Band representation

Figure 6. A flat band representation of electron (left) and hole (right) injection from an excited dye molecule on a silver halide surface. The first step in both processes is excitation of the dye molecule. The second step is the transfer of the electron (left) that was excited up to the LUMO into the conduction band before radiative or radiationless decay. In the case of hole injection (right), the hole remaining in the HOMO after excitation is transferred to the valence band (equivalently an electron is transferred to the HOMO from the valence band).

It is apparent that as the momentum p increases, the finite difference spectrum deviates more and more from the correct value. It is usually assumed that acceptable accuracy with the FD method is obtained when at least 10 points are used per wave period. This means also using 10 points per unit volume in phase space. The finite difference algorithms are based on a local polynomial approximation of the wave function and therefore the convergence of the method follows a power law of the form (Aq)n, where n is the order of the finite difference approximation. This semilocal description leads to a poor spectral representation of the kinetic energy operator, which will be true as well, for other banded representations of the kinetic energy operator such as the... 

Figure 13.2 Energy-band representations of materials (a) insulators, (b) intrinsic semiconductors, (c) n-type extrinsic semiconductors, (d) p-type extrinsic semiconductors, (e) metals and (f) semimetals. The innermost filled energy bands are omitted in parts (c) and (d)...

Energy band representation of SnOj showing the different contributions to the work function of CO sensing in dry air conditions. Following symbols are used for the different parameters ... 

Energy bands representation for a p-type SMOX material. 

The main problem in explaining the physicochemical nature of cobalt monosilicide is to find a correlation between the crystal-chemical scheme proposed above and the band representations describing the electrical properties. The comparatively small values of the effective masses of the carriers in CoSi suggests that the overlapping bands responsible for its thermoelectric properties are fairly wide (about 1 eV). From this, it must be assumed that 4p and 4pj, levels of cobalt atoms form molecular bonding orbitals with six adjacent metal atoms which, when the degeneracy is removed in the crystal, form two bands overlapping by 0.05 eV. 

Let us return to our example of the induced representation (c, oig) of the space group Oj formed by 2s functions of oxygen atoms in the perovskite strnctnre. Using the full gronp irreps from the site [16] we calcnlate the snbdnction freqnencies (3.66) for belonging to K set points of Brillouin zone and write the indnced band representation (c,aig) in the k basis in the form / (l+3+)R(5+)M(4+5 )X(l+3 4 ). The labels of the small irreps of the Uttle groups are taken from [17],... 

In Tables 3.15 and 3.16 we give the notations of induced representations in fc-basis for symmetry points of the BriUouin zone. We include only the band representations for upper valence bands of all the crystals under consideration. 

Table 3.15. Band representations of space groups and 0 for upper valence bands of...

Table 3.16. Band representations of upper valence bands in SrZrOs crystal induced from ...

In our examples, aU the induced irreps are simple, excluding the BR corresponding to the 6-sheeted lower valence subband (see Fig. 3.3). This band representation is a composite one as it is formed by two simple band representations d,aig) and (6, t u) induced by 0 2s- and Sr 4pstates, respectively. Analysis of the space symmetry of crystalline orbitals is used to consider the possible centers of localization of chemical bonding in crystals. This task requires the Wannier-function definition and is considered in the next section. 

Symmetry of Localized Orbitals and Band Representations of Space Groups... 

As the third example of the band representations generated we consider SrZrO crystal. The configurations of valence electrons in free atoms are ... 

In our example, the BR corresponding to the 6-sheeted valence band is a composite one as it is formed by two simple band representations (d,aig) and (h.fiu) induced... 

In accordance with the theory of induced (band) representations the corresponding Wannier functions in the silicon crystal (four per unit cell) are centered at the middle of the bonds between the nearest Si atoms (Wyckoff position c with site group Gc = Dsa) and transform according to the irrep aig of the site group Dsa, see Sect. 3.3.1. 

We shall illustrate the discussed procedure using the example of the SrZrOs crystal with the cubic perovskite structure. In Table 9.16 band representations of space group Oj induced by those atomic states that take part in the valence band formation are given. Each line of this table dehnes the connection of the q-basis for site-symmetry groups Oh Si), Dih 0) with the fe-basis at the symmetry points of BZ, see also Sect. 3.3. 

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