chain
Following a similar approach, we are now going to consider the electronic structure of the hypothetical regular chain (NaCl)nin which the chlorine atoms provide five electrons through their (3p)Cl orbitals and the sodium atoms
Fig. 6.10
Hypothetical chain with formula NaCl discussed in the text.
provide one electron through the(3s)Na orbital (Fig. 6.10). In this qualitative approach we will neglect the secondary role played by the empty(3p)Naand filled (3s)Cl orbitals. We will qualitatively build the band structure of the system mostly through the use of symmetry-based arguments.
6.3.1 Group of the different k points
This system is stable under application of the symmetry operations of the space groupG,which is the direct product of the translation groupTnand the point group (D∞h)N a0 leaving the sodium atom of the reference cell invariant.4
4Here we have placed the origin of the chain at a sodium atom of the reference cell. Of course, it could have been equiv- alent to place it on the chlorine atom of
the reference cell. Consequently, the appropriate group for pointsand X is(D∞h)N a0. and X points
Looking at the Bloch orbitals for these points, schematically represented in Fig. 6.11, their symmetry properties and thus the associated symmetry labels are quite clear. Thus, among all of these orbitals, only BO(3s)Na(X) and BO(3pz)Cl (X)may interact. In all other cases the Bloch orbitals may be identified with the crystal orbitals of the system.
kpoints other thanand X
For all otherk points, the group to be used isC∞v. Looking at the character table for this group it is clear that the(3s)Naand(3pz)Clare bases for the same representation,+, whereas the(3px)Cland(3py)Clorbitals provide a basis for the representation. Consequently, the Bloch orbitals generated by the (3px)Cland(3py)Clorbitals are degenerate at anykpoint and do not interact with BO(3s)Na(k) and BO(3pz)Cl(k). In contrast, the Bloch orbitals BO (3s)Na(k) and BO(3pz)Cl(k)may interact at anykpoint.
Summary
The symmetry properties of the Bloch orbitals and the crystal orbitals of the (NaCl)nchain at differentkpoints are summarised in Table 6.3.
6.3.2 Bands associated with σ -type overlaps
We begin this section by looking at the two bands generated by the(3s)Naand (3pz)Clorbitals that are involved inσ-type interactions.
Fig. 6.11
Bloch orbitals for theand X points.
Table 6.3 Symmetry labels for the Bloch orbitals centered on Na (BONa) and Cl (BOCl) and for the crystal orbitals (CO) of the (NaCl)nchain.
kpoint Symmetry
BONa BOCl CO
label
+g BO(3s)Na — 1 non-bonding
orbital
+u — BO(3pz)Cl 1 non-bonding
orbital
u — BO(3px)Cl 2 non-bonding
BO(3py)Cl degenerate orbitals
X +g BO(3s)Na BO(3pz)Cl
2 orbitals resulting from the interaction of two Bloch orbitals
X g — BO(3px)Cl 2 non-bonding
BO(3py)Cl degenerate orbitals
ka=0,1/2 + BO(3s)Na BO(3pz)Cl
2 orbitals resulting from the interaction of two Bloch orbitals
ka=0,1/2 — BO(3px)Cl 2 non-bonding
BO(3py)Cl degenerate orbitals
Fig. 6.12
Energies of the BO(3s)Na(k)and BO(3pz)Cl(k)Bloch orbitals.
Energy of the BO(3s)Na(k)and BO(3pz)Cl(k)orbitals
In a H¨uckel-type approach only those interactions between nearest neighbour atoms are considered. Since the periodic fragments Clnand Nan are not built from nearest neighbour atoms of the (NaCl)nchain, the energies of the Bloch orbitals BO(3s)Na(k) and BO(3pz)Cl(k) do not change withkand they keep the same value as for the isolated(3s)Naand(3pz)Clorbitals, respectively.
Band structure when thens−npz interaction is taken into account At this point we will explore how the Bloch orbitals BO(3s)Na(k) and BO(3pz)Cl(k) can interact. At the point, the two orbitals BO(3s)Na() and BO(3pz)Cl()cannot interact because they are of different symmetry. In con- trast, they can interact at any other point including X. Thus, the two non- interacting bands (dotted lines in Fig. 6.13) repel each other, and this is more effective the farther it is from, thek point under consideration. This is so because the overlap between the Bloch orbitals BO(3s)Na(k) and BO(3pz)Cl(k) increases withkain the interval [0, 1/2]. The bonding band is essentially built from orbitals centred on the chlorine atoms whereas the upper antibonding band is essentially made of orbitals centred on the sodium atoms. A schematic band structure summarising the situation is shown Fig. 6.13.
6.3.3 Complete band structure
Now we can combine the results of the previous sections and build the com- plete band structure for the (NaCl)n chain. The easiest way to proceed is to remind ourselves that the two degenerate non-bonding π-type bands are
Fig. 6.13
Partial band structure for (NaCl)n (+-type bands). The energies of the BO(3s)Na(k)and BO(3pz)Cl(k)are shown as dotted lines. The arrows are used to highlight the effect of the interaction between the two Bloch orbitals.
Fig. 6.14
Band structure of (NaCl)n.
flat and have the energy of an isolated(3p)Cl orbital. This leads us to place these twoπ bands at the top of the σ-bonding lower band. Since we have neglected the contribution of the low-lying 3sorbitals of chlorine, there are six electrons to fill these bands. The Fermi level therefore lies at the top of the third band. This completes the information needed to draw the band structure (see Fig. 6.14). It is clear that the system is not metallic because there is a clear band gap at the Fermi level. Now it is interesting to analyse the electronic density of the NaCl chain and compare it with those of isolated Na and Cl atoms. In the chain there are four electrons in theπ-type bands localised on the(3px)Cland (3py)Clorbitals, and two electrons on the 1+band. This is a filled bonding band, mostly based on the(3pz)Clorbitals. Consequently, in the chain there is on average less than one electron in the vicinity of the sodium atom and more than five in the neighbourhood of the chlorine: the chain is thus strongly ionic.
The situation is in fact completely analogous to that in the NaCl molecule. An obvious consequence of this situation is that the band gap at the Fermi level for this type of chain will increase with the electronegativity of the anionic partner and with the electropositive character of the cationic partner.