RKKY Interaction for the Spin Polarized Electron Gas

TL;DR

This paper extends the RKKY interaction model to spin-polarized electron gases, revealing additional tensor interactions and beating patterns due to broken symmetry, with explicit formulas for 2D and 3D systems.

Contribution

It introduces a generalized RKKY interaction including tensor terms in spin-polarized gases, highlighting effects of broken symmetry beyond traditional models.

Findings

Tensor interaction $ ightarrow$ non-zero in spin-polarized gas

Beating pattern in RKKY oscillations due to two Fermi momenta

$R^{-3}$ decay preserved in 3D case

Abstract

We extend the original work of Ruderman, Kittel, Kasuya, and Yosida (RKKY) on the interaction between two magnetic moments embedded in an electron gas to the case where the electron gas is spin polarized. The broken symmetry of a host material %(here we have a broken time reversal symmetry) introduces the Dzyaloshinsky-Moriya (DM) vector and tensor interaction terms, in addition to the standard RKKY term, so that the net interaction energy has the form: $ {\cal H} = J \vec S_1 \cdot \vec S_2 + \vec D \cdot \vec S_1 \times \vec S_2 + \vec S_1 \cdot \stackrel{\leftrightarrow}{\Gamma} \cdot \vec S_2 $. We find that for the spin-polarized electron gas, a non-zero tensor interaction $\stackrel{\leftrightarrow}{\Gamma}$ is present in addition to the scalar RKKY interaction $J$, while $\vec D$ is zero due to the presence of inversion symmetry. Explicit expressions for these are derived for the electron gas both in 2D and 3D. The RKKY interaction exhibits a beating pattern, caused by the presence of the two Fermi momenta $k_{F\uparrow}$ and $k_{F\downarrow}$, while the $R^{-3}$ distance dependence of the original RKKY result for the 3D electron gas is retained. This model serves as a simple example of the magnetic interaction in systems with broken symmetry, which goes beyond the RKKY interaction.