Phonon mediated spin-spin interactions

Abstract: Indirect long range interactions between localized magnetic moments are in metals mediated by itinerant electrons. In insulators and semi-conductor, such interactions need to be small, if not negligible, due to the absence of mediating carriers. The existence of magnetically ordered insulators, for instance, metal-oxides, is therefore an everlasting source for proposals of various mechanisms that may support the order. Here, phonon mediated interactions between localized magnetic moments is considered as a mechanism that can provide quantifiable symmetric and anti-symmetric anisotropic spin-spin interactions. It is demonstrated that while a symmetric anisotropic interaction exists for all types of phonons, the existence of anti-symmetric anisotropic interactions requires broken inversion symmetry. The latter mechanism may explain the weak ferromagnetic order observed in chiral, e.g., CuO and CoO compounds. Furthermore, the interaction is nearly independent of the temperature at low temperature while approaches a linear growth at high. Spatially, the interactions have an oscillatory power law decay with the inter-nuclei distance.

• The paper presents a detailed theory showing that phonons mediate both symmetric anisotropic and antisymmetric (Dzyaloshinskii–Moriya) spin-spin interactions in insulating and semiconducting materials.
• The framework demonstrates that while symmetric exchange is generic, the antisymmetric interaction emerges only in non-centrosymmetric lattices, highlighting the role of lattice geometry.
• Temperature plays a crucial role, with phonon contributions increasing with thermal population, enabling tunable magnetic textures relevant for spintronics and multiferroic applications.

Phonon-Mediated Spin-Spin Interactions in Insulating and Semiconducting Materials

Introduction

This work addresses the longstanding question of how indirect, long-range spin-spin interactions emerge in systems lacking itinerant electrons, such as insulators and semiconductors. Historically, indirect interactions in metals have been understood as RKKY-type coupling mediated by itinerant electrons, while superexchange and double-exchange provide the standard mechanisms for magnetic order in electron-poor systems. However, recent observations of weak ferromagnetism in chiral metal oxides, where superexchange would normally result in strictly antiferromagnetic order, highlight deficiencies in these paradigms. This paper constructs a robust, general theory of indirect exchange mediated by phonons—collective excitations of nuclear motion—showing that phonons can generate both symmetric and antisymmetric anisotropic spin-spin couplings, depending sensitively on the underlying lattice symmetry.

Theoretical Framework for Spin-Phonon Coupling

The model system comprises localized spins (magnetic moments) embedded in an insulating or semiconducting host. The electronic degrees of freedom are spatially localized, motivating a description in terms of localized orbital electrons and nuclear displacements. The spin-electron and electron-phonon interactions are combined to yield an effective spin-phonon coupling, formulated through a tensor $\calT^\text{sc}$ encoding spin-charge susceptibility, which remains finite in nearly all realistic oxides and semiconductors due to nonvanishing spin-orbit coupling.

Expressing the nuclear displacement operator in terms of phonon creation and destruction operators, and exploiting the inherent coupling between the spin and nuclear subsystems, enables a Hamiltonian treatment in which the phononic reservoir dynamically mediates spin-spin interactions.

Kubo-Type Formulation of Phonon-Mediated Exchange

The spin-spin interaction is derived by expanding the partition function to bilinear order in the localized magnetic moments. The resulting effective action contains an interaction tensor $\calD(x,x')$, which, through contractions with the local spin operators, describes spin exchange mediated solely by virtual or real phonon excitations.

The central result is the decomposition of $\calD$ into symmetric ($\mathbb{S}$) and antisymmetric ($\mathbb{A}$) tensors. Notably:

• Symmetric anisotropic interaction: Exists generically for all types of phonons and all lattice symmetries. Unlike electron-mediated exchange, the isotropic Heisenberg term does not uniquely emerge; the interaction is always anisotropic in spin space.
• Antisymmetric Dzyaloshinskii–Moriya-type interaction: This component is strictly contingent on broken inversion symmetry (e.g., chirality of the lattice or presence of chiral phonons). Its magnitude vanishes identically for inversion-symmetric structures, emphasizing the fundamental role of lattice geometry in the emergence of complex magnetic textures.

