Inversion of magnon lifetime of ferromagnetic and exchange resonance modes in ferrimagnets

Domain: physics
Published: 2026-02-11
Editorial outcome: minor_revision

Abstract
Antiferromagnetically coupled ferrimagnets exhibit both ferromagnetic resonance and exchange resonance modes. The antiferromagnetic exchange resonance mode, characterized by a higher magnon frequency than the ferromagnetic resonance mode, holds promise for fast spintronic applications. However, as higher magnon frequencies are typically associated with shorter magnon lifetimes, the exchange resonance mode is expected to decay more rapidly than the ferromagnetic resonance mode, leading to challenges for long-lived information transfer and coherent dynamics. Here we demonstrate that this inverse relationship between frequency and lifetime can be broken in ferrimagnets with two inequivalent magnetic sublattices. Using time-resolved magneto-optical Kerr effect spectroscopy on CoGd, we observe that the exchange resonance mode exhibits a longer magnon lifetime than the ferromagnetic resonance mode near the angular momentum compensation point. Our theoretical and simulation models reveal that this inversion of magnon lifetime arises from the inequivalence in magnetic damping of the two sublattices. The unique combination of higher frequency and longer lifetime in the exchange resonance mode of ferrimagnets highlights its potential for high-speed and energy-efficient spintronic devices.

Introduction
For a damped harmonic oscillator, the frequency (\omega) and the lifetime (\tau) are given by \omega = \sqrt{k/m} - 1/\tau^2 and \tau = 2m/b, respectively, where m is the mass of the object, k is the spring constant, and b is the damping constant. In systems with weak damping (\omega \gg \sqrt{k/m}), a higher frequency is typically associated with a shorter lifetime, through the mass (m). This typical relationship also applies to magnetization oscillation: antiferromagnets exhibit higher resonance frequencies but shorter lifetimes than ferromagnets due to the strong antiferromagnetic exchange interaction, which results in an effective mass for antiferromagnetic dynamics1. Therefore, achieving both higher frequency and longer lifetime for antiferromagnetic dynamics compared to ferromagnetic dynamics is a major challenge. In this work, we show that this challenge can be overcome in antiferromagnetically coupled ferrimagnets with inequivalent magnetic sublattices.

Ferrimagnets exhibit intriguing phenomena around the magnetization compensation or the angular momentum compensation points, such as fast domain wall motion2, spin-torque tunability3,4, and energy-efficient switching5–7. These phenomena arise from the combined effect of antiferromagnetic exchange coupling between the sublattices and finite net magnetic moment or angular momentum. As a result, ferrimagnets combine features of both ferromagnets and antiferromagnets. In ferrimagnets, there are two distinct eigenmodes: the ferromagnetic resonance (FMR) mode and the antiferromagnetic exchange resonance (EXR) mode. Unlike the FMR mode in the GHz range, the EXR mode typically operates in the sub-THz range8–10, making it promising for next-generation spintronic devices requiring fast operation11. While previous studies have focused on the frequency characteristics of ferrimagnetic resonance modes8–10, the magnon lifetime remains largely unexplored, despite its significance for understanding the energy dissipation of magnetic systems and implementing the practical device operation for coherent information transfer.

In this article, we investigate both magnon frequencies and lifetimes of the two modes for a ferrimagnetic thin film Co1−xGdx. Time-domain spectroscopy reveals a longer magnon lifetime of the high-frequency EXR mode than that of the low-frequency FMR mode near the angular momentum compensation point. This observation is supported by theoretical analysis and numerical simulations, demonstrating the important role of mode-dependent damping torques in ferrimagnets. These results challenge the conventional view that the magnon lifetime is shorter for higher-frequency modes and provide a foundation for optimizing ferrimagnetic devices for high-speed, energy-efficient magnon applications, as well as sub-THz ferrimagnetic spin oscillators.

Results
Magnetization and angular momentum compensation points of CoGd
We study a prototypical rare-earth (RE) transition-metal (TM) ferrimagnet Co1−xGdx with compositions varying from Co83Gd17 to Co65Gd35 (Methods). The full structure consists of Si/SiO2 (substrate)/Ta(1nm)/Pt(6nm)/Co1−xGdx(7nm)/TaOx(2nm), as shown in Fig. 1a. The bottom Ta layer, grown directly on the Si/SiO2 substrate, acts as a buffer layer to enhance adhesion and improve surface roughness. The next Pt layer establishes the perpendicular magnetic anisotropy (PMA) of the CoGd ferrimagnetic layer, as confirmed by the magnetic-field-angle-dependent vibrating sample magnetometer (VSM) results of these samples (Supplementary Note 1). Finally, the TaOx layer on top acts as a protective capping layer to prevent oxidation of the CoGd ferrimagnetic layer. The two compensation points of antiferromagnetically coupled ferrimagnets are the magnetization compensation point (Ms,TM = Ms,RE) and the angular momentum compensation point (Ms,TM/\gammaTM = Ms,RE/\gammaRE), where Ms,i and \gammai (i = TM, RE) are the magnetization and gyromagnetic ratio of sublattice i, respectively. These two compensation points can be tuned by composition variation or temperature change.

