Kondo effect, resonance

The model of a degenerate gas of spin polarons suggests that if the direct or RKKY interaction between moments is weak and EF too great to allow ferromagnetism then the moments might all resonate between their various orientations. This would mean that it is possible in principle to have a heavily doped magnetic semiconductor or rare-earth metal in which there is no magnetic order, even at absolute zero. This possibility is discussed further in Section 8 in connection with the Kondo effect. 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

Using this local moment model, and using band theory or its variations, a number of workers have been able to formulate expressions which represent the measured magnetic data reasonably well, at least for the case where well-localized moments are developed on the solute atoms (II, 18). However, considerably more data has become available on other properties of dilute alloys, including data on resistivity and specific heat, neutron scattering, various magnetic resonance experiments, Moss-bauer measurements, Kondo effect, and the like. Measurements have been extended also to alloys of many other systems besides those involving the platinum metals. 

It has been pointed out early that the ESR of local moments could be used to observe the Kondo effect directly by studying the resonance absorption of the Kondo impurity itself Spencer and Doniach (1967) calculated the g-shift and Walker (1968) the relaxation rate due to the Kondo effect. Both contributions are rather small and experimentally hard to measure. This is partly due to the large residual linewidths typically found in Kondo alloys. 

Allen and Martin (1982) and Lavagna et al. (1982, 1983) proposed that the y-a transition in Ce was related to the Kondo effect. Since Kondo systems have densities of states with two peaks, one very close to the Fermi level and one below, this offered a possible explanation of the two peaks in the photoelectron spectra, assuming that the photoabsorption process would transfer the structure in the density of states to the emitted spectrum. Detailed calculations of a photoelectron spectrum were not, however, carried out. The Kondo (or Abrikosov-Suhl) resonance in the density of states can be obtained from the Anderson Hamiltonian in the limit U CO (Lacroix 1981). For a Fermi level in the center of a valence band of width 2D and a constant density of states coupled by a constant matrix element V to an Nf-fold degenerate localized level at energy Sf below two peaks can arise in the density of states of the coupled system at low temperatures if is not too small. As... 

The main equation for the d-electron GF in PAM coincides with the equation for the Hubbard model if the hopping matrix elements t, ) in the Hubbard model are replaced by the effective ones Athat are V2 and depend on frequency. By iteration of this equation with respect to Aij(u>) one can construct a perturbation theory near the atomic limit. A singular term in the expansions, describing the interaction of d-electrons with spin fluctuations, was found. This term leads to a resonance peak near the Fermi-level with a width of the order of the Kondo temperature. The dynamical spin susceptibility in the paramagnetic phase in the hydrodynamic limit was also calculated. 

The CeAlj compound shows a VltW value of 1.2JK /Ce atom, but the C /T ratio increases up to 2 JK"VCe atom at 0.5 K, which signals the transition to the coherent coupling of the Kondo resonances in a Kondo lattice (Bredl et al. 1984). Under an applied field the C nlT maximum value is reduced to 1.7 J K /Ce atom for H = 4 T and lightly shifted to lower temperatures (see also Bredl et al. 1984). The effect of pressure on /ltCO) is similar to that in CeCug, reducing its value to 0.55 JK VCe atom under 8.2 kbar, but it is significant that the C /T maximum disappears under only 0.4 kbar of pressure. 

However, in the case of a dilute alloy with unstable-moment inpurities (e.g., Ce " ) many-body effects lead to the existence of narrow Kondo resonance states above the Fermi level as discussed in detail in sect. 4. This leads to a strongly energy-dependent scattering rate for conduction electrons E E) = 1/t( ) that is directly proportional to the density of many-body resonance states (see fig. 46). The energy scale for the t E) dependence is now the Kondo temperature which can be comparable with T. Therefore, in this case one should expect that MAQO and dHvA amplitudes may deviate from the LK formula. Experimentally this effect has been quite elusive. One of the rare cases where it was actually observed is the dilute alloy Laj Ce Bg (x = 0.10), see Thalmeier et al. (1987). Figure 41 shows the T-dependence of MAQO amplitudes for a very small extremal area with F = 6.5T or... 

The first equation shows that the center of the quasiparticle bands lies at an energy above the original conduction band Fermi level. This should be compared with the position Tg = aT of the Kondo resonance in the impurity model. The second equation simply results from the charge constraint QflN = g, which is now enforced only on the average in contrast to the impurity model. The occupation f(0) is obtained by setting equal to the actual value of 1 IN. The quasiparticle bands are the result of a hybridization with an effective strength K Here r iT) is the average fraction of sites without f occupation,... 

For the non-symmetrical Anderson model with C/ + 2ef 0, one has in the Coqblin Schrieffer and Kondo Hamiltonians in addition to the exchange a potential scattering term. The Kondo resonance is no longer at the Fermi energy ep = 0 but is shifted. This shift leads to a smaller resistivity p(T), the maximum of which, however, is still at T = 0. The potential scattering has a dramatic effect on the thermoelectric power, which vanishes in the symmetric case C/ + 2ef = 0, but has a huge peak near 7k for C/ 4- 2ef 0. 

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