Anderson localization transition in thin films of gadolinium

\textit{In situ} temperature-dependent transport studies have been performed on a series of gadolinium (Gd) films deposited onto sapphire substrates having sheet resistance $R_{0}$~$\equiv $~$R_{xx}$(5K) varying over the range 4011~$\Omega $ ($\sim $35{\AA}) to 132 K$\Omega $ ($<$ 20 {\AA}). The disorder strength, as measured by $R_{0}$, is sufficiently high so that quantum corrections to the classical Boltzmann conductivity are no longer observed. In this region of moderately strong disorder, we find a temperature-dependent conductivity of the form \textit{$\sigma $}($T)$~ = \textit{A~+BT}$^{p}$ where $A$ and $B$ are disorder-dependent constants and $p$ is a power with value 0.4. We find that $A$ is proportional to (1-$R_{0}/R_{c})^{s}$ where the conductivity exponent $s~$=1 and the critical resistance $R_{c }$= 22.7 k$\Omega $. This change in sign of $A$ with unity exponent at critical disorder describes the critical regime of an Anderson localization transition[1] with the temperature-dependent localization length sufficiently small so that the Gd films can be considered to be in the 3D regime, rather than the 2D regime where metallic behavior does not occur [2]. [1] Lee {\&} Ramakrishnan, RMP 57, 287 (1985); Belitz {\&} Kirkpatrick, RMP 66, 261 (1994) [2] Abrahams, Anderson, Licciardello {\&} Ramakrishnan, PRL 42, 673 (1979)