Disorder and Localization

In 1958, Anderson [9] showed that localization of electronic wavefunctions occurs if the random component of the disorder potential is large with respect to the bandwidth of the system, as shown in the schematic diagram in Fig. 3.1. The mean free path ( ) in a system with bandwidth B, random potential Vo, and interatomic distance a is given by 

The ratio Vo/B determines the transition from coherent diffusive propagation of wavefunctions (delocalized states) to the trapping of wavefunctions in random potential fluctuations (localized states). If I Vo, then the electronic states are extended with large mean free path. By tuning the ratio Vq/B, it is possible to have a continuous transition from extended to localized states in 3D systems, with a critical value for Vq/B. Above this critical value, wave-functions fall off exponentially from site to site and the delocalized states cannot exist any more in the system. The states in band tails are the first to get localized, since these rapidly lose the ability for resonant tunnel transport as the randomness of the disorder potential increases. If Vq/B is just below the critical value, then delocalized states at the band center and localized states in the band tails could coexist. 

Strongly localized, l a l — Mean free path a — Interatomic distance 

The critical energy that separates localized from extended states is called the mobility edge (Ec). Mott pointed out that, as the extent of disorder 

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A description of diffuse scattering that allows for short-range substitutional disorder and local atomic displacements that accompany the SRO can be obtained by expanding the exponential in the general diffraction equation in terms of powers of displacement... 

We have determined the dependence of y, y, o, and p on the temperature in PHT. Fig. 2 shows that the strongly T-dependent a is mainly due to y(T). When the temperature of a PHT-FET increases over 340 K, which is about the glass transition temperature for PHT, the mobility and conductivity decrease, Fig. 2. This has b eij interpreted to arise from some structural changes, such as conformons, and/or the outdiffusion of acceptor dopants, Fig. 2 (c). The conformons cause further disorder and localization of the states. The hopping rates decrease rapidly with decreasing wave function decay lengths. 

Fig. 2. Enthalpy of solution of CaMigSi20g-CaAl2Si0g clinopyroxenes in Pb2B205 melt at 970 K. Brackets show experimental determination of Newton, Charlu and Kleppa (recent work). Lines show trends predicted by the "ordered", "completely disordered" and "local charge balance" models of Wood (1975) given the heats of solution of the pure end-members.

At a given ideal composition, two or more types of defects are always present in every compound. The dominant combinations of defects depend on the type of material. The most prominent examples are named after Frenkel and Schottky. Ions or atoms leave their regular lattice sites and are displaced to an interstitial site or move to the surface simultaneously with other ions or atoms, respectively, in order to balance the charge and local composition. Silver halides show dominant Frenkel disorder, whereas alkali halides show mostly Schottky defects. 

The surface reconstruction of Au(110) is more rapid than that of Au(l 1 l)andAu(100).257,467,504-514,516-518Au(533)andAu(311), localized in the [(110)-(100)] zone, and Au(221) and (331), localized in the [(111)-(110)] zone, exhibit stable terrace step structural arrangements largely free from disordering and facetting 485 Au(210) and (410), localized in the [(100)-(110)] zone, display only a short-range structural order related to the especially open nature of these faces. 

In spite of the absence of periodicity, glasses exhibit, among other things, a specific volume, interatomic distances, coordination number, and local elastic modulus comparable to those of crystals. Therefore it has been considered natural to consider amorphous lattices as nearly periodic with the disorder treated as a perturbation, oftentimes in the form of defects, so such a study is not futile. This is indeed a sensible approach, as even the crystals themselves are rarely perfect, and many of their useful mechanical and other properties are determined by the existence and mobility of some sort of defects as well as by interaction between those defects. Nevertheless, a number of low-temperamre phenomena in glasses have persistently evaded a microscopic model-free description along those lines. A more radical revision of the concept of an elementary excitation on top of a unique ground state is necessary [3-5]. 

Table 8.53 shows the main features of XAS. The advantages of EXAFS over diffraction methods are that the technique does not depend on long-range order, hence it can always be used to study local environments in amorphous (and crystalline) solids and liquids it is atom specific and can be sensitive to low concentrations of the target atom (about 100 ppm). XAS provides information on interatomic distances, coordination numbers, atom types and structural disorder and oxidation state by inference. Accuracy is 1-2% for interatomic distances, and 10-25 % for coordination numbers. 

Effect of off-diagonal dynamic disorder (off-DDD). The interaction of the electron with the fluctuations of the polarization and local vibrations near the other center leads to new terms VeP - V P, Vev - Vev and VeAp - VAPd, VA - VAd in the perturbation operators V°d and Vfd [see Eqs. (14)]. A part of these interactions corresponding to the equilibrium values of the polarization P0l and Po/ results in the renormalization of the electron interactions with ions A and B, due to their partial screening by the dielectric medium. However, at arbitrary values of the polarization P, there is another part of these interactions which is due to the fluctuating electric fields. This part of the interaction depends on the nuclear coordinates and may exceed the renormalized interactions of the electron with the donor and the acceptor. The interaction of the electron with these fluctuations plays an important role in processes involving solvated, trapped, and weakly bound electrons. 

Additional effect of diagonal dynamic disorder. The variations of the electron densities near the centers A and B due to polarization fluctuations and local vibrations lead to changes in the interaction of the electron with the medium and, hence, to changes in the shape of the potential energy surfaces Ut and Uf as compared... 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

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