Role of spin–orbit coupling in spin-to-lattice energy conversion in ferrite nanoparticles

Abstract

We report on the influence of spin–orbit coupling (SOC) in governing spin-to-lattice energy conversion in ferrite nanoparticles (MnFe₂O₄, Fe₃O₄, and NiFe₂O₄). Using infrared thermography, ferromagnetic resonance measurements, and numerical temperature-profile fitting based on an analytical thermal model, we successfully identify and clarify distinct energy conversion pathways. Specifically, we decompose the process into three sequential steps: field-to-spin energy transfer, spin precession and relaxation, and spin-to-lattice heat conversion. Our quantitative analysis reveals that spin-to-lattice conversion efficiencies are found to be approximately 21%, 18%, and 16% for the Fe, Ni, and Mn ferrites, respectively. These values closely follow the trends in the estimated Landé g-factors and SOC strengths. While SOC plays a dominant role in determining spin-to-lattice conversion efficiency, we emphasize that the total heat output also depends on intrinsic spin relaxation power, as evidenced from the high thermal response of MnFe₂O₄ despite its relatively weaker SOC. Our findings provide a quantitative framework for understanding SOC-mediated spin-to-lattice thermal conversion and offer material design guidelines for spin-caloritronics, nanoscale thermal management, and energy dissipation applications.

Introduction

Ultrafast and efficient energy conversion at the nanoscale is critical to the advancement of spintronic^1–3, magnonic^4,5, and spin-caloritronic^6,7 technologies. Bosonic quasiparticles such as magnons and phonon—and their interactions with electrons—play crucial roles in spin dynamics and energy transport in magnetic systems. These interactions enable a range of functionalities^8–13, from renewable energy harvesting to miniaturized thermal management for high-performance computing^14–20. Recent demonstrations of angular momentum transfer from spins to phonons, often described in the context of the Einstein–de Haas effect, highlight the potential of spin–lattice coupling in nanoscale energy conversion^21–23, opening new opportunities for spin-based thermal control and next-generation spintronic applications.

Ferrimagnetic nanoparticles—including various spinel ferrites^24–26 and yttrium iron garnet (YIG)^27–29—have attracted growing interest not only for their favorable magnetic properties (e.g., superparamagnetism and geometry-sensitive spin dynamics)^30–32 but also for their proven ability to mediate spin–lattice coupling and facilitate spin-to-heat conversion^24–29. These ferrimagnetic insulators offer an ideal platform for studying spin-phonon interactions, and associated heat generation mechanisms, as their negligible free-electron contribution to dissipation enables clearer observation of thermal effects arising from spin dynamics. This platform has enabled key advances in understanding magnetoelastic wave behavior²⁶ and spin-phonon coupling phenomena^33,34, providing valuable insight into how spin excitations drive lattice vibrations and contribute to thermal energy dissipation^5,35–37.

More recently, experimental studies have highlighted spin precession as a particularly effective mediator of heat generation and transport^38–41, stimulating interest in its role across diverse excitation modes and nanostructured geometries. In our earlier work, we identified spin precession under ferromagnetic resonance (FMR) condition as a direct and highly efficient mechanism for converting magnetic energy into heat via spin-relaxation^42–46. This process has demonstrated heating efficiencies that are several orders of magnitude higher than those achieved by conventional Néel or Brownian relaxation mechanisms^44–46. Despite these developments, the direct conversion of spin-precession energy into heat remains incompletely understood. In particular, the quantitative influence of intrinsic material parameters such as spin–orbit coupling (SOC) strength on spin-to-lattice energy transfer efficiency remains insufficiently understood. Addressing this challenge requires an integrated experimental and theoretical effort to uncover the fundamental multistep energy transfer process within confined magnetic nanostructures.

In this study, we develop a multistep energy-conversion model that isolates spin-to-lattice transfer from the overall magnetic field–to–heat process, as captured by time-resolved temperature measurements. Using three spinel ferrite nanoparticles—MnFe₂O₄, NiFe₂O₄, and Fe₃O₄—with systematically varying SOC strength, we investigate how SOC affects spin-to-lattice conversion efficiency. Through a combination of FMR spectroscopy, infrared thermography, micromagnetic simulation, and numerical thermal modeling, we reveal a clear correlation between SOC strength and spin-to-lattice conversion efficiency. Importantly, we also demonstrate that total heat output is not governed by SOC alone, but also by intrinsic spin relaxation power, as evidenced by the strong thermal response observed in MnFe₂O₄ despite its relatively weak SOC. These findings provide a comprehensive framework for understanding SOC-governed spin–lattice coupling and guiding the rational design of magnetic materials for spin-caloritronic, nanoscale thermal management, and energy dissipation applications.

