S−dModel with Nonspherical Fermi Surface

Abstract

A study is made of the effects of a nonspherical and multiply connected Fermi surface on the properties of the Ruderman-Kittel-Kasuya-Yosida $s\ensuremath{-}d$ interaction between localized spins and the conduction electrons. At a critical value of the conduction-electron concentration ${\overline{N}}_{e}$, the Fermi surface touches a number of Brillouin-zone faces, giving rise to a number of necks as in copper and nickel. It is shown that, in this region of electron concentration, the paramagnetic Curie temperature $\ensuremath{\theta}$ exhibits a kink associated with a discontinuity in the exchange-stiffness parameter $D\ensuremath{\propto}\ensuremath{\Sigma}{\mathbf{R}}^{2}J(\mathbf{R})$, where $J(\mathbf{R})$ is the indirect-exchange parameter. These effects are expected to be more pronounced in materials with a large number of necks in the multiply connected Fermi surface. We choose the simplest possible model consistent with the crystal symmetry and the Bragg condition at the Brillouin-zone faces, i.e., a tight-binding approximation for the conduction electrons; but a number of results may be shown to be model-independent. We also neglect interband transitions, but these are shown not to affect the long-range component of the conduction-electron polarization. The long-range effects are shown to be model-independent because they are related only to the level-density function $\ensuremath{\rho}(\ensuremath{\epsilon})$ and its derivatives at the Fermi level, whereas the short-range effects are shown to be sensitive to the details of the band structure. The paramagnetic temperature $\ensuremath{\theta}$ can be separated into a model-independent term related to $\ensuremath{\rho}({\ensuremath{\epsilon}}_{F})$ and a model-dependent term related to short-range effects. In the region corresponding to the spherical Fermi surface, $\ensuremath{\theta}$ is nearly proportional to ${{\overline{N}}_{e}}^{\frac{1}{3}}$, and the model-dependent contribution to $\ensuremath{\theta}$ is relatively small, at least for the simple cubic structure. It is also shown that the exchange moments $〈{\mathbf{R}}^{2}〉\ensuremath{\propto}{\ensuremath{\Sigma}}_{\mathbf{R}} {\mathbf{R}}^{2}J(\mathbf{R})$ and $〈{\mathbf{R}}^{4}〉\ensuremath{\propto}{\ensuremath{\Sigma}}_{\mathbf{R}} {\mathbf{R}}^{4}J(\mathbf{R})$ are model-independent and related to the level-density function $\ensuremath{\rho}(\ensuremath{\epsilon})$ and its first and second derivatives at the Fermi level. In the neighborhood of the critical concentration, relatively large positive values of $\frac{〈{\mathbf{R}}^{4}〉}{({a}^{2}〈{\mathbf{R}}^{2}〉)}$ can be obtained for the simple cubic case, in contrast to the negative values of $\frac{〈{\mathbf{R}}^{4}〉}{〈{\mathbf{R}}^{2}〉}$ obtained by Kasuya in the limit ${k}_{F}a\ensuremath{\rightarrow}0$.