Weak disorder

In the presence of weak disorder, one should consider an additional contribution to the resistivity due to weak localisation resulting from quantum interference effects and/or that due to Coulomb interaction effects. A single-carrier weak localisation effect is produced by constructive quantum interference between elastically back-scattered partial-carrier-waves, while disorder attenuates the screening between charge carriers, thus increasing their Coulomb interaction. So, both effects are enhanced in the presence of weak disorder, or, in other words, by defect scattering. This was previously discussed for the case of carbons and graphites [7]. 

Typical magnetoconductance data for the individual MWCNT are shown in Fig. 4. At low temperature, reproducible aperiodic fluctuations appear in the magnetoconduclance. The positions of the peaks and the valleys with respect to magnetic field are temperature independent. In Fig. 5, we present the temperature dependence of the peak-to-peak amplitude of the conductance fluctuations for three selected peaks (see Fig. 4) as well as the rms amplitude of the fluctuations, rms[AG]. It may be seen that the fiuctuations have constant amplitudes at low temperature, which decrease slowly with increasing temperature following a weak power law at higher temperature. The turnover in the temperature dependence of the conductance fluctuations occurs at a critical temperature Tc = 0.3 K which, in contrast to the values discussed above, is independent of the magnetic field. This behaviour was found to be consistent with a quantum transport effect of universal character, the universal conductance fluctuations (UCF) [25,26]. UCFs were previously observed in mesoscopic weakly disordered... 

As we saw in Section 3.2, in the absence of disorder (A(x)=Ao), the electron spectrum has a gap between the energies =-Aq and e=+Ao- Disorder gives rise to the appearance of electron states inside the gap, although for weak disorder a pseudogap still exists. Using the phase formalism [471, Ovchinnikov and Erikhman derived a losed expression for the integrated average density of states (J of the... 

From Eq. (3.23) it is clear that at weak disorder (g 1) the density of states close to the middle of the pseudogap is strongly suppressed. The reason for this is that a large fluctuation of A(jc) is required in order to create an electron state with energy e < Aq. This makes it possible to apply a saddle-point approach to study the typ-... 

As has already been mentioned in the discussion of the stacking model, such equations are particularly useful for the analysis of nanostructured material with weak disorder in order both to assess the perfection of the material and to discriminate among lattice and stacking models (cf. Sect. 8.8.3). 

For K = 0, high temperatures, but weak disorder we adopt an alternative method by mapping the (classical) one-dimensional problem onto the Burgers equation with noise [26]. With this approach one can derive an effective correlation length given by... 

In the presence of weak disorder the free energy density of this stripe domain state is given by... 

Summary. We study how the single-electron transport in clean Andreev wires is affected by a weak disorder introduced by impurity scattering. The transport has two contributions, one is the Andreev diffusion inversely proportional to the mean free path i and the other is the drift along the transverse modes that increases with increasing . This behavior leads to a peculiar re-entrant localization as a function of the mean free path. 

Here we report how the single electron transport in Andreev wires at low temperatures T impurity scattering assuming that inelastic processes are negligible. The Andreev wire is clean in the sense that the mean free path is much longer than the wire diameter, 3> a. 

In the last two sections we analyzed spectral and relaxation properties of 3D and 2D strong dipolar excitons in high-quality crystals at low temperatures in terms of the strong excitonic coherence of band width 500 cm l, preserving the properties of the quasi-ideal crystal structure (what we called the intrinsic surface-bulk system) in the presence of weak disorder A... 

In what follows, we present in Section IV.A a theory of the effects of weak disorder on the retarded interactions of 2D strong dipolar excitons, and in Section IV.B we analyze the effects of stronger disorders on the coulombic interactions, calculating the density of states and absorption spectra in 2D lattices, in the framework of various approximations of the mean-field theory. 

For weak disorders, the correction Xk tends to zero. Now, introducing the resolvent of the crystal without disorder (V — 0),... 

Thus, we study the binary alloy (cf. Section IV.B) with the intermolecular interactions W of Section I. We are in the case of weak disorder A W, e.g. of a virtual crystal with one polariton band, and for A W we have separation into two bands, around vA and vB, each one embodying a polariton broadened by the disorder. 

