The RKKY Interaction

Indirect exchange couples moments over relatively large distances. It is the dominant exchange interaction in metals where there is little or no direct overlap between neighbouring magnetic electrons. It therefore acts through an intermediary which in metals are the conduction electrons (itinerant electrons). This type of exchange was first proposed by Ruderman and Kittel[22] and later extended by Kasuya and Yosida to give the theory now generally know as the RKKY interaction.

The interaction is characterised by a coupling coefficient, $j$, given by[23]

$$j\left( \boldsymbol{R}_{l} - \boldsymbol{R}_{l^{\prime}} \right) ... ...\left( 2k_{F}\vert\boldsymbol{R}_{l} - \boldsymbol{R}_{l^{\prime}}\vert \right)$$ (5.8)

where $k_{F}$ is the radius of the conduction electron Fermi surface, $R_{l}$ is the lattice position of the point moment, $\epsilon_{F}$ is the Fermi energy and

$$F(x) = \frac{x\cos x - \sin x}{x^{4}}$$ (5.9)

The RKKY exchange coefficient, $j$, oscillates from positive to negative as the separation of the ions changes and has the damped oscillatory nature shown in Figure 5.1. Therefore, depending upon the separation between a pair of ions their magnetic coupling can be ferromagnetic or antiferromagnetic. A magnetic ion induces a spin polarisation in the conduction electrons in its neighbourhood. This spin polarisation in the itinerant electrons is felt by the moments of other magnetic ions within range, leading to an indirect coupling.

Figure 5.1: Variation of the indirect exchange coupling constant, $j$, of a free electron gas in the neighbourhood of a point magnetic moment at the origin $r=0$.

In rare-earth metals, whose magnetic electrons in the 4$f$ shell are shielded by the 5$s$ and 5$p$ electrons, direct exchange is rather weak and insignificant and indirect exchange via the conduction electrons gives rise to magnetic order in these materials.