To find out the two compensation points, we conduct temperature-dependent anomalous Hall effect (AHE) measurements of Co78Gd22, as shown in Fig. 1b. The hysteresis loop reversal between 160 K and 165 K confirms the presence of magnetization compensation temperature (TM) in this range. Near TM, the slope changes at high field indicate a spin-flip transition12. The resistance RAHE and coercivity Hc are extracted from the AHE results (upper panel of Fig. 1c). The sign change of RAHE and the divergence of Hc confirm the TM near 165 K.

Furthermore, we measure the saturation magnetization (Ms) of Co78Gd22 with 1 K temperature resolution under a 500 mT out-of-plane field by temperature-dependent VSM. The resulting Ms–T curve under this field-cooling condition is presented as symbols in the bottom panel of Fig. 1c. The minimum point at which Ms = 0 indicates TM = 165 K. The temperature dependence of Ms of Co78Gd22 is fitted (blue curve) to decouple sublattice magnetizations, and the total angular momentum Atotal of Co78Gd22 is analytically obtained (black curve) by considering the different Landé g factors of Co (gCo = 2.2) and Gd (gGd = 2) (Supplementary Note 1). From the vanishing point of the total angular momentum, TA is estimated to be 250 K for Co78Gd22. Similarly, we conduct VSM and polar magneto-optical Kerr effect (MOKE) characterization of various Co1−xGdx films at room temperature. The extracted minimum Ms and maximum Hc (Fig. 1d) indicate the presence of a magnetization compensation point (XM) at Gd ~ 25% at room temperature6. The angular momentum compensation composition (XA) is estimated to be 23% of Gd, in accordance with previous reports13.

Two ferrimagnetic resonance modes detected in TR-MOKE
We conduct time-resolved magneto-optical Kerr effect (TR-MOKE) spectroscopy on a Co-rich sample, Co78Gd22, to investigate the magnetization dynamics, with the experimental setup illustrated in Fig. 2a. The pump pulse initiates ultrafast demagnetization of Co78Gd22 and triggers the precession motion of magnetization, with the two eigenmodes schematically depicted in Fig. 2a. The magnetization dynamics are subsequently detected by the probe pulse through the polarization rotation of the reflected probe pulse. The time delay between pump and probe pulses is controlled by the light path difference, allowing for the measurement of the time-domain dynamics (Methods and Supplementary Note 2). With this experimental setup, we detect both FMR and EXR modes in Co78Gd22 simultaneously under an external magnetic field of 50 mT (Fig. 2b). The fast Fourier transform (FFT) results (Fig. 2c) highlight the two modes at 13 GHz (FMR mode) and 110 GHz (EXR mode). Because the applied external in-plane field is less than 200 mT, which only causes a negligible tilt of Néel vector and sublattice canting14–16 (Supplementary Note 3), we therefore consider the Néel vector as aligned along the out-of-plane direction. Magnetic-field-dependent TR-MOKE results indicate the EXR mode is independent of the external magnetic field, while the FMR mode is influenced by the external magnetic field (Supplementary Note 4).

We then perform TR-MOKE spectroscopy on Co78Gd22 at varying temperatures to investigate magnetization dynamics around TA. Due to laser-induced heating, the actual sample temperature is about 20–25 K higher than the TR-MOKE system recorded temperature (Supplementary Note 5). For consistency, we use the system temperature throughout the following discussion. Accordingly, the compensation points at TM = 165 K and TA = 250 K appear at system temperatures of approximately 140 K and 225 K, respectively. We first measure the EXR mode at various temperatures with zero magnetic field (Fig. 3a). The EXR mode frequency is approximately 110 GHz, reaching a minimum near TA (ref. 8). Additionally, we measure the FMR mode with an in-plane magnetic field of 200 mT at various temperatures (Fig. 3b). In contrast to the EXR mode, the FMR mode exhibits a maximum frequency near TA (refs. 17, 18), as summarized in Fig. 3c.

To quantify the TR-MOKE results, we extract the magnon frequency, magnon lifetime, and effective damping of both modes as a function of the temperature from the time-domain oscillation components with a damped sine fitting \thetaKerr = Ae^{-t/\tau}sin(2\pi ft + \phi), where \thetaKerr is the measured oscillating Kerr signal, A is the oscillation amplitude, \tau is the magnon lifetime, f is the magnon frequency, and \phi is the phase. The extracted parameters are shown in Fig. 3c–e. As mentioned above, the FMR mode reaches a frequency peak around TA, while the EXR mode shows a shallow dip around TA (Fig. 3c). The large frequency gap between the EXR mode and FMR mode distinguishes them from each other, resembling previous experiments on ferrimagnets8,10.

From the damped sine fitting of the signals, we obtain the magnon lifetime (\tau) of the two modes (Fig. 3d). For temperatures far away from TA, the EXR mode shows a shorter magnon lifetime than the FMR mode, which is in line with the conventional understanding that a higher-frequency mode decays more rapidly. However, near TA, the EXR mode shows a longer magnon lifetime than the FMR mode, despite its much higher frequency.

To further validate this behavior, we conduct composition-dependent TR-MOKE spectroscopy on Co1−xGdx at room temperature (Supplementary Note 6). We also confirm that the magnon lifetime inversion occurs near the composition XA (i.e., Gd 23%), corresponding to angular momentum compensation XA. Therefore, both temperature- and composition-dependent measurements consistently show that the EXR mode exhibits a longer magnon lifetime than the FMR mode near the angular momentum compensation condition, establishing an equivalence between the two datasets and reinforcing our observations.