Symmetry and Spatial Decay

The explicit dependence of the antisymmetric exchange on the sine of the relative phase (originating from $\sin(\bfq\cdot\bfR)$) manifests the requirement for broken inversion symmetry: only in chiral or otherwise non-centrosymmetric structures will the antisymmetric tensor survive. This is in direct analogy to the classical theory of Dzyaloshinskii–Moriya interactions for electrons.

The spatial decay of both symmetric and antisymmetric components exhibits oscillatory power-law behavior with internuclear distance $R$, obeying $1/R^{d-1}$ for acoustic phonons in $d$ dimensions. Significantly, the sign and type of the interaction (ferro- or antiferromagnetic) can oscillate with $R$ in symmetry-broken lattices due to interference effects ($\calD(x,x')$0 and $\calD(x,x')$1 terms). For optical phonons, the interaction remains long-ranged and further deviates from strict power-law decay. The formalism encompasses the full tensorial and oscillatory character of these exchange processes.

Thermal Properties of the Phonon-Mediated Exchange

A salient result is the nontrivial temperature dependence of the interaction. Unlike electron-mediated exchange, which is largely insensitive to temperature well below the band gap, the phononic contribution grows almost linearly with increasing temperature above a certain threshold (well above the low-temperature regime where $\calD(x,x')$2). This is a direct consequence of the increasing thermal population of relevant phonon modes. In contrast, at low temperatures, the interaction saturates at a minimal, temperature-independent value.

This theoretical prediction is consistent with experimental reports of enhanced magnetic ordering (e.g., increasing coercive field) in metal-organic and chiral compounds at elevated temperatures, suggesting that phonons, rather than purely electronic processes, play a decisive role in stabilizing and modulating these magnetic phases.

Implications for Magnetic Order and Chiral Magnetism

The presence of a generic symmetric anisotropic exchange implies that in centrosymmetric insulators, phonon mediation will typically favor collinear antiferromagnetic order. However, in systems lacking inversion symmetry, both ferro- and antiferromagnetic exchanges arise, and the antisymmetric component induces canting of the moments, creating weakly noncollinear or canted magnetic textures. This provides a physically robust explanation for weak ferromagnetism observed in certain chiral oxides (e.g., CuO, CoO) where traditional mechanisms are insufficient.

These results suggest that tailored manipulation of lattice symmetry (through strain, controlled chiral crystal growth, or interface engineering) and thermal excitation could allow for tunable magnetic anisotropy and novel topological spin textures in insulating magnets—relevant for spintronics, quantum computation, and designer multiferroic materials.

Outlook and Future Directions

Experimental validation of these predictions can proceed via measurement of the distance dependence of interaction between magnetic moments, using, for example, molecular adsorbates on insulating substrates. Ab initio simulations, including explicit consideration of spin-orbit-phonon coupling in chiral systems, would help quantify interaction strengths and anisotropies for comparison with experiment.

The theoretical framework provided here lays a foundation for future investigations into dynamical control of magnetism using phonon engineering (via optical excitation, strain, or non-equilibrium driving), as well as for the interpretation of noncollinear magnetism in complex oxides where electronic mechanisms are either weak or absent.

Conclusion

The analytic theory presented rigorously demonstrates the robustness and ubiquity of phonon-mediated spin-spin interactions in insulating and semiconducting materials. The framework captures the emergence of symmetric anisotropic exchange for arbitrary lattice symmetry, as well as antisymmetric (Dzyaloshinskii–Moriya-type) exchange exclusively in non-centrosymmetric settings. The interactions persist over long distances, oscillate as a function of spatial separation, and increase strongly with temperature due to the Bose population of phonons. These mechanisms offer a unified explanation for both conventional and exotic ordered magnetic phases observed in low-carrier-density systems, with implications for future materials engineering and the control of magnetic order via lattice degrees of freedom.

For a detailed account of the derivation and full analytic results, see "Phonon mediated spin-spin interactions" (2604.14731).