We will now consider a linear periodic system Anwhere A is an atom bearing the valence orbitalsns,npx,npy, andnpz(see Fig. 6.5). We are going to build the band structure of this system by using, as far as possible, its symmetry
This system is stable under the operations of theGspace group, which is the direct product of theTntranslation group and the(D∞h)0group leaving invari- ant the atom of the reference cell,A0. Consequently, the appropriate group for and X is the(D∞h)0group. For any other point, the appropriate group is C∞v, which is a subgroup of(D∞h)0containing the symmetry operations of (D∞h)0leaving every cell of the chain invariant.
6.2.2 Symmetry of the different Bloch orbitals
and X points
Looking at the representation of the Bloch orbitals forand X it is easy to determine their symmetry labels according to the(D∞h)0group (see Fig. 6.6 and Appendix for the character table of theD∞hsymmetry group). The bases {BOnpx(),BOnpy()}and{BOnpx(X),BOnpy(X)}belong to theusymme- try, whereas the Bloch orbitals BOns() and BOns(X)are of+g symmetry, and the BOnpz() and BOnpz(X)orbitals belong to theu+ symmetry. Con- sequently, the different orbitals BOnpx(k), BOnpy(k), BOnpz(k),and BOns(k) cannot interact at theand X points. Thus the crystal orbitals of the system are nothing more than these Bloch orbitals. The energy ordering of the crystal orbitals atis clearcut:
E(BOns()) <E(BOnpx())= E(BOnpy()) <E(BOnpz())
Fig. 6.6
Bloch orbitals of the Anchain corresponding to theand X points.
Fig. 6.7
Energies of the BOnsand BOnpzBloch orbitals for theand X points corresponding to the two different situations considered in the text.
In contrast, at the X point it is not possible to decide with certainty how the antibonding BOns(X)orbital should be positioned with respect to the bonding BOnpz(X)orbital. The energy ordering of these orbitals depends on the energy of the atomic orbitals ns andnp. We need to consider the two possibilities shown in Fig. 6.7:
• Case (a): if the energy difference between thens andnp is large and the overlap is moderate, the BOns(X)orbital, despite its antibonding character, lies lower in energy than the BOnpz(X)bonding orbital.
• Case (b): if the energy difference between thensandnpis small and/or the overlap is large, the BOns(X)orbital, because of its antibonding character, lies higher in energy than the BOnpz(X)bonding orbital.
In both cases, the Bloch orbitals{BOnpx(X),BOnpy(X)}are associated with a higher energy than the{BOns(X),BOnpz(X)}orbitals (except for very special situations in case (b)).
kpoints other thanand X
Since each symmetry operation of the C∞v group leaves every atom of the chain invariant, we can directly consider the properties of the ns,npx,npy, and npz orbitals in the C∞v group. Examination of the character table of this group (see Appendix) tells us that the ns andnpz orbitals are bases for the representation +, whereas the npx andnpy orbitals are a basis for the representation. Consequently, the Bloch orbitals generated by thenpx and npy orbitals keep their degeneracy at every point and do not interact with the
and X u
BOnpx,BOnpy 2 degenerate orbitals
ka=0,1/2 + BOns,BOnpz
2 orbitals resulting from the interaction of two Bloch orbitals
ka=0,1/2
BOnpx,BOnpy 2 degenerate orbitals
BOns(k) and BOnpz(k) Bloch orbitals. In contrast, the Bloch orbitals BOns(k) and BOnpz(k) can interact at anykpoint other thanand X.
Summary
The main results described in the two previous sections are summarised in Table 6.2.
6.2.3 Bands associated with σ -type overlaps
We will start by considering the two bands generated by thensandnpzorbitals that are associated withσ-type overlaps.
