Diploma - Max Planck Institute for Solid State Research

4.1 Overview – properties and classification 25
Figure 4.2: overview of the Brillouin Zone: (left) k z = 0 section of the BZ – the green color
represents the BZ of the bodycentered structure (SPG 139), the red color the simple tetragonal
cell (SPG 123); neglecting body-centered basis symmetry, bands get backfolded; respective
directions are marked by the blue and yellow lines (right) 3D representation of the BZ of SPG
139 (grey) and 123 (red); the section of the left hand side is depicted correspondingly
(either at the center or at the boundary), which has to be large enough, in order
that the overlap of the wavefunctions of adjoined atoms is negligible – resulting
in a model, quasi 2D material with no k z dispersion. The amount of layers to
construct this configuration is a compromise between computational time needed
(because the number of atoms per unit cell rises) and the information about the
surface / bulk relation which is intended to be obtained.
Since only the Bravais lattice accounts for the BZ, one can furthermore reduce the
needed amount of information (and thereby the computational time) by using the point
symmetry which results in the irredicuble wedge of the BZ used finally for the calculations.
Setups for the afore mentioned configurations and a note on the difference
between lattice symmetry and point symmetry are depicted in fig. 4.1.
In principle one can distinguish (dependent on the level of localization) between bulk
states (delocalized), surface resonances (increased probability at the surface) and surface
states (localized at the surface corresponding to two-dimensional states). Using a unit
cell m-times repeated in z-direction results in respective backfolding of bands inside the
BZ (with “reduced” k z dispersion), because it decreases with the same factor the unit
cell increases. Extending the procedure to m → ∞ maps all dispersion with respect to
k z onto the k x × k y plane resulting in a quasi two-dimensional representation of the
bulk band structure, the so-called projected bulk band structure. This process can be
imagined as reflecting the band structure subsequently along a mirror plane – which is
shown for the transition from SPG 139 to SPG 123 in fig 4.3c. It should be noted, that
the projected band structure does not depend on a surface or an exponential decay of
the wavefunction into the vacuum because mathematically it contains the same amount
of information as the bulk band structure. However, in an ideal bulk the smallest