Muon diffusion and trapping in aluminum and dilute aluminum alloys: Experiments and comparison with small-polaron theory

Abstract

In order to examine whether the small-polaron theory can correctly describe the motional behavior of light interstitials in metals at low temperatures, we have systematically investigated the diffusion of positive muons in Al and in Al doped with substitutional impurities of various concentrations in the temperature range between 30 mK and 100 K. In pure Al ( 1 ppm overall impurity content) the muon was found to be mobile down to 30 mK. For doped Al, the diffusional behavior exhibits two regimes: (i) Above 1 K maxima in the depolarization rate evolve, indicating a static muon in the peak regions. The position of the peaks (between 15 and 50 K) is characteristic for each kind of impurity while the height and width depend on the impurity concentration. In $\mathrm{Al}{\mathrm{Mn}}_{x}$ the muon occupies a tetrahedral site in this regime. (ii) Below 1 K the depolarization increases again with decreasing temperature and increasing impurity concentration. The temperature dependence exhibits a universal behavior independent of impurity concentration and the kind of impurity. Here the muon site is octahedral. Both regimes are interpreted in terms of diffusion-limited trapping at the impurities: (i) Above 1 K the diffusion is phonon assisted and entails capture rates which increase with concentration and temperature. The peak positions are determined by the escape processes from the traps which occur at characteristic temperatures for each impurity. A quantitative analysis with a two-state model for diffusion in the presence of traps revealed a linear concentration dependence of the trapping rate. A linear temperature dependence for the diffusion coefficient was found which is in strong disagreement with the ${T}^{7}$ prediction of conventional small-polaron theory. This linear behavior is characteristic for a one-phonon process, which should be effective only if alternating transitions between octahedral and tetrahedral sites occur. (ii) Below 1 K, the apparent trapping rate increases with decreasing temperature, implying a faster diffusion mechanism at low temperatures. The most appealing interpretation is a trapping, preceded by a fast coherent diffusion, which is limited by muon-electron scattering. Again, in contrast to conventional small-polaron theory, which predicts ${D}_{\mathrm{coh}}\ensuremath{\propto}{T}^{\ensuremath{-}9}$, we find proportionality to ${T}^{\ensuremath{-}0.6}$ for this rate.