In this chapter, the Car-Parrinello method is applied in an investigation of the energetics of surfaces. As described in Chapter 2, in order to make the calculation tractable, periodic boundary conditions are used. This is necessary in order to apply Bloch's theorem. This implies that Bloch's theorem cannot be applied in the direction perpendicular to the surface. Therefore an infinite cell or a non-plane wave basis set would be required in order to describe the wavefunction in that direction. This problem is easily avoided by the use of a periodic supercell that reintroduces the periodicity which is required to carry out the calculation with a plane wave basis set and Bloch's Theorem.
A surface still has periodicity in the plane of the surface, but it loses this periodicity perpendicular to the plane of the surface. In order to reintroduce the required periodicity into the calculation, the supercell contains a crystal slab and a region of vacuum. This is illustrated schematically in Figure 6.1.
Figure 6.1: Diagram representing a supercell geometry for a surface of a
bulk solid. The supercell is the area enclosed by the dashed
lines.
To ensure that the results of such an ab initio calculation accurately describe an isolated surface, the vacuum regions must be wide enough so that the faces of the adjacent surfaces do not interact across the vacuum region. But the surfaces could also interact through the bulk of the crystal, therefore this region must also be thick enough so that this interaction is also eliminated.
By use of this supercell method it is now possible to employ the
ab initio calculations using a plane wave basis set. Throughout the
following calculations, the wavefunctions were expanded in a basis set
with an energy cut of 200eV. The ionic positions are relaxed under the
influence of Hellmann-Feynman forces until the calculated forces are
below 0.1eV/Å. The supercells are found to be large enough that
sampling of the Brillouin Zone is required only at the 
point.
Total energy convergence with respect to k-point sampling is
checked by repeating the calculation with a set of 4 special
k-points. The difference is surface energies is found to be
negligible. The basic unit cell parameter is taken as being 6.54Å as
found from the previous pseudopotential calculations in Chapter 3. All
surfaces in this work are treated using supercells containing a slab of
BC8 Si (and region of vacuum) to which periodic boundary conditions are
applied. A sufficiently large supercell must be constructed so as to
isolate the two surfaces from each other. In all cases considered, the
dimension of the supercell perpendicular to the surface is 3 cubic unit
cell lengths, 19.62Å, and the thickness of the vacuum region is 1
unit cell. In all cases, it is found that the atoms in the two furthest
layers from the surface remain in their initial positions and their
bondlengths also remain unchanged, implying that sufficient bulk
material has been included.