Peierls distortions

One-dimensional crystals are subject to a distortion that leads to alternating short and long bonds. The half-filled valence band thereby develops a gap at the Fermi level. We have met this situation already for polyenes. The result is easily calculated using the Hiickel model with bond-length-dependent coupling p. 

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  

The second-order Jahn-Teller term is the last of the terms of the right member. This term may be calculated using the expressions given in Equation 16.36 for the wave functions. Since Ej Eg, the denominator is negative, but has to compete 

FIGURE 16.5 Oligomer of polyene. Geometry-optimized CC bond distances are given, (a) The neutral molecule and (b) The positive ion. 

If there is one electron and one orbital per site, the band will be half-filled. We allow the period to be doubled (from a to 2a). We are free to move every second nucleus to decrease its distance to one neighbor and to increase the distance to the other neighbor. This leads to energy savings due to the second term in Equation 16.40, based on the distortion of the bands in the vicinity of the new gaps. 

The combination of s-orbitals could also be demonstrated for a linear string of hydrogen atoms. Otherwise, hydrogen does not form a stable metallic solid, neither a one- nor three-dimensional one. The chain separates into individual hydrogen molecules. A similar 

The reason for the bond separation is called Peierls distortion. In the case of hydrogen atoms, this leads to a complete decay of the chain into individual hydrogen molecules under normal conditions. MetaUic hydrogen is only possible under extreme conditions, e.g., on the planet Jupiter. If the chains consist of -CH- units (poly-acetylene), Peierls distortion leads to an asymmetry of the neighboring bonds. 

Poly-acetylene is an insulator, but it can be transferred by oxidation into an electronic conductor. These intrinsically conducting polymers will be described in Chapter 11. 

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SN)x are free to move under the influence of an applied potential difference and thus conduction occurs along the polymer chain. The S-N distances in the chain are essentially equal, consistent with a delocalized structure. The increase in conductivity with decreasing temperature is characteristic of a metallic conductor. The predicted Peierls distortion is apparently inhibited by weak interactions between the polymer chains (S—S = 3.47-3.70 A S—N = 3.26-3.38 A.). ... 

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 ... 

Band structure for a chain of H atoms. Left, with equidistant atoms right, after PEIERLS distortion to H2 molecules. The lines in the rectangles symbolize energy states occupied by electrons... 

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. 

The structure of MnP is a distorted variant of the NiAs type the metal atoms also have close contacts with each other in zigzag lines parallel to the a-b plane, which amounts to a total of four close metal atoms (Fig. 17.5). Simultaneously, the P atoms have moved up to a zigzag line this can be interpreted as a (P-) chain in the same manner as in Zintl phases. In NiP the distortion is different, allowing for the presence of P2 pairs (P ). These distortions are to be taken as Peierls distortions. Calculations of the electronic band structures can be summarized in short 9-10 valence electrons per metal atom favor the NiAs structure, 11-14 the MnP structure, and more than 14 the NiP structure (phosphorus contributes 5 valence electrons per metal atom) this is valid for phosphides. Arsenides and especially antimonides prefer the NiAs structure also for the larger electron counts. 

Regular jc-stack Peierls distorted 7E-stack p-type doping (a) (b) (c)... 

Figure 4 Schematic band structures for (a) a regular n-stack (b) a Peierls distorted n-stack and (c) a Peierls distorted n-stack after p-type doping...

Figure 9 (Top) schematic of bistability in 1,3,2-dithiazolyl radicals arising from a solid-solid phase transition between regular and Peierls distorted n-stacks (bottom) free energy diagram of the two structural phases present...

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. 

Poly(4-phenoxybenzoyl-1,4-phenylene) (PPBP), sulfonated, 23 718 Polyacetal, antioxidant applications, 3 121 Polyacetaldehyde, 1 103 Polyacetal fiber, 13 392 Polyacetylene, 7 514-515 26 953 conduction in, 7 527 22 208 molecular structure of, 22 211 optical band gap, 7 529t Peierls distortion in, 22 203, 208 room temperature conductivity, 7 532 synthesis of, 22 213... 

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... 

Figure 10 (a) Splitting apart of the half-filled band on the asymmetric distortion of Figure 8a. This is the higher of the two bands of Figure 9(a). Notice the change in orbital character as a result of a Peierls distortion for the platinum chain, (b) Plot of orbital energy vs k, showing the orbitals most strongly affected.

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-... 

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