The electronic structure of BC8 and ST12 materials will now be
considered. The method of energy minimisation
at sets of special *k*-points as described in Chapter 2 is at first sight
unsuitable for general band structure calculations since

- unoccupied states do not contribute to the total energy in the minimisation calculation since they do not contribute to the ground state electronic charge density.
- the minimisation calculation has to be done at a set of special
*k*-points rather than, for example, on lines of high symmetry where conventional band structures are shown.

constructing *H* each time from the previously found charge density will
give the correct energy eigenvalues. Note that the eigenvalues have to
be found by diagonalisation since methods such as conjugate gradients
only find linear combinations of eigenvalues and eigenvectors.

The above method can also be used to obtain energies of the unoccupied
bands. However there are problems which prove difficult to overcome
using density functional theory. Density functional theory gives a
variational method of finding the ground state charge density only.
Therefore the above method of evaluating the energy eigenvalues gives
the energy of an empty excited state when there are *N* electrons still
in the ground state (for an *N* electron system). The actual excited
state should be evaluated from *N*-1 electrons in their ground state
with one electron in the excited state under consideration. The charge
density for this state cannot be found using density functional theory.
As a consequence, a band gap found by this method is usually
underestimated by up to 50%.

The band structures of silicon in the diamond, BC8 and ST12 structures are shown in Figures 4.26, 4.27 and 4.28.

**Figure 4.26:** Electronic band structure of silicon in the diamond structure
along several high symmetry lines in the FCC Brillouin zone. The
calculations predict a semiconducting structure with a band gap of
0.59eV.

**Figure 4.27:** Electronic band structure of silicon in the BC8 structure
along several high symmetry lines in the BCC Brillouin zone. The
calculations predict a semimetallic structure.

**Figure 4.28:** Electronic band structure of silicon in the ST12 structure
along several high symmetry lines in the tetragonal Brillouin zone. The
calculations predict a semiconducting structure with a band gap of
0.7eV.

It is found that silicon in the diamond structure is semiconducting with
an indirect band gap of 0.589eV along the line
( ). This is in agreement with other *ab initio*
calculations[65] although, as expected, the band gap is
underestimated with experimental measurements giving a gap of
1.1eV. In contrast, BC8 silicon is found to be semimetallic. This
occurs at the *H* point in agreement with empirical pseudopotential
calculations[66] where band energies are fitted to experimental
measurements. The lowest energy conduction band is found to drop below
the highest point of the valence band (the *H* point) along the
( ) line by 0.046eV and along the *G* line
( ) by 0.071eV. Silicon ST12 is found also to be
semiconducting with an indirect gap of 0.7eV between the and
*Z* points.

Thu Oct 31 19:32:00 GMT 1996