Energy of the BOns(k) and BOnpz(k) Bloch orbitals
Knowing the shape of the Bloch orbitals atand X (see Figs 6.6 and 6.7), we can represent their energies for anykpoint (dotted lines in Fig. 6.8). The energy of the BOns(k)orbital increases when going fromto X whereas the energy of the BOnpz(k)orbital decreases along the same path. In case (a) (Fig. 6.7) the two dotted lines representing the energy of the Bloch orbitals BOns(k)and BOnpz(k)cannot cross (Fig. 6.8a). However, in case (b) there is a crossing (Fig. 6.8b).
σ-type band structure
Let us now switch to the interaction of the Bloch orbitals BOns(k) and BOnpz(k). This interaction is stronger as the difference in energy between the two Bloch orbitals decreases and the overlap between the orbitals increases.
In view of the nil overlap at the points and X, the mixing between the two orbitals will be as much effective as the energies of the two orbitals become similar and as we are farther away from the and X points. Let us begin by considering case (a). The two dotted lines representing the energy of the Bloch orbitals are never very close to each other at points far from X (Fig. 6.8a). Consequently, the mixing between these two Bloch orbitals will be quite modest, i.e. the two dotted lines are only slightly pushed away from each other at any point other thanand X. As a result, the lower band will be slightly stabilised and the upper band will be slightly destabilised. The crystal
Fig. 6.8
Band structure diagrams associated with the+-type bands of An. The energies of the Bloch orbitals BOns(k)and BOnpz(k)are represented as dotted lines, while the continuous lines represent the energy of the crystal orbitals resulting from the interaction between these two Bloch orbitals. Cases (a) and (b) refer to Figs 6.7a and b, respectively.
orbitals of the lower band are essentially built from orbital BOns(k)whereas the crystal orbitals of the upper band are mainly built from orbital BOnpz(k). In contrast, in the second case (Fig. 6.8b), the Bloch orbitals BOns(k)and BOnpz(k) interact quite strongly near the crossing of the two dotted lines:
there is an avoided crossing. The lower band resulting from this interaction is essentially built from orbital BOns(k)forkpoints near,and from orbital BOnpz(k)forkpoints near X. Forkpoints near the crossing between the two dotted lines, the crystal orbitals of the two bands result from a strong mixing of the two Bloch orbitals. Thus, in that case, the nature of the predominant Bloch orbital for a given band changes on the way fromto X. The main results of the analysis of this section are summarised in Fig. 6.8.
6.2.4 Complete band structure
At this point we need to guess where the degenerate bands involving orbitals npxandnpymust occur with respect to the other bands. These bands are of symmetry for anykpoint other thanand X,and ofu symmetry at these two points. Consequently, these bands can cross all other bands, which are of +symmetry. On the other hand, in Section 3.2.4 we saw that the Bloch orbitals associated with the points withka= ±1/4 (see Fig. 3.11) are non- bonding and thus their energy coincides with that of the orbital from which they are generated. As a result, the curves representing the energies of the Bloch orbitals generated by thenpx,npy, andnpz orbitals cross at the points
Fig. 6.9
Band structure diagrams for An corresponding to the two different situations considered in Fig. 6.7. The dotted lines give the energy dependence of the Bloch orbitals BOns(k)and BOnpz(k).
with ka= ±1/4, since the np atomic orbitals are degenerate. Because the lateral overlap between orbitals npx or npy is less effective than the axial overlap between the ns and npz orbitals, the width of theπ-type bands is smaller than that of the+bands. The present discussion leads us to the two band structures of Fig. 6.9, on which the energies of the Bloch orbitals BOns(k) and BOnpz(k) have also been represented as dotted lines.
This analysis clearly shows that it is possible to determine the band structure of a chain of high symmetry following an approach that is entirely similar to that followed to obtain the electronic structure of a symmetric molecule.
The only difference stems from the fact that, for the chain the interactions between the different orbitals must be analysed for everykpoint. When there is some uncertainty concerning the relative position of two bands, i.e. when simple qualitative reasoning such as that used in our example may not be used, then one must rely on some kind of computation even if it is very simple, for instance the extended H¨uckel approach. Of course, this is also the case for molecules (see for instance ref. [3].).