The effective damping, defined as \alphaeff = 1/2\pi f\tau, of the EXR mode is smaller than that of the FMR mode (Fig. 3e), indicating a potential merit of utilizing the EXR mode for device applications. In particular, the longer magnon lifetime of the EXR mode near TA and XA leads to the lowest effective damping near the angular momentum compensation conditions. The domain wall velocity is estimated based on our TR-MOKE results, as shown in Fig. 3f (Supplementary Note 7), in which the maximum at 225 K corresponds to TA and is in line with our observations.

Theoretical analysis of the magnon lifetime in ferrimagnets
We employ the staggered Landau–Lifshitz–Gilbert (LLG) equation to analyze and understand ferrimagnetic dynamics. Using the Néel vector n = (mTM − mRE)/2, where mTM and mRE are unit magnetization vectors of the ferrimagnet sublattices, the dynamics of collinear ferrimagnets can be described as (Supplementary Note 8)1,19–25:
\delta s n_ - \rho n \times n€ - \alpha s+ n \times n_ - \alpha s- 2 s / \rho n \times (n€ \times n) + n \times f n = 0 (1)
f n = K n z z^ - 4 \rho D2 / s2 n z D D (2)
where \delta s = sTM − sRE is the net spin density, s = sTM + sRE is the total spin density, and si = Mi/\gammai is the spin density of sublattice i, which is tuned by temperature or composition variations in experiments. Here, \rho is the inertial term associated with antiferromagnetic dynamics of n, which is defined as \rho = s2d2/8Aex in two-spin model, where d is the lattice parameter and Aex is the exchange stiffness. In addition, \alpha s+ = \alpha TM sTM \pm \alpha RE sRE, \alpha is the Gilbert damping constant of sublattice i, K is the easy-axis magnetic anisotropy of the Néel vector along the z axis, D is the Dzyaloshinskii–Moriya interaction (DMI) constant26–28 between two sublattices, and D is the DMI vector, defined as D = r \times d, where r is the unit vector along the inversion symmetry breaking direction, and d is the unit vector of lattice displacement direction. We note that the DMI-dependent term of Eq. (2) arises from finite-sized atomic lattices where the periodic boundary condition is lifted (Supplementary Note 9).

In the absence of damping, the resonance frequencies can be obtained from Eq. (1):
\omega RE \pm = \sqrt{\delta s^2 + 2\sigma 1 \pm \sigma 2} = 2\rho (3)
where \sigma 1 = \rho K + 2 \rho D2/s2 and \sigma 2 = \sqrt{\delta s^4 + 4\delta s^2 \sigma 1 + 16 \rho D2/s4}. Here, \omega RE+ corresponds to the resonance frequency of the EXR mode, and \omega RE- corresponds to that of the FMR mode. At angular momentum compensation (\delta s = 0), Eq. (3) simplifies to \omega RE+ = \sqrt{K/\rho + 4 D2/s2} and \omega RE- = \sqrt{K/\rho}, where the DMI induces a frequency gap between the two modes in antiferromagnetic systems29,30. We plot Eq. (3) as solid lines in Fig. 4a, demonstrating agreement with the two-spin simulation results, shown as symbols (Methods).

We next derive the magnon lifetimes of the two resonance modes from Eq. (1), considering terms up to the first order of the Gilbert damping constants:
\tau \pm = -1/\omega I \pm = M 0 \rho \sigma 2 / @ \alpha s+ \sigma 2 \pm 2 \delta s^2 \alpha s- \delta s \sigma 2 \pm 2 \delta s^2 + 2\sigma 1 (6)^{-1} A (4)
where \tau+ corresponds to the magnon lifetime of the EXR mode, and \tau- corresponds to that of the FMR mode. In Fig. 4b, we plot \tau for the case where \alpha RE = \alpha TM. We find that at angular momentum compensation (\delta s = 0), the lifetimes are equal: \tau+ = \tau- = 2\rho/\alpha s+. However, away from angular momentum compensation, the EXR mode (\tau+) exhibits a shorter lifetime than the FMR mode (\tau-). For \alpha RE = \alpha TM, therefore, \tau+ is either equal to or always smaller than \tau-, in line with the conventional understanding that a higher frequency mode decays more rapidly.

This conventional understanding breaks down when \alpha RE \neq \alpha TM (i.e., \alpha RE \gg \alpha TM)31. Figure 4c shows \tau as a function of \delta s in this case. We find that at angular momentum compensation, the relation \tau+ = \tau- = 2\rho/\alpha s+ remains valid even in this case. However, for small negative \delta s, the EXR mode (\tau+) exhibits a longer lifetime than the FMR mode (\tau-). This inversion of magnon lifetimes between the EXR and FMR modes is consistent with our experimental observations (Fig. 3d).

In this range of \delta s, the magnon lifetimes are approximated as:
\tau \pm \approx 2\rho/\alpha s+ \pm 2\alpha s- \delta s \rho (2\sigma 1 - \sigma 2) / (\alpha s+^3 \sigma 2^2) (5)
Equation (5) demonstrates that the magnon lifetime inversion is directly associated with \alpha s-. These results suggest that the difference in damping constants between the two sublattices induces an asymmetric behavior in the magnon lifetime near the angular momentum compensation point, leading to a crossing of magnon lifetimes between the EXR and FMR modes, where the EXR mode attains a longer lifetime. Similarly, for the case \alpha RE \ll \alpha TM, magnon lifetime inversion occurs for \delta s > 0 near the angular momentum compensation point (Supplementary Note 10).

