Conductivity, electronic Anderson model

The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

The next step in the theory is to calculate the conductivity above and below the mobility edge. In the Anderson model, locali2ed states are defined by a decreasing probability that the electron diffuses a larger distance from its starting point. Mott and Davis (1979 Chapter 1) prove that the dc conductivity in the localized states is zero at T = 0 K. They use the Kubo-Greenwood formula for the conductivity,... 

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 ... 

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... 

Mfn = 549.5 g/mol (Fa)2PF6.) Solid curves calculated temperature dependence for thermally-activated paramagnetism (t.a.p.) and for the paramagnetism of the conduction electrons with an effective energy gap of 2Aeff(T) in the quasi-metallic state for T> Tp, according to the Lee-Rice-Anderson model (L-R-A). From [29]. 

Edelstein (1968, 1970) has adopted the Coqblin-Blandin model for cerium but with an emphasis on the effect of spin compensation. This point of view was that the decrease or loss of the magnetic moment in a-Ce and the slightly decreased moments in y- and /3-Ce (based on Lock s data) was due to the antiferromagnetic polarization of conduction electrons around the 4f virtual state. Edelstein supported this idea in part by noting an apparent T dependence of the magnetic susceptibility of mixed phase cerium samples above 13 K (Lock, 1957). This temperature dependence had been suggested by Anderson (1967) for the spin compensation contribution to the susceptibility for Kondo alloys. However, subsequent susceptibility measurements of single phase a-Ce (see... 

For —Sf A, U A and U + 2ef = 0 the Anderson Hamiltonian, eq.(1), can be transformed into the Coqblin-Schrieffer (CS) Hamiltonian. In this Kondo limit charge fluctuations are completely suppressed and the model describes an effective 4f-electron spin j which interacts via exchange with the conduction electrons... 

Near the Ce or Yb end of the R series, the 4f level thus approaches the Fermi level in energy and the 4f electrons hybridize more strongly with the conduction electrons with the kinetic energy E. This f-hybridized coupling constant is denoted by V. A theoretical treatment for such a system is called the periodic Anderson model (Anderson 1961). The parameters E, V, Ef and U predominantly control the dynamics of tiie system. These values depend actually on the crystal structure. The relation between flie magnetic ordering temperature and the distance between the Ce (or U) atoms is known as a Hill plot (Hill 1970). 

The second approach (Allen and Martin 1982) to include coupling to the lattice is to assume that the hybridization matrix element Fkf between the conduction electrons and the 4f electrons varies as the volume varies. This form of coupling receives justification (Freeman et al. 1988) iiom band theoretic treatments of cerium and its compounds that show that the cell volume can decrease when the 4ficonduction-band l bridization increases, without significant associated change in f-count. One way to include this in the Anderson model is to assume that the volume dependence of Fkf is reflected in a dependence of the characteristic (Kondo) temperature Tk on V. In the Kondo limit, where the occupation number nf 1 does not vary with temperature. 

Kondo models are based on the single-impurity Anderson (1961) model, which describes an f shell that is bathed in a Fermi sea of conduction electrons. The Hamiltonian, H, can be written as the sum of three terms. 

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). 

Anderson and Grantscharova conducted one of the first studies on the CO oxidation on Pt using a molecular orbital theory with a simple molecular model.113 They used an atom superposition and electron delocalization molecular orbital (ASED-MO) method to investigate the electrochemical oxidation of the adsorbed CO on the Pt anodes. They found that the interaction of CO(ads) with the oxidant OH(ads) was effective only at high surface coverage. 

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