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In magnetic thin films, large excitation amplitudes can lead to the onset of the first order Suhl instability, where the driven uniform mode undergoes three magnon scattering to two modes of half the frequency and equal and opposite wavevectors. Although experimental studies of the steady-state behavior are established, a comprehensive study of the nonequilibrium behavior over a large range of excitation powers is lacking. We have developed an experimental technique for sensitive detection of the time evolution of the FMR amplitude over five orders of magnitude in power, which provides a highly comprehensive understanding of the three-magnon scattering process. Applying this technique to patterned 3μm thick YIG thin films, we observe strong agreement with our results from micromagnetic simulations of the Suhl instability. We have developed a semianalytical model to explain the three magnon scattering process. This model shows quantitative agreement with experiment and simulation, including a predicted additional regime corresponding to power-dependent nonlinear oscillatory behavior on the order of 50 MHz. Additional observed effects will also be discussed.&lt;br/&gt;&lt;br/&gt;&lt;i&gt;This work was supported by SMART, a center funded by nCORE, a SRC program sponsored by NIST, DARPA under Grant W911NF-17-0100, MINT at Minnesota, and the NSF XSEDE through Allocation No. TG-ECS200001. We acknowledge the use of MSI resources.&lt;/i&gt;

![](https://assets.underline.io/uploads/markdown_image/1/image/1206326856094257491a60e9c5fd8367.jpg)

Time-resolved Nonlinear Magnetization Dynamics in YIG Thin Films

2022 Joint MMM-INTERMAG

Time-resolved Nonlinear Magnetization Dynamics in YIG Thin Films

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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MMM 2020

In magnetic thin films, large excitation amplitudes can lead to the onset of the first order Suhl instability, where the driven uniform mode undergoes three magnon scattering to two modes of half the frequency and equal and opposite wavevectors. Although experimental studies of the steady-state behavior are established, a comprehensive study of the nonequilibrium behavior over a large range of excitation powers is lacking. We have developed an experimental technique for sensitive detection of the time evolution of the FMR amplitude over five orders of magnitude in power, which provides a highly comprehensive understanding of the three-magnon scattering process. Applying this technique to patterned 3μm thick YIG thin films, we observe strong agreement with our results from micromagnetic simulations of the Suhl instability. We have developed a semianalytical model to explain the three magnon scattering process. This model shows quantitative agreement with experiment and simulation, including a predicted additional regime corresponding to power-dependent nonlinear oscillatory behavior on the order of 50 MHz. Additional observed effects will also be discussed. This work was supported by SMART, a center funded by nCORE, a SRC program sponsored by NIST, DARPA under Grant W911NF-17-0100, MINT at Minnesota, and the NSF XSEDE through Allocation No. TG-ECS200001. We acknowledge the use of MSI resources.

![](https://assets.underline.io/uploads/markdown_image/1/image/1206326856094257491a60e9c5fd8367.jpg)

Time-resolved Nonlinear Magnetization Dynamics in YIG Thin Films

technical paper

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Spin-torque oscillator (STO) is a class of emerging spintronic devices that can find applications in wireless communication systems in the form of radio frequency (rf) generator1,2 and rf detector3. The STO offers a compatibility with CMOS technology and high frequency tunability2. However, the single STO output is not sufficient for the rf applications. In order to overcome this, the mutual synchronization of STOs is critical for communication, energy harvesting and neuromorphic applications. The short range magnetic coupling-based synchronization has spatial restrictions (few μm), whereas the long-range electrical synchronization using vortex STOs has limited frequency responses in hundreds MHz (&lt; 500 MHz), restricting them for on-chip GHz-range applications. Here, we demonstrate electrical synchronization of four non-vortex uniformly-magnetized STOs at 2.4 GHz WiFi band4, resolving the frequency-area quandary for designing STO based on-chip systems. We use a single dc source to synchronize canted free layer STOs, thereby simplifying the synchronization scheme. Due to mutual synchronization, STOs show a power enhancement and linewidth reduction [Fig. 1a,b]. When injection-locked with an external rf source, the synchronised STOs show a wide locking range around 2.4 GHz [Fig. 1c,d], which can be useful for neuromorphic computing. Under injection locking, synchronized STOs demonstrate an excellent time-domain stability and improved phase noise performance [Fig. 1e,f]. Our results highlight the significance of electrical topology (series vs. parallel) while designing an on-chip STOs system.