We note that the term associated with \alpha s- in Eq. (1) was previously considered negligible, except in studies focusing on cross-sublattice damping32–36. However, when the damping constant difference between sublattices is significant, \alpha s- becomes non-negligible, leading to extraordinary dynamical effects, particularly in relation to magnon lifetime behavior. In addition, previous ultrafast studies of the temporal processes in ferrimagnets mainly investigated longitudinal relaxation channels on femtosecond timescales37–39, whereas our TR-MOKE results probe the coherent transverse eigenmodes whose magnon lifetimes are governed by the coupled damping–exchange relation above.

Discussion
We explain the microscopic origin of magnon lifetime inversion through mode-dependent damping torques. Figure 4d and e schematically illustrate the FMR and EXR modes, respectively, for \delta s < 0 and \alpha TM \ll \alpha RE so that \alpha s- < 0. These figures show how the two damping torques, sgn(\delta s) \alpha s+ n \times n_ (green arrow) and sgn(\delta s) \alpha s- 2 s / \rho n \times (n€ \times n) (red arrow), act differently depending on the resonance mode. The coefficient sgn(\delta s) implies these damping torques are inverted depending on the sign of the net spin density. The black arrows represent the effective field of each mode, whereas the gray arrow represents n.

In Fig. 4d, the dominant effective field inducing the FMR mode arises from the PMA energy density. As a result, n precesses around the PMA field, Heff,FMR. Considering the time derivatives of n, both sgn(\delta s) \alpha s+ n \times n_ and sgn(\delta s) \alpha s- 2 s / \rho n \times (n€ \times n) point to the same direction, acting as damping torques in the FMR mode. In contrast, in Fig. 4e, the effective field generating the EXR mode arises from the exchange energy. Consequently, n processes around the exchange field Heff,EXR, which is proportional to -\rho n€. Similar to the FMR mode, sgn(\delta s) \alpha s+ n \times n_ acts as a damping torque. However, sgn(\delta s) \alpha s- 2 s / \rho n \times (n€ \times n) now points in the opposite direction to sgn(\delta s) \alpha s+ n \times n_, effectively functioning as an anti-damping torque. Therefore, sgn(\delta s) \alpha s- 2 s / \rho n \times (n€ \times n) serves as a mode-dependent damping torque, providing the microscopic origin for magnon lifetime inversion.

The EXR mode frequency scales with the square root of the product of the anisotropy and exchange fields, while the FMR mode frequency depends only on the anisotropy field. Since the exchange field is generally much stronger, the EXR mode frequency is much higher, making it attractive for ferrimagnetic spintronics11. However, the magnon lifetime (\tau) of the FMR mode scales inversely with the anisotropy field, whereas that of the EXR mode scales inversely with the exchange field. Consequently, when both sublattices have equal damping constants, the EXR mode exhibits a shorter magnon lifetime than the FMR mode due to its higher frequency. Our study, however, reveals that when the damping constants of the two sublattices differ substantially, the EXR mode can exhibit a longer magnon lifetime than the FMR mode near the angular momentum compensation condition, thereby overcoming this limitation. This finding suggests that engineering sublattice-specific damping offers a promising strategy to achieve both high frequency and extended magnon lifetime in the EXR mode. The combination of higher frequency and longer magnon lifetime of the EXR mode could also result in substantially lower effective damping compared to the FMR mode (Fig. 3e), further enhancing the viability of the EXR mode for future spintronic applications, such as the sub-THz oscillators40,41 and circuits42.

Methods
Sample fabrication and characterization
We deposited film stacks of Ta(1nm)/Pt(6nm)/Co1−xGdx(7nm)/TaOx(2nm) films onto oxidized Si substrates via magnetron sputtering at room temperature with a base pressure below 5 × 10−8 torr (ATC, AJA International). The Co1−xGdx layer was grown by Co and Gd co-sputtering, with Co sputtering power fixed at 120 W and Gd sputtering power varied from 40 to 120 W to achieve different Gd concentrations. The Ta layers at the bottom and top enhance adhesion and prevent oxidation, respectively. The composition of Co1−xGdx was analyzed using energy-dispersive X-ray spectroscopy (EDX). The hysteresis loops of the films were characterized using Kerr microscopy (TTT-02-Kerr Microscope, TuoTuo Technology). Vibrating sample magnetometry (VSM) measurements were conducted over a temperature range of 60–300 K using a physical property measurement system equipped with a VSM option (DynaCool, Quantum Design).

Time-resolved magneto-optical Kerr measurements
The laser source was an 80 MHz Ti:sapphire femtosecond oscillator (Mai Tai BB, Spectra-Physics) emitting pulses centered at 800 nm with a pulse duration < 80 fs. The original laser pulse was split into a pump and a probe pulse by a beam splitter. The pump pulse was then focused onto a \beta-barium borate (BBO) crystal to convert it to 400 nm, while the probe pulse was directed onto the sample after passing through a mechanical delay stage to control the optical path difference. Both pump and probe pulses were linearly polarized and focused onto the sample surface at normal incidence using a 50× objective lens, which formed a polar MOKE configuration. The typical fluences of the pump and probe pulses were 2 and 0.9 mJ cm−2, respectively. The sample was placed in an optical cryostat (Cryostation S50, Montana Instruments) for magnetic-field-dependent and temperature-dependent measurements.