Results and discussion

FMR-induced temperature rise and thermal balance model

The experimental setup for real-time temperature measurements of ferrite nanoparticles under FMR is illustrated in Fig. 1a (see Methods for details). A static magnetic field (H_DC) is applied along the microstrip line, while an alternating magnetic field (H_AC) is applied perpendicular to it. The temperature response of the nanoparticles is recorded via infrared thermal imaging during pulsed radio-frequency (RF) excitations.

Fig. 1.

(a) Schematic of the experimental setup for real-time temperature measurement of ferrite nanoparticles. An in-plane AC magnetic field (H_AC) is applied perpendicular to the axis of the microstrip line, while a static field (H_DC) is applied along the stripline axis. Infrared emission from the nanoparticle region is monitored using an IR thermal imaging system. (b) Time-dependent temperature profiles of three different ferrite nanoparticles under FMR excitation conditions (f_RES = 3 GHz; H_DC optimized for each material). The AC field amplitude was set to H_AC = 17.6 Oe with a pulse duration of t_pulse = 0.1 s. Each curve represents the average of three independent measurements. For comparison, the temperature response of the bare microstrip line is also shown (yellow symbols). A rapid temperature drop occurs after the RF pulse field ends at 0.1 s.

Figure 1b presents the time-dependent temperature profiles of three different materials (MnFe₂O₄, Fe₃O₄, and NiFe₂O₄) nanoparticles under each FMR condition with f_AC = 3 GHz, using an RF field pulse of t_Pulse = 0.1 s and H_AC = 17.6 Oe. To keep FMR conditions, H_DC was fixed at 0.103 T, 0.097 T, 0.098 T, respectively. Distinct temperature rises are observed during the excitation window, followed by cooling once the RF input is turned off at t = 0.1 s. Although the microstrip line itself undergoes minor heating (see Fig. 1b) due to Joule and dielectric losses, its temperature rise under identical pulsing conditions is limited to ~ 8 K and has negligible impact on the nanoparticle temperature increase. The corresponding energy dissipation is estimated to be ~ 0.007 MW/kg—less than 1% of the total observed heating—indicating that the dominant heat source is spin relaxation within the nanoparticles under FMR.

Most promisingly, despite identical RF power and pulse duration, the nanoparticles exhibit clear material-dependent differences in their thermal response. Specifically, MnFe₂O₄ shows the highest peak temperature rise, followed by Fe₃O₄ and then NiFe₂O₄. These temperature increases occur on a sub-second timescale—typically within 0.1 s—which is more than three or four orders of magnitude faster than heating rates achieved through conventional Néel and Brownian relaxation mechanisms in magnetic hyperthermia⁴⁴. This rapid heating is a direct consequence of ultrafast spin precession under FMR on the nanosecond time scale, where magnetic energy is efficiently absorbed by resonant spin precession and dissipated via intrinsic spin relaxation, as reported in Refs.^44–46.

We focus on the material-dependent differences in temperature rise under FMR conditions, which suggest a strong influence of intrinsic magnetic properties on the efficiency of energy conversion along distinct pathways—specifically, field-to-spin, spin relaxation, and spin-to-lattice conversions, as will be described in the next section. To quantitatively interpret these observations, we apply a thermal energy balance model that links the experimentally measured temperature profiles to the underlying energy conversion processes. The temperature variation is governed by the energy conservation equation⁴⁶:

1

where is the input power to the nanoparticle ensemble from the applied RF magnetic field, is the heat loss toward the surrounding environment. is the specific heat capacity per unit mass of the nanoparticles. As shown in Fig. 1b, the temperature evolution can be divided into two regimes. During RF field excitation (0 < t ≤ 0.1 s), the system continuously absorbs magnetic energy, and the temperature rise is governed by the balance of input power and thermal losses described in Eq. (1). The slope of the temperature rise and the maximum peak attained reflect the interplay between , which vary with the magnetic and thermal characteristics of the nanoparticles. After the RF field is turned off (t > 0.1 s), the power input vanishes (Q_in = 0), and the cooling behavior is governed by =.

In this model, the power loss term Q_out accounts for both radiative and convective heat dissipation through the surface of the nanoparticles to the surrounding environment. Conductive losses to the copper microstrip line are negligible, primarily due to the minimal physical contact between the spherical nanoparticles and the flat substrate, as well as the temperature gradient favoring dissipation from high to cooler regions. Therefore, total heat loss is given by the sum of the radiative and convective contributions, as Q_out (t) = Q_Rad (t) + Q_Conv (t);

2

3

where σ is the Stefan–Boltzmann constant, is the emissivity of the nanoparticle surface, and h_air ( 5 W m⁻² K⁻¹)⁵⁸ is the convective heat transfer coefficient. A_NP and denote the total geometric surface area and total mass of the nanoparticle ensemble, respectively. In practical systems, however, not all of the nanoparticle surface contributes equally to heat exchange. Surface inaccessibility due to particle agglomeration, partial embedding, or limited airflow between particles can significantly reduce the effective surface area. To account for this, we introduce an effective surface area as defined by A_eff = k_Area⋅A_NP, where k_Area is a dimensionless correction factor (typically < 1) representing the fraction of the total nanoparticle surface that actively participates in radiative and convective cooling. Incorporating A_eff ensures a more realistic evaluation of heat dissipation, especially under densely packed or substrate-supported nanoparticle configurations.

