Peierls transition theory

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]. 

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]. 

The effects of an external periodic potential VQ with Q = 2kP on an electronic-Peierls transition at T = TP have been investigated in detail by Hansen and Carneiro [51] within a mean-field theory. They have found in this case that the Peierls transition is somewhat smeared out around TP, but not depressed. The modified gap 2A(7) is larger than the original Peierls gap 2AP(7) below TP and it does not vanishes above TP. Thus only a change in the slope of A(7) at TP recalls in this case the underlying transition [28,51]. If VQ is small enough [V"Q < 0.1AP(0)], then A(T) varies dramatically around TP with dMdl maximum, at TP. One also gets in this case the two limits... 

H. E. Stanley Introduction to phase transitions and critical phenomena 32. A. Abragam Principles of nuclear magnetism 27. P. A. M. Dirac Principles of quantum mechanics 23. R. E. Peierls Quantum theory of solids... 

Essential parts of the BCS theory can be taken over in the mean-field theory of the Peierls transition if one replaces the Debye energy tkoo in the superconductor by the Fermi energy Ep in the metal. Since Eplhrop, 10 -100, the Peierls phase transition temperatures are considerably higher than the critical temperatures of BCS superconductors, ii) The frequency of the phonons which are responsible for the Peierls transition has, for T > the temperature dependence... 

The recent experimental confirmation of the existence of one-dimensional metallic systems has led to a rapid increase in the experimental and theoretical study of these conducting systems. The objective of this section is to acquaint the reader with the physical basis of the concepts currently being used to explain the experimental results. Emphasis is given to the development of one electron band theory because of its central importance in the description of metals and understanding the effects of lattice distortion (Peierls transition), electron correlation, disorder potentials, and interruptions in the strands. It... 

Theoretical studiesof the Peierls distortion showthat within mean field theory its presence can be a function of temperature (261, 347, 349). This leads to a transition from a band metal to a band semiconductor/insulator as the temperature is lowered below the transition temperature, Tp, termed the Peierls transition. The characterization of the metal-insulator transition in KsPt(CN)4-Bro.3(H20)s as a Peierls transition has triggered much of the increased work on the theory of the Peierls transition. 

Fig. 15. Schematic illustration of the temperature dependence of the Peierls transition in mean field theory for a one-dimensional metal. Above Tp the temperature dependence of the energy of the 2kp phonon is illustrated. Below Tp, the temperature dependence of the energy gap A is illustrated. The energy scale above and below Tp is not the same.

We can expect that in these highly conducting polymers electrons are much less localized and that they will reflect certain features of inorganic metals. But the chain-like structure of the conjugated polymers will give rise to a very anisotropic behaviour and some aspects of one-dimensionality will show up, such as the metal-insulator transition (Peierls-transition / /). The most exciting speculation is probably that on the existence of mobile conjugation defects, which can be described as non-linear excitations and are often referred to as solitons /5/ (Fig. U). These defects share many properties with non-linear excitations in other fields of physics and offer an interdisciplinary connection all the way from chemistry to elementary particle physics and field theory /6/. 

In the Mott-Hubbard theory on the other hand, it is shown that there exists an instability in the narrow-band electronic structure (Peierls instabihty ) and if the bandwidth decreases below a critical value, a sudden transition (Mott transition) takes place toward a complete localized situation. In this approach, it is assumed, in fact, that band magnetism does not exist and one has to deal only with 2 classes of materials... 

About one decade after the development of band theory, two Dutch industrial scientists at the NV Philips Corporation, Jan Hendrik de Boer (1899-1971) (de Boer was later associated with the Technological University, Delft) and Evert Johaimes Willem Verwey (1905-1981), reported that many transition metal oxides, with partially filled bands that band theory predicted to be metallic, were poor conductors and some were even insulating (de Boer and Verwey, 1937). Rudolph Peierls (1907-1995) first pointed out the possible importance of electron correlation in controlling the electrical behavior of these oxides (Peierls, 1937). Electron correlation is the term applied to the interaction between electrons via Coulombs law. 

The marked effects of disorder in pseudo-ID systems have been clarified by both experiment and theory. These include (a) transport in the absence of collective effects (INV 1, INV 13), (b) the role of impurities in pinning incommensurate CDW s and the effect on charge transport (F3), and (c) the relative effect of impurities on the Peierls and superconducting transition temperature. 

Landau-Ginzbuig coefficients 282 Landau-Ginzburg Hamiltonian, critical 303 Landau-Khalatnikov mechanism 564 Landau-Lifshitz theory, phase transitions 366 Landau-Peierls instabilities 285, 647 Langmuir-Blodgett film, atomistic simulations 85 Laplace equation 445... 

ABSTRACT. The general concepts of quasi-one-dimensional conducting polymers are introduced including the role of band theory, electron-phonon interactions, the Peierls ground state, and commensurability. The dominant defect states present upon doping, solitons, polarons and bipolarons, are discussed. Application of these concepts to polyaniline is made with emphasis on the mechanism for the insulator-to-metal transition. 

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