Applications of a linear response formalism to the Kondo system

The electrical conductivity of a metallic system containing a very low concentration of paramagnetic impurities (commonly known as a Kondo system) is investigated by applying the linear response formalism. Starting from the s-d exchange Hamiltonian, the Nagaoka equations are re-formulated in the presence of a uniform electromagnetic field and a finite concentration of impurities. These self-consistent, closed set of equations are solved diagrammatically and it is shown that for s-wave scattering only, or more generally if only a single phase shift of the exchange interaction is non zero, the vertex corrections drop out. The calculations of conductivities at high and low temperatures, compared to the Kondo temperature where the perturbation theory breaks down, within the response formalism then give the same results as those of Nagaoka. Our investigation also substantiates the approximation that the Kondo system relaxes with a characteristic time provided the concentration of impurities is so low that the interaction between the impurities can be neglected.

The same formalism has been used to calculate analytically the frequency-dependent conductivity at low temperatures. Because of the approximations involved in the calculations, our result is valid for frequencies very much less than 10 Hz. The real and imaginary parts of the complex conductivity are written down in terms of a dimensionless parameter in the 'pure' and ’dirty’ limits (concentration of impurities, c ~ 10-5 or more). In contrast with the numerical calculations of Murata and Wilkins, the real part of the conductivity does not show a peak with respect to the frequency. In the ’pure' limit, the conductivities can be expressed as a universal function of impurity-concentration, and in the 'dirty' limit they are independent of it. These results are in agreement with the calculations of Murata and Wilkins. The predicted peak has not yet been observed experimentally.

The basic equations are derived again, taking an extended exchange interaction, with finite concentration of impurities but this time in the absence of the electric field. By following the same procedure, these equations have been solved for s and p wave scattering together. The vertex function for high temperatures has been evaluated from an integral equation formed through the Ward's identity. The electrical conductivity at high temperatures is calculated and a third order term in the exchange interaction is found in the coefficient of the leading logarithmic term as a correction. This term occurs due to the interference of s and p wave scattering and it has been shown that for a particular ratio of the strengths of s and p wave scattering, the leading logarithmic term can vanish giving an entirely new high temperature behaviour. The next highest order term has been calculated, and it is shown that this term does not vanish for the particular ratio when the leading term vanishes. Some experimental results are discussed in the light of the present theory. The approximate expression for the conductivity at low temperatures has been found. At absolute zero temperature, the effect of inclusion of p wave scattering is a significant reduction in the conductivity. For weak p-wave scattering, the conductivity at low temperatures is found to depend on p-wave scattering only.