This modeling framework provides the basis for the quantitative extraction of energy conversion coefficients and material-specific spin–thermal conversion properties, as discussed in the following sections.

Three-step energy transfer model

To interpret the temperature rise observed in ferrite nanoparticles under RF magnetic field excitation, we model the energy flow from magnetic input to thermal output using a thermal energy balance framework. Specifically, to evaluate the magnetic power input term Q_in, we introduce a three-step serial energy conversion model that describes how magnetic field energy is converted into heat via spin dynamics (Fig. 2). This model separates the magnetic-to-thermal energy conversion process into the three physically distinct stages. In the first stage, the applied alternating magnetic field H_AC(t) excites the magnetization of the ferrite nanoparticles, including coherent spin precession at the FMR condition. The efficiency of RF energy absorption into the spin system is characterized by the field-to-spin conversion coefficient, k_Field→Spin. This value is inherently much smaller than 1 due to the geometric mismatch between the spatial extent of the AC magnetic field and the volume occupied by the nanoparticle ensemble; the RF field is distributed over a large region while the particles occupy only a small fraction of it, only a limited portion of the total field energy is absorbed. Techniques such as field focusing may improve this coupling efficiency in future device architectures.

Fig. 2.

Schematic diagram of the three-stage energy conversion model for FMR-induced heating in ferrite nanoparticles. The process consists of: (i) Field-to-spin conversion, where the resonant AC magnetic field (H_AC) excites coherent spin precession; (ii) Spin relaxation, which dissipates precessional energy through intrinsic damping; and (iii) Spin-to-lattice transfer, where dissipated spin energy is converted to lattice vibrations via spin–orbit coupling, leading to temperature rise (ΔT_NP). The total power input (Q_in) is described as the product of the steady-state resonant dissipation rate () and two material-specific conversion coefficients: and k_{Spin→Lattice}.

In the following stage, the spin precession undergo intrinsic relaxation through mechanisms such as Gilbert damping and magnon–magnon scattering. These relaxation processes result in the dissipation of spin-precession energy, Q. The dissipated energy under FMR can be quantified by the steady-state resonant energy dissipation rate per particle, . This value can be determined by the damping constant, saturation magnetization, and magnetic anisotropy terms, as well as the field parameters, as reported according to an analytical form⁵³.

In the final stage, a portion of the dissipated spin energy is transferred to the lattice via SOC, which mediates the conversion of angular momentum into phononic excitations. This spin-to-lattice energy transfer results in local heating and is described by the spin-to-lattice conversion efficiency coefficient, k_{Spin→Lattice}. The magnitude of this coefficient depends on SOC strength and governs the degree to which spin energy contributes to observable temperature rise in the nanoparticle system.

By combining these three step processes, the net magnetic energy input contributing to heat generation can be expressed as: , where reflects intrinsic spin relaxation power at equilibrium states under FMR. To simplify analysis, we define an overall coefficient as k_RES​ = k_Field→Spin​ × k_{Spin→Lattice}​, so that the input power becomes . This expression links the spin-relaxation power with the experimentally determined conversion coefficient, which varies across materials depending on the intrinsic SOC strength.

Evaluation of from micromagnetic simulations

To evaluate the total magnetic power input Q_in contributing to the nanoparticles’ temperature increase, it is essential to first quantify the resonant energy dissipation rate per particle, denoted as . This parameter represents the steady-state rate at which spin-precession energy is lost through intrinsic relaxation mechanisms—primarily Gilbert damping—when the system is driven under FMR conditions. Importantly, is not directly measurable by experiment, as it reflects ultrafast, nanoscale energy dissipation that occurs within individual particles. Therefore, this quantity can be determined through micromagnetic simulations based on the Landau–Lifshitz–Gilbert (LLG) equation, solved under zero-temperature, single-domain (single macrospin) conditions^53–60. Details of the simulation methodology—including mesh resolution, boundary conditions, and numerical solvers—are provided in the Methods section and in Refs.^44–46,53.

