The Peierls Distortion

The curve for the energy dependence as a function of k in Fig. 10.4 has a positive slope. This is not always so. When p orbitals are joined head-on to a chain, the situation is exactly the opposite. The wave function y/0 = Z.X, is then antibonding, whereas yKja is bonding (Fig. 10.5). 

Different bands can overlap each other, i.e. the lower limit of one band can have a lower energy level than the upper limit of another band. This applies especially to wide bands. 

Band structure for a chain of p orbitals oriented head-on 

The model of the chain of hydrogen atoms with a completely delocalized (metallic) type of bonding is outlined in the preceding section. Intuitively, a chemist will find this model rather unreal, as he or she expects the atoms to combine in pairs to give H2 molecules. In other words, the chain of equidistant H atoms is expected to be unstable, so it undergoes a distortion in such a way that the atoms approach each other in pairs. This process is called Peierls distortion (or strong electron-phonon coupling) in solid-state physics  

The very useful chemist s intuition, however, is of no help when the question arises of how hydrogen will behave at a pressure of 500 GPa. Presumably it will be metallic then. 

Up until now, in this section, we have placed all of the electrons, paired, in the lower half of the energy band. There is, of 

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The one-dimensional chain of hydrogen atoms is merely a model. Flowever, compounds do exist to which the same kind of considerations are applicable and have been confirmed experimentally. These include polyene chains such as poly acetylene. The p orbitals of the C atoms take the place of the lx functions of the H atoms they form one bonding and one antibonding n band. Due to the Peierls distortion the polyacetylene chain is only stable with alternate short and long C-C bonds, that is, in the sense of the valence bond formula with alternate single and double bonds ... 

The Peierls distortion is not the only possible way to achieve the most stable state for a system. Whether it occurs is a question not only of the band structure itself, but also of the degree of occupation of the bands. For an unoccupied band or for a band occupied only at values around k = 0, it is of no importance how the energy levels are distributed at k = n/a. In a solid, a stabilizing distortion in one direction can cause a destabilization in another direction and may therefore not take place. The stabilizing effect of the Peierls distortion is small for the heavy elements (from the fifth period onward) and can be overcome by other effects. Therefore, undistorted chains and networks are observed mainly among compounds of the heavy elements. 

The octet principle, primitive as it may appear, has not only been applied very successfully to the half-metallic Zintl phases, but it is also theoretically well founded (requiring a lot of computational expenditure). Evading the purely metallic state with delocalized electrons in favor of electrons more localized in the anionic partial structure can be understood as the Peierls distortion (cf. Section 10.5). 

The Gillespie-Nyholm rules can be applied with the aid of this formulation. The occurrence of both kinds of building blocks in Li2Sb, chains and dumbbells, shows that in this case the Peierls distortion contributes only a minor stabilization and is partially overridden by other effects. The Peierls distortion cannot be suppressed that easily with lighter elements. 

We saw in Section 12.2.3.1 that the presence of additional chalcogen atoms in BEDT-TTF/TCNQ promotes interstack interactions, suppressing the Peierls distortion and imparting upon the salt increased dimensionality compared to TTF/TCNQ. The result of including a different chalcogen into the TTF/TCNQ structure is shown in Table 2. Despite losing donor efficiency compared to TTF (Table 1) the TCNQ complexes of m/trans-diselenadithiafulvalene (DSDTF, 55/56) and TSF show an improvement in conductivity when two or four selenium atoms are incorporated. The reduced metal-insulator transition suggests that this effect is also caused by a suppression of the Peierls distortion. Increased Se-Se interstack contacts add dimensionality to the structure and limit the co-facial dimerisation typical of Peierls distortion. Wider conduction bands are afforded from the improved overlap of diffuse orbitals. 

In contrast, in the SSH model, the electrical bandgap arises because of the alternation between single and double carbon-carbon bonds, a signature of the Peierls distortion in a ID system. When a perfect ID chain of equidistant carbon atoms is considered, the electronic structure resulting from the electronic coupling between the atomic Pz-orbitals is that of a half-filled n band, implying a metallic... 

The temperature of the metal-to-insulator transition in TTF—TCNQ is 53 K. For systems with increased interchain coupling, the transition temperature for the onset of metallic conduction increases roughly as the square of the interaction between the chains. This behavior is true as long as the coupling between chains remains relatively weak. For compounds with strong interactions between stacks, the material loses its quasi-ID behavior. Thus, the Peierls distortion does not occur even at low temperatures, and the materials remain conductive. 

Lio.5tPt S2C2(CN)2 2]-2H20.129 X-Ray studies reveal that the [Pt(mnt)2]05 anions are stacked face-to-face along the a axis of the unit cell to form a fourfold distorted linear chain. This can be regarded as the Peierls distorted state analogous to that found in Rbi 67[Pt(C204)2] H20. The room temperature conductivity (cr ) is 1 Q 1 cm-1 and the temperature dependence is that of a semiconductor.129... 

Peierls was careful to point out that his conclusion was not complete because it assumed the validity of the adiabatic approximation. This approximation cannot be strictly valid in the case of a metal because of the close spacing of energy levels, and thus the motion of the nuclei must be taken into account in a more rigorous treatment of the problem. Peierls result is based on a simple one-electron treatment of the problem in which electron-electron interactions are neglected. Such electron-electron interactions mix states above and below the gap in a manner somewhat analogous to that of raising the temperature and so also affect the tendency to distort. Consequently, a more sophisticated analysis is needed before one can draw any definite conclusions on the stability of a particular system against the Peierls distortion. 

Phase transitions. Low-dimensional conductors undergo several types of specific structural phase transitions, such as the Peierls distortion (electron-phonon coupling), the spin-Peierls distortion (spin-phonon coupling), anion-ordering transitions, and so on. These first have to be detected and then measured and understood. However, the foregoing distortions may be very small and difficult to observe, and up to now, only a few lattice distortions have been fully measured and described. 

The origin of these transformations is very difficult to investigate. Yet it appears that the optical study should be very helpful for this purpose. An analysis of T dependence of the phase phonon absorptions at 317 and 253 cm -1 show that the 54 K metal-insulator transition is driven by the Peierls distortion on the TCNQ sublattice, whereas the distortion on the TTF chains increases markedly around 49-K phase transition [100]. It is a typical example of a close relationship between the optical properties of organic conductors and a molecular mechanism of the phenomena that occur in the material. 

The Peierls distortion plays a crucial role in determining the structure of solids in general. The one-dimensional pairing distortion is only one simple example of its workings. Let s move up in dimensionality. 

Exercise 6.4. Consider a crude, but useful, model of polyacetylene (CH),, consisting of the ideal zig-zag (all-trans) chain shown below in which all the C-C bond distances are equal. Draw its TT-type band structure and show that it is Peierls unstable. Show that, unlike the H-chain case, the Peierls distortion does not correspond to a doubling of the unit cell. 

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