Observation of spin-wave mediated Altshuler-Aronov and weak localization corrections to the conductivity in thin films of gadolinium

We present a study of quantum corrections to the conductivity tensor of thin ferromagnetic gadolinium films. Using the sheet resistance as a measure of disorder, \textit{in situ} magnetotransport studies were performed on a series of gadolinium films deposited onto sapphire substrates having sheet resistance $R_{0 }\equiv $~$R_{xx}$ (5K) varying over the range 428~$\Omega $ ($\sim $135{\AA}) to 4011 $\Omega $ ($\sim $35 {\AA}). For temperatures $T$~$<$~30 K and $R_{0} \quad <$ 4011 $\Omega $, we observe the simultaneous presence of two types of quantum correction to the Drude conductivity, $\sigma =\sigma _{Drude} +\Delta \sigma _{SpinWaveMediated} +\Delta \sigma _{WL} $. The characteristic feature of the first correction is an approximately linear increase with temperature of conductivity, and we attribute this as a spin-wave mediated Altshuler-Aronov correction to conductivity. The second correction to the Drude conductivity comes from weak localization, with a characteristic logarithmic temperature dependence of conductivity with a prefactor ${e^2} \mathord{\left/ {\vphantom {{e^2} {2\pi ^2\hbar }}} \right. \kern-\nulldelimiterspace} {2\pi ^2\hbar }$ in 2D. We observe a breakdown of this behavior at a sheet resistance $R_{0}$ =4011 $\Omega $, which is very close to the quantum of resistance, $\hbar \mathord{\left/ {\vphantom {\hbar {e^2\approx 4100\Omega }}} \right. \kern-\nulldelimiterspace} {e^2\approx 4100\Omega }$.