May 19, 2008

Fig. 1: This is a prototype figure. Note that
clicking on this figure brings up the high-resolution
original. The computer code that generated these
bands is available here. I plotted
using gnuplot. The plot file is shown
here. I used gimp to shrink the figure,
after first fattening its lines. I also shrank the
labels at the bottom less. I did this by shrinking
a second copy of the figure, excising its labels and
pasting them in. |

This is a report template that shows the band
structure of graphene. The equation plotted is the last one below.
In this equation the three vectors **r**_{ν}, given by

are the translation vectors taking one carbon to its
three near neighbors. Note that there are two inequivalent atoms in the
unit cell. The translation vectors taking this second atom to its near
neighbors are thus - **r**_{ν}. The bravais lattice

where

is the test of all translations from an atom to any equivalent atom. To this bravais lattice there is a corresponding reciprocal lattice defined by

where

The crystal momentum **q** of an electron is
defined only modulo addition of any reciprocal lattice vectors. We thus
need consider the states only in the first brillouin zone, the set of
all crystal momenta **q** closer to the origin than to any other
reciprocal lattice vector. The brillouin zone in this case is a
hexagon. We cannot plot the bands everywhere, so we just plot them on a
representative "tour" of the brillouin zone. The choice of tour is
somewhat arbitrary. You do what you think makes the situation clear.
The first leg of my tour is a straight line from the zone center
(**Γ**) to the hexagon vertex (**K**), from there to the mid
point of the face (**M**), and from there back to the origin
(**Γ**). The special points are given by

The band energies are given by

The technology I use for equations is to make a LaTeX
source, output to PostScript using **dvips** and then using
**gimp** to chop out the equation from the PostScript and convert it
to .gif format. Under my version of **gimp** there is a choice of
resolution when importing PostScript. I choose 300 dpi because my
version of **dvips** prints at 300 dpi. The graphic thus created is
very crisp - pure black-and-white, no gray. It is, however, at
inappropriate resolution for Internet use. You can, if you wish, put a
scaling command into your "img" entry for the equation, but I have
chosen just to shrink the equation image to 1/4 its original linear
dimension. This factor of 1/4 comes from screen resolution, which is
typically 72 dpi. You must be careful in shrinking to minimize
resolution degradation. I have found by experiment that this is best
done by the 3-stop process (1) convert image to grayscale, (2) scale
image and (3) re-convert to indexed with 256 colors. It is important to
store equations in .gif format rather than .jpg because the latter tends
to put speckle in the whitespace, due to the compression algorithm. An
example of an original, uncompressed is shown below. The original LaTeX
source for these equations is available here.