Muon-spin relaxation in Au Fe and Cu Mn spin glasses

Zero- and longitudinal-field muon-spin relaxation (ZF \ensuremath{\mu}SR and LF \ensuremath{\mu}SR) measurements have been applied to dilute-alloy spin glasses AuFe(1.0 and 1.4 at %) and CuMn(1.1, 3, and 5 at. %). A stochastic theory of muon-spin relaxation has been formulated, and an analytic form of zero-field (ZF) muon-spin-relaxation function has been derived to describe the depolarization of muon spins in static, dynamic, and coexisting static and dynamic random local fields from magnetic impurities. The observed experimental results for the muon-spin-relaxation function ${G}_{z}$(t) in all the specimens have been well explained by the model functions based on the present theory. Depolarization of muon spins due to the rapidly fluctuating dynamic local field has been indicated by the ``root-exponential'' shape of ${G}_{z}$(t) observed above the susceptibility-cusp temperature ${T}_{g}$. The spin correlation time ${\ensuremath{\tau}}_{c}$ of Mn (or Fe) moment, deduced from the dynamic depolarization rate of muon spins, shows a rapid change from ${\ensuremath{\tau}}_{c}$\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}12}$ sec at T\ensuremath{\gtrsim}2${T}_{g}$ towards ${\ensuremath{\tau}}_{c}$\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}9}$ sec at T\ensuremath{\sim}${T}_{g}$ in all the specimens. Coexistence of static and dynamic random fields at each muon site has been found by ZF and LF \ensuremath{\mu}SR below ${T}_{g}$. The average amplitude ${\mathrm{scra}}_{s}$ of static random fields, determined precisely by ZF \ensuremath{\mu}SR, attains finite values only below ${T}_{g}$, and increases towards the full amplitude ${\mathrm{scra}}_{0}$ at T=0. ${\mathrm{scra}}_{s}$/${\mathrm{scra}}_{0}$ is proportional to the static polarization of Mn (or Fe) moment, and its temperature dependence has been found to be consistent with a M\"ossbauer-effect study in AuFe. The squared quantity (${\mathrm{scra}}_{s}$/${\mathrm{scra}}_{o}$${)}^{2}$ of ZF \ensuremath{\mu}SR shows good agreement with the static order parameter Q(T) determined by ac-susceptibility (${\ensuremath{\chi}}_{\mathrm{ac})}$ and by neutron-spin-echo (NSE) measurements for cut-off pieces of the same specimen. In view of the fundamental difference of the hyperfine-field--type from the susceptibility-type measurements, this comparison reveals that spin freezing is characterized by a rather homogeneous amplitude of static spin polarization among different spins in spin glasses, i.e., ``homogeneous freezing.'' The combined results of ZF \ensuremath{\mu}SR, ${\ensuremath{\chi}}_{\mathrm{ac}}$ and NSE measurements in AuFe and CuMn spin glasses have demonstrated rapid slowing down of impurity spin fluctuations above ${T}_{g}$, and the appearance of long-time, persisting static spin polarization below ${T}_{g}$.