![](https://assets.underline.io/uploads/markdown_image/1/image/5c47ef17602057fa745ff11bcef50d6b.jpg)

Fig. 1. a,b Oscillation power (P) and linewidth (△f) versus numbers of oscillators (N) in parallel (a) and series (b), respectively. Solid lines are linear fit and N−1 scaling for the power and linewidth, respectively. c,d STO frequency (f0) versus radio frequency (frf) response for a single oscillator (red) and four oscillators (blue) at Prf = −20 dBm for the parallel (c) and series (d) connection. The locking range defines the region where f0 ~ frf. e,f Phase noise for 1 ms time-domain trace in parallel (e) and series (f). The solid black and blue lines in (f) shows the 1/f2 and 1/f3 phase noise, respectively.

Electrically connected spin-torque oscillators array for 2.4 GHz WiFi band transmission

Understanding how spin currents flow across metal-semiconductor interfaces at femtosecond timescales has impacts on ultrafast spintronics, data processing and storage applications. However, the possibility to directly access the propagation of spin currents on these time scales has been hampered by the simultaneous lack of both ultrafast site selective magnetic sensitive probes and tailored metal-semiconductor interfaces. By means of FEL-based core-resonant Kerr spectroscopy, we report direct experimental evidence of spin currents across a Ni/Si interface in terms of different magnetodynamics at the Ni M2,3 and Si L2,3 absorption edges. This further allows us to extract the propagation velocity of the spin current in Si as 0.2 nm/fs.

![](https://assets.underline.io/uploads/markdown_image/1/image/ba4eb9360a1db64b8e33f4c491757403.jpg)

Fig. 1: a) The sample stack consists of a Si substrate (blue), a thin Si3N4 passivation layer (grey), 7 nm of Ni film (red) and 2 nm of Ag capping layer (yellow). The white arrow represents the in-plane magnetization of the Ni film. The FEL pulses at the Si L2,3 edge (blue arrow) and at the Ni M2,3 edge (red arrow) are also depicted. The IR laser pulses at 1.55 eV hit the sample at a time delay t-t0, before the FEL pulses arrival. The penetration depths of the pump and the probes are represented by the length of the arrows. The fraction of the IR laser energy absorbed by each layer is reported above the probe arrow. b) Scheme of the FEL RMOKE set-up at MagneDyn. RMOKE was probed in longitudinal configuration at an angle of incidence of 45°. The polarization ε of the incoming FEL pulses (blue arrow) is in the scattering plane, while the Kerr rotation of the polarization of the reflected pulse is indicated as θ.

![](https://assets.underline.io/uploads/markdown_image/1/image/388e11ba365f236752de3629d9784f15.jpg)

Fig. 2: RMOKE magnetic hysteresis in degrees of Kerr rotation at the Ni M2,3 edge (a) and at the Si L2,3 edge (b). Empty and filled circles curves represent the unpumped and pumped hysteresis after 0.5 ps (Ni) and 0.3 ps (Si) (colored star markers in c). c) relative change of the site resolved magnetization M (Ni - red and Si - blue dots) as a function of time delay measured in saturation at 550 mT. The solid lines represent the tting results, from which we extract the characteristic times for demagnetization and recovery. The difference (ΔM/M)js is also shown (gray pentagons).

All-Optical Spin Injection in Silicon Revealed by Element Specific Time-Resolved Kerr Effect

In the last decade, charge to spin conversion and spin-orbit torques have been extensively studied to manipulate the magnetization for efficient data storage devices functionalities. The charge-spin conversion is governed by spin-orbit interactions such as spin Hall effect in the bulk of heavy metals and Rashba effect at their interfaces[1]. Recently, substantial increase in the magnetic torques has been observed with Cu/CuOx or Cu/Al2O3 light element interfaces. The observation of magnetic torques in the absence of spin-orbit interaction can be explained in the framework of orbital currents at Cu/CuOx interface[2-3]. Furthermore, as the orbital effects are believed to be one order larger than the spin-orbital coupling effects, we expect larger torques efficiency by utilizing the orbital currents. To aim this, we have measured the damping-like (HDL) and field-like (HFL) effective fields in SiOx/Co(2)/Pt(x)/Cu/CuOx(3) and SiOx/Co2)Pt(x)/AlOx(1) samples (and where x=2-6 nm) using second harmonic Hall measurement technique. Here, all the samples possess an in-plane magnetic anisotropy. As the orbital current does not carry spin (S=0), it is important to convert them into spin-current to interact with magnetization via s-d exchange interactions to promote the torques. Therefore, Pt is used to convert the orbital current into the spin current. Although the HFL is found to be negligible in all the samples, one observes that the HDL increases linearly with the current density in Pt, like expected for the current induced torque. Fig. 1 shows HDL for 107 A/cm2 current density in Pt. However, we do not observe significant change for AlOx, a two-fold increase is observed in Co(2)/Pt(4)/Cu/CuOx(3) compared to our reference Co(2)/Pt(4) sample, and where the exact spin current profile has been analyzed. As proposed in the recent literature [2], we attribute this enhancement in torques to the generation of orbital current at Cu/CuOx interface.