Numerical simulations
We carried out numerical simulations based on the Landau–Lifshitz–Gilbert (LLG) equation for a two-spin system, m_i = -\gamma i m i \times Heff,i + \alpha i m i \times m_i, where Heff,i = H + Hk m z,i z^ - Hex,i m j + H m \times D (i = TM, RE, j = RE, TM), Hk = K / (1/2 Ms,TM + Ms,RE), Hex,i = 2Aex / Ms,i d2, HD,TM = D / Ms,TM, HD,RE = -D / Ms,RE, and D = y^. \gamma i = gi \mu B / \hbar is the gyromagnetic ratio, gi is the Landé g factor, \alpha i is the damping parameter, K0 = K/4 is the PMA energy density of each sublattice, Aex is the exchange stiffness, d is the lattice constant, Ms,i is the saturation magnetization, and D is the DMI constant. We assume that the effective field from PMA, Hk, is identical for both sublattices. We use Aex = 1.0 × 10−12 J/m, D = 0.9 mJ/m2, K = 2.5 × 105 J/m3, gTM = 2.2, gRE = 2.0, and d = 0.4 nm43. For numerically determining the resonance frequency and magnon lifetime, we first apply an external field (\mu 0H) along the in-plane direction of 10 mT. Once the system reaches equilibrium, we tilt polar angles of the magnetizations by 0.1 degrees and track their trajectories over time. Then we fit the x-component of n using a damped sine function to get \tau and \omega RE \pm values: nx(t) = A e^{-t/\tau} sin(2\pi ft + \phi). For the numerical results in Fig. 4, we use varying values of Ms,TM = (1100 - y) and Ms,RE = (1000 + 3y) kA/m.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The article entitled “Inversion of magnon lifetime of ferromagnetic and exchange resonance modes in ferrimagnets” is\ndevoted to magnon lifetime investigation using magneto-optical time-resolved measurements. The authors show that in the\nvicinity of the angular momentum compensation point, the lifetime of the high-frequency (“exchange”) mode increases.\nWithout a doubt, extending the lifetime of high-frequency magnon modes is an important topic for\nmagnon-based devices. However, I can not recommend the article for publication in Nature Communications in the present form. Please see the\ncomments below.\n1. The manuscript lacks clarity regarding the role of Ta, Pt, and TaOx in the structure. Will the observed effects be applicable\nto the structure made of pristine Co_xGd_1-x alloy?\n2. The authors investigated the magnetization dynamics, specifically magnon lifetime, of the Co78Gd22 alloy. The alloy's\nmagnetization compensation point (Tm) is around 140 K, while the angular momentum compensation point (Ta) is around\ntemperature. It is unclear why the Co78Gd22 composition was chosen for the study, as the sample required cooling to\nachieve the desired compensation points. I would also recommend adding plots of M_Co(T) and M_Gd(T) to Supplement.\n3. More precise temperature-dependent VSM and MOKE measurements are needed to determine the magnetization\ncompensation point (Fig. 1d shows that magnetization does not reach zero). To determine Ta, temperature-dependent\ndomain-wall velocity measurements could be performed [Nature materials, 16(12), 1187-1192 (2017), Nature\nnanotechnology, 14(3), 232-236 (2019)].\n4. The structure under consideration possesses PMA, as demonstrated by MOKE and VSM measurements in out-of-plane\nconfiguration (at 0 magnetic field net magnetization is oriented normally to the sample plane). However, time-resolved\nmagnetization dynamics measurements were performed in an in-plane configuration of the external magnetic field. In the\ncase of an in-plane configuration, a non-collinear magnetic ground state is achieved near Tm [Physical Review B, 100(6),\n064409 (2019), Journal of Magnetism and Magnetic Materials, 623, 172968 (2025), Physical Review B, 108(17), 174442\n(2023)]. Given that Tm is close to Ta, I believe the sample's magnetic phase diagram should be measured and calculated to\nensure that the ground state is collinear.\n5. The temperature of the sample is not indicated in the Fig. 2. The size of the plotted magnetization oscillations depicted in\nFig. 2b is too small; the high-frequency mode can not be well resolved.\n6. The magnetization oscillations shown in Fig. 3a,b are too small. I believe that for EXR and FMR modes, different\ntimescales could be used to improve visibility. I also recommend adding fitted curves used for signal approximation and\noscillation parameter extraction (such as frequencies, lifetime, etc.) to Figs. 2a, 3a, and b.\n7. Magnetization dynamics measurements do not provide adequate temperature resolution. As the authors attempt to\npersuade the audience that the magnon lifetime of the exchange magnon mode increases near Ta, increased temperature\nresolution is critical near this temperature point. The current version of the manuscript only contains data for 180 K and 210\nK.\n8. There is a misprint on Fig. 3c,d,e. (I believe, that Xa should be replaced by Ta).\n9. The oscillations shown in Fig. 3a were measured with no external magnetic field. Could the authors provide a MOKE\nimage of the magnetic domains to confirm that the pump and probe beams are smaller than the domains?\n10. To properly compare experimental and simulated results, I recommend plotting them on the same graph. The simulated\nand experimental FRM modes have different frequencies. The FMR mode in Fig. 3c has a frequency of about 30 GHz, while\nthe simulated FMR mode in Fig. 4a has a frequency of around 80 GHz.