The Peierls model

In this chapter we describe the consequences of electron-phonon coupling in the absence of electron-electron interactions. The celebrated model for studying this limit is the so-called Su-Schrieffer-Heeger model (Su et al. 1979, 1980), defined in Section 2.8.2. In the absence of lattice dynamics this model is known as the Peierls model. We begin by describing the predictions of this model, namely the Peierls mechanism for bond alternation in the ground state and bond defects in the excited states. Finally, we reintroduce lattice dynamics classically and briefly describe amplitude-breathers. 

Formally, a is the electron-phonon coupling parameter defined by eqn (2.38), but it is often convenient to regard it as a semiempirical parameter. Notice that a positive value of A corresponds to a reduction in the bond length, and vice versa. It is this term in that couples the electrons to the lattice, and corresponds to eqn (2.40) with / = 0. A plays the role of an order parameter, whereby a nonzero value indicates a broken symmetry. 

K is the spring constant of the cr-bonds and dr is the average change in bond length relative to the a-bond reference value due to the 7r-electrons (see eqn (2.45)). 

The expectation value of the bond-order operator is a measure of the strength of that bond, as illustrated by the simple example of ethylene. Modelling this by two TT-orbitals with two electrons shared between them it is easily shown that the bonding molecular orbital has a bond-order value of +1, while the antibonding molecular orbital has a bond-order value of —1. Thus, a larger bond-order value implies a stronger bond. 

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A second major difficulty with the Peierls model is that it is elastic and therefore conservative (of energy). However, dislocation motion is nonconservative. As dislocations move they dissipate energy. It has been known for centuries that plastic deformation dissipates plastic work, and more recently observations of individual dislocations has shown that they move in a viscous (dissipative) fashion. 

As early as 1938, internal friction in vibrating zinc crystals was observed at strain amplitudes as small as 10The friction was attributed (with good cause) to dislocation motion (Read, 1938). This strongly indicated that the Peierls model could not be accepted as being quantitative. 

The Peierls model explains why a chain of unsaturated carbon atoms with one conduction electron per atom does not exhibit metallic properties. If all the atoms are spaced at eqnal distance, a, the basic cell in reciprocal space is the Brillouin zone in the interval -nlawave vector). With one electron per atom, the band would be half-filled and hence the chain would exhibit metallic behaviour. A periodic distortion of the chains, commensurate with the original structure, generates an -fold super-structure and reduces the Brillouin zone to -nlnaunit cell. The effect of the distortion is to open a gap at the boundaries k = n/na of the new Brilliouin zone (Figure 1.1). Therefore, if only states below the new gap are... 

The Su-Schrieffer-Heeger model in the limit of static nuclei is known as the Peierls model. This is defined and discussed in Section 4.2. 

Fig. 11.15. The fractional change in transfer integrals of poly(para-phenylene) from the uniform valne, t, in the noninteracting limit. The electron-phonon parameter used in the Peierls model (eqn (4.1) is A = 0.12. The labels refer to the bonds shown in Fig. 11.16. Only the upper rung of bonds are shown. Notice that the change in transfer integrals is opposite to the change in bond lengths.

As discussed in Chapter 4, the noninteracting limit in the adiabatic approximation is described by the Peierls model (defined in Section 4.2). The ground and excited state structures are easily obtained via the Hellmann-Feynman procedure, described in Section 4.4. 

Table 11.6 The relaxation energies of the IBiu state and polaron for para-phenylene oligomers (in eV) calculated from the Peierls model (eqn (4-1)) (t = 2.514 eV and A = 0.12 ...

Phason lines are linear defects with a [010] line direction, which can move along the [0 01] direction. It is unlikely, however, that phason lines move as a whole, that is, that the complete line performs a vertex jump in one single step. Although this has not been investigated in detail, it is a plausible assumption that phason lines move by a mechanism involving sequential jumps of small portions of the line, that is, by the formation of kinks and their subsequent movement along the line. This is in full analogy to the Peierls model, which describes dislocation motion by a kink-pair mechanism [36]. 

The effect of driving shear stresses on the dislocations are studied by superimposing a corresponding homogeneous shear strain on the whole model before relaxation. By repeating these calculations with increasing shear strains, the Peierls barrier is determined from the superimposed strain at which the dislocation starts moving. 

The Burgers vectors, glide plane and ine direction of the dislocations studied in this paper are given in table 1. Included in this table are also the results for the Peierls stresses as calculated here and, for comparison, those determined previously [6] with a different interatomic interaction model [16]. In the following we give for each of the three Burgers vectors under consideration a short description of the results. 

For the deformation of NiAl in a soft orientation our calculations give by far the lowest Peierls barriers for the (100) 011 glide system. This glide system is also found in many experimental observations and generally accepted as the primary slip system in NiAl [18], Compared to previous atomistic modelling [6], we obtain Peierls stresses which are markedly lower. The calculated Peierls stresses (see table 1) are in the range of 40-150 MPa which is clearly at the lower end of the experimental low temperature deformation data [18]. This may either be attributed to an insufficiency of the interaction model used here or one may speculate that the low temperature deformation of NiAl is not limited by the Peierls stresses but by the interaction of the dislocations with other obstacles (possibly point defects and impurities). 

Excellent agreement between experiment and onr calculations is obtained when considering the low temperature deformation in the hard orientation. Not only are the Peierls stresses almost exactly as large as the experimental critical resolved shear stresses at low temperatures, but the limiting role of the screw character can also be explained. Furthermore the transition from (111) to (110) slip at higher temperatures can be understood when combining the present results with a simple line tension model. 

Monte Carlo simulations [17, 18], the valence bond approach [19, 20], and g-ology [21-24] indicate that the Peierls instability in half-filled chains survives the presence of electron-electron interactions (at least, for some range of interaction parameters). This holds for a variety of different models, such as the Peierls-Hubbard model with the onsite Coulomb repulsion, or the Pariser-Parr-Pople model, where also long-range Coulomb interactions are taken into account ]2]. As the dimerization persists in the presence of electron-electron interactions, also the soliton concept survives. An important difference with the SSH model is that neu-... 

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 ... 

In this contribution, we review our recent work on disordered quasi-one-dimen-sional Peierls systems. In Section 3-2, we introduce the basic models and concepts. In Section 3-3, we discuss the localized electron stales in the FGM, while, in Section 3-4, we allow for lattice relaxation, leading to disorder-induced solitons. Finally, Section 3-5 contains the concluding remarks. 

The SSH model (Eq. (3.2)) is, essentially, the model used by Peierls for his discussion of the electron-lattice instability [33]. Its ground state is characterized by a non-zero expectation value of the operator. 

This model, which is sometimes referred to as the Fluctuating Gap Model (FGM) [42], has been used to study various aspects of quasi-one-dimensional systems. Examples arc the thermodynamic properties of quasi-one-dimensional organic compounds (NMP-TCNQ, TTF-TCNQ) [271, the effect of disorder on the Peierls transition [43, 44, and the effect of quantum lattice fluctuations on the optical spectrum of Peierls materials [41, 45, 46]. 

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 ... 

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... 

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. 

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... 

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