Peierls coupling

A couple of theoretical studies [5,19] have hitherto attempted to estimate the Peierls transition temperature (Tp) for metallic CNT. A detailed theoretical check with respect to the stability of metallic wavefunction in tube (5, 5) has also... 

It will be intriguing to theoretically examine the possibility of superconductivity in CNT prior to the actual experimental assessment. A preliminary estimation of superconducting transition temperature (T ) for metallic CNT has been performed considering the electron-phonon coupling within the framework of the BCS theory [31]. It is important to note that there can generally exist the competition between Peierls- and superconductivity (BCS-type) transitions in lowdimensional materials. However, as has been described in Sec. 2.3, the Peierls transition can probably be suppressed in the metallic tube (a, a) due to small Fermi integrals as a whole [20]. 

In the weak-coupling limit unit cell a (, 0 7a for fra/u-polyacetylene) and the Peierls gap has a strong effect only on the electron states close to the Fermi energy eF-0, i.e., stales with wave vectors close to . The interaction of these electronic states with the lattice may then be described by a continuum, model [5, 6]. In this description, the electron Hamiltonian (Eq. (3.3)) takes the form ... 

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 most striking implication of the electron lattice coupling in ID chains is the appearance of the semiconducting state the equal bond ID lattice (metallic state) is unstable (33) with respect to a lattice distorsion and this so called static Peierls instability is the origin of the opening of the intrinsic band gap at the edge of the B.Z. with an infinite density of states there and the presence of band alternation. 

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... 

Although the anisotropy of the complexes electrical conductivity was not mentioned, the three-dimensional nature of the complex facilitates a metal-like temperature dependence of its conductivity down to 4K where a = 105 2 lcm l (room-temperature a is a respectable 300 2 1cm 1). The absence of a metal-insulator transition down to 4K shows that the Peierls instability has been successfully avoided by increasing the interstack coupling. 

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. 

Semiempirical methods which involve an explicit quantum mechanical treatment of the -electrons by PPP-type theories, coupled with a classical a-com-pression energy, have been employed by Paldus et al. to investigate the general question of Peierls distortion in polyenes and in large [n]-annulenes.22-25 The results show that the r-energy tends to be... 

Findlay, S.E.G., Pace, M.L., Lints, D., Cole, J.J., Caraco, N.F., and Peierls, B. (1991) Weak coupling of bacterial and algal production in a heterotrophic ecosystem the Hudson River estuary. Limnol. Oceanogr. 36, 268-278,... 

Cu(tfd)2 units by 0.015 A in an orthogonal direction to fill in the space vacated by the TTF molecules. This existence of a progressive dimerization in a regular chain of molecules of spin 1/2 along with the observed activated temperature dependence of the magnetic susceptibility below Tc has been interpreted in terms of a spin-Peierls instability77-78) in a one-dimensional antiferromagnetically coupled chain. 

At high temperature, TTF TCNQ is metallic, with a(T) oc T-2 3 since TTF TCNQ has a fairly high coefficient of thermal expansion, a more meaningful quantity to consider is the conductivity at constant volume phonon scattering processes are dominant. A CDW starts at about 160K on the TCNQ stacks at 54 K, CDW s on different TCNQ chains couple at 49 K a CDW starts on the TTF stacks, and by 38 K a full Peierls transition is seen. At TP the TTF molecules slip by only about 0.034 A along their long molecular axis. 

Coupled ID electronic and magnetic properties have been reported in (Per)[M(mnt)2] complexes (131). Some of these systems undergo simultaneous Peierls and spin-Peierls transitions, but the existence of a real interplay is not yet established. This work was reviewed in Sections II and III. 

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. 

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. 

Recently, the spectral study of DMTM(TCNQ)2 phase transition was performed [60]. The salt is a quarter-filled organic semiconductor containing segregated chains of TCNQ dimers and DMTM counterions. This material undergoes an inverted Peierls transition, which has tentatively been explained in terms of a crystal-field distortion. It was shown that the experimental values of unperturbed phonon frequencies and e-mv coupling constants are nearly independent of temperature. The dimer model fails to reproduce the phonon intensities and line shapes and underestimates the coupling constants, whereas the CDW model produces better results... 

Peierls pointed out in 1955 that a one-dimensional metallic chain is not stable at T = 0 K, against a periodic lattice distortion of wave vector 2kF, as the result of electron-phonon coupling, opening a gap 2A at the Fermi level. From this fact a collective electronic state results called a charge density wave (CDW). In the limit where U, the intrasite Coulomb repulsion, is infinite, since a given k state cannot be occupied by more than one... 

The spin-Peierls 2kF instability usually occurs when U is big and is originated by spin-phonon interactions. As in the CDW case, in real systems, the interchain Coulomb interactions couple the SDW and may lead to phase transitions at T > 0. 

It may be recalled here that the Peierls transition is basically a onedimensional effect coming from the divergent response in one dimension of the electron system at 2kF. However, because of the fluctuations, any transition is possible only at 0 K in one dimension and not at the temperature Tup predicted by mean-field theory. It is then an effect of the (small) interchain coupling to restore a transition temperature lower than TMF but finite. When the interchain coupling becomes too large (under high pressure, for instance) the one-dimensional character is lost and the Peierls transition is suppressed [2,3]. 

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