# RKKY interaction

RKKY stands for Ruderman-Kittel-Kasuya-Yosida and refers to a coupling mechanism of nuclear magnetic moments or localized inner d or f shell electron spins in a metal by means of an interaction through the conduction electrons.

The RKKY interaction was originally proposed by M.A. Ruderman and Charles Kittel of the University of California, Berkeley as a means of explaining unusually broad nuclear spin resonance lines that had been observed in natural metallic silver. The theory uses second-order perturbation theory to describe an indirect exchange coupling whereby the nuclear spin of one atom interacts with a conduction electron via the hyperfine interaction, and this conduction electron then interacts with another nuclear spin thus creating a correlation energy between the two nuclear spins. (Alternatively, instead of nuclear spins coupling to conduction spins via the hyperfine interaction, another scenario is for inner electron spins to couple to conduction spins via the exchange interaction.) The theory is based on Bloch wavefunctions, and is therefore only applicable to crystalline systems. The derived exchange interaction takes the following form:

$H(\mathbf{R}_{ij}) = \frac{\mathbf{I}_i \cdot \mathbf{I}_j}{4} \frac{\left | \Delta k_m k_m \right |^2 m^*}{(2 \pi )^3 R_{ij}^4 \hbar^2} \left [ 2 k_m R_{ij} \cos( 2 k_m R_{ij} ) - \sin( 2 k_m R_{ij} ) \right ]$

where H represents the Hamiltonian, $R_{ij}$ is the distance between the nuclei i and j, $\mathbf{I}_i$ is the nuclear spin of atom i, $\Delta k_m k_m$ is a term that represents the strength of the hyperfine interaction, $m^*$ is the effective mass of the electrons in the crystal, and $k_m$ is the wave vector of the conduction electrons. In crystalline materials, the wave vectors of conduction electrons are very close to the Fermi surface.

Tadao Kasuya of Nagoya University later proposed that a similar indirect exchange coupling could be applied to localized inner d-electron spins interacting via conduction electrons. This theory was expanded more completely by Kei Yosida of the University of California, Berkeley to give a Hamiltonian that describes (d-electron spin)-(d-electron spin), (nuclear spin)-(nuclear spin) as well as (d-electron spin)-(nuclear spin) interactions. Van Vleck clarified some subtleties of the theory, particularly the relationship between the first and second order perturbative contributions.

Perhaps the most significant application of the RKKY theory has been to the theory of giant magnetoresistance (GMR). GMR was discovered when the coupling between thin layers of magnetic materials separated by a non-magnetic spacer material was found to oscillate between ferromagnetic and antiferromagnetic as a function of the distance between the layers. This ferromagnetic/antiferromagnetic oscillation is one prediction of the RKKY theory.[1][2]

## References

1. M.A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954).
2. T. Kasuya, Prog. Theor. Phys. 16, 45 (1956).
3. K. Yosida, Phys. Rev. 106, 893 (1957).
4. J. H. Van Vleck. Reviews of Modern Physics 34, 681-686 (1962).
5. A. Blandin and J. Friedel, J. phys. rad. 20, page 160-168 (1959)
6. Quantum Theory of Solids, 2ed. pp 360–366, C. Kittel, Wiley 1987
1. ^ S.S.P. Parkin and D. Mauri, Physical Review B Vol. 44 7131-7134 (1991)
2. ^ Y. Yafet, Physical Review B Vol. 36 3948-3949 (1987)