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RKKY interaction
RKKY interaction
RKKY interaction
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The RKKY interaction is a long-range interaction between magnetic moments in a metal. The energy oscillates with distance, decaying as . The oscillations are caused by the interaction of the magnetic moments with the conduction electrons in the metal.
A schematic diagram of 4 electrons scattered by 4 magnetic atoms far apart. Each atom is at the center of decaying electron waves. The electrons mediate the interactions among the atoms, whose poles can flip because of the influence of other atoms and the surrounding electrons. Reproduced from [1] and [2].

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: 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 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]

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RKKY interaction

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The Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction is an indirect exchange mechanism that couples two localized magnetic moments through the itinerant conduction electrons in a metallic host, leading to a long-range oscillatory magnetic interaction whose sign and magnitude depend on the distance between the moments. First theoretically described by Malvin A. Ruderman and in 1954 as an indirect coupling of nuclear magnetic moments via conduction electrons in metals, the interaction was independently extended by Tadao Kasuya in 1956 and Kei Yosida in 1957 to explain the coupling between localized electron spins and conduction electrons in ferromagnetic and antiferromagnetic metals. In its standard form for a three-dimensional free-electron gas, the RKKY interaction energy between two magnetic impurities separated by distance R\mathbf{R} is proportional to J(R)∝cos⁡(2kFR)−2kFRsin⁡(2kFR)R4J(\mathbf{R}) \propto \frac{\cos(2k_F R) - 2k_F R \sin(2k_F R)}{R^4}, where kFk_F is the Fermi wavevector, reflecting an with period π/kF\pi / k_F that alternates between ferromagnetic (parallel alignment) and antiferromagnetic (antiparallel alignment) preferences, and a power-law decay that scales as 1/R31/R^3 at large distances. This form arises from second-order in the s-d exchange model, where the conduction electrons' spin susceptibility at wavevector 2kF2k_F induces the Ruderman-Kittel . In lower dimensions, such as two-dimensional systems like , the decay exponent changes (e.g., to 1/R31/R^3 without oscillations in undoped cases), and the interaction can exhibit sublattice-dependent ferromagnetic or antiferromagnetic . The RKKY interaction plays a crucial role in understanding magnetic ordering in various condensed matter systems, including diluted magnetic semiconductors where it mediates carrier-induced , heavy-fermion compounds exhibiting complex magnetic phases, and layered magnetic structures with interlayer . In modern contexts, it influences phenomena like stabilization in chiral magnets and spin interactions in topological materials, such as Weyl semimetals, where band structure modifications alter its range and .

Background

Conduction electrons and

In simple metals, the conduction electrons behave as a nearly free gas of fermions, forming a degenerate at low temperatures where is much smaller than the , leading to occupation of states up to the according to the . This model treats the valence electrons as delocalized plane waves moving through a uniform positive background of ions, neglecting periodic lattice effects for a basic understanding of metallic properties. Key characteristics of this Fermi gas include the Fermi wavevector kF=(3π2n)1/3k_F = (3\pi^2 n)^{1/3}, where nn is the electron number density, defining the radius of the Fermi sphere in momentum space. The Fermi energy is EF=ℏ2kF22mE_F = \frac{\hbar^2 k_F^2}{2m}, typically on the order of several electron volts for metals like copper or sodium, representing the chemical potential at absolute zero. The density of states at the Fermi level, N(EF)=3n2EFN(E_F) = \frac{3n}{2E_F}, quantifies the number of available electron states per unit energy interval, crucial for response functions in metals. The enforces that only states below EFE_F are occupied [at T](/page/AT&T) = 0, resulting in a step-like Fermi-Dirac distribution that suppresses low-energy excitations and enhances susceptibility to perturbations near the . This leads to Pauli paramagnetism, where the spin susceptibility χ\chi is proportional to N(EF)N(E_F), reflecting the alignment of spins in a magnetic without orbital contributions dominating. The static spin susceptibility for non-interacting free electrons is captured by the Lindhard function χ(q)\chi(q), which describes the response to a wavevector qq perturbation and exhibits a logarithmic singularity in its at q=2kFq = 2k_F, arising from the sharp and nesting of particle-hole excitations. This feature highlights the enhanced susceptibility for momenta connecting antipodal points on the Fermi sphere, influencing indirect exchange mechanisms in metallic hosts.