At resonance and under the steady-state regime, the energy absorbed from the RF field is exactly balanced by the energy dissipated via spin relaxation. This condition allows to be extracted from the simulated magnetization dynamics in response to a rotating AC magnetic field. Figure 3 shows the simulated time evolution of the instantaneous dissipation rate for different rotating field amplitudes. The simulations were carried out using experimentally determined material parameters—including the saturation magnetization M_s​, Gilbert damping constant α, anisotropy constant K₁, and exchange stiffness A_ex​—as summarized in Table 1. As shown in Fig. 3, the simulated dissipation rate rapidly converges to a steady-state value within tens of nanoseconds for all three materials. This indicates that the absorbed magnetic energy is balanced by spin relaxation losses, confirming that the system has reached dynamic equilibrium. The plateau value of each curve corresponds to the steady-state resonant dissipation rate, , which serves as a material-specific parameter for quantifying energy conversion efficiency in subsequent thermal modeling. Figure 3b presents the simulated values of as a function of the square of the applied AC magnetic field amplitude, for three different ferrite materials: MnFe₂O₄ (blue squares), Fe₃O₄ (red circles), and NiFe₂O₄ (green downward triangles). In factor, is reported to be analytically expressed in terms of materials parameters⁵³:

4

where ρ is the material density and is the gyromagnetic ratio (= rad MHz/Oe ). This expression holds in the linear regime, provided that the excitation field satisfies ⁵³. In our experiments and simulations, this inequality holds for all samples, confirming the absence of nonlinear FMR effects. As expected from Eq. (4), the simulated values of increase linearly with for all materials, as shown in Fig. 3b. However, the slopes differ significantly among the three ferrite compositions. Specifically, MnFe₂O₄ shows the stiffest slope, followed by Fe₃O₄ and NiFe₂O₄.This trend reflects differences in their intrinsic magnetic properties—most notably, the ratio of M_s/α, which governs the spin-relaxation power at resonance. These results indicate that while ​ is the dominant external parameter, the spin energy dissipation is strongly modulated by material-specific magnetic characteristics. These findings establish a quantitative basis for linking spin relaxation to thermal energy conversion and form a key input to the efficiency analysis presented in the following sections.

Fig. 3.

(a) Time evolution of the instantaneous spin energy dissipation rate Q_res (t), simulated under FMR conditions (f_RES = 3 GHz at various rotating field amplitudes). The static magnetic field H_DC for each ferrite was chosen to satisfy FMR. (b) Steady-state resonant dissipation rate plotted as a function of for MnFe₂O₄ (blue squares), Fe₃O₄ (red circles), and NiFe₂O₄ (green triangles), based on the simulation data from panel (a).

Table 1.

Extracted values of the Gilbert gyromagnetic ratio , damping constant , and internal magnetic field H_int for MnFe₂O₄, Fe₃O₄, and NiFe₂O₄ nanoparticles, providing the intrinsic material parameters used in the modeling and simulation.

Materials	[MHz/Oe]	Gilbert damping,	Hint [Oe]	Ms [emu/g]
MnFe2O4	2.764 0.004	0.114 0.007	124.8 2.3	114
Fe3O4	2.712 0.003	0.180 0.003	174.4 1.9	101
NiFe2O4	2.751 0.003	0.134 0.008	160.7 1.4	90

Numerical fitting to temperature profiles

Figure 4 shows the experimentally measured thermal responses (open colored symbols) and the corresponding numerical fits (dashed lines) for MnFe₂O₄, Fe₃O₄, and NiFe₂O₄ nanoparticles. These results were obtained under two sets of FMR conditions: (a) four AC magnetic field amplitudes (H_AC​ = 7.0, 9.9, 13.1, and 17.6 Oe), corresponding to input powers ranging from 10 to 63.1 W with a fixed pulse duration of 0.1 s; and (b) four pulse durations (0.03, 0.05, 0.1, and 0.5 s) at a fixed value of H_AC = 17.6 Oe. All measurements were conducted at a constant excitation frequency f_AC = 3 GHz. As shown in Fig. 4a, increasing H_AC​ results in a faster and higher temperature rise near 0.1 s, attributed to enhanced magnetic energy absorption. The maximum temperature increase () reached values of 50, 38, and 29 K, for Mn, Fe and Ni ferrites, respectively. As shown in Fig. 4b, similarly, extending the pulse duration prolongs the heating period and results in larger values of 94, 63, and 49 K for Mn, Fe and Ni ferrites, respectively.

Fig. 4.

(a) Real-time temperature rise profiles of ferrite nanoparticles under different AC field amplitudes (H_AC = 7.0, 9.9, 13.1, and 17.6 Oe) with a fixed pulse duration (t_pulse = 0.1 s). (b) Temperature profiles at fixed AC field amplitude (H_AC = 17.6 Oe) for different pulse durations (t_pulse = 0.03, 0.05, 0.1, and 0.5 s). All measurements were conducted under FMR condition at f_AC = 3.0 GHz.

