7.3 General remarks concerning Peierls distortions
7.3.2 Second-order Peierls distortions
Second-order Peierls distortionsare associated with interactions between non- degenerate crystal orbitals, which provide the driving force for the stabilisation afforded by such a distortion. They are associated with changes in the point group symmetry but do not modify the translational properties of the system.
The possible distortions for polyacene that are considered in this chapter provide examples of this type of process. For simplicity we will assume that bands 1a2 and 2b2 are degenerate at X, i.e. we will neglect the weak 1–4 interactions. This does not introduce any serious flaws in our discussion. The first distortion we have considered (Fig. 7.11) allows the mixing at X of the COb2u(X)and COb3g(X)crystal orbitals of the regular structure, as well as of the COau(X)and COb1g(X)ones. More generally, such a distortion allows the interaction of the levels of the 1a2and 2a2bands as well as those of the 1b2and 2b2all along the Brillouin zone, but especially around X. These interactions are at the origin of the appearance of a band gap and thus of the lowering of the electronic energy of the system.
The second distortion considered (Fig. 7.14) allows the mixing of the COb2u(X)and COau(X)crystal orbitals of the regular structure as well as the COb3g(X)and COb1g(X)ones. More generally, such a distortion avoids any band crossing inside the Brillouin zone. These interactions also lead to the opening of a band gap at the Fermi level.
For the two distortions considered, the two crystal orbitals of regular poly- acene lying at the Fermi level, which are essentially non-bonding (COb3g(X) and COau(X)), become slightly bonding and slightly antibonding, respectively, after the distortion (Fig. 7.12). The two distortions considered for regular polyacene allow the interaction of non-degenerate crystal orbitals: this is the fingerprint of a second-order Peierls distortion that does not change the trans- lational properties of the system.
In general, such second-order Peierls distortions are associated with lower energy gains than those associated with first-order Peierls distortions, simply because they result from weaker orbital interactions as a consequence of the larger energy difference between the interacting orbitals.
Exercises
(7.1) Outline different Lewis structures for polyacene.
(7.2) Draw a schematic representation of theπ-type molec- ular orbitals ofcis-butadiene and give their occupation
in the fundamental state. What are the symmetry prop- erties of these orbitals?
(7.3) Is there a direct product of groups{P,O},other than that discussed in Section 7.1.2, that is appropriate to generate the space group of regular polyacene?
(7.4) Draw an energy diagram with the energies of the differ- ent C4H2fragment orbitals.
(7.5) What is the difference between the fragment orbitals of Exercise (7.4) and theπ molecular orbitals ofcis- butadiene?
(7.6) Indicate a different set of fragment orbitals that could have been chosen to build the band structure of poly- acene.
(7.7) Establish an interaction diagram for the Bloch orbitals of polyacene at,showing how they lead to the dif- ferent crystal orbitals. Draw an energy diagram of the different crystal orbitals.
(7.8) Draw the energies of the different crystal orbitals at X in the diagram you produced in Exercise (7.7).
(7.9) Redraw the qualitative band structure of Fig. 7.8 by using thePgroup obtained in Exercise (7.3). What are the main conclusions of the exercise?
(7.10) Determination of the band structure of polyacene dis- torted as in Fig. 7.14. We will follow a similar pro- cedure as in Section 7.2.1, i.e. we will assume that the electronic structure of regular polyacene is already known and treat the distortion as a perturbation leading to the mixing ofπorbitals, which in the regular struc- ture could not interact.
(a) What symmetry operations disappear when going from the regular (Fig. 7.1) to the distorted (Fig. 7.14) system?
(b) Is the band crossing shown in Fig. 7.8 possible in the distorted system?
(c) How will the crystal orbitals for regular polyacene, which are shown in Fig. 7.8, be modified as a result of the distortion?
(d) Show where the new Fermi level lies. Show that the nature of the new occupied crystal orbitals at and X demonstrate that there is a strengthening of the C–C bonds, which become shorter in the new structure.