![](https://assets.underline.io/uploads/markdown_image/1/image/919726e4cd4ba2cd060a27d54a1bd6b4.jpg)

Fig1. Comparison of HDL in various capping layers as a function of Pt thickness. The Blue (Black) arrow shows the increase in HDL in Co/Pt/Cu/CuOx (Co/Pt/Cu/AlOx) samples due to the orbital current at Cu/CuOx (Cu/AlOx) interface.

Enhanced Torques in Co/Pt Systems via Orbital Current Contribution

Gilbert damping remains among the foremost properties impacting the design of spintronic devices. There are many cases where the observed damping is larger than the predicted damping, which is usually based on considerations of electronic structure (such as the Kamberský torque-correlation model [1]). It is therefore of great importance to identify alternative damping mechanisms in order to obtain more accurate results. Two-magnon scattering is one such mechanism, but this can usually be distinguished from Gilbert damping due to its nonlinear dependence on frequency or by orienting the magnetization normal to the film plane [2]. In this talk we discuss how magnetoelastic coupling can cause a substantial enhancement of the overall Gilbert damping in a series of Fe0.7Ga0.3 thin films [3]. In our case, the effect is exaggerated due to the exceptionally high magnetostriction found in FeGa alloys (measured in these films as ~100 ppm at room temperature). The effect is based on a model of energy loss through phonon relaxation: Phonons are driven by magnetization dynamics through magnetoelastic coupling, and the relaxation of these phonons leads to a form of Gilbert damping (see Fig. 1) [4]. We show this by presenting the orientational and temperature dependence of the Gilbert damping in total of seven Fe0.7Ga0.3 thin films with thicknesses ranging from 17 to 70 nm. The orientation dependence of the Gilbert damping demonstrates that the magnetoelastic Gilbert damping is mitigated for out-of-plane magnetization due to confinement effects from the interfaces of the film, leading to a very large anisotropy (up to a factor of 10 at room temperature). The temperature dependence demonstrates both the effects of the magnetostriction and the phonon viscosity. The damping increases significantly at low temperatures due to the increasing magnetostriction. Importantly, however, we observe a nonmonotonic temperature dependence of the Gilbert damping. As the temperature decreases from ~75 K to 0, the damping decreases due to the reduction in the phonon relaxation rate (main panel of Fig. 2). This is attributed to the influence of the temperature dependence of the phonon viscosity, which increases linearly with temperature (inset of Fig. 2). It is thereby shown that the competing effects of magnetostriction, which decreases with increasing temperature, and phonon relaxation lead to the observed nonmonotonic temperature dependence. We conclude by emphasizing that, in general, the magnetoelastic damping may constitute a significant portion of the total Gilbert damping, particularly in materials with high magnetostriction. This work was supported by SMART, a center funded by nCORE, a Semiconductor Research Corporation program sponsored by NIST.

![](https://assets.underline.io/uploads/markdown_image/1/image/5c9a5677720ac744236b40def5e52359.jpg)

Fig. 1: Schematic diagram of the magnetoelastic damping process with magnetization M(t) lying in the film plane. The lattice displacement u(t) is driven by the magnetostriction λ, leading to energy loss due to the presence of phonon viscosity (represented by the wavy orange arrows).

![](https://assets.underline.io/uploads/markdown_image/1/image/cd4033452c06dedfa3fbed5997c11741.jpg)

Fig. 2: Main panel, left ordinate: Magneloelastic Gilbert damping as a function of temperature for the 21 nm (blue triangles), 57 nm (red circles), and 70 nm (black squares) Fe0.7Ga0.3 films. Main panel, right ordinate: The square of the magnetostriction λ as a function of temperature, interpolated from data given in Ref. [5]. Inset: Phonon viscosity, defined as the ratio of αme to λ2 shown as a function of temperature for the 21 nm (blue triangles), 57 nm (red circles), and 70 nm (black squares) Fe0.7Ga0.3 films.