To quantitatively evaluate the magnetic-to-thermal energy conversion efficiency, we performed numerical fitting of the observed time-variable temperature profiles. The fitting procedure consisted of two sequential steps. First, we fitted the cooling segments of the temperature curves (t > t_pulse ​), during which the input power Q_in​ is zero. In this regime, temperature decay is governed solely by radiative and convective heat losses, as described by Eqs.  (2) and (3). In this fitting step, importantly, k_Area was not assumed but rather directly extracted from experimental data to account for non-ideal thermal contact due to nanoparticle agglomeration or partial surface contact with the substrate. The extracted k_Area​ values were remarkably consistent across all three ferrites, ranging from 5.20 × 10⁻⁵ to 6.87 × 10⁻⁵, reflecting the similar particle sizes (~ 12 nm) and thermal boundary conditions. The small magnitude of k_Area​ suggests that only a small fraction of the geometric surface area is actively involved in radiative and convective heat transfer.

Once k_Area was obtained, the heating segments (0 < t ≤ t_pulse) were then fitted using energy balance equation (Eq. 1) together with , in order to obtain the fit values of . Here we used values obtained from micromagnetic simulations under the same FMR conditions (see Fig. 3). The fitting results of k_RES, summarized in Table 2, show distinct material dependence: MnFe₂O₄ exhibits the highest efficiency (7.45 10^–5), followed by Fe₃O₄ (6.30 10^–5) and NiFe₂O₄ (5.03 10^–5). Although these values appear small, this is primarily due to the fact that the nanoparticles occupy only a small portion of the spatial volume where the RF field is distributed.

Table 2.

Summary of fitted values for k_RES and k_Area, and spin-to-lattice energy conversion efficiencies (k_Spin-Lattice) for each ferrite, along with the Landé g-factor derived from FMR data. SOC strength (λ) was estimated using the g-factor and crystal field splitting energy (Δ) from literature. The correlated trends among k_Spin-Lattice, g-factor, and λ highlight the role of SOC in governing spin-to-lattice energy conversion efficiency.

Materials	kRES (10–5)	KField-Spin [%]	kSpin-Lattice [%]	g-factor		SOC strength () [eV]
MnFe2O4	7.45	4.65 0.16	16.0 0.6	2.083 ± 0.004	1.784	0.075
Fe3O4	6.30	2.95 0.20	21.1 1.1	2.213 0.003	2.282	0.217
NiFe2O4	5.03	2.77 0.13	18.2 0.9	2.181 ± 0.004	2.483	0.235

The fitting procedure we used here was applied to the four experimental datasets per material (one dataset for four H_AC values and the other for four pulse durations). This approach yielded excellent agreement between the model and experimental data, with an average root-mean-square error (RMSE) of 0.48 K. All physical constants (see Table 1) used in the model were fixed based on independent measurements or reliable literature values, ensuring that the fitted values of k_RES and k_Area are robust and physically meaningful.

Determination of from FMR measurements

As discussed earlier, the overall conversion coefficient k_RES​ can be decomposed into two components as k_RES = k_Field→Spin​ k_{Spin→Lattice}. ​The first term k_Field→Spin​ can be estimated from FMR measurements based on transmission loss. To determine k_Field→Spin​, we conducted FMR measurements on MnFe₂O₄, Fe₃O₄, and NiFe₂O₄ nanoparticles. Figure 5a presents the transmission spectra ∣ΔS21∣| as a function of static magnetic field H_DC and AC frequency f_AC​, revealing clear resonance bands centered around 3 GHz. Baseline subtraction using the signal at H_DC = 3.0 kOe was applied to eliminate background losses and isolate the magnetic absorption component.

Fig. 5.

(a) Color maps of the measured scattering parameter magnitude |ΔS₂₁| plotted as a function of static magnetic field H_DC and AC frequency f_AC for MnFe₂O₄, Fe₃O₄, and NiFe₂O₄ nanoparticles. Diagonal absorption bands indicate FMR. (b) Frequency-dependent ΔS₂₁ spectra (open green circles) measured at FMR for each material, with Lorentzian fits (black lines). Resonance frequencies are centered around 3 GHz. The depth of each dip corresponds to the fraction of absorbed RF power, serving as a quantitative measure of magnetic energy absorption.

Figure 5b shows the frequency-resolved ΔS₂₁ spectra at resonance for each material. Lorentzian fits to the absorption dips yield − 0.207 ± 0.007 dB for MnFe₂O₄, − 0.130 ± 0.009 dB for Fe₃O₄, and − 0.122 ± 0.006 dB for NiFe₂O₄. These correspond to relative energy absorption rates of 4.65 ± 0.16%, 2.95 ± 0.20%, and 2.77 ± 0.13%, respectively. These relative absorption rates are used as estimates for k_Field→Spin in low-power excitation conditions. This approach is justified by the following: (i) Microstrip dielectric losses were independently determined to be < 1% of the total dissipation. (ii) All FMR measurements were conducted under steady-state, low-power, and well-shielded conditions. (iii) Reflection losses were minimized through careful microstrip design. Lorentzian line shapes of the ΔS₂₁​ dips confirm that resonance absorption arises from uniform-mode magnetic precession.

