Low-Temperature Resistivity of Dilute Magnetic Impurities in the Presence of Internal Fields

The variation of the low-temperature resistivity in the presence of internal fields is examined for dilute concentrations of magnetic impurities in a nonmagnetic metal host. The relaxation times are calculated in the second Born approximation for two different internal fields: one arising in a system in which long-range order exists, and another in which the magnetic impurities interact via a Ruderman-Kittel-Kasuya-Yosida interaction. In the latter case the internal field $H$ is a random variable whose probability distribution $P\left(H\right)\mathrm{}$ can, in principle, be obtained. Using an Ising-like probability distribution, it is predicted that the change in the very low-temperature resistivity $\Delta \rho \mathrm{}\left(T\right)\mathrm{}$ is, except for a small $\ln T$ term, linear in $T$. This is in agreement with experiment for Au-0.1% Fe, where the experiment was performed at sufficiently low temperatures. More generally we find that $\Delta \rho \mathrm{}\left(T\right)\mathrm{}$ is approximately proportional to that part of the low-temperature specific heat which arises from the magnetic disordering of the impurities in their internal fields. The "width" of the probability distribution function obtained from the low-temperature specific-heat measurements gives the slope $m$ of $\Delta \rho \mathrm{}\left(T\right)\mathrm{}$ in rather good agreement with experiment. This is additional evidence that the excess specific heat in these alloys arises from a magnetic disordering of the impurities. The slope of the resistivity is, from our theory, approximately independent of the impurity concentration and the exchange interaction $J$ at sufficiently low temperatures. For higher temperatures we obtain a resistivity maximum at a temperature proportional to the impurity concentration. This maximum arises from the suppression of the Kondo $\ln T$ term by the presence of internal fields. For concentrations of the order of 1%, the maximum as well as the minimum disappears, and the resistivity decreases monotonically as the temperature is lowered. The behavior of the resistivity as a function of the impurity concentration, the strength and the sign of the $s-d$ interaction, the impurity spin, and the temperature is discussed. It is proposed that low-temperature resistivity measurements be used to probe the behavior of the probability distribution $P\left(H\right)\mathrm{}$ of the fields near $H=0\mathrm{}$. The present results apply only to temperatures much greater than the Suhl-Abrikosov resonance temperature.