# RKKY interaction

In the physical theory of spin glass magnetization, the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction models the coupling of nuclear magnetic moments or localized inner d- or f-shell electron spins through conduction electrons. It is named after Malvin Ruderman, Charles Kittel, Tadao Kasuya, and Kei Yosida, the physicists who first proposed and developed the model.

Malvin Ruderman and Charles Kittel of the University of California, Berkeley first proposed the model to explain unusually broad nuclear spin resonance lines in natural metallic silver. The theory is an indirect exchange coupling: the hyperfine interaction couples the nuclear spin of one atom to a conduction electron also coupled to the spin of a different nucleus. The assumption of hyperfine interaction turns out to be unnecessary, and can be replaced equally well with the exchange interaction.

The simplest treatment assumes a Bloch wavefunction basis and therefore only applies to crystalline systems; the resulting correlation energy, computed with perturbation theory, takes the following form:

${\displaystyle H(\mathbf {R} _{ij})={\frac {\mathbf {I} _{i}\cdot \mathbf {I} _{j}}{4}}{\frac {\left|\Delta _{k_{m}k_{m}}\right|^{2}m^{*}}{(2\pi )^{3}R_{ij}^{4}\hbar ^{2}}}\left[2k_{m}R_{ij}\cos(2k_{m}R_{ij})-\sin(2k_{m}R_{ij})\right]{\text{,}}}$
where H represents the Hamiltonian, Rij is the distance between the nuclei i and j, Ii is the nuclear spin of atom i, Δkmkm is a matrix element that represents the strength of the hyperfine interaction, m* is the effective mass of the electrons in the crystal, and km is the Fermi momentum.[3] Intuitively, we may picture this as when one magnetic atom scatters an electron wave, which then scatters off another magnetic atom many atoms away, thus coupling the two atoms' spins.[2]

Tadao Kasuya from Nagoya University later proposed that a similar indirect exchange coupling could occur with localized inner d-electron spins instead of nuclei.[4] 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.[5] J.H. Van Vleck clarified some subtleties of the theory, particularly the relationship between the first- and second-order perturbative contributions.[6]

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.[7][8]

## References

1. ^ Stein, Daniel L. (July 1989). "Spin Glasses". Scientific American. 261 (1): 52–59. doi:10.1038/scientificamerican0789-52. ISSN 0036-8733.
2. ^ a b Stein, Daniel L.; Newman, Charles M. (2013). Spin glasses and complexity. Primers in complex systems. Princeton Oxford: Princeton University Press. Figure 4.4. ISBN 978-0-691-14733-8.
3. ^ Ruderman, M. A.; Kittel, C. (1954). "Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons". Physical Review. 96 (1): 99–102. Bibcode:1954PhRv...96...99R. doi:10.1103/PhysRev.96.99.
4. ^ Kasuya, Tadao (1956). "A Theory of Metallic Ferro- and Antiferromagnetism on Zener's Model". Progress of Theoretical Physics. 16 (1): 45–57. Bibcode:1956PThPh..16...45K. doi:10.1143/PTP.16.45.
5. ^ Yosida, Kei (1957). "Magnetic Properties of Cu-Mn Alloys". Physical Review. 106 (5): 893–898. Bibcode:1957PhRv..106..893Y. doi:10.1103/PhysRev.106.893.
6. ^ 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): 681–686. Bibcode:1962RvMP...34..681V. doi:10.1103/RevModPhys.34.681.
7. ^ 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.
8. ^ Yafet, Y. (1987). "Ruderman-Kittel-Kasuya-Yosida range function of a one-dimensional free-electron gas". Physical Review B. 36 (7): 3948–3949. Bibcode:1987PhRvB..36.3948Y. doi:10.1103/PhysRevB.36.3948.