Kondo-lattice model

An alternative approach to accounting for the maxima in the temperature dependence of p is based on the Kondo-lattice model (Lavagna et al. 1982). The periodic array of independent Kondo impurities, described by the single-ion Kondo temperature TK, provides a proper description at elevated temperatures, while a coherent state yielding a drop of the resistivity is attained when the system is cooled to below another characteristic temperature coh- Although this approach is suitable particularly for Ce compounds where the Kondo regime was identified inequiv-ocally, the coherence effects are probably significant also in narrow-band actinide materials, as indicated by an extreme sensitivity of the lower-temperature decrease of the resistivity to the presence of impurities. 

Keywords Anderson model, Kondo lattice, magnetic susceptibility, strong correlation. 

The Kondo-lattice Hamiltonian conserves total spin and being an interacting model is nontrivial to solve. However, as with the conjugated systems, it is possible to solve finite Kondo chains efficiently by employing the VB method. The VB... 

The exact diagonalization method has been widely exploited in the study of polyenes as well as small conjugated molecules. It has also been employed in studying spin systems and systems with interacting fermions and spins such as Kondo lattices. These studies have been mainly confined to low- dimensions. The exact diagonalization techniques also allow bench-marking various approximate many-body techniques for model quantum cell Hamiltonians. 

The method which held the promise of overcoming the difficulty of exploding dimensionalities is the renormalization-group technique in which one systematically throws out the degrees of freedom of a many-body system. While this technique found dramatic success in the Kondo problem [62], its straightforward extension to intereicting lattice models was quite inaccurate [63]. 

It is suggestive that the narrow Kondo resonance states of individual 4f impurities will form heavy quasiparticle bands in a periodic lattice of 4f ions. A satisfactory microscopic theory of heavy-band formation has yet to be developed. The Hamiltonian of eq. (107) can be generalized to the lattice by introducing a Bose field h,- at every lattice site. However, in this model it is no longer practicable to restrict to physical states with = 1 at every site. The most successful approach so far consists in a mean-field approximation for the Bose field (Coleman 1985, 1987, Newns and Read 1987) that is valid for large N and r < It can be applied both for the impurity and the lattice model. It starts from the observation that in the limit with QJN= fixed, the rescaled... 

Fig. 27. Resistivity p(T) (a), thermoelectric power S(T) and Lorenz number L T) (b), calculated with LNCA techniques for a sixfold degenerate Anderson lattice model in the Kondo regime (Cox and Grewe 1988). Impurity results, scaled with concentration are shown for comparison. The resistivity results exhibit the logarithmic increase with decreasing temperature and the coherence-derived decrease below T = to the residual value due to impurities, which is quadratic in the Fermi liquid regime T < T. S(T ) is positive definite for the simple model situation chosen.

The lanthanide compounds are usually treated in magnetism by an f-localized model, but show various interesting phenomena such as valence fluctuations, gap states, Kondo lattice, and heavy electrons. These originate from the 4f electrons in the lantiianide compounds, which are either bound to the lanthanide atoms or delocalized, indicating... 

Electrical resistivity. The best known feature caused by the Kondo interaction (in the presence of crystal-field splitting) is the In T dependence of the resistivity in combination with a maximum in the vicinity of the overall crystal-field splitting. This behaviour has been explained by Cornut and Coqblin (1972) within the scope of their model, applying a perturbation calculation. However, below a certain temperature - the Kondo temperature, - this calculation fails to describe the observed behaviour, since many-body effects become important. In the case of a Kondo lattice in the low-temperature region the large-Af approximation is appropriate to account for the deduced T -dependence of p T), as well as the maximum centred roughly around the Kondo temperature. 

The individual variants of the lattice model differ fi om each other in the way the spatial distribution of the molecules of the individual components is taken into account. The simplest solution is the Bragg-Williams (B-W) approach which assumes a random distribution of molecules within the bulk phase. The thermodynamical meaning of this assumption is that the mixture is regular. In the adsorption layer, however, it is only in two dimensions (i.e., within the individual sublayers that a statistical distribution of molecules is assumed). Pioneering work in this field was published by Ono [92-94] and Ono and Kondo [95,96]. The method was later applied to the description of L/G interfaces by Lane and Johnson [97] and later taken up by Altenberger and Stecki [98]. Analytic isotherm equations have also been derived from the above... 

The PAM was simulated for a wide variety of fillings and parameters. Results are presented here for [% = 1.5, V = 0.6, and i7ff = 2.0, F = 0.5 (measured in units of t which is considered to be a few electronvolts, the t5 ical bandwidth of conduction electrons in metals). In order to model the Ce-based Kondo lattice materials, the correlated f-band is placed below the Fermi level (so f Ri 1) and the conduction band filling is adjusted by vaT3dng the Fermi level. Since the model is particle-hole symmetric when /if=n 1 may be inferred firom these results. Thus, beginning at =Hk T = 10, we choose Cf and eless than unity. When the temperature is changed, f - is kept fixed and the chemical potential varied to conserve the toted number of electrons. For the results presented here, the variation of /if from one is less than a few percent. Results firom simulations of the SIM are presented for comparison. 

Low temperature conductivity properties are calculated with some theoretical models. These assume an insulating phase for stoichiometric TmSe and use a periodic Anderson-Hamiltonian, for example, Coqblin etal. [21,22], or use a one-dimensional Kondo-lattice Hamiltonian, Jullien etal. [23]. 

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