Disorders Anderson model

Fig. 1.20 Density of states in a disordered system (a) in the Anderson model (Fig. 1.17) (i) shows the behaviour without disorder, simple cubic lattice, and (ii) that with disorder ...

In the three-dimensional problem, it will be noticed from (71) that in X oJ2 the density of states and the diffusion coefficient occur in the denominator, as they do also in the expression given by Kawabata (1981). If the disorder broadens the band, as will occur in the Anderson model if V0 > B9 then (75) should be modified to... 

The critical disorder strength needed to localize all the states via the Anderson model is ... 

Fig. 1.8. The Anderson model of the potential wells for (a) a crystalline lattice and (b) an amorphous network. is the disorder potential.

In a semiconductor, as discussed in the previous section, localisation can also occur as the width of the allowed energy band is reduced, and this was defined in terms of a limiting mobility. The Anderson model shows that disorder can lead to localisation in metals as well as semiconductors. In metals, since conduction is due only to electrons within a partially filled band, the energy in the band tail that separates localised from delocalised electron states is termed the mobility edge. The onset of localisation in a metal occurs at a minimum conductivity. This can be seen as follows. For an electron at the Fermi energy its mean free path, l, is just the scattering time, r, multiplied by the electron velocity at the Fermi energy, vF. Then, from Equations (4.1) and (4.2) it follows that ... 

An efficient terminator technique is certainly desirable in the application of recursion methods to the study of disordered systems. It has been shown recently that a self-consistently determined terminator can be fruitfully applied to calculate the electronic states in the Anderson model and to evaluate the vibrational spectrum of lattices with isotopic disorder. The basic idea is to extend the procedures discussed in Section IV to ensemble averages. In this case a useful generalization of Eq. (4.5), satisfied by the terminator t(E), is... 

An extensive discussion of experiments on exciton transport in isotopically disordered crystals and numerical simulations of this phenomenon in the framework of a percolation model may be found in the review paper by Kopel-mann (20). A more recent review of this field, including the discussion of the Anderson model, may be found in the book by Pope and Swenberg (21). 

Mott, N.F., Electrons in disordered structures. Advances in Physics, 1967. 16 p. 49 Moura, F.A.B.F. and M.L. Lyra, Delocalization in the ID Anderson model with long-range correlated disorder. Physical Review Letters, 1998. 81 p. 3735... 

Mott has shown in 1967 that in an Anderson model for a disordered metal the conductivity cannot be arbitrarily small if E is above E. The reason is that the position of E, the mobility edge, is given by a balance between the electronic overlap / of wave functions centered at adjacent atomic sites and the disorder potential Fq. For a mobility edge the former energy is a definite fraction of the latter and hence the lower limit of the conductivity is always the minimum metallic conductivity Onjin — 200 D cm" , the value depending somewhat on assumptions about unknown constants. This concept... 

Since the pioneering work of Anderson and Mott it is known that the wave functions of disordered systems are localized. The degree of this so-called Anderson localization increases with increasing disorder. Anderson s original argument was based on a simple model Hamiltonian. In a number of cases different authors > carried out numerical studies on disordered systems still using different simplified model Hamiltonians. 

Long-range disorder inside polymers, even if local order of the crystallites forming fibrils can be observed [25]. These polymers can be ma-croscopically considered as amorphous materials [26,27] characterized by localized states in band tails and with random distribution of potential wells where electronic sites are linked (Anderson model). 

Orientational disorder and packing irregularities in terms of a modified Anderson-Hubbard Hamiltonian [63,64] will lead to a distribution of the on-site Coulomb interaction as well as of the interaction of electrons on different (at least neighboring) sites as it was explicitly pointed out by Cuevas et al. [65]. Compared to the Coulomb-gap model of Efros and Sklovskii [66], they took into account three different states of charge of the mesoscopic particles, i.e. neutral, positively and negatively charged. The VRH behavior, which dominates the electrical properties at low temperatures, can conclusively be explained with this model. 

To conclude, we can draw an analogy between our transition and Anderson s transition to localization the role of extended states is played here by our coherent radiant states. A major difference of our model is that we have long-range interactions (retarded interactions), which make a mean-field theory well suited for the study of coherent radiant states, while for short-range 2D Coulombic interactions mean-field theory has many drawbacks, as will be discussed in Section IV.B. Another point concerns the geometry of our model. The very same analysis applies to ID systems however, the radiative width (A/a)y0 of a ID lattice is too small to be observed in practical experiments. In a 3D lattice no emission can take place, since the photon is always reabsorbed. The 3D polariton picture has then to be used to calculate the dielectric permittivity of the disordered crystal see Section IV.B. 

Figure 7.3. (a) In the Kronnig-Penney model, the potential at each lattice site in a monatomic crystalline solid is of the same depth, (b) The random introduction of impurity atoms in the crystalline lattice produces variation in the well depths, known as diagonal, or Anderson, disorder, (c) Amorphous (noncrystalline) substances have unevenly spaced potential wells, or off-diagonal disorder. 

Fig. 7.15. Illustration of the Anderson localization model showing atomic potentials and the shape of the band, with and without the disorder.

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. 

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