Theory of the interaction between electrons and the two-level system in amorphous metals. III. Experimentally observable quantities

In the two immediately preceding papers a theory of interaction between electrons and two-level systems (TLS) has been developed. According to that theory, the electron scattering on the TLS is resonant scattering below a characteristic crossover temperature $T_k$. This resonant scattering contributes to the lifetime $T_1$ of the TLS due to a Korringa-type mechanism. The relaxation time $T_1$ has been calculated by using the renormalized enhanced-coupling constants. Making use of the different ultrasound experimental data, we have given the effective coupling strengths for different alloys. The average coupling is small in PdCuSi and NiP alloys to form a resonant state. For PdZr and NbZr alloys, however, the averaged coupling strengths are too small—only by less than a factor of 2—to have a resonant state. Thus in these alloys a portion of the TLS may have sufficiently large coupling to have resonant scattering if a distribution for the coupling is assumed. The electrical resistivity, as a function of temperature, is calculated in detail. At the crossover temperature a logarithmic temperature dependence with a negative coefficient is found for one decade of the temperature. At lower temperature a crossover between the logarithmic and Fermi-liquid-type behaviors is suggested; therefore the resistivity near $T=0\mathrm{}$ must behave as $\Delta R\sim (1-a{T}^2)$. This overall behavior is in agreement with the available experimental data. The amplitude of the resistivity maximum arount $T=0\mathrm{}$ is calculated, and the experimentally observed values can be explained by assuming reasonable densities for the TLS. It has been suggested that an enhanced density of state may be due to the renormalization (reduction) of the energy splitting of the TLS. That enhancement is necessary to explain the amplitude of the resistivity maximum at $T=0\mathrm{}$ if only a small portion of the TLS has sufficiently large coupling to form the resonant state. Finally, the inelastic inverse scattering lifetime ${\tau }_{\mathrm{in}}$ for conduction electrons is calculated. Assuming a strong coupling case, the amplitude of the ${\tau }_{\mathrm{in}}$ obtained is of the same order of magnitude supported by experimental data. The assumption of dealing with the strong coupling case offers a possibility of resolving the discrepancy concerning the shortness of ${\tau }_{\mathrm{in}}$ relevant in localization theory. Finally, it is emphasized that a systematic study combining the ultrasound, resistivity, and electron inelastic scattering lifetime measurements may justify the applicability of the present theory to real metallic glasses.