Electrical Resistivity of Ferromagnetic Metals at Low Temperatures

In a ferromagnetic metal the electrical resistance is caused by scattering of conduction electrons by phonons and by spin waves, the latter case arising from the exchange interaction between the conduction electrons and the localized magnetic electrons. Because these metals have two overlapping bands at the Fermi energy, an $s$ band and a $d$ band, the scattering in either case may occur within a single band or may involve $s-d$ transitions. Furthermore, both phonon and spin-wave umklapp processes can take place. The spin-wave contribution to $\rho$, often called the "spin-disorder resistivity," has been investigated for Fe, Co, Ni, and Gd. At the lowest temperatures (below 10°K for Fe, Co, and Ni) this varies as $T^2$ and is reasonably well described by the single-band theory of Kasuya and others. However, it was found that at higher temperatures the main source of resistance comes from the scattering of $s$ electrons into holes in the $d$ band. A model consisting of spherical energy bands with Fermi momenta $k_{F1\mathrm{}}$ and $k_{F2\mathrm{}}$ has been used to study the normal processes, neglecting spin-wave umklapp processes. A consequence of the model, which is thought to be qualitatively correct, is that $s-d$ transitions require spin waves whose wave vectors exceed the radial distance between the two Fermi spheres. Thus, the $s-d$ transition mechanism is ineffective at very low temperatures. Representative calculations indicate that at higher temperatures this contribution to $\rho$, which has previously been overlooked, is about an order of magnitude larger than that arising from single-band scattering.