condensed form. It is clear, e.g., that this structure has no band gap. |
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It would be sufficient for most
purposes to know the En(k) curves - the dispersion
relations - along the major directions of the reciprocal lattice
(n is the band index). |
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This is exactly what is done when real band
diagrams of crystals are shown. Directions are chosen that lead from the center
of the Wigner-Seitz unit cell - or the Brillouin zones in the more generalized
picture - to special symmetry points. These points are labeled according to the
following rules: |
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- Points (and lines) inside the Brillouin zone are denoted with Greek letters.
- Points on the surface of the Brillouin zone with Roman letters.
- The center of the Wigner-Seitz cell is always denoted by a G
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For cubic reciprocal lattices, the points with a
high symmetry on the Wigner-Seitz cell are the intersections of the Wigner
Seitz cell with the low-indexed directions in the cubic elementary cell. |
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We use the following
nomenclature: (red for fcc,
blue for bcc): |
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The intersection point with the [100] direction is called X
(H); the line G—X is called D.
The intersection point with the [110] direction is called K
(N); the line G—K is called S.
The intersection point with the [111] direction is called L
(P); the line G—L is called L.
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The picture above already used this
kind of labelling. Since the tracing of the dispersion curve can be done on
different levels - corresponding to the 1st, second, etc. Brillouin zone
- the points are often indexed with the number of the Brillouin zone they
use. |
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This may look like this (after
"Yu") |
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The top pictures show the elementary cell of the
diamond lattice or of the
ZnS type
lattice; the lower left picture the Bravais lattice of the fcc type
and the primitive (non-cubic)
lattice which is an equally valid, if less symmetric, representation of the
fcc lattice.. |
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The lower right picture shows the
cubic reciprocal lattice of the
cubic fcc lattice (which is a bcc
lattice) and the Wigner-Seitz cells (identical with the first Brillouin zone)
which also represent the reciprocal lattice |
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We now can draw the band diagrams
along all kinds of lines - not only from G to
some point on the Brillouin zone, but also from point to point, e.g. from
L to K or to some other points not yet labeled. An example for
the fcc structure and the free electron gas
approximation is shown below. |
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The first Brillouin zone with the proper indexing
of the relevant points and some dispersion parabola along prominent directions
are shown. The picture is taken from
Hummel's
book. |
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The indexing of the various branches is a bit
more complicated than in the illustration example for reasons explained
below. |
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Contemplate this picture a bit and
than ask yourself: |
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- Do I find this picture alarming ? ("Gee, if even the most simple
situation produces such a complicated structure, I'm never going to understand
it)
- Do I find this picture exciting? ("Gee, what a wealth of information
one can get in a simple diagram if you pick a smart way of
representation").
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Yes, it is a bit confusing at first. But do not
despair: If you need it, if you work with it, you will quickly catch on! |
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It is standard praxis, to join the
single diagram at appropriate points and to draw band diagrams by showing two
branches starting from G to major points and
to continue from there as already practiced
above. |
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The band diagram of Si, e.g., then assumes
its standard form: |
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The indexing of the major points in the Brillouin
zone is more complex than described so far - it is more than just an band
index. This reflects the fact that there is no unique choice of the G point, or that the the band structure allows certain
symmetry operations without changing. The indexing follows rules of group
theory displaying the symmetries, but shall not be described here. |
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The band structure as shown in this
standard diagram contains a tremendous amount of information; at this level it
is, e.g., evident, that: |
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Si has a band gap of about 1.1
eV. |
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Si is an indirect semiconductor because
the maximum of the valence band (at G) does
not coincide with the minimum of the conduction band (to the left of
X). |
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There is, however, a lot more
information encoded in this diagram, as we will see later. |
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© H. Föll (Semiconductor - Script)
