Pair anisotropy in disordered magnetic systems

Accurate modelling of magnetism is pivotal for elucidating the microscopic origins of magnetic phenomena in functional materials. However, for a specified class of materials, such as random dilute ferromagnets or alloys, the reliance on simplifying assumptions, such as single-ion anisotropy, limits the accuracy of existing spin models. In such systems, there is a significant probability of the formation of nearest-neighbor magnetic ion pairs or higher order clusters, whose presence breaks the local symmetry of otherwise isolated magnetic species. Here, we introduce the concept of pair-induced uniaxial anisotropy and demonstrate how nearby atoms influence each other’s anisotropic behavior. This effect is investigated in the dilute magnetic semiconductor Ga$_{1-x}$Mn$_x$N, by means of density functional theory calculations. The inclusion of pair anisotropy in the atomistic spin simulations significantly improves the agreement between simulated and experimental magnetization curves, in contrast to models that consider only single-ion anisotropy.

💡 Research Summary

The paper addresses a fundamental limitation of conventional spin models for disordered dilute magnetic systems, namely the reliance on a single‑ion anisotropy description. In materials such as dilute magnetic semiconductors (DMS) or random alloys, a substantial fraction of magnetic ions form nearest‑neighbour (NN) pairs or small clusters, which break the local symmetry assumed in single‑ion models. The authors introduce the concept of “pair‑induced anisotropy” and demonstrate its relevance for the wurtzite dilute magnetic semiconductor Ga₁₋ₓMnₓN.

First, a statistical analysis shows that for Mn concentration x≈6 % the probability that a Mn ion has at least one Mn NN is about 52 %; the probability of exactly one NN is 36.5 %, while larger clusters occur with ≈16 % probability. Hence, even at modest doping the single‑ion approximation is inadequate.

Using density‑functional theory (DFT) with the projector‑augmented wave method and the PBE‑GGA functional, the authors compute the structural and electronic properties of (i) a supercell containing a single Mn ion, and (ii) supercells containing two Mn ions placed at NN positions, both in‑plane and out‑of‑plane. For the isolated Mn, both a trigonal distortion (parameter t) and a Jahn‑Teller (JT) distortion (parameters j₁, j₂) are present, lowering the local symmetry from tetrahedral to tetragonal. In contrast, when a second Mn is introduced, the JT distortion disappears; instead, the Mn–N bond that bridges the two Mn atoms shortens markedly, producing a pronounced anisotropic distortion aligned with the Mn–Mn axis. This symmetry breaking lifts the orbital degeneracy of the Mn d‑states, suppressing the JT effect and generating a new uniaxial anisotropy component.

Non‑collinear spin‑orbit DFT calculations are performed by rotating the Mn magnetic moments through spherical angles (θ, φ) while keeping the crystallographic c‑axis as the reference. The total‑energy landscape ΔE(θ, φ) is fitted to a phenomenological spin Hamiltonian. For a single Mn the Hamiltonian contains a trigonal term (K_TR S_z²) and a JT term (K_JT (S·e_JT)²). For Mn–Mn pairs an additional term K_pair (S₁·e_pair)(S₂·e_pair) is required, where e_pair is the unit vector along the Mn–Mn bond. The extracted K_pair values are positive and depend on the pair orientation, confirming that the pair axis introduces a strong uniaxial anisotropy.

These parameters are incorporated into atomistic spin simulations (Monte‑Carlo) of macroscopic magnetization curves M(H). When only single‑ion anisotropy is used, simulated curves deviate significantly from low‑temperature experimental data, especially in the approach to saturation. Inclusion of the pair‑induced anisotropy term dramatically improves the agreement, reducing the discrepancy by more than a factor of two. This demonstrates that even a few‑meV pair anisotropy energy can dominate the macroscopic magnetic response in dilute systems.

The authors justify the use of plain PBE‑GGA rather than DFT + U or hybrid functionals: while the latter improve the band gap, they introduce inconsistencies in non‑collinear spin‑orbit calculations. Since the study focuses on relative energy differences governing anisotropy (order of meV), PBE provides a robust and consistent framework.

Finally, the paper argues that pair‑induced anisotropy is a generic feature of disordered magnetic materials. It should be considered in modeling dilute magnetic semiconductors, 2D van‑der‑Waals magnets with random spin sites, spin glasses, and cluster‑based magnets. In Ga₁₋ₓMnₓN, the ability to tune single‑ion anisotropy via the inverse piezoelectric effect makes the pair contribution especially relevant for electric‑field‑controlled spin‑tronic devices. By quantitatively linking microscopic pair geometry to macroscopic magnetic behavior, the work establishes a new paradigm for accurate spin‑model construction in disordered magnetic systems.