# 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 is the J/t >> 1 limit of the double exchange interaction.

The RKKY interaction was originally proposed by Malvin Ruderman and Charles Kittel of the University of California, Berkeley[1], 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 through 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 through the hyperfine interaction, another scenario is for inner electron spins to couple to conduction spins through 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:

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

where H represents the Hamiltonian, ${\displaystyle R_{ij}}$ is the distance between the nuclei i and j, ${\displaystyle \mathbf {I} _{i}}$ is the nuclear spin of atom i, ${\displaystyle \Delta _{k_{m}k_{m}}}$ is a matrix element that represents the strength of the hyperfine interaction, ${\displaystyle m^{*}}$ is the effective mass of the electrons in the crystal, and ${\displaystyle k_{m}}$ is the Fermi momentum.

Tadao Kasuya from Nagoya University later proposed that a similar indirect exchange coupling could be applied to localized inner d-electron spins interacting through conduction electrons.[2] This theory was expanded more completely by Kei Yosida of the UC Berkeley, to give a Hamiltonian that describes (d-electron spin)–(d-electron spin), (nuclear spin)–(nuclear spin), and (d-electron spin)–(nuclear spin) interactions.[3] J.H. Van Vleck clarified some subtleties of the theory, particularly the relationship between the first- and second-order perturbative contributions.[4]

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.[5][6]

## References

1. Ruderman, M. A.; Kittel, C. (1954). "Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons". Physical Review. 96: 99. Bibcode:1954PhRv...96...99R. doi:10.1103/PhysRev.96.99.
2. Kasuya, Tadao (1956). "A Theory of Metallic Ferro- and Antiferromagnetism on Zener's Model". Progress of Theoretical Physics. 16: 45. Bibcode:1956PThPh..16...45K. doi:10.1143/PTP.16.45.
3. Yosida, Kei (1957). "Magnetic Properties of Cu-Mn Alloys". Physical Review. 106 (5): 893. Bibcode:1957PhRv..106..893Y. doi:10.1103/PhysRev.106.893.
4. Van Vleck, J. H. (1962). "Note on the Interactions between the Spins of Magnetic Ions or Nuclei in Metals". Reviews of Modern Physics. 34 (4): 681. Bibcode:1962RvMP...34..681V. doi:10.1103/RevModPhys.34.681.
5. Parkin, S. S. P.; Mauri, D. (1991). "Spin engineering: Direct determination of the Ruderman-Kittel-Kasuya-Yosida far-field range function in ruthenium". Physical Review B. 44 (13): 7131. Bibcode:1991PhRvB..44.7131P. doi:10.1103/PhysRevB.44.7131.
6. Yafet, Y. (1987). "Ruderman-Kittel-Kasuya-Yosida range function of a one-dimensional free-electron gas". Physical Review B. 36 (7): 3948. Bibcode:1987PhRvB..36.3948Y. doi:10.1103/PhysRevB.36.3948.