Temperature-dependent Ruderman-Kittel-Kasuya-Yosida interaction.

Abstract

We develop a method that provides an analytical expression of the free-electron spin susceptibility for arbitrary temperature. The result can be further reduced by using the low-temperature approximation, which shows that the oscillation decays exponentially as exp(-\ensuremath{\pi}${\mathit{T}}^{\mathcal{'}}$${\mathit{k}}_{\mathit{Fr}}$) in the long-range limit, and the wave number of the oscillation varies as ${\mathit{k}}_{0}$\ensuremath{\sim}${\mathit{k}}_{\mathit{F}}$[1-(${\mathrm{\ensuremath{\pi}}}^{2}$/12)${\mathit{T}}^{{\mathcal{'}}^{}2}$], where ${\mathit{T}}^{\mathcal{'}}$ is the normalized temperature with respect to the Fermi energy. The result contradicts what Darby has proposed, especially in the long-range region. On the other hand, in the short-range region, where the Sommerfeld expansion method is valid, numerical study shows that Darby's result agrees with ours fairly well.