Importantly, the FMR absorption strengths differ noticeably among materials. MnFe₂O₄ shows the highest k_Field→Spin, despite having the lowest damping constant (α = 0.114) and moderate saturation magnetization (Mₛ = 0.53 T). In contrast, NiFe₂O₄, with higher α (= 0.134) and lower Mₛ (= 0.37 T), exhibits the weakest absorption. These trends reflect how intrinsic magnetic parameters—particularly α and Mₛ—govern magnetic susceptibility and field-coupling strength at resonance.

Furthermore, the consistency of temperature profile fitting across different excitation conditions, using a single value of k_Field→Spin per material, supports the validity of this FMR-derived absorption ratio. These values, summarized in Table 2, serve as crucial inputs for obtaining another coefficient k_{Spin→Lattice}. Once k_Field→Spin is known, k_{Spin→Lattice} can be extracted using the experimentally fitted values of k_RES​, as will be discussed in the following section.

Determination of k_{spin-to lattice} and the Role of SOC

With both ​ (obtained from temperature-profile fitting) and (derived from FMR absorption spectra) independently determined, we extracted the spin-to-lattice energy conversion efficiency, ​ for each ferrite nanoparticle. The results, summarized in Table 2, reveal a clear material-dependent trend: Fe₃O₄ shows the highest conversion efficiency (21.1 ± 1.1%), followed by NiFe₂O₄ (18.2 ± 0.9%) and MnFe₂O₄ (16.0 ± 0.6%). This trend suggests the ability of the spin system to transfer spin energy to the lattice is governed by intrinsic properties, particularly the Landé g-factor and the strength of SOC. The Landé g-factors was extracted from FMR measurements using the Zeeman-based relation ⁵⁸, where is the Bohr magneton. This relation holds under quasi-isotropic conditions. Although superparamagnetic nanoparticles exhibit non-zero internal anisotropy fields, thermal energy at room temperature significantly exceeds their magnetic anisotropy energy, leading to rapid thermal averaging of the magnetization direction. This minimizes the influence of static anisotropy fields, validating the use of the Zeeman-based relation for estimating g-factors. As shown in Fig. 6, the extracted ​ increase monotonically with the g-factor. Specifically, Fe₃O₄, which has the largest g-factor (2.213), shows the highest ​, whereas MnFe₂O₄, with the smallest g-factor (2.083), exhibits the lowest.

Fig. 6.

Comparison of spin-to-lattice energy conversion efficiency (k_{Spin−Lattice}) (black triangles, left axis), Landé g-factor (red diamonds, right axis), and estimated spin–orbit coupling (SOC) constant λ (blue hexagons, right axis) for MnFe₂O₄, Fe₃O₄, and NiFe₂O₄. The consistent trends across all three parameters support the interpretation that spin–orbit coupling (SOC), as reflected in the g-factor, governs the efficiency of spin-to-lattice energy transfer.

Furthermore, the g-factor is related to SOC strength via the relation^64–66:

where Δ is the crystal field splitting energy. Using representative Δ values reported for bulk spinel ferrites with octahedral coordination such as—1.7 eV for MnFe₂O₄, 2.2 eV for Fe₃O₄, and 2.4 eV for NiFe₂O₄, we estimated to be approximately 0.075 eV, 0.217 eV, and 0.235 eV, respectively. While these are approximate values due to potential nanoparticle effects (e.g., surface disorder, confinement, or strain), they correlate well with the experimentally determined values of . These results reinforce the interpretation that SOC governs the efficiency of spin-to-lattice energy transfer. However, we note that the g-factor, while experimentally accessible, is not a unique indicator of SOC strength, as it can also be affected by extrinsic factors such as structural defects or crystal field asymmetry. Thus, both the g-factor and estimated λ values are used here as qualitative indicators to compare relative SOC strength across the three ferrite compositions used in this study.

Discussion

The three-step energy transfer framework developed in this study provides a clear foundation for understanding the material dependence of FMR-driven heating in ferrite nanoparticles. This model decomposes magnetic-to-thermal energy conversion into three physically distinct stages:(i) RF field energy absorption by the spin system ) (ii) intrinsic spin relaxation characterized by the steady-state resonant dissipation rate and (iii) spin-to-lattice energy transfer ().

Among these, the field-to-spin conversion efficiency , estimated from FMR transmission loss measurements, is relatively low—ranging from 2.77 to 4.65% across the studied materials. This reflects not only intrinsic differences in magnetic absorption but also geometric limitations inherent to our experimental setup. Specifically, the nanoparticles occupy only a small fraction of the RF magnetic field distribution and tend to agglomerate, limiting effective field overlap. Despite these constraints, the observed differences between materials (e.g., MnFe₂O₄ vs. NiFe₂O₄) imply that intrinsic magnetic parameters such as damping constant, permeability, and saturation magnetization still meaningfully affect the relative absorption efficiency under identical experimental conditions.

