A half-filled band

Equation (43) assumes a specific value for Sq- A more general result follows from the fact that the Fourier transforms in (41) will be small, except for small values of o)p so that, as expected, to obtain high probabilities of charge transfer, Eq should either be in the solid band or, at least, not lie very far from the band. An interesting result, showing the dependence of P, on Eq, can be obtained for the one-dimensional model of section 3.3 with a half-filled band, V(t) of exponential form, P — co) = 0 and V% XB. This is... 

For Aat = 1, corresponding to a half-filled band, kj = njla and for Aat = 2, corresponding to a filled band, = r/a (see Fig. 1.28). In fact for CTSs Mt = Q-Fet us consider briefly the rather simple polyacetylene molecule, -(CH) , in which each carbon is o bonded to only two neighbouring carbons and one hydrogen atom with one n electron on each carbon (the orbital pointing perpendicularly to the chain direction). If the carbon bond lengths were equal, with one n electron per formula unit, it would imply a metallic state E < 0) as discussed above. However, neutral polyacetylene is a semiconductor with an energy gap of approximately 1.5 eV. The reason for this discrepancy is discussed next. ... 

There are two competing mechanisms to produce an insulating state for the half-filled o band. One is through valency disproportionation, and this is the case for BaBiOs, i.e., 2BiIV — Bira + Biv. If BiIV had not disproportionated in BaBiOj, we would expect this compound to be metallic because it would have a half-filled band. Many have been confused by this disproportionation description because of confusion about the meaning of valent states as opposed to real charges. We will come back to the subject later. 

In what follows we should bear in mind that the generation of a diamagnetic metallic state (irrespective of whether it is a superconductor or not) will not be favored by a half-filled band of electrons. Either a Peierls distortion or the generation of an antiferromagnetic insulating state will result, with a ferromagnet being less likely for the reasons discussed. Superconductivity in these materials is in fact only observed if electrons are removed, or (less commonly to date) added to the half-filled band. Considerable effort is underway to theoreti-... 

The factor 0.03 is deduced here for s-state and a half-filled band, and simple diagonal disorder (Fig. 1.17). We believe that it has much wider validity, and can be applied, at any rate in a theory of non-interacting electrons (as for instance in a semiconductor) to any form of disorder or for p- or d-states. Our confidence depends on the success of the scaling theory of Abrahams et al. (1979), which will be outlined in Section 13. 

Fig. 7.21 Bonding of a mole of lithtum atom 2s orbitals to form a half-filled band. Heavy shading indicates the filled portion of the band, the top of which is called the Fermi level.

The nomenclature for band filling is that a filled band has two electrons (or holes) per site (with spin up and spin down) a half-filled band has only one electron (or hole) per site a quarter-filled band has one electron (or one hole) per two sites. 

We can see this in the case of a half-filled band by following the arguments of Schulz. There is strong short-range antiferromagnetic order... 

As mentioned in Section II, LRO in two dimensions can exist only for a real order parameter, that is, for CDW in a half-filled band. This would be the case for BOW in the polymers or the Peierls state, which would be stabilized by transverse hopping or interchain coupling. This is also the case of the CDW state of the n = 1 two-dimensional Hubbard model. All other types of instabilities, such as those treated in the RPA previously in Section V, require three-dimensional coupling to stabilize any LRO. 

Figure 2 (a) Linearized electron dispersion in the Luttinger approximation (A) and diagrammatic representation of elementary interactions g, g4 (B). Solid and dashed lines represent electrons near kF and - kF, respectively. The g3 interaction exists only in case of a half-filled band (b) Diagrammatic representation of the Cooper pair susceptibility A(q, to) and the density wave susceptibility II (2kF + q, second order (D) which shows the mixture between Cooper and Peierls channels. 

A dependence close to a linear law is observed down to 100 K. At low temperature, both the thermal expansion and the pressure coefficient are small. Therefore, the constant-volume temperature dependence of the resistivity does not deviate from the quadratic law observed under constant pressure. At this stage it is interesting to stress that the theory of the resistivity in a half-filled band conductor [63], including the strength of the coulombic repulsions as derived from NMR data (Section III.B), should lead to a more localized behavior than that observed experimentally in Fig. 14. 

The theoretical problems raised by the study of CPs will be discussed below as the need arises. Many special properties of CPs are related to their quasi-one-dimensional character for instance, the large influence of disorder, the importance of residual three-dimensional coupling, and the importance of electron-phonon interactions, which, among other consequences, manifests itself in the case of a half-filled band by the occurrence of the Peierls instability. Much of the early theoretical work was concerned with PA, which, as we shall see, is peculiar among presently known CPs by having a degenerate ground state (see Section IV.B). 

Alternation of the CC bond lengths along the chain and the existence of a large energy gap are well-established facts in PA (see Chapter 12, Section II.C.2). However, since each carbon atom contributes one tt electron, there is at first sight no obvious reason why CC bonds should not be equivalent. If they were, and taking into account the electron spin, the tt electrons should generate a half-filled band such a material is a metal. If there is bond alternation, the one-dimensional unit cell is doubled and a gap opens at the Brillouin zone boundary the material is a semiconductor. 

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. 

Before leaving this section, we need to tell you a very important point that beginners often forget. As in the case of Jahn-Teller instability, Peierls instability occurs for particular electron counts. For example, dimerization (a distortion leading to the doubling of the elementary unit cell parameter) is expected to occur only in the case of a half-filled band (or nearly half-filled if the material is not stoichiometric), i.e., Peierls instability depends on band population. 

Figure 7.7. The Peierls distortion of a one-dimensional metallic chain, (a) An undistorted chain with a half-filled band at the Fermi level (filled levels shown in bold) has an unmodulated electron density, (b) The Peierls distortion lowers the symmetry of the chain and modulates the electron density, creating a CDW and opening a band gap at the Fermi level, (c) The Fermi surface nesting responsible for the electronic instability.

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