Weak localization

From our transport measurements, we can conclude that at low temperatures, the conductivity of the bundle of buckytubes shows two-dimensional weak localization behavior and the MR is negative above 60 K the MR is positive and increases approximately... 

Fig. 4. (a) Magnetic field dependence of the high- and low-temperature MR, respectively. The solid lines are calculated using a simple two-band model for (a) and the 2D weak localization theory for (b) (after Song et o/.[16]). 

Benzyl alcohol (C6H5CH2OH). This has antibacterial and weak local anaesthetic properties and is used as an antimicrobial preservative at a concentration of 2%, although its use in cosmetics is restricted. 

Thus, since intramolecular bonding interactions in the solid are much stronger than relatively weak i n termo1ecu1 ar van der Vaals interactions, each molecular unit is essentially an independent source of nonlinear response, arrayed in an acentric cystal structure, and coupled to its neighbors mainly through weak local fields. In the rigid lattice. gas approximation, the macroscopic susceptibility X is expressed as... 

Mention should be made here of recent attempts by Piepho, Schatz and Krausz (46) to give a general interpretation of intervalence bandshapes in terms of a Hamiltonian equivalent to that of eq 6. They use vibronic eigenfunctions (following the method of solution of Merrifield (47)) rather than adiabatic Born-Oppenheimer (ABO) functions. Thus, the aim is to interpret an observed spectrum in terms of one vibrational coupling mode, which is antisymmetric. Their analysis of the spectrum of the Creutz-Taube ion yields a value of 0 of 1.215, i.e., a rather weakly localized ground state. Using their assumed unperturbed... 

One can show (Ericksen, 1975) that the homogenous state w(x) = Wi corresponding to a deeper minimum of g(w Oo) is the global minimizer of the functional (2.1) another homogenous configuration w(x) = Wo (the metastable state), is only a weak local minimizer... 

Ishizaki Y, Izumi M, Ohnuld H, Kalita-Lipinska K, Imakubo T, Kobayashi K (2001) Formation of two-dimensional weak localization in conducting Langmuir-Blodgett films. Phys RevB63 134201/l-5... 

These effects are described as weak localization . Experimental evidence for the effects predicted for amorphous metals is described in Chapter 10. It will be seen that the effects are only significant when l is small, and thus typically for noncrystalline materials. It is remarkable that, as a consequence of (51), inelastic scattering increases the conductivity. 

The assumption that a is proportional to the square of the density of states is normally deduced only in the regime Edwards cancellation theorem. As shown in Chapter 1, Section 6.3, however, this is valid only if we can write vf=h l dE/dk and deduce vf ccg 1, which means a uniform expansion of the band. This is certainly not so for weak localization. For the singularity predicted by Altshuler and Aronov, any correction to dE/dk will be zero if averaged over a range kBTabout EF. We consider then that if a>%,... 

Section 6, but a mathematical treatment has not, as far as we know, been given. However the results shown in Fig. 7.5 make it very probable that it is so, if weak localization is present, though it appears unlikely without. 

Figure 7.7 shows that at low temperatures hopping conduction is observed, giving further evidence of weak localization. 

Weak local nucleus that is stabilized by long-range interactions to give large extended nucleus... 

An X-ray photoelectron spectroscopic study of Ni(DPG)2I showed no evidence of trapped valence or any appreciable change in the charge on the metal upon oxidation.97 The site of partial oxidation and hence the electron transport mechanism is still unclear but one explanation of the relatively low conductivity is that the conduction pathway is metal centred and that the M—M distances are too long for effective orbital overlap. Electron transport could be via a phonon-assisted hopping mechanism or, in the Epstein—Conwell description, involve weakly localized electronic states, a band gap (2A) and an activated carrier concentration.101... 

When deriving Eqs. (2) we neglected possible weak localization corrections that may originate from quantum interference of electron waves. This approximation is legitimate if a magnetic field is applied as in Ref. [1] or dephasing is strong due to inelastic processes. 

The theory [15] cannot be applied to InO directly. But just as the divergence of the weak localization quantum corrections point to the Anderson localization, quantum corrections here point to the SIT. This returns us to the problem of low dimensions. Fluctuations are larger in the systems with low dimensions, but the difference is only quantitative and we may expect SIT... 

Transport of electrons along conducting wires surrounded by insulators have been studied for several decades mechanisms of the transport phenomena involved are nowadays well understood (see [1, 2, 3] for review). In the ballistic regime where the mean free path is much longer than the wire lengths, l 3> d, the conductance is given by the Sharvin expression, G = (e2/-jrh)N, where N (kpa)2 is the number of transverse modes, a, is the wire radius, a Fermi wave vector. For a shorter mean free path diffusion controlled transport is obtained with the ohmic behavior of the conductance, G (e2/ph)N /d, neglecting the weak localization interference between scattered electronic waves. With a further decrease in the ratio /d, the ohmic behavior breaks down due to the localization effects when /d < N-1 the conductance appears to decay exponentially [4]. 

Figure 4.22. Polariton solutions for a 3D mixed crystal in the mean-polarizability approximation (4.117). In strong local field (A3), one obtains a resonance of the virtual crystal cAwA + cBwB another solution, strongly shifted, exists at low frequencies. On the contrary, in weak local fields (A,), the frequencies of the pure A and B crystals, slightly shifted, are solutions. We note that for cB - 0, one of the solutions tends, for any strength of the local field A, to ojb, which is the frequency of B unshifted by the interaction with the lattice A.

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... 

Contour plots of two-dimensional functions help illustrate these concepts. In general, the equation f(x) — y defines a surface in R"+I. When n - 2, the plane curves corresponding to various values of y generate contour plots (or maps) of a function. Figure 4 shows the contour plots for the two-dimensional functions discussed before. Note, for the first, the two stationary points corresponding to a minimum and saddle point. For the second, note the region of weak local minima. The contour plots are shaded so that darker areas correspond to higher function values. 

Up to now we have discussed two extreme limits, the band picture on the one hand, and strong localization associated with interruptions in the metallic chains on the other. In fact, from work on thin metallic films and metallic glasses it is known that there is an intermediate region, that of weak localization. This occurs when the mean free path for elastic scattering (Lel) is only somewhat larger than, or comparable with, that for inelastic processes (Lin). In the first approximation there are corrections to the Boltzmann transport formula which depend on the ratio Lin/Lel in different ways for one-, two-, and three-dimensional materials. Weak localization... 

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