Even in typical disordered metals, the classical model for MR breaks down due to quantum corrections to conductivity, especially at low temperatures [13]. In the presence of weak disorder, carriers get localized by repeated back-scattering due to constructive quantum interference, and this is called weak localization (WL). A weak magnetic field can destroy this interference process and delocalize the carrier. As a result, a negative MR (resistivity decreases with field, usually less than 3%) can be observed at temperatures around 4 K. Another quantum correction to low temperature conductivity is due to e-e interaction contributions. This is mainly due to the fact that carriers interact more often when they diffuse slowly in random disorder potentials. The resistivity increases (usually less than 3%) with field due to e-e interaction contributions. Hence, the total low-field magnetoconductance (MC, Act) due to additive contributions from WL and e-e interactions is given by... 

Thus weak disorder may be the main source of localization and conjugation length limitation. The variations in electronic properties such as absorption will then not always follow laws such as those given in Section II.C N 1 variations are well verified for polyenes, but then would not give the correct extrapolation for N — °°. Weak disorder also localizes excitations for instance, photoexcited charge carriers. It will induce the presence of polarons even if they were not bound states of the perfect chain, or of bipolarons, modify the transport properties, and increase the lifetime of charge carriers against recombination [98]. 

In a second step interchain coupling is taken into account. Interchain transport can now take place. Two limiting cases have to be considered. In the case of weak disorder (the chains are reasonably parallel to each other), interchain coupling, by reintroducing some three-dimensional features, can prevent one-dimensional effects, such as one-dimensional localization, which reduce conductivity. If Ae (the mean-square deviation of the on-site energy) is a measure of the disorder, the condition for the onedimensional localization to be removed is... 

Figure 7.4. (a) A schematic DOS curve showing localized states below a critical energy, fj, in the conduction band. Conduction electrons are localized unless the Fermi energy is above E. (6) In weakly disordered metals, a pseudogap, forms over which states are localized around the Fermi energy, owing to an overlap between the valence band and conduction band tails. 

Cb is the Boltzmann conductivity given by Eq. (7.51), is the momentum at the Fermi energy and C is a constant estimated to be near unity. This equation is derived as a description of the effects of weak disorder on the conductivity well above E(. and is extended to describe the conductivity at the mobility edge. The first term on the right hand side is the usual Boltzmann conductivity to which a factor g is added for the same reason as in Eq. (7.52). The second term in the bracket describes the effects of multiple scattering on the electron. Briefly, the amplitude of the wavefunction contains a sum over the scattering terms a, such that Za, = 1. Only the first order term a, contributes to the conductivity so that o is proportional to. 

One of our goals for studying TauD was to correlate our spectroscopic results with the structural information available from the X ray crystal structure of the ternary complex prepared under anaerobic conditions. Subsequently, this information could be used as a foundation to study similarities in catalytic mechanisms between different members of this enzyme class. Because the catalytic mechanism involves the displacement of bound water molecules from the Fe(II) cofactor, we studied samples of Fe(II)NO-TauD without cosubstrates and with just the aKG added to determine if FIYSCORE could be used to follow this chemistry. Figure 16(a) shows the FIYSCORE spectrum at g = 4.00 for Fe(II)NO TauD in aqueous buffer without the two cosubstrates. Two water molecules should be coordinated to the Fe(II)NO center under these conditions. The HYSCORE shows a weak, disordered system of overlapping H arcs for this sample, which most likely reflects a high degree of... 

We will discuss these fracture properties of disordered solids, modelled by the random percolation models, and concentrate on their statistics, given by the cumulative failure strength distribution F a) under stress a, and the most probable fracture strength erf of such samples. We will discuss separately the cases for weak disorder p 1) and strong disorder p Pc)-The scaling properties of <7f near p pc and the nature of the competition between the percolation and extreme statistics here, will be discussed in detail. 

Except for perfect crystals, the fracture surfaces in solids are never very smooth. In fact, it seems now to be established experimentally that for weakly disordered solids, the surfaces formed during fracture processes are very rough, and on a mesoscopic scale (which is much larger than the atomic scale but smaller than the macroscopic sample size) they are observed to have self-affine properties (Mandelbrot et al 1984, Bouchaud et al 1993 a,b, Roux 1994). These have recently been observed using various fractographic investigations. By self-affinity of the fracture surface, we mean that the surface coordinate in the direction perpendicular to the crack or fracture x — y) plane has the scaling property such that... 

---