In contrast, the second and third energy-conversion stages show stronger material dependence. The steady-state resonant dissipation rate ​obtained from micromagnetic simulations, reflects how efficiently spin-precession energy is converted into thermal power. This quantity is strongly influenced by both α and M_s. As analytically derived, materials with lower damping constant and higher saturation magnetization (e.g., MnFe₂O₄) show larger ​thereby converting more spin-precession energy into heat per unit time.

The third-stage efficiency, ​, displays a clear trend with material composition. Fe₃O₄ exhibits the highest value (21.1 ± 1.1%), followed by NiFe₂O₄ (18.2 ± 0.9%) and MnFe₂O₄ (16.0 ± 0.6%). This trend correlates with the Landé g-factors extracted from FMR measurements and with SOC strengths estimated from crystal field relation. The consistent correlation among ​, the g-factor, and the estimated SOC strength (λ) supports the role of SOC-mediated magnon–phonon coupling in spin-to-lattice energy transfer.

While SOC strongly influences ​, the overall efficiency of magnetic-to-thermal energy conversion also depends on how much energy reaches the spin system in the first place. To identify the dominant limiting factor in total heat generation, we compared the typical magnitudes of the three conversion pathway efficiencies across all materials: k_Field→Spin∼ 2.77–4.65%, k_{Spin→Lattice} ~ 16–21%, and ​ ~ 0.1–0.15 MW/kg. This comparison clearly indicates that RF field absorption, governed by k_{Field→Spin,} is the primary bottleneck in magnetic-to-thermal energy-conversion stage. While spin-to-lattice transfer is indeed governed by SOC, its efficiency remains relatively high across all three ferrites, especially when compared to the small fraction of the available RF power that is actually transferred into the spin system. Therefore, enhancing overall heating performance requires primarily improving k_Field→Spin​, for instance, by optimizing nanoparticle dispersion and localized field concentration. These findings highlight the need to combine intrinsic material selection (e.g., for high SOC or favorable g-factors) with structural and device-level engineering to control and enhance thermal output in spin-caloritronic systems.

Although this work focused on three representative spinel ferrite nanoparticles, the framework and conclusions are broadly applicable to a wide range of magnetic materials and nanostructures. The three-step model is readily extendable to systems with complex spin textures, including magnetic vortices^67,68 or skyrmions⁶⁹, which may exhibit unique dissipation behavior. Additionally, geometrical modifications—such as using nanowires⁷⁰, disks^38,72,73, or thin films^39,40,73,74—can alter by reshaping field overlap and magnetic anisotropy.

Parallel strategies to enhance k_{Spin→Lattice} ​may involve compositional tuning or incorporating interfaces with heavy metals^75–77 to strengthen SOC. Beyond SOC-driven processes, other factors such as magnon–magnon scattering⁷⁸, interfacial strain^79,80, and phonon bottleneck effects⁷⁵ may also influence thermal response. Decoupling these contributions will be essential for the advancement of nanoscale heat management technologies based on spin dynamics.

Conclusion

We demonstrated that the spin-to-lattice energy conversion efficiency varies in a manner that correlates with SOC strength, within the limits of a semi-quantitative analysis. By integrating FMR measurements, temperature profiling, and a multistep energy-conversion model, we semi-quantitatively evaluated the role of SOC in governing spin-precession-driven heat generation.

Although SOC plays a critical role in spin-to-lattice conversion, it is not the sole factor determining total heat output. For example, MnFe₂O₄, despite having the weakest SOC, exhibits the highest temperature rise due to its intrinsically greater spin-relaxation power. This highlights the complementary influence of damping, saturation magnetization, and SOC in defining overall thermal performance.

This study presents a robust framework for understanding and optimizing spin-mediated thermal energy conversion at the nanoscale. The ability to tailor SOC through compositional or structural design offers a promising strategy for enabling tunable thermal control in spintronic and magnonic devices.

Our findings not only advance the understanding of spin–lattice dissipation mechanisms, but also provide practical design principles for developing spin-caloritronic materials and thermally responsive magnetic systems. Together, these results bridge fundamental spin dynamics with applied materials engineering, establishing a predictive platform for thermal management, localized heating, and energy conversion in magnetic nanostructures.

Methods

Sample fabrication

Monodisperse ferrite nanoparticles with a silica coating (MFe₂O₄, where M = Mn, Fe, or Ni) were synthesized using a high-temperature thermal decomposition method in organic solvents under an argon atmosphere (see Refs. 47–50 for synthetic details). The nanoparticles exhibited a uniform spherical morphology and average core diameter of ~ 12 nm and a size dispersion of less than 8%. To enhance thermal insulation and dispersion stability, a conformal silica shell (12 nm) was subsequently formed via a reverse microemulsion process. Each experimental batch was prepared with a fixed nanoparticle mass of 1 mg to ensure consistent comparisons across different measurements.

