Pair anisotropy in disordered magnetic systems

TL;DR

The paper identifies pair-induced uniaxial anisotropy as a crucial mechanism in disordered magnetic systems, arising from nearest-neighbor magnetic ion pairs that break local symmetry. By combining first-principles DFT calculations with atomistic spin simulations, it shows that pair anisotropy, modeled as a bond-directed uniaxial term, significantly improves agreement with experimental magnetization curves for Ga$_{1-x}$Mn$_x$N at $x=7.9\%$, compared to models relying solely on single-ion anisotropy. The study reveals geometry-dependent anisotropy parameters and JT suppression in Mn pairs, highlighting the need for pair-aware spin Hamiltonians in disordered magnets. Collectively, these results provide a general framework for predictive multiscale modeling of spin dynamics in dilute magnetic semiconductors and related disordered systems.

Abstract

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.

Figures (5)

• Figure 1: Investigated wurtzite Ga$_{1-x}$Mn$_x$N structure. (a) The $3\times2\times2$ supercell of GaN containing a single Mn ion. (b) Close-up of Mn and its four nearest-neighbor nitrogen anions. Distances labeled $A$, $B$, $C$, $\alpha$, $\beta$, and $\gamma$ represent different N--N separations. (c) Top view of (b). Parenthetical values indicate changes in these distances due to trigonal and Jahn-Teller distortions in pm (see Eq. \ref{['eq:jt_para']} in the main text). (d) and (e) A single Mn ion is surrounded by 12 nearest-neighbor Ga cations, which can be potential sites for other Mn substitution. This geometry gives rise to six symmetry-equivalent Mn--Mn in-plane pairs (example in d) and six out-of-plane pairs (example in e). Panels (d) and (e) illustrate Mn ions and their nearest-neighbor nitrogen environments.
• Figure 2: Trigonal and Jahn--Teller (JT) distortions and their effect on the density of state (DOS) calculations. (a) Ideal tetrahedral configuration of the Mn ion with its four nearest-neighbor nitrogen anions (no trigonal or JT distortion). Dashed lines are guides for the eye in illustrating the original cubic symmetry, which is modified in panels (b) and (c). (b) Trigonal distortion along the c-axis of GaN. (c) Jahn-Teller distortion reducing the symmetry from tetrahedral to tetragonal. (d) Partial density of states (PDOS) of the Mn ion and total density of states (TDOS) of the supercell with one Mn ion in wurtzite GaN with optimized GaN atomic positions and lattice parameters (only trigonal distortion present). (e) PDOS and TDOS after full relaxation of both atomic positions and lattice parameters, incorporating Jahn-Teller distortion. Colored lines in the inset of (d, e) show PDOS contributions from different Mn $3d$ orbitals. Solid and dashed lines represent spin-up and spin-down states, respectively. The black vertical line at $E = 0$ indicates the Fermi level.
• Figure 3: Magnetocrystalline energy of the supercell containing a single Mn ion. Panels (c) and (d) show the energy variation $\Delta E(\theta,\phi)$ obtained from DFT (orange lines) as a function of spherical angles $\phi$ and $\theta$ of Mn spin, respectively. Blue lines represent fits to the spin Hamiltonian model defined in Eq. \ref{['eq:ani_model']}. Mn spin orientations are illustrated in (a) and (b), corresponding to rotations over $\phi$ and $\theta$, respectively.
• Figure 4: Magnetocrystalline energy of the supercell containing the in-plane (left panel) and out-of-plane (right panel) Mn--Mn pair. DFT results (orange) and spin Hamiltonian fit (blue) for both Mn spins rotations over $\phi$ and $\theta$ are shown in (c-d) and (g-h). The spin Hamiltonian model is defined in Eq. \ref{['eq:ani_model_pairs']}. Panels (a), (b), (e) and (f) illustrate the corresponding Mn spins rotation geometries. The position of second Mn ion is represented as a purple star relative to the first one (centered, not shown).
• Figure 5: Experimental and simulation results of magnetization and thermoremanent magnetization for the Ga$_{1-x}$Mn$_x$N sample with $x=7.9\%$. Magnetization measurements (blue dots) are performed at $T$ = 2 K for two magnetic field orientations: (a) in-plane ($\mathbf{H} \perp \mathbf{c}$) and (b) out-of-plane ($\mathbf{H} \parallel \mathbf{c}$). Insets show close-up on the weak magnetic field regime. For the clarity of the presentation only one branch of the hysteresis loop is shown. (c) The in-plane thermoremanent magnetization. The numerical atomistic spin model simulation results are obtained using single-ion anisotropy (SIA, green diamonds) and pair anisotropy (PA, red squares) approaches.