References
1. For qualitative discussions of the band structure of polyacene see for instance:
(a) M.-H. Whangbo, R. Hoffmann, and R. B. Woodward,Proc. Roy. Soc. London, A366, 23, 1979; (b) J. K. Burdett,Prog. Sol. State Chem., 15, 173, 1984; (c) M.
Kertesz, Y. S. Lee, and J. P. Stewart,Int. J. Quant. Chem., XXXV, 305, 1989.
8
We will now analyse the electronic structure of selected inorganic chains and the relationship between this structure and the conduction properties. We have chosen to study two examples in detail and to treat three additional systems as exercises. It is essential that after considering the two detailed analyses of 1D examples in this chapter the reader also attempts the exercises. This is the best prelude to consideration of the more complex 2D and 3D materials discussed in the following chapters. Here we will follow the approach outlined at the end of Chapter 6 (see Section 6.5). First, we will establish the band structure of the system under consideration, then we will consider the possible structural distortions that may stabilise the system and finally we will consider how the resulting band structure may give a hint on the transport properties of the material.
8.1 KCP
Let us start by considering the electronic structure of K2[Pt(CN)4]Br0.3ã3H2O, which is usually referred to as KCP. It is the best known of the inorganic metallic chains built on the basis of square-planar tetracyanoplatinate units, with general formula Cx[Pt(CN)4]AyãzH2O, where C is a cation and A is an anion. [1] KCP is a typical example of a real compound (i.e. a 3D material).
It is made of structurally independent chains with formula (Pt(CN)14.7−)n,in between which the K+ cations, Br− anions and water molecules reside. The system may be considered 1D because the interchain interactions are very weak. These materials also provide a nice illustration of the notion of doping.
Doping of a material is the process by which the number of electrons is modified, by adding either a donor or an acceptor. For instance, K2[Pt(CN)4]ã
3H2O is a non-doped compound whereas KCP is doped because of the addition of 0.3 bromines per platinum atom. These bromine atoms are found in isolation between the chains of KCP. Since the bromine atoms are more electronegative than the platinum chains, they are found as Br− anions. Thus doping of the platinum chains results in a removal of electrons, i.e. an oxidation. Conse- quently, every Pt(CN)4fragment of KCP bears a charge of –1.7 instead of –2.0 in the non-doped material. The materials of this family, Cx[Pt(CN)4]AyãzH2O,
Fig. 8.1
Structure of the Pt(CN)4chain: (a) staggered structure observed in KCP; (b) eclipsed structure of the model chain.
may be doped with anions (A = Br−, Cl−, F−, FHF−, etc), cations (C = K+, Cs+, NH+4, etc.) or both simultaneously. The conductivity properties of these materials depend on the degree of doping. Thus the non-doped compound, K2[Pt(CN)4]ã3H2O, is an insulator whereas KCP is metallic at room temper- ature and becomes semiconducting at lower temperatures, as a result of the development of a Peierls distortion.
Even if doping does not induce major modifications in the structure of these materials, there is a considerable shortening of the Pt–Pt bond1as well as a
1The Pt–Pt bond length is 3.47 ˚A in the
non-doped compound but 2.87 ˚A in KCP. change in the relative conformation of two adjacent Pt(CN)4units (eclipsed in the non-doped compound but staggered in KCP). From a structural viewpoint, KCP is built from (Pt(CN)14.7−)nchains generated by a repeat unit consisting of two Pt(CN)4square-planar units in a staggered configuration and a repeat vectora(see Fig. 8.1a). Before considering the staggered chain we will study the eclipsed chain shown in Fig. 8.1b. The latter is generated by a vectora(a=
a/2) and a repeat unit of just one (Pt(CN)14.7−) unit. Then we will examine the influence of the CN−ligands’ orientation on the conductivity of the system. In principle, since the Pt–Pt distances are of the order of 3 ˚A, the ligand orbitals do not overlap. Hence, the conductivity process must essentially involve only the platinum orbitals. Consequently, the ligand orientation must play a secondary role in the conductivity. Thus, a study of the band structure of the structurally simpler eclipsed model should be a convenient starting point to build the band structure for KCP.