\n11. Please, evaluate the effect of laser heating on the sample and system.\n12. In Supplement Note 1, Fig. 1, there are insets with \"200\". What does this mean?", "reviewer_2": "This manuscript investigates the damping of magnon modes in exchange coupled ferrimagnets. It finds a comparatively long\nlived exchange-resonance mode of the antiferromagnetic magnon in the MOKE spectra, and can convincingly explain the\nresults in the framework of a two-sublattice LLG model. (I would not necessarily call it a \"theory\" in the abstract.)\nThis is a well written manuscript. (The only misprint I noticed was \"long-live information transfer\" in the abstract.) The\nexperiments are carefully done and interpreted, and the details of the numerical modeling are presented in sufficient detail\nthat the can be understood from the text and the supplement. The result is interesting in its own right. I agree with the authors\nthat the effect that they have discovered has so far, to the best of my knowledge, gone unnoticed by the community. Once\nyou read about it, their result almost becomes intuitively clear and it should become part of the common knowledge on the\nmagnetization dynamics in these systems. It can essentially be published \"as is\".\nI have two minor comments. As far as I can tell the quantity rho (\"inertial term associated with the antiferromagnetic\ndynamics\") is never defined. Also in the supplement, there are factors of A_x and A_ex with various typesetting conventions.\nThe authors should clean this up.\nThe coupling of sublattices has also been investigated on ultrashort timescales, see, e.g., Mentink PRL 108 (2012), Ostler\nNat. Comm. 3 (2012), Baral PRB 91 (2015). There the distinction between timescales of relaxation (alpha) and exchange is\nnot as clear as in their case. Maybe the authors can relate their results to the ones I mentioned."}, "author_response": "-------------------------------------------------------------------\nResponse to Reviewer #1\n-------------------------------------------------------------------\nThe article entitled “Inversion of magnon lifetime of ferromagnetic and exchange resonance\nmodes in ferrimagnets” is devoted to magnon lifetime investigation using magneto-optical\ntime-resolved measurements. The authors show that in the vicinity of the angular momentum\ncompensation point, the lifetime of the high-frequency (“exchange”) mode increases. Without\na doubt, extending the lifetime of high-frequency magnon modes is an important topic for\nmagnon-based devices. However, I can not recommend the article for publication in Nature\nCommunications in the present form. Please see the comments below.\nReply: We appreciate the reviewer’s acknowledgement of the significance of our work. We\nalso thank the reviewer for the careful review and valuable comments. In light of these concerns,\nwe have performed additional experiments and calculations to address them, which will be\nshown in the point-by-point responses. All the modifications based on the reviewer’s comments\nare highlighted in red in the revised manuscript.\n1. The manuscript lacks clarity regarding the role of Ta, Pt, and TaOx in the structure. Will the\nobserved effects be applicable to the structure made of pristine Co_xGd_1-x alloy?\nReply: Thank you for raising this important point. The full film structure, as shown in Fig. 1a\nand Methods, is Si/SiO (substrate)/Ta (1 nm)/Pt (6 nm)/Co Gd (7 nm)/TaO (2 nm). Each\n2 1−x x x\nlayer serves a specific role in forming high-quality ferrimagnetic structure with perpendicular\nanisotropy (PMA). The bottom Ta layer, grown directly on the Si/SiO substrate, acts as a buffer\n2\nlayer to enhance adhesion and improve surface roughness. The next Pt layer establishes the\nPMA of the ferrimagnetic layer. Then, CoGd is grown on Pt and forms PMA, which is the\ncondition discussed in our manuscript. Finally, the TaO layer on top acts as a protective\nx\ncapping layer to prevent oxidation of the CoGd ferrimagnetic layer. This multilayer architecture\nhas been widely adopted in previous studies to achieve high-quality PMA rare-earth–transition-\nmetal (RE–TM) ferrimagnets, such as CoGd [Phys. Rev. Lett. 118, 167201 (2017), Nat.\nElectron. 3, 37–42 (2020), Nat. Mater. 19, 980–985 (2020)], GdFeCo [Appl. Phys. Lett. 110,\n242405 (2017)], and GdTbCo [Phys. Rev. B 105, 054410 (2022)].\nRegarding the applicability of our results to pristine Co Gd alloys, our main\n1−x x\nobservation remains valid because our model considers only intrinsic contributions in\nferrimagnetic systems, such as exchange, DMI, anisotropy, and damping, instead of any other\nextrinsic contribution.\ntext. “We study a prototypical rare-earth (RE) transition-metal (TM) ferrimagnet Co Gd with\n1−x x\ncompositions varying from Co Gd to Co Gd (Methods). The full structure consists of\n83 17 65 35\nSi/SiO (substrate)/Ta (1 nm)/Pt (6 nm)/Co Gd (7 nm)/TaO (2 nm), as shown in Fig. 1a.