Half-filled Peierl systems

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 origin of these effects has been debated. One possibility is the Peierls instability [57], which is discussed elsewhere in this book In a one-dimensional system with a half-filled band and electron-photon coupling, the total energy is decreased by relaxing the atomic positions so that the unit cell is doubled and a gap opens in the conduction band at the Brillouin zone boundary. However, this is again within an independent electron approximation, and electron correlations should not be neglected. They certainly are important in polyenes, and the fact that the lowest-lying excited state in polyenes is a totally symmetric (Ag) state instead of an antisymmetric (Bu) state, as expected from independent electron models, is a consequence... 

The DOS associated with the band structure of 97, with one main group element of group 15 per lattice site, must have the block form 98. There are five electrons per atom, so if the s band is completely filled, we have a half-filled p band. The detailed DOS is given elsewhere.74 What is significant here is what we see without calculations, namely, a half-filled band. This system is a good candidate for a Peierls distortion. One pairing up all the atoms along x, y, and z directions will provide the maximum stabilization indicated schematically in 99. 

Infinite linear polyenes show a bond alternation between successive long and short C-C bonds [1], a consequence of the Peierls theorem on the nonexistence of one-dimensional metals [2], This Peierls distortion (or instability) is very important both from a theoretical and a practical point of view, being a typical example of a metal-insulator transition [3]. Consider an infinite chain of equally spaced sites -(CH)-, each of them bearing one electron in a single valence orbital. In this case we have a half-filled band and the system has metallic character. If we distort the chain into an alternating sequence of short and long bonds -(CH=CH)-, the half-filled band splits into a lower one completely filled and an upper empty band, separated by a gap. This dimerized polyacetylene is an insulator. 

Our discussion so far has focused on polyacetylene and related examples. The broad results however, are transfciable to many other systems. Algebraically, for example, our discussion applies equally well to the case of a one-dimensional chain of hydrogen atoms bearing l.v orbitals. Recall the one-to-one correspondence between the orbitals of finite molecules and their polyene analogs. With one electron pet-atom, Figure 13.3 indicates a half-filled band foi the geometry 13.34. This will distort (in a Peierls fashion) to a solid composed of H2 molecules (13.35) as chemical... 

The resulting band is only half-filled (metallic regime) because each of the carbon atoms offers one electron, and the number of COs is equal to the number of carbon atoms (each CO can accommodate two electrons). Therefore, the Peierls mechanism (Fig. 9.11) is bound to enter into play, and in the middle of the band, a gap will open. The system is, therefore, predicted... 

Figure 1.1 One-dimensional electronic system with a half-filled band band structure (a) before and (b) after Peierls distortion, where E is energy and Ep is the Fermi energy...

Bond alternation in polyenes is an example of a more general theorem for one-dimensional crystals called Peierls theorem. This theorem applies to systems like polyenes, where there is one orbital and one electron per atom, that is, a half-filled band. The theorem was first stated in 1955 by Rudolf Peierls. In the case of one-dimensional systems and half-filled bands, we use a proof by Lionel Salem. The Hamiltonian is expanded in a Taylor series for a geometry with equal bond lengths ... 

Geometry optunizatiou by semi-empirical methods gives the bonding pattern seen in Figure 16.5. Each N atom corresponds to one CH unit in polyacetylene (PA). In PA, each CH unit contributes one % orbital and one electron. The valence band is half filled and the systan PA is therefore subject to Peierls distortiou. In (SNj, on the other hand, the sp hybridized S atom contributes two electrons to the jt-system. The jt-system of (SN)x is therefore three-quarters filled and not subject to auy Peierls distortion. However, three-quarter filling leads to other peculiarities, as we will see next. 

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