Ferromagnetic Resonance in a Magnetically Dilute Percolating Ferromagnet: An Experimental and Theoretical Study

TL;DR

This work tackles percolating ferromagnetism in the dilute semiconductor Ga$_{1-x}$Mn$_x$N with $x \approx 8\%$ by combining ferromagnetic resonance (FMR) and SQUID magnetometry, and by linking these static and dynamic probes with an atomistic spin model implemented via the stochastic Landau–Lifshitz–Gilbert framework. The authors extract a common set of microscopic parameters that describe both the magnetization curves and the FMR response, revealing a robust uniaxial anisotropy driven by Mn$^{3+}$ single-ion effects and a small negative magnetocrystalline anisotropy constant $K_2$, with an easy axis perpendicular to the $c$-axis. They observe FMR signals from $T \sim 9$ K up to 70 K, while strong line broadening below ~$9$ K eliminates detectable FMR, indicating the presence of non-percolating ferromagnetic clusters well above $T_{\mathrm{C}}$, whose magnetization tracks the bulk $M(T)$. The atomistic simulations reproduce the main trends of both static and dynamic data without parameter tuning, underscoring FMR as a robust diagnostic for spin-diluted, percolating ferromagnets and offering insights for spintronic applications in dilute FM semiconductors.

Abstract

Ferromagnetic resonance (FMR) serves as a powerful probe of magnetization dynamics and anisotropy in percolating ferromagnets, where short-range interactions govern long-range magnetic order. We apply this approach to Ga$_{1-x}$Mn$_x$N ($x \simeq 8$\%), a dilute ferromagnetic semiconductor, combining FMR and superconducting quantum interference device magnetometry. Our results confirm the percolative nature of ferromagnetism in (Ga,Mn)N, with a Curie temperature $T_{\mathrm{C}} = 12$ K, and reveal that despite magnetic dilution, key features of conventional ferromagnets are retained. FMR measurements establish a robust uniaxial anisotropy, dictated by Mn$^{3+}$ single-ion anisotropy, with an easy-plane character at low Mn content. While excessive line broadening suppresses FMR signals below 9 K, they persist up to 70~K, indicating the presence of non-percolating ferromagnetic clusters well above $T_{\mathrm{C}}$. The temperature dependence of the FMR intensity follows that of the magnetization, underscoring the stability of these clusters. We quantitatively describe both FMR and SQUID observables using atomistic spin model operating on a common set of parameters. The level of agreement, achieved without tuning parameters between datasets, demonstrates the robustness and practical applicability of the approach in capturing the essential physics of spin-diluted, percolating ferromagnets. This study advances the understanding of percolating ferromagnetic systems, demonstrating that FMR is a key technique for probing their unique dynamic and anisotropic properties. Our findings contribute to the broader exploration of dilute ferromagnets and provide new insights into percolating ferromagnetic systems, which will be relevant for spintronic opportunities.

Figures (9)

• Figure 1: (Color online) Comparison of experimental (solid symbols) and simulated with the atomistic spin model (open symbols) magnetization curves for (Ga,Mn)N film with $x_{\mathrm{Mn}}=7.9$%. (a) Temperature dependence of the thermo-remnant magnetization (TRM) for the in-plane orientation of $H$ (full green circles). The initial field-cooled (FC) magnetization measured during cooling at $H_{\mathrm{FC}} = 3$ kOe is shown as solid blue squares. The temperature at which TRM vanishes defines the Curie temperature, $T_{\mathrm{C}}$, of the film. Open circles represent results from atomistic spin model and Monte Carlo simulations. (b) Magnetization as a function of magnetic field, $H$, measured at $T=5$ K for two field orientations: perpendicular to the $c$-axis (full squares) and along the $c$-axis (full diamonds), and results of the atomistic spin model added as matched open symbols. The inset provides an expanded view of the low-field region.
• Figure 2: (Color online) (a) Selected, temperature dependent ferromagnetic resonance spectra measured with magnetic field $\mathbf{H}$ applied along the out-of-plane, [0001] crystallographic axis of the (Ga,Mn)N film. The spectra are shifted vertically for clarity. Black dashed lines are fits of Eq. \ref{['Eq:Lorentzian']}. (b) Temperature dependence of the ferromagnetic moment determined by double integration of the FMR signals (pentagons), compared with the magnetization of the (Ga,Mn)N film measured in the same geometry at several magnetic fields $H$ (indicated in the graph).
• Figure 3: (Color online) Temperature $T$ dependence of the magnetic susceptibility, $\chi(T) = M(T)/H$, measured for the (Ga,Mn)N film in magnetic fields between 1 and 10 kOe, and plotted in a double logarithmic scale (small colored symbols). Black pentagons represent the temperature dependence of the FMR intensity normalized by the resonance field $I/H_R$. The dashed magenta line indicates the Curie law behavior, i.e. the proportionality to $1/T$, whereas the solid black line follows $\chi(T) \propto T^{-1.23}$ at high temperatures, as expected for a random paramagnet with ferromagnetic correlations.
• Figure 4: (Color online) (a) Scheme of the FMR measurements geometry emphasizing the coordinate system used. The applied dc magnetic field vector $\mathbf{H}$ lies in the $(1\bar{1}00)$ plane and the azimuthal angle $\varphi_H$ is counted from the [0001] (X) axis. The orientation of the magnetization vector $\mathbf{M}$ is given by the azimuthal $\varphi$ and polar $\theta$ angles, the latter counted from the $[1\bar{1}00]$ (Z) axis. In principle, $\mathbf{H}$ and $\mathbf{M}$ are not collinear. (b) Selected, angularly dependent ferromagnetic resonance spectra recorded with the magnetic field rotated in the $(1\bar{1}00)$ plane at $T=12$ K. Symbols indicate resonance field values. (c) Angular dependence of the resonance fields $H_R$ for the magnetic field $\textbf{H}$ rotated in the $(1\bar{1}00)$ plane at $T=12$ K. Symbols represent experimental data, the solid line is a fit.
• Figure 5: (Color online) (a) Temperature dependence of the resonance fields $H_R$ for two magnetic field orientations: $\mathbf{H}\parallel[0001]$ (squares) and $\mathbf{H}\parallel[11\bar{2}0]$ (circles). The dashed line marks the field corresponding to $g=2$. The inset shows the temperature dependence of the effective g-factor. (b) Temperature dependence of the anisotropy field $H_2$ and the demagnetization field $2\pi M$ (squares and circles, respectively). The latter is determined using the magnetization data shown in the inset of Fig. \ref{['Fig:FMR_T']} for $H=4$ kOe. (c) Simulation results depicting the temperature dependence of $H_R$ for two magnetic field orientations: $\mathbf{H}\parallel[0001]$ (open squares) and $\mathbf{H}\parallel[11\bar{2}0]$ (open circles). (d) Temperature dependence of the uniaxial magnetocrystalline anisotropy energy $K_2$, obtained from (b). The solid line is a guide for the eye.