Size Dependence In The Disordered Kondo Problem

We study here the role randomly-placed non-magnetic scatterers play on the Kondo effect. We show that spin relaxation effects (with time τ o s ) in the vertex corrections to the Kondo self-energy lead to an exact cancellation of the singular temperature dependence arising from the diffusion poles. For a thin film of thickness L and a mean-free path l, disorder provides a correction to the Kondo resistivity of the form τ o s /(kFLl 2) lnT that explains both the disorder and sample-size depression of the Kondo effect observed by Blachly and Giordano (PRB 51, 12537 (1995)). PACS numbers:72.10.Fk, 72.15.Nj, 75.20.Hr Typeset using REVTEX 1 At low temperatures, the resistivity of a metal alloy acquires a logarithmic temperature dependence [1] in response to spin-flip scattering between local magnetic impurities and the conduction electrons. This behaviour persists down to a temperature (the Kondo temperature, Tk) where the magnetic impurities and conduction electrons begin to condense into singlet states. While perturbation theory is sufficient to establish the existence of the lnT term, its presence ultimately signifies that perturbation theory is breaking down. Alternatively, spin-flip scattering between conduction electrons and localized magnetic centers has a singular frequency (ω) dependence. Magnetic impurities are not alone in this respect. It is well-known that even non-magnetic impurities can generate a singular (lnω in d=2) frequency dependence in the conductivity [2]. In a sample containing both magnetic and non-magnetic impurities, the question arises: which singularity ultimately wins or can the interplay between the singularities lead to a suppression of either localization or the Kondo effect? In this letter, we resolve these questions. The motivation for this study is two-fold. First, while there have been numerous treatments of this problem [4][7], a clear consensus has not been reached. To illustrate, Everts and Keller [4] were first to show that non-magnetic scattering contributes a 1/ √ T to the Kondo self-energy that dominates the Kondo lnT singularity at low temperatures in d=3. Bohnen and Fisher [5] argued, however, that such a term would not survive in the conductivity. More recently, Ohkawa, Fukuyama, and Yosida [6] showed that disorder results in a singularity of the form T d/2−2 in the conductivity. At low temperatures this singularity dominates the Kondo lnT . As a result, these groups conclude that static disorder can mask the Kondo resistivity as T → 0. On the experimental side, Blachly and Giordano [8] recently measured the conductivity in a series of thin films containing magnetic as well as non-magnetic impurities. They found no evidence for the T d/2−2 singularity but observed instead a suppression of the Kondo resistivity as the strength of the disorder increased. Earlier experiments by Korn [9] also failed to observe the T d/2−2 singularity but observed instead an enhancement in the Kondo resistivity. The point of agreement between these experiments is that disorder couples non-trivially to the Kondo effect and ultimately modi2 fies the coefficient of the lnT dependence. Given the strong dimensional dependence of the localization transition, disorder could eventually lead to a sample size dependence of the Kondo effect. At the outset, we set aside the still controversial issue (ref. 8c) of the sample size dependence and focus on the seemingly straightforward problem of the role non-magnetic disorder plays in the Kondo effect. The new wrinkle we introduce in this problem is the feedback effect spin scattering has on localization. While it is standard to consider the direct influence of localization on the Kondo effect, the reverse effect has not been included [10]. Nonetheless, it is well-known that electron scattering by disordered Heisenberg spins introduces a cutoff of the diffusion pole in both the particle-hole (diffuson) and particleparticle (Cooperon) channels except for the S = 0 particle-hole channel [11]. When spinscattering is included in the diffusion propagators, the fate of the T d/2−2 singularity rests on whether the S = 0 particle-hole propagator contributes to the Kondo self-energy. We show explicitly it does not. The starting point for our analysis is a model Hamiltonian H = Ho +Hsd that contains both normal impurities