Although direct magnetization versus temperature (M–T) or field cooling/zero-field cooling (FC/ZFC) measurements were not performed, the superparamagnetic behavior at room temperature was inferred from a coherent set of independent indicators. These include the small particle size (~ 12 nm), negligible remanence and coercivity, and high magnetic susceptibility observed via vibrating sample magnetometry (VSM).

Owing to the superparamagnetic nature of the nanoparticles, the magnetization rapidly approaches saturation at relatively low fields. Therefore, the magnetization values measured at 1 kOe were taken as effective saturation magnetization, yielding 114 emu/g for MnFe₂O₄, 101 emu/g for Fe₃O₄, and 90 emu/g for NiFe₂O₄. These magnetic characteristics suggest minimal interparticle dipolar coupling in the absence of an external field. Moreover, due to the nanoscale size, long-range electron hopping between Fe²⁺ and Fe³⁺ ions is significantly suppressed, supporting the insulating nature of the particles.

FMR and thermal measurements

FMR spectra were measured using a two-port vector network analyzer (VNA) system. The external static magnetic field H_DC​ was swept from 0 to 2.4 kOe, while the transmission coefficient S₂₁ was recorded over a frequency range spanning 1 MHz to 10 GHz. From these measurements, key magnetic parameters—including the resonance frequency f_RES​, Gilbert damping constant α, internal field H_int​, and Landé g-factor—were extracted. All measurements were performed at room temperature.

For thermal characterization, real-time temperature monitoring was conducted using an infrared (IR) camera (A655sc, FLIR) with a frame rate of 5 ms and a view area of 640 × 120 pixels, yielding an effective spatial resolution of approximately 340 μm per pixel, based on the pixel size, lens field of view, and a working distance of ~ 50 cm. The nanoparticles were dispersed in distilled water and drop-cast onto a copper microstrip line, with a static magnetic field applied perpendicular to the substrate surface. To ensure accurate thermal readouts, surface emissivity values were carefully calibrated using reference points from boiling water and ice, and were set as follows: 0.85 (Fe₃O₄), 0.74 (MnFe₂O₄), and 0.83 (NiFe₂O₄). The emissivity of copper was fixed at 0.6 based on literature-reported values. These calibrations helped minimize infrared measurement artifacts and enabled precise temperature tracking during excitation. RF field excitation was provided by a signal generator (E8257D, Agilent) in combination with a power amplifier (5193E, Ophir), generating AC magnetic fields of up to 20 Oe.

Micromagnetic simulations

To investigate spin-precession dynamics and intrinsic energy dissipation in ferrite nanoparticles under ferromagnetic resonance (FMR), we performed micromagnetic simulations using the Finite-Element MicroMagnetics Environment (FEMME, version 5.0.9). This software numerically solves the Landau–Lifshitz–Gilbert (LLG) equation using a finite-element method, allowing for accurate modeling of three-dimensional magnetization dynamics in nanoscale magnetic structures. All simulations were conducted under a zero-temperature approximation to exclude thermal noise, thereby enabling a focused examination of deterministic relaxation behavior. The computational mesh used a spatial resolution of ~ 2 nm, which is smaller than the magnetic exchange length, ensuring both numerical stability and adequate resolution of spin texture evolution. Time integration of the LLG equation was carried out using a fourth-order Runge–Kutta algorithm with a time step of 0.1 ps.

Simulation parameters for each ferrite composition were selected based on experimentally measured values and reliable literature sources. For MnFe₂O₄, we used a static external magnetic field of 1040 Oe, a saturation magnetization Ms = 0.53 T (measured at H_DC = 1 kOe), a Gilbert damping constant α = 0.114, an exchange stiffness of 13.2 pJ/m, and a first-order magnetic anisotropy constant K₁ =  − 1.1 × 10⁵ erg/cm³. The same exchange stiffness value was applied to Fe₃O₄ and NiFe₂O₄, given the absence of well-established material-specific data for these materials. For Fe₃O₄, the simulation parameters were: H_DC = 970 Oe, M_s = 0.52 T, α = 0.180 and /cm³. For NiFe₂O₄, we used , M_s = 0.37 T, α = 0.134, and K₁ =  − 0.3 × 105 erg/cm³.

Author contributions

S. K. K. and Y. K. conceived the main idea and designed the overall experimental concept. Y. K. and H. A. contributed to the experimental measurements. Y. K. performed the micromagnetic simulations. Y. K., H. A., and S. K. K interpreted the data. S. K. K. supervised the project and wrote the manuscript, based on the tutorial thesis of Y. K. (submitted in February 2025).

Data availability

Data supporting this study’s findings are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.