LDA-gap

Most of the results presented in Table 9 do not take into account selfenergy corrections, which are necessary in order to describe, in a proper way, the one-particle excited states. In the fourth column of Table 9 we report the GW corrected band-gaps, for the smallest Ge-NWs in the [111],[110] directions, and for all the [100] Ge-NW. A complete discussion on this part can be found elsewhere [121], We can see (Table 9, fifth column) that the effect of the GW correction is an opening of the DFT-LDA gap, by an amount which... 

In Table 11.21 the calculated bandgaps (BG) for slabs are compared. It is clearly seen that in all cases the BG for SrZrOs systems is wider. This agrees with the larger ionicity of SrZrOs in comparison with SrTiOs. One can notice that in the case of SrTiOs the BG for the model I is markedly greater than that for models II and III. The substantial reduction of BG for the TiOs surface relative to the bulk value is due to an extended shoulder in the valence-band electronic DOS (see below). The same picture has been found in DFT PW calculations [829], in spite of the fact that the LDA gaps are half of others (see Table 11.21). 

Fig. 6 Experimental and calculated optical gaps (LDA, TDLDA, and GtF/BSE). The LDA gap is the energy difference between two Kohn-Sham eigenvalues. Reprinted with permission from ref. 52. Copyright 2008 by the American Physical Society.

Fig. 5. Calculated electronic structure by the LDA method of (a) an isolated Cuo molecule and (b) fee solid Cuo where the direct band gap at the X-point is 1.5 eV [60],...

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

The upper VB for ratile is composed of the 02p orbitals and is 5.52 eV wide. The lower 02s band has a width of 2.08 eV. These numbers agree well with the experimental values of 5.4 and 1.9 eV, respectively. 1 The calculated direct energy gap of 1.83 eV is in good agreement with other LDA results and is smaller than the experimental value of 3.0 eV. 1... 

Table 1. Energy band gaps of diamond and silicon calculated by FLAPW-LDA and FLAPW-SIC schemes. The experimental values [34] are also shown. Units are in eV.

For excited state calculations, significant progress has been made based on the GW method first introduced by Hybertsen and Louie. [29] By considering quasi-partide and local field effects, this scheme has allowed accurate calculations of band gaps, which are usually underestimated when using the LDA. This GW approach has been applied to a variety of crystals, and it yields optical spectra in good agreement with experiment. 

The bandstructure of fee aluminium is shown in Fig. 5.9 along the directions and TL respectively. It was computed by solving the Schrodinger equation selfconsistently within the local density approximation (LDA). We see that aluminium is indeed a NFE metal in that only small energy gaps have opened up at the Brillouin zone boundary. We may, therefore, look for an approximate solution to the Schrodinger equation that comprises the linear combination of only a few plane waves, the so-called NFE approximation. 

Table 6 Absorption and emission gaps calculated as HOMO- -LUMO differences within DFT-LDA and GW ...

Table 9 DFT-LDA electronic gaps in Ge-NWs and Si-NWs are reported respectively in the third and fifth column, quasi-particle gaps are reported for Ge-NWs in the fourth column. All values are in eV...

Figure 40 The imaginary part of the dielectric function, 62. for the Si[i]-SiC>2(0 01) fully passivated system (-) compared with the 62 of the Si[i]- SiC>2(0 01) system with an O vacancy at the interface (- --). A shift of 0.8 eV higher in energy has been applied in order to overcome the LDA underestimation of the gap.

As for the fundamental gap of semiconductors, the LDA is known to underestimate its width, typically from half to two thirds of experimental values, while the Hartree-Fock gap is again much worse, for example more than five times larger than the experimental value in the case of silicon. In the case of covalent-bond materials including carbon, the LDA band structure is in general expected to be accurate, while the fundamental gap value should be considered to be larger in semiconductors than the LDA value. 

Fig.2 3. LDA electronic energy levels of C q (/eft panel) and the band structure of the fee C q crystal (right panel). In the band structure panel, energy is measured from the valence band top at the X point. The optically allowed transitions with excitation energies less than 6 eV in an isolated C60 are indicated by arrows, b Band structure of the fee C60 around the fundamental energy gap [15]...

The electronic structure of this tetragonal phase is semiconducting (Fig. 10b). Interestingly, the LDA fundamental gap value is 0.72 eV, being smaller than that of the fee C60. Introducing sp3 carbon into the originally sp2 carbon lattice does... 

The LDA electronic band structure of the ACB-stacking rhombohedral polymer was reported to be semiconducting with an indirect fundamental gap. The... 

The LDA electronic structure for the one-dimensional orthorhombic polymer phase shows that it is also a semiconductor, while a rather strong chain-orientation dependence is also found. The LDA fundamental gap value also depends on the chain orientation, while the value itself is larger than that of two-dimensional polymers [41]. 

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