Electrons Anderson localization

Anderson localization is the localization of electrons on low-dimensional materials, which is induced by the irregularity of the periodic potential field [43]. Figure 17 gives a schematic representation of Anderson localization of a particle in one-dimensional box. The same is true for an electron on a polymer skeleton. A localized state in a completely periodic... 

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. 

We discuss in this section the effect of short-range interaction on the Anderson-localized states of a Fermi glass described in Chapter 1, Section 7, and in particular the question of whether the states are singly or doubly occupied. Ball (1971) was the first to discuss this problem. In this section we consider an electron gas that is far on the metal side of the Wigner transition (Chapter 8) the opposite situation is described in Chapter 6, where correlation gives rise to a metal-insulator transition. We also suppose that Anderson localization is weak (cca 1) otherwise it is probable that all states are singly occupied. 

Our problem is to estimate A . It is the mean repulsive energy of a pair of charges at a distance or1 from each other. This will depend on the effective dielectric constant of the electron gas. This should be large for weak Anderson localization and will effectively screen out the repulsion, except when both electrons are in the same atom. We therefore write... 

Throughout we make use of the pseudogap model outlined in Chapter 1, Section 16- A valence and conduction band overlap, forming a pseudogap (Fig. 10.1). States in the gap can be Anderson-localized. A transition of pure Anderson type to a metallic state (i.e. without interaction terms) can occur when electron states become delocalized at EF. If the bands are of Hubbard type, the transition can be discontinuous (a Mott transition). 

We note that at the consolute point the conductivity is still metallic, the appearance of an activation energy e2 occurring for somewhat lower concentrations. The reason for this, in our view, is as follows. The consolute point should occur approximately at the same concentration as the kink in the free-energy curve of Fig. 4.2, namely that at which the concentration n of carriers is of order given by n1/3aH 0.2. Above the consolute point there is no sudden disappearance of the electron gas as the concentration decreases its entropy stabilizes it, so metallic behaviour extends to lower concentrations, until Anderson localization sets in. Conduction, then, is due to excited electrons at the mobility edge, as discussed above. 

Attempts to take into account both localization and percolation or, in other words, to allow for quantum effects in percolation go back to Khmel-nitskii s pioneer paper [68]. The experimental attempts to study quantum effects in conductivity close to the percolation threshold have been undertaken in Refs. [69-71]. The physical sense of these results is stated in Ref. [71] and could be described as follows. The percolation cluster is non-uniform it includes both big conductive regions ( lakes ) and small regions (weak links or bottlenecks) which connect lakes to each other. On approaching the percolation threshold from the metallic side of the transition, these weak links become thinner and longer, and at x = xc the cluster breaks or tears into pieces just in such areas. As a result, exactly these conditions start to be sufficient for the electron localization. Thus, a percolation provokes an Anderson localization in bottlenecks of the percolation cluster. Sheng and collaborators [36,37,72] tried to take into account the influence of tunneling on conductivity for systems in the vicinity of the percolation transition. Similar attempts have been made in papers [38,56]. The obtained results prove that the possibility of tunneling shifts the percolation threshold toward smaller x values and affects material properties in its vicinity. 

It has been seen in the previous section that the ratio of the onsite electron-electron Coulomb repulsion and the one-electron bandwidth is a critical parameter. The Mott-Hubbard insulating state is observed when U > W, that is, with narrow-band systems like transition metal compounds. Disorder is another condition that localizes charge carriers. In crystalline solids, there are several possible types of disorder. One kind arises from the random placement of impurity atoms in lattice sites or interstitial sites. The term Anderson localization is applied to systems in which the charge carriers are localized by this type of disorder. Anderson localization is important in a wide range of materials, from phosphorus-doped silicon to the perovskite oxide strontium-doped lanthanum vanadate, Lai cSr t V03. 

The model also assumes that the disorder is uniform. The disorder is introduced at the beginning of the calculations by the Anderson localization criterion, but the effects of disorder on the transfer of the electron from site to site is not considered at a microscopic level. Scaling theory, which is described next, considers the microscopic disorder and reaches some diflerent conclusions. 

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)) ... 

Apparently, in the near future there will be developed (a) a detailed theory of surface excitons not only at the crystal boundary with vacuum but also at the interfaces of various condensed media, particularly of different symmetry (b) a theory of surface excitons including the exciton-phonon interaction and, in particular, the theory of self-trapping of surface excitons (c) the features of surface excitons for quasi-one-dimensional and quasi-two-dimensional crystals (d) the theory of kinetic parameters and, particularly, the theory of diffusion of surface excitons (e) the theory of surface excitons in disordered media (f) the features of Anderson localization on a surface (g) the theory of the interaction of surface excitons of various types with charged and neutral particles (h) the evaluation of the role of surface excitons in the process of photoelectron emission (i) the electronic and structural phase transitions on the surface with participation of surface excitons. We mention here also the theory of exciton-exciton interactions at the surface, the surface biexcitons, and the role of defects (see, as example, (53)). The above list of problems reflects mainly the interests of the author and thus is far from complete. Referring in one or another way to surface excitons we enter into a large, interesting, and yet insufficiently studied field of solid-state physics. 

We have supposed that the states are localized from E to E. and extended outside it. Anderson (1958) has shown that electrons in localized states cannot diffuse at T = 0°K. At finite temperatures they presumably can contribute to the conductivity only by phonon assisted hopping. We therefore expect a fall in (E) near E or E. of at least several orders of magnitude, as shown in Figure 3.3b. The energies E and E. are therefore mobility edges (Cohen (1969)). Evaluation of Eq. (3.4) with n (E) and (E)... 

---