Spin-wave-mediated quantum corrections to the conductivity of thin ferromagnetic films of gadolinium

We present a study of quantum corrections to the conductivity of thin ferromagnetic gadolinium films. In situ magnetotransport measurements were performed on a series of thin films with thickness $d<150\text{Å}$. For sheet resistances $R_0\le 2840\Omega$ and temperatures $T\le 30\text{K}$, we observe a localizing, linear in temperature contribution to the conductivity in addition to the logarithmic temperature dependence expected from well-known quantum corrections in two dimensions. We show that such a linear $T$ dependence can arise from a spin-wave-mediated Altshuler-Aronov-type correction.

Figure 1

(Color online) Temperature dependence on a logarithmic scale of ${\sigma }_{xx}$ for two Gd thin films having sheet resistances $R_0=428\Omega$ (open squares, left hand axis) and $R_0=2840\Omega$ (open triangles, right hand axis). The solid-line fits for each curve are obtained using Eq. (1).

Figure 2

(Color online) Dependence of the fitting parameters $\left(P_1,P_2,P_3,p\right)$ defined in Eq. (1) on the disorder parameter $R_0$ for 16 thin-film samples. The horizontal dashed lines in panels (b) and (d) represent, respectively, the averaged values of $P_2$ and $p$, whereas the curved dashed line in panel (c) is a guide to the eyes.

Figure 3

(Color online) Normalized relative changes ${\Delta }^N\left({\sigma }_{xy}\right)$, defined in Eq. (2), for two different sheet resistances. For comparison we also show ${\Delta }^N\left(R_{xx}\right)$ and ${\Delta }^N\left(R_{xy}\right)$.

Figure 4

Spin-wave contributions to the longitudinal conductivity. Solid lines are impurity averaged Green’s functions, broken lines are “diffusons,” and wavy line represents the effective spin-wave-mediated interactions. There are two diagrams of type (a) and four diagrams of type (b)