Theoretical studies on impurity models of mixed valent systems

Abstract

In the first part of this thesis, consisting of the first 3 chapters, calculations of thermodynamic properties like susceptibility, magnetisation and specific heat, and of the single particle spectral density are discussed, for two models of magnetic and mixed valent impurities in metals: (1) the U = ? Anderson model and (2) the Coqblin–Schrieffer model. These calculations are done using 1/N expansion techniques, N being the orbital degeneracy of the impurity. The aim is to assess the viability of 1/N as a small expansion parameter, and the efficacy of the various 1/N expansion techniques in providing a systematic and useful prescription for predicting the properties of mixed valent and Kondo systems, for which these two models are considered to be applicable. The different formulations of the expansion techniques that we have studied are:

The Self Consistent Approximation or the NCA, The Direct Expansion, and The Functional Integral Method. Various technical difficulties faced by them are clearly analysed and discussed.

Chapter I is the introductory chapter where the two models are introduced and our studies, presented in detail in Chapter II and III, motivated. The slave boson representation for the two models, the mapping between them in some regimes of their parameters and the question of imposition of the U = ? constraint, a feature which distinguishes the various formulations of the 1/N expansion techniques, are discussed. The most important consequences of large orbital degeneracy as brought out in the N ? ? limit are summarised. They are compared with the exact results available from the Bethe Ansatz for the thermodynamic properties of interest. No exact results are available for the spectral density, but what is known about them from earlier calculations is summarised. In Chapter II, the NCA and the direct expansion are discussed in detail. A review of the encouraging finite temperature results available in literature is followed by a detailed study of the T = 0 behaviour. Serious pathologies in the magnetisation and spectral function, absent in a finite temperature calculation, are pinpointed and analysed. In Chapter III, the functional integral method is discussed in detail. A serious limitation of the method is that the leading order (i.e., the N ? ? limit) calculation gives rise to a spurious phase transition at a finite temperature TcT_cTc? even for a single impurity. Formally, fluctuation effects are known to destroy this phase transition but in practice, remnants of the phase transition in any finite order calculation make a calculation of physical properties through the transition temperature TcT_cTc? rather difficult. A possible prescription for getting over this difficulty is outlined. Some preliminary results are presented, which seem to indicate that it could be a good working prescription. The calculations presented in the first part of the thesis provide a basis for discussing pressure-induced phase transitions in mixed valent systems using an approximation that views them as a collection of independent impurities. This is the subject of Chapter IV. The effects of pressure are modelled by giving a volume dependence to the Anderson model parameters, ? the hybridisation width and ?f_ff? the f-level position. For the impurity free energy, results in the N ? ? limit from the Functional Integral Method as discussed in Chapter III are used. This work extends considerably the earlier work in this direction, namely the Kondo–Volume–Collapse model applicable to phase transitions in Cerium systems. In particular the inclusion of a volume-dependent shift of the f-level position leads to a theory which should be a useful starting point for understanding the continuous and discontinuous valence transitions in Sm, Eu and Yb-based mixed valent systems. Notwithstanding the usefulness of the independent impurity picture of mixed valent systems, impurity interactions surely have to be taken into account. It is useful to study these in the simpler context of a two-impurity problem. In Chapter V of this thesis, one specific manifestation of these interaction effects, namely the spin–spin correlation ?S1?S2?\langle S_1 \cdot S_2 \rangle?S1??S2??, is discussed. This correlation function is calculated for the two-impurity Kondo problem using perturbative thermodynamic scaling theory. The calculations are carried out for all temperatures, and for values of kFRk_F RkF?R such that the RKKY interaction III dominates the Kondo effect. When I?TKI \gg T_KI?TK? the impurity spins are locked ferromagnetically at low temperatures whence ?S1?S2??1/4\langle S_1 \cdot S_2 \rangle \approx 1/4?S1??S2???1/4. When ?I?TK-I \gg T_K?I?TK? there is an antiferromagnetic locking of the impurities and ?S1?S2???3/4\langle S_1 \cdot S_2 \rangle \approx -3/4?S1??S2????3/4. The deviations of ?S1?S2?\langle S_1 \cdot S_2 \rangle?S1??S2?? from these limiting values are calculated for varying kFRk_F RkF?R (except close to the nodes of the RKKY interaction where ?I??TK|I| \ll T_K?I??TK? whence non-perturbative methods are required to get ?S1?S2?\langle S_1 \cdot S_2 \rangle?S1??S2??).