Localized magnetic moments in metals

Localized magnetic moments in metals originate from unpaired electrons in the partially filled d-shells of impurities or f-shells of rare-earth impurities embedded in non-magnetic host metals such as (Cu) or (Au). In these dilute alloys, the impurity atoms, like (Mn) in Cu or cerium (Ce) in Au, retain their atomic-like electronic configuration due to strong intra-atomic repulsion, leading to well-defined spin moments that do not delocalize into the host lattice. This localization contrasts with the itinerant nature of conduction electrons in the pure metal, where d- or f-electrons would hybridize broadly. The interaction between a single such localized impurity spin and the surrounding conduction electrons is described by the Kondo model Hamiltonian: H=JKS⋅s,H = J_K \mathbf{S} \cdot \mathbf{s}, where S\mathbf{S} is the spin operator of the impurity, s\mathbf{s} is the spin operator of the conduction electron at the impurity site, and JK>0J_K > 0 represents the antiferromagnetic exchange coupling strength. This s-d exchange model captures the essential physics of spin-flip scattering processes without invoking direct orbital overlap between the localized moment and conduction band states. In dilute alloys, these localized spins are separated by distances much larger than the spatial extent of their wavefunctions, precluding direct exchange interactions that rely on orbital overlap; instead, any between distant moments must be mediated indirectly by the itinerant conduction electrons. Experimental evidence for these localized moments includes enhancements in the of the host metal, often following a Curie-like dependence at higher temperatures due to the free spins, as observed in Cu-Mn alloys where the impurity contribution exceeds the Pauli susceptibility of pure Cu. Additionally, resistivity anomalies manifest as a minimum in the electrical resistance at low temperatures, arising from the temperature-dependent of conduction electrons by the impurity spins, a hallmark seen in dilute alloys like Au-Fe or Cu-Cr.

Historical development

Original proposals

The RKKY interaction was first proposed in 1954 by Malvin A. Ruderman and in their seminal paper addressing the indirect coupling of nuclear magnetic moments in metals. Motivated by observations of (NMR) in metals, particularly the broad linewidths reported in experiments on metallic samples, they sought to explain how hyperfine interactions between conduction electrons and nuclear spins could lead to long-range effects. This work built on earlier NMR studies in molecules that suggested indirect spin-spin couplings, extending the concept to the metallic environment where conduction electrons mediate interactions between distant nuclei. The key insight of Ruderman and was that a second-order perturbation process, arising from the hyperfine interaction, induces an effective oscillatory coupling between nuclear spins separated by distances on the order of the Fermi wavelength. In this mechanism, a nuclear spin polarizes the surrounding conduction electrons, which in turn influence another distant nuclear spin, resulting in an indirect exchange that decays with but oscillates in sign. Their derivation assumed a for the conduction band, treating the electrons as non-interacting and the nuclear spins as classical moments to simplify the calculation. This proposal emerged during the post-World War II boom in , a period marked by rapid advancements in understanding electronic properties of materials, driven by growing interest in magnetic alloys and their applications in . Ruderman and noted limitations in their model, including the neglect of electron-electron interactions and the classical treatment of spins, which would later be addressed in extensions to quantum and interacting systems.

Independent formulations

In 1956, Tadao Kasuya extended the original Ruderman-Kittel proposal by applying it to the between localized d-electron spins in metals, mediated by conduction electrons. Building on Zener's model of metallic magnetism, Kasuya derived a long-range exchange-type interaction that arises from the s-d between itinerant s-electrons and localized d-spins, enabling the explanation of both ferromagnetic and antiferromagnetic ordering in transition metals and alloys. This formulation emphasized the role of indirect exchange in dilute systems where direct overlap between localized moments is negligible. Independently in 1957, Kei Yosida refined the calculation of the interaction's effects on the paramagnetic susceptibility of conduction electrons in dilute alloys, such as Cu-Mn. Yosida demonstrated that the induced spin polarization oscillates with distance from the impurity and decays as 1/r³, leading to ferromagnetic coupling for certain separations and antiferromagnetic coupling for others, depending on the position relative to the . This oscillatory behavior highlighted the interaction's tendency to favor parallel or antiparallel alignments based on inter-impurity spacing. The acronym RKKY, denoting Ruderman-Kittel-Kasuya-Yosida, emerged from these concurrent developments, encapsulating the unified framework for conduction-electron-mediated coupling of localized magnetic moments. These independent formulations established the RKKY interaction as a cornerstone for interpreting magnetic properties in dilute alloys.

