## RKKY interaction

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The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction is the electron-mediated long-distance exchange coupling between localized magnetic moments in metals. It was first proposed by M.A. Ruderman and C. Kittel [1] as a mechanism of the unexpectedly strong broadening of the nuclear spin resonance in tin, thallium and silver. Later on, in the works by T. Kasuya [2] and K. Yosida [3] it was applied to the localized spins of atoms with a partially filled inner d-shell.
The model. The RKKY interaction is mediated by the electrons partially occupying the conduction band with a single-particle dispersion relation $E(k)$. The conduction electrons experience a local spin-conserving interaction with localized spins $\vec S_1, \dots , \vec S_N$ positioned at points $R_1,\dots , R_N$ inside the metal. The Hamiltonian of the system is
where $c^\dagger _{s\mathbf k} (c_{s\mathbf k})$ is the creation (annihilation) operator of an electron with spin $s$ and the Bloch momentum $\mathbf k$, $\vec \sigma$ is the vector whose components are the three Pauli matrices, and $\Delta _{\mathbf k\mathbf k'}$ is the system-specific form-factor defining the interaction between the localized spin and the conduction electrons. The long distance RKKY interaction is dominated by the conduction electrons close to the Fermi surface and therefore only depends on $\Delta =\Delta _{k_F k_F}$. Regarding $\Delta$ as a small parameter, one can find the interaction between localized spin using the second-order perturbation theory. In a three-dimensional system the perturbative expression for the interaction between two localized spins is given by
In this expression $k_F$ is the Fermi momentum of conduction electrons and it is assumed that with a reasonable accuracy $E(\mathbf k)= \hbar ^2 k^2/2m_*$ for $k \le k_F$ (for the discussion of systems with non-spherical Fermi surfaces see [4]).
A peculiar property of the RKKY exchange, is that it experiences oscillations with the period $\pi /k_F$ as a function of distance between localized spins. These are known as Friedel oscillations and play an important role in applications such as the giant magnetoresistance (GMR) and spin glasses.
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]^ D. I. Golosov and M. I. Kaganov, J. Phys.: Condens. Matter 5, 1481-1492 (1993).
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