\n2 1−x x x\nThe bottom Ta layer, grown directly on the Si/SiO substrate, acts as a buffer layer to enhance\n2\nadhesion and improve surface roughness. The next Pt layer establishes the perpendicular\nmagnetic anisotropy (PMA) of the CoGd ferrimagnetic layer, as confirmed by the magnetic-\nfield-angle-dependent vibrating sample magnetometer (VSM) results of these samples\n(Supplementary Note 1). Finally, the TaO layer on top acts as a protective capping layer to\nx\nprevent oxidation of the CoGd ferrimagnetic layer.”\n2. The authors investigated the magnetization dynamics, specifically magnon lifetime, of the\nCo78Gd22 alloy. The alloy's magnetization compensation point (Tm) is around 140 K, while\nthe angular momentum compensation point (Ta) is around 200 K. According to experimental\ndata (see Fig. 1c), the authors also have samples with compensation points close to room\ntemperature. It is unclear why the Co78Gd22 composition was chosen for the study, as the\nsample required cooling to achieve the desired compensation points. I would also recommend\nadding plots of M_Co(T) and M_Gd(T) to Supplement.\nReply: Thanks for the insightful comments. We choose the sample Co Gd to achieve three\n78 22\nrepresentative ranges in the same sample: Gd-rich (T < 200 K), around angular momentum\ncompensation, and Co-rich region (T > 240 K). It allows us to demonstrate the angular\nmomentum compensation as the key factor determining the magnon lifetime inversion (τ <\nFMR\nτ ), while the other two Gd-rich and Co-rich regions act as control experiments (τ > τ ).\nEXR FMR EXR\nGiven the operating range of our cryostat (4–340 K), using the sample Co Gd with T at low\n78 22 A\ntemperature provides the systematic experiments to illustrate our results across all three\nregimes.\nAs the reviewer suggested, the desired compensation point can also be achieved at room\ntemperature in the Co Gd sample. To further confirm our observation, we additionally study\n76 24\nCo Gd sample at room temperature to confirm the magnon lifetime inversion phenomenon.\n76 24\nWe conduct field-dependent TR-MOKE measurement at room temperature, with the time-\ndomain results shown in Fig. R1a. The magnon frequency, lifetime, and effective damping are\nsummarized in Fig. R1b–d, respectively. The magnon lifetime inversion (τ < τ ) remains\nFMR EXR\nvalid across all tested magnetic fields, demonstrating that our observation persists at room\ntemperature when the Co and Gd composition is appropriately tuned. This further highlights\n2\nthe relevance of our observation for potential room-temperature magnonic device applications.\nThese new data have been added to Supplementary Note 4.\nFig. R1. (Supplementary Fig. 8) Magnetic-field-dependent magnetization dynamics of\nCo Gd . a, Field-dependent TR-MOKE results of two modes. b–d, Extracted parameters\n76 24\nfrom the time-domain signals: magnon frequency (b), magnon lifetime (c), and effective\ndamping (d). The red line represents the EXR mode while the gray line represents the FMR\nmode.\nThe temperature dependence of sublattice magnetizations follows the formula M(T) =\ni\nM(0)(1 − T/T )βi, where i = Co or Gd, M(T) is the sublattice magnetization at temperature T,\ni c i\nM(0) is the sublattice magnetization at zero temperature, T is the Curie temperature, and β is\ni c i\nthe exponent of the sublattice magnetization temperature dependence. We note that both\nsublattices share the same T due to their coupling. The total magnetization M is calculated\nc s\naccording to [Nat. Mater. 16, 1187–1192 (2017)]\nM (T) = |M (T) − M (T)| = |M (0)(1 − T/T )βGd − M (0)(1 − T/T )βCo|. (R1)\ns Gd Co Gd c Co c\nThe total M (T) of Co Gd is experimentally measured as a function of temperature by VSM\ns 78 22\nunder a 500 mT field cooling condition (symbols in Fig. R2a). By fitting Eq. (R1) to the\nexperimentally measured M (curve in Fig. R2a), we decouple the two sublattices and extract\ns\nparameters T = 400 K, M (0) = 1260 emu cm−3, β = 0.45, M (0) = 1100 emu cm−3, and β\nc Gd Gd Co Co\n3\n= 0.19. Thus, the obtained temperature dependence of M (T) and M (T) is plotted in Fig. R2b,\nCo Gd\nwhich matches previous reports.\nIn addition, the total angular momentum A (T) is analytically obtained using\ntotal\nA (T) = |A (T) − A (T)| = |[M (0)/γ ](1 – T/T )βGd – [M (0)/γ ](1 – T/T )βCo|, (R2)\ntotal Gd Co Gd Gd c Co Co c\nwhere γ (γ ) is the gyromagnetic ratio of Gd (Co) defined as γ = g μ /ℏ (γ = g μ /ℏ),\nGd Co Gd Gd B Co Co B\nBohr magneton μ = 9.274×10−24 J T−1, and reduced Planck’s constant ℏ = 1.055×10−34 J s. By\nB\nconsidering the different Landé g factor of Co (g = 2.2) and Gd (g = 2), we calculate A (T),\nCo Gd total\nA (T), and A (T) by using the parameters extracted above and plot in Fig. R2c. From the\nGd Co\nvanishing point of the total angular momentum, T is estimated to be 250 K.\nA\nThe detailed calculation is added to Supplementary Note 1, and the plots are also added\nto the main Fig. 1c with the following explanation:\n“The temperature dependence of M of Co Gd is fitted (blue curve) to decouple sublattice\ns 78 22\nmagnetizations, and the total angular momentum A of Co Gd is analytically obtained\ntotal 78 22\n(black curve) by considering the different Landé g factors of Co (g = 2.2) and Gd (g = 2)\n(Supplementary Note 1).”