Theoretical derivation

Perturbation theory framework

The RKKY interaction arises as an indirect exchange coupling between two localized magnetic moments in a metal, mediated by the conduction electrons. The theoretical foundation is provided by second-order applied to the Kondo-like Hamiltonian describing the local exchange between the localized spins and the itinerant electron spins. Specifically, the perturbation Hamiltonian is given by H′=J1S1⋅s(r1)+J2S2⋅s(r2),H' = J_1 \mathbf{S}_1 \cdot \mathbf{s}(\mathbf{r}_1) + J_2 \mathbf{S}_2 \cdot \mathbf{s}(\mathbf{r}_2), where S1\mathbf{S}_1 and S2\mathbf{S}_2 are the localized spin operators at positions r1\mathbf{r}_1 and r2\mathbf{r}_2, J1J_1 and J2J_2 are the local exchange couplings, and s(r)\mathbf{s}(\mathbf{r}) represents the spin density operator of the conduction electrons at position r\mathbf{r}. In this framework, the unperturbed system consists of non-interacting conduction electrons filling states up to the Fermi level, with the localized spins treated as fixed in the initial state. The second-order energy correction to the ground-state energy, which captures the effective interaction between the localized spins, is ΔE(2)=∑f≠0∣⟨f∣H′∣0⟩∣2E0−Ef,\Delta E^{(2)} = \sum_{f \neq 0} \frac{|\langle f | H' | 0 \rangle|^2}{E_0 - E_f}, where ∣0⟩|0\rangle denotes the unperturbed ground state, ∣f⟩|f\rangle are the excited states of the conduction electrons, E0E_0 is the ground-state energy, and Ef>E0E_f > E_0 are the energies of the intermediate states. This energy shift results in an effective Heisenberg-type interaction of the form Heff=JRKKYS1⋅S2H_{\rm eff} = J_{\rm RKKY} \mathbf{S}_1 \cdot \mathbf{S}_2, where the coupling constant JRKKYJ_{\rm RKKY} encapsulates the indirect exchange mediated by the electrons. The virtual excitations in this process involve the creation of electron-hole pairs in the conduction band, specifically those with total spin 1 that can couple to the scalar product S1⋅S2\mathbf{S}_1 \cdot \mathbf{S}_2. These pairs are generated across the : the interaction with the first localized spin polarizes the electron spins, creating a virtual spin excitation, which then scatters to interact with the second spin, thereby linking the two moments. The plays a crucial , as the energy denominators in the perturbation sum emphasize contributions from states near the , where Ef−E0E_f - E_0 is small. This perturbative approach relies on several key assumptions to its validity. The local exchange couplings must satisfy J1,2≪EFJ_{1,2} \ll E_F, where EFE_F is the , to justify treating H′H' as a small perturbation relative to the of the conduction electrons. Additionally, the concentration of localized magnetic impurities is assumed to be sufficiently low such that interactions between more than two moments can be neglected, avoiding higher-order effects or multiple processes. These conditions align with the dilute limit typical of many metallic systems hosting such interactions.

Calculation in free electron model

In the perturbation theory framework outlined previously, the effective exchange interaction between two localized spins separated by distance r\mathbf{r} is given by J(r)∝∑qχ(q)eiq⋅rJ(\mathbf{r}) \propto \sum_{\mathbf{q}} \chi(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}}, where χ(q)\chi(\mathbf{q}) is the static spin susceptibility of the conduction electron gas. For the three-dimensional free electron model, the susceptibility χ(q)\chi(\mathbf{q}) is the Lindhard function, which accounts for the response of the Fermi sea to a spin perturbation. The Lindhard function takes the form χ(q)=N(EF)[12+1−u24uln⁡∣1+u1−u∣]\chi(q) = N(E_F) \left[ \frac{1}{2} + \frac{1 - u^2}{4u} \ln \left| \frac{1 + u}{1 - u} \right| \right]
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