Random potential

PautmeierL, Richert R, Bassler H (1991) Anomalous time-independent diffusion of charge-carriers in a random potential under a bias field. Phil Mag B 63 587... 

Richert R, Pautmeier L, Bassler H (1989) Diffusion and drift of charge-carriers in a random potential - deviation from Einstein law. Phys Rev Lett 63 547... 

Figure 6.33 (a) Anderson localization due to disorder in site potentials. For comparison, potentials in a regular lattice are also shown in (b), fV is one-electron band-width in the absence of the random potential Vq. Localization is determined by (fV/y) ratio. 

All the preceding mechanisms of the carrier packet spread and transit time dispersion imply that charge transport is controlled by traps randomly distributed in both energy and space. This traditional approach completely disregards the occurrence of long-range potential fluctuations. The concept of random potential landscape was used by Tauc [15] and Fritzsche [16] in their models of optical absorption in amorphous semiconductors. The suppressed rate of bimolecular recombination, which is typical for many amorphous materials, can also be explained by a fluctuating potential landscape. 

Protection of quantum states from the influence of noise is important. It has been shown that the alternating transport of a EEC generated by the fast-forward driving field suppresses the influence of a fluctuating random potential on the EEC [47], The EEC is kept undisturbed for a longer time than is characteristic of the simple trapping with a stationary potential because the effective potential, which the quanmm state feels, becomes uniform when the transport velocity is sufficiently large. 

Fig. 1.17 Potential energy used by Anderson (1958) (a) without a random potential and (b) with such a potential. The density of states is also shown.

In the case discussed here a Mott transition is unlikely the Hubbard U deduced from the Neel temperature is not relevant if the carriers are in the s-p oxygen band, but if the carriers have their mass enhanced by spin-polaron formation then the condition B U for a Mott transition seems improbable. In those materials no compensation is expected. We suppose, then, that the metallic behaviour does not occur until the impurity band has merged with the valence band. The transition will then be of Anderson type, occurring when the random potential resulting from the dopants is no longer sufficient to produce localization at the Fermi energy. 

Fig. 46. Nonlinear circuit with random potential generator.

Hu results from the effects of impurities with random potential strength Ui and positions x. The potential strength is characterized by / = 0 and UiUj = U mp5itj, and includes a forward and a backward scattering term proportional to po and pi, respectively. The disorder average of the impurity potential U(x) follows then to be given by U(x) = 0 and... 

At finite temperatures thermal fluctuations wipe out the random potential, which leads to the pinning of the CDW at t = 0 and K < K . Thus, there... 

For co = 0, this equation reduces to (14), whereas dm/dz = 0 reproduces (23). Mathematically, Eq. (24) is a well-known random-potential eigenvalue problem, which can be solved numerically or by transfer-matrix methods [5, 153],... 

Figure 14 shows some examples. An interesting point is that all modes are localized [105], as one expects from the quantum-mechanical analog of a one-dimensional electron gas in a random potential [157]. Alternatively, from a localized tight-binding point of view, micromagnetic delocalization... 

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. 

Fig. 3.1. Random potentials vs. bandwidth, wavefunctions and density of states...

Irradiation of the crystal by electrons or neutrons is the simplest way of introducing the defects by controlled means. Optical properties of organic conductors are sensitive to changes in the electron distribution induced by irradiation defects (i.e., their spectra are sensitive to the localization of the carriers, due to random potentials in the environment of the defects). The electronic absorption spectra give information on the density of charge carriers and their localizations as well as on the electronic energy levels. 

Let us note, however, that it has been proposed recently that the disorder-induced one-dimensional localization could not be effective in particular cases namely, if the sites with random potential can be associated by pairs, or dimers. In that case the random dimer model shows that there should exist an energy spectrum of electrons that can propagate freely [31]. 

Anderson 1958). The crystal is described by an array of identical atomic potential wells and the corresponding band of electronic states is broadened to a band width B by the interaction between atoms. The disordered state is represented by the same array of sites to which a random potential with average amplitude is added. Anderson showed that when VJB exceeds a critical value, there is zero probability for an electron at any particular site to diffuse away. All of the electron states of the material are localized and there is no electrical conduction at zero temperature. 

In addition to Mott-Hubbard localization, there is another common source of electron localization, which arises when a lattice is under a random potential (e.g. a random distribution of alkali metal ions in alkali metal containing transition metal oxides). For a metal, a practical consequence of a random potential is to open a band gap at the Fermi level. Insulating states induced by random potentials are referred to as Anderson localized states (see Anderson Localization)) ... 

It is clear that the introduction of a critical volume fraction is a step toward dealing with percolation on a continuum. To this end Zallen and Scher (1971) considered the motion of a classical particle in a random potential, V r), and introduced a function, which detines the fraction of space accessible to particles of energy E. The connection with percolation is in the fact that, for energies such that 4>(E) > c(.E), there are infinitely extended volumes of allowed (V < T) space. The critical value c is identified with 0.15 for d = 3, and delocalized states appear above c ... 

Anderson s simple model to describe the electrons in a random potential shows that localization is a typical phenomenon whose nature can be understood only taking into account the degree of randomness of the system. Using a tight-binding Hamiltonian with constant hopping matrix elements V between adjacent sites and orbital energies uniformly distributed between — W/2 and W/2, Anderson studied the modifications of the electronic diffusion in the random crystal in terms of the stability of localized states with respect to the ratio W/V. 

In this paper, we studied the glass transition and the localization-delocalization transition in a disparate-size hard-sphere mixture from a dynamical viewpoint. For Cl = 0.5, the existence of the delocalized phase of the small particles is confirmed by investigating the frequency-dependent diffusion constant. Near the glass transition, we found an additional quasi-elastic structure in " q,uj) and Di(u>) at small o), which suggests that the diffusion mechanism of the small particles would change from the liquid-like diffusion to a slow diffusion in a random potential. 

The early conductivity model of Stevels (1957) and Taylor (1956, 1959) is in a sense a random potential energy model. It is assumed in this model that the ions experience randomly varying potential energy which is due to the presence of a random structure. For the d.c. conduction, the... 

Experimentally one generally distinguishes several spin- lattice (longitudinal) relaxation times, Tipd, Tig, Tip etc.. In glasses due to the presence of random potential energy barriers, the relaxation times exhibit a distribution. In general, they can be expressed as ... 

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A (CH), polymer chain, however, is considered to consist of an array of about 500 carbon atoms (Shirakawa et al., 1980), which may reduce the localization syndrome described above. The localization characteristics of the electronic wave function have been studied with respect to the (CH), model consisting of 300 carbon atoms with the diagonal of the off-diagonal disorder of the potentials (Tanaka et al., 1983a). The results have shown that the eigenfunction near the Fermi level is relatively strong against the temptation to localize caused by the existence of the random potentials. 

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