\nFig. R2. (Supplementary Fig. 2) Magnetization and angular momentum compensation in\nCo Gd . a, Temperature-dependent M of Co Gd under a 500 mT field cooling condition.\n78 22 s 78 22\nSymbols represent experimental data, and the green line represents fitting results based on Eq.\n(R1). b, Calculated sublattice-specific magnetization and total magnetization from the\nextracted parameters based on Eq. (R1). T = 165 K is determined by the crossing temperature\nM\nat M = 0. c, Calculated sublattice-specific angular momentum and total angular momentum\ns\nfrom the extracted parameters based on Eq. (R2). T = 250 K is determined by the crossing\nA\ntemperature at A = 0.\ntotal\n3. More precise temperature-dependent VSM and MOKE measurements are needed to\ndetermine the magnetization compensation point (Fig. 1d shows that magnetization does not\nreach zero). To determine Ta, temperature-dependent domain-wall velocity measurements\n4\ncould be performed [Nature materials, 16(12), 1187-1192 (2017), Nature nanotechnology,\n14(3), 232-236 (2019)].\nReply: We thank the reviewer for this valuable suggestion. Following this, we conduct\nadditional temperature-dependent VSM measurements with finer temperature steps to provide\nmore information in addition to the M–H results in main Fig. 1 and Supplementary Fig. 1.\nSpecifically, under a 500 mT out-of-plane field, we measure the saturation magnetization (M )\ns\nof Co Gd with 1 K temperature resolution. The resulting M–T curve under this field-cooling\n78 22\ncondition is presented in the above Fig. R2a. The minimum point at which M = 0 indicates T\ns M\n= 165 K.\nIn addition to the room-temperature MOKE measurements, we conduct temperature-\ndependent anomalous Hall effect (AHE) measurements to further demonstrate the hysteresis\nloop reversal and to cross-check the T of Co Gd , as shown in Fig. R3a. The hysteresis loop\nM 78 22\nreversal between 160 K and 165 K confirms the existence of T in this range. Near this\nM\ntemperature range, the slope changes at high field indicate a spin-flip transition [Phys. Rev. B\n100, 064409 (2019)]. The resistance R and coercivity H are extracted from the AHE results.\nAHE c\nThe sign change of R in Fig. R3b and the divergence of H in Fig. R3c confirm the T near\nAHE c M\nFig. R3. Temperature-dependent AHE results of Co Gd with out-of-plane fields. a,\n78 22\nHysteresis loops measured by AHE as a function of temperature. b–c, Extracted parameters\nfrom the hysteresis loops: resistance R (b) and coercivity H (c).\nAHE c\nFig. R4. (main Fig. 1) Co Gd magnetic property around the compensation points. a,\n1−x x\nFilm structure. The inset shows the compensated sublattices of Co and Gd magnetic moments.\nb, Temperature-dependent AHE results of Co Gd with out-of-plane fields. c, Temperature-\n78 22\ndependent magnetic property of Co Gd . Upper: The resistance R (left) and coercivity H\n78 22 AHE c\n(right) of Co Gd film at various temperatures. Bottom: The saturation magnetization M (left)\n78 22 s\nand calculated total angular momentum A (right) of Co Gd film at various temperatures.\ntotal 78 22\nThe blue curve shows the magnetization compensation temperature (T ) and the black curve\nM\nshows angular momentum compensation temperature (T ). d, M and H of various Co Gd\nA s c 1−x x\nfilms at room temperature extracted from polar MOKE and VSM results. The bottom blue\narrow shows the magnetization compensation point (X ) and the bottom black arrow shows\nM\nthe angular momentum compensation point (X ).\nA\nThe domain wall velocity is widely studied as experimental evidence of T . Previous\nA\nworks achieve the real-time domain wall detection via the anomalous Hall effect (AHE) [Nat.\nMater. 16, 1187–1192 (2017), Phys. Rev. Lett. 122, 127203 (2019)]. We have estimated the\ndomain wall velocity by our TR-MOKE experiments of the ferromagnetic resonance. By\nchanging the temperature, the domain wall velocity V is expressed by [Nat. Mater. 16, 1187–\n1192 (2017)]\n𝑉 =\n!(#!$!%#!$\")('!('\")\n𝐻 , (R3)\n(#!$!%#!$ \")\"%($!($\")\" )**\n6\nwhere w is the domain wall width, α is the damping constant of sublattice i, S is the angular\ni i\nmomentum of sublattice i, M is the magnetization of sublattice i, and H is the effective field.\ni eff\nWe rewrite the formula as\n𝑉 =\n! ('!('\")\n𝐻 . (R4)\n($!%!&$!%\") % (%!(%\") ($!($\") )**\n(%!(%\") ($!%!&$!%"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors addressed all of the comments made during the first round of the review process. I'd like to underline that the\nauthors performed extensive additional experiments and theoretical computations to strengthen and enrich the work. I\ngreatly appreciate the work they did. I now recommend the manuscript entitled \"Inversion of magnon lifetime of ferromagnetic\nand exchange resonance modes in ferrimagnets\" for publication in Nature Communications.", "reviewer_2": "The authors have addressed the detailed comments of Referee 1 and those that I had in the first round. The have done\nadditional measurements and responded in detail. I believe the manuscript has been further improved by these changes and\nI recommend publication."}, "author_response": ""}]

Decision letter