Band Structures

 

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

1. 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.
2. 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.

Both of these problems are easily overcome. The self consistent charge density obtained by the minimisation procedure contains all the ground state information (assuming the calculation has been done at k-point convergence). Thus for any k-point a minimisation of the Rayleigh-Ritz ratio

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.