Peierls model

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. 

Comparison with the Holstein-Peierls Model and Transport... 

The ab initio calculations of polaron mobility on the basis of the Holstein-Peierls model including a nonlocal electron-lattice coupling (Hannewald and Bobbert [12,13] see references to earlier work therein) reproduced the temperature-... 

Group method was applied for dynamic aL and yL using ID Hubbard and U-V models and a Hubbard-Peierls model Hamiltonian in the case of linear polyenes. 

The symmetrized Density Matrix Renormalisation Group Procedure145 using a Hubbard-Peierls model Hamiltonian has given for a linear polyene chain quite satisfactory results for aL(static) and for yL(coa co, co2, ) (coa = co + a>2 + ) and for (yL/jV)(static). 

Pang and Liang have calculated the charge excitation gap for the Hubbard-Peierls model, which is defined for a neutral chain of N sites (number of electrons Ng = N) as Eg N) E Ne = N + l)+E Nc = N — 1) — 2E Ne = N) [105]. We note that this is not the optical gap in the extended Hubbard-Peierls model, since the definition excludes the contribution due to Coulomb attraction between electron and hole. Instead, this quantity becomes our definition of charge excitation band gap which constitutes the continuum edge, the reason being, taht at the band edge, the electron and hole are uncorrelated... 

For P = 0, as expected for the Hubbard-Peierls model, the binding energy Ef, is always calculated to be a small negative quantity 0 1/N) for any U and 6. This excellent agreement with the physical picture serves as a check on our numerical scheme. 

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. 

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. 

We start this investigation by treating the electronic degrees of freedom within the Born-Oppenheimer approximation, where the nuclear degrees of freedom are static, classical variables. The 7r-electron model that describes both electron-electron and electron-phonon interactions in the Born-Oppenheimer approximation is known as the Pariser-Parr-Pople-Peierls model. This is described and its predictions are analyzed in the following sections. Chapter 10 will deal with quantum phonons in an interacting electron model, specifically for trans-polyacetylene. 

Thus, the Pariser-Parr-Pople-Peierls model is defined as... 

First, we examine the relaxed and vertical energies of the Pariser-Parr-Pople-Peierls model as a function of the interaction strength. These transition energies are illustrated in Fig. 7.4. We first note the crossover in the vertical energies of the and states as a function of C (as already discussed in Chapter... 

Table 7.1 The vertical and relaxation energies (in eV) of a site linear polyene calculated from the Pariser-Parr-Pople-Peierls model with extrinsic dimerization (t = 2.5 eV, U = 10.0 eV, A = 0.1, and <5e = O.i ...

In the next section we describe how the semiempirical Pariser-Parr-Pople-Peierls model - a correlated 7r-electron model with electron-lattice coupling -quantitatively predicts the excitation spectrum of polyene oligomers, while it qualitatively predicts the spectrum for frons-polyacetylene thin films. 

The predictions of the Pariser-Parr-Pople-Peierls model (defined by eqn (7.1)) as a function of the Coulomb interaction strength for the linear polyene structure were described in Chapter 7. In this section we discuss the predictions taken from Barford et al. (2001) of the model for the particular parameter set relevant for trans-polyacetylene. 

Fig. 10.4. The staggered, normalized bond dimerization, S , (defined in eqn (4.27)) as a function of bond index from the centre of the chain of various states of trans-polyacetylene calculated from the Pariser-Parr-Pople-Peierls model. 1 A+ (crosses), (squares), 1 5 (triangles), 2 (diamonds) andpolaron (circles).

The adiabatic approximation is widely accepted as being applicable to the electronic states of conjugated polymers. As described above, solutions of an adiabatic Hamiltonian (namely, the Pariser-Parr-Pople-Peierls model) agree remarkably well with experimental observations for short polyenes. A linear extrapolation in inverse chain length of the experimental observations coincide with the experimental observations of the energies of the and states in thin... 

The Pariser-Parr-Pople-Peierls model is the adiabatic limit of this model, taken by setting M oo and treating the nuclear displacements classically. However, now we intend to quantize the nuclear degrees of freedom. To do this... 

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.

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