Electronic Bandstructures

The electronic bandstructure of a solid shows the eigenvalues associated with the valence and conduction bands along specific directions in the Brillouin zone of that particular crystal structure. One of the most important reasons for computing bandstructures is to determine the band gap, i.e. the difference between the highest valence band and the lowest conduction band energies, since this can provide insight into potentially useful materials for optical device applications. However band gaps calculated from Kohn-Sham eigenvalues using the LDA and the GGA notoriously underestimate the results obtained from experiment. In fact this is one property where the GGA provides no consistent improvement over the LDA, and is sometimes worse. Discrepancies between DFT and experiment are in the range $50 \%$ to $100 \%$.

Problems arise in calculating band gaps because the Kohn-Sham eigenvalues do not have a strict physical interpretation (except for the highest occupied level [120,121]) and also because the exact exchange-correlation potential exhibits a discontinuity when the number of electrons in the system passes through an integer [121,122,123], which is not described by the usual continuum description [37] provided by the LDA and the GGA. The size of the discontinuity therefore sets a limit on the accuracy that can be achieved with continuum functionals, although its exact value in solids is unknown. It is generally believed to be a significant proportion of the actual gap [124,125], however Städele et al. [126] recently suggested that it may be smaller than originally thought. From a purely pragmatic viewpoint, it would be highly advantageous to develop functionals that improve upon the particularly poor description yielded at present, and so give band gaps in reasonable agreement with experiment.

To determine the bandstructure of a material, a single-point energy calculation is first performed at a specified crystal geometry to obtain the self-consistent groundstate density. This fixes the form of the Kohn-Sham Hamiltonian which is then solved to give the corresponding Kohn-Sham eigenvalues. The eigenvalues are computed at a greater number of $k$-points, along specific directions in the Brillouin zone, than the ones used in the single-point energy calculation. Fig. 3.3 shows the Brillouin zone associated with the diamond/zinc-blende structure, along with the $k$-point path used in the bandstructure calculations performed here.

Figure 3.3: The primitive cell (purple dashed lines) and corresponding first Brillouin zone (black lines) for the diamond/zinc-blende structure. The positions of high symmetry in the Brillouin zone are indicated by the brown characters - G represents the $\Gamma$ point, and the $k$-point path used in the bandstructure calculations is also shown by the blue dashed lines.

Table 3.4: Minimum electronic band gaps (in eV) calculated using the LDA, PW91, and HCTH at optimised lattice constants. Values in parentheses are band gaps calculated at experimental lattice constants.

Table 3.4 displays minimum band gaps calculated at optimised lattice constants; values calculated at experimental lattice constants are given in parentheses. At optimised lattice constants, the HCTH band gaps are larger than LDA and PW91 for all systems except Ge, GaN, and GaAs. At experimental lattice constants, the HCTH band gaps are generally larger than at the optimised lattice constants as a result of lattice contraction. The PW91 and HCTH bandstructures for Si are superimposed in Fig. 3.4. In general the main differences between the two GGAs occur for the conduction bands away from the $\Gamma$ point, so although the gap increases with HCTH, the shift in energy is not uniform across the Brillouin zone.. The conduction band minimum (CBM) for Si correctly occurs at a point along the $\Gamma$-X axis in the Brillouin zone for all three functionals. The HCTH functional decreases the valence band width, i.e. the difference between the lowest highest valence band energies, with respect to PW91 by $\sim 0.2$ eV.

In Ge, the CBM and valence band maximum (VBM) touch at the $\Gamma$ point for both HCTH and PW91, therefore giving no gap. The systems SiC, AlP, and AlAs have indirect band gaps for all three functionals, with the CBM correctly occuring at the $X$ point in all cases. Each functional also correctly yield direct gaps for GaAs, although are considerably underestimated - the GGAs give the worst agreement for this system.

Figure 3.4: The bandstructure of bulk Si calculated using HCTH (solid lines) and PW91 (dashed lines). The top valence bands have been aligned at the $\Gamma$ point (zero energy).

Considering the III-V nitride semiconductors, each functional correctly predicts GaN to have a direct transition at the $\Gamma$ point, although LDA gives a larger gap than both PW91 and HCTH at the optimised lattice constant. AlN has an indirect band gap occurring at the X-point with each functional, despite the direct nature found by experiment. The band gap at the $\Gamma$ point for AlN is calculated to be $4.62$, $4.13$ and $4.13$ eV for LDA, PW91 and HCTH respectively, at the optimised lattice constant.