Magnetoelastic Gilbert Damping in Magnetostrictive Fe0.7Ga0.3 Thin Films

The generation of phonons by ferromagnetic resonance in magnetic thin films and subsequent propagation of phonons into the substrate is an effect known as phonon pumping [1-2]. This effect can be utilized in spintronic devices that rely on magnetoelastic coupling for information transfer. The phonon pumping effect is most efficient in the perpendicular configuration, and can therefore be suitably integrated in devices composed of PMA materials. Here we demonstrate the pumping of phonons by ferromagnetic resonance in a series of [Co(0.8 nm)/Pd(1.5 nm)]n multilayers (n = 6, 11, 15, and 20) with large magnetostriction and perpendicular anisotropy. The effect is shown using broadband ferromagnetic resonance over a range of temperatures (10 to 300 K), where a resonant damping enhancement is observed at frequencies corresponding to standing wave phonons across the multilayer. The strength of this effect is strongly dependent on temperature—approximately a factor of 4 larger at 10 K compared to room temperature (see figure). Lastly, we demonstrate that the damping enhancement coincides with a shift in the ferromagnetic resonance field, and further that it is a complementary effect that can be predicted from the resonant damping enhancements. We emphasize that [Co/Pd]n multilayers are an ideal platform for studying this effect due to their large magnetostriction and perpendicular anisotropy. Importantly, the PMA does not show significant dependence on multilayer thickness, so devices can be engineered that have phonon pumping resonances at zero applied field. This work was supported by SMART, a center funded by nCORE, a Semiconductor Research Corporation program sponsored by NIST.
References [1] S. Streib, H. Keshtgar, and G. E. W. Bauer, Damping of Magnetization Dynamics by Phonon Pumping, Phys. Rev. Lett. 121, p.027202 (2018).
[2] K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hurdequint, N. Beaulieu, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Coherent long-range transfer of angular momentum between magnon Kittel modes by phonons, Phys. Rev. B 101, p.060407(R) (2020).

![](https://assets.underline.io/uploads/markdown_image/1/image/3704a35d4e1ee87bd33e2993e6fa34b3.jpg)

Ferromagnetic resonance linewidth enhancements due to phonon pumping at different temperatures for the [Co/Pd]11 multilayer. Linewidth contributions from Gilbert damping and inhomogeneous broadening are excluded. Vertical dashed lines at 23 and 46 GHz show positions of phonon pumping resonances.

Anomalous Temperature Dependence of Phonon Pumping by Ferromagnetic Resonance in [Co/Pd]n Multilayers with Perpendicular Anisotropy

Advancing the understanding of magnetic relaxation, often discussed in terms of “damping,” is essential for energy-efficient spintronic memories, oscillators, and spin-wave circuits. BCC Fe is an elemental ferromagnet with attractive properties for these applications. Here, we examine magnetic relaxation – i.e., intrinsic and extrinsic damping – in two series of sputter-grown, polycrystalline Fe thin films with distinct structural properties: (1) strongly textured, smooth Fe films grown on Cu and (2) weakly textured, rough Fe films grown on Ag. Out-of-plane ferromagnetic resonance (FMR) measurements reveal intrinsic Gilbert damping parameters of ≈ 0.0024 for all Fe films thicker than 4 nm, regardless of their structural properties, as shown in Fig. 1. This damping parameter range is in quantitative agreement with that for epitaxial Fe [1]. The remarkable invariance with microstructure strongly suggests that intrinsic Gilbert damping is a local property, averaged over a short length scale of a few nm, with limited impact from grain boundaries and film roughness. The in-plane FMR linewidths of the Fe films exhibit pronounced nonlinear frequency dependencies, indicating the presence of strong extrinsic damping. Figure 2 shows the extrinsic linewidth obtained by subtracting the intrinsic Gilbert contribution ΔHG (isotropic for Fe [1,2]) from the total in-plane linewidth ΔHIP. To fit our experimental data, we have used a grain-to-grain two-magnon scattering model [3-5] with two types of correlation functions proposed for modeling self-affine and mounded surfaces [6], which capture inhomogeneities in the film. However, neither of the two correlation functions is able to reproduce the data quantitatively. Our finding points to the need to further examine the fundamental impact of film microstructure on extrinsic damping.

![](https://assets.underline.io/uploads/markdown_image/1/image/1036e2b51a2ec70732f50783b672e744.jpg)


Fig. 1. Out-of-plane Gilbert damping αOOP as a function of film thickness.

![](https://assets.underline.io/uploads/markdown_image/1/image/0dd999a01cf02f7a52d306323d0872e3.jpg)


Fig. 2. In-plane extrinsic FMR linewidth vs frequency for (a) Cu/Fe (8 nm) film and (b) Ag/Fe (25 nm) film. Black squares represent experimental linewidth data. Dashed blue and solid red curves represent the fitted curves using correlation functions proposed for modeling self-affine and mounded surfaces, respectively.

Intrinsic and Extrinsic Damping in Polycrystalline Fe Thin Films

Spin transport across interfaces is fundamental to many spintronic devices. To fully exploit interfacial phenomena, such as spin-orbit torques and spin-pumping, in ferromagnetic (FM) and heavy metal systems (HM) the properties of both the HM layer and the interface must be considered, particularly in the case of perpendicularly magnetised systems [1]. Previously we showed the evolution of spin transport as a function of FM and HM thicknesses [2] and the role of insulating layers [3]. Debates remains regarding the physical basis of interfacial spin transport and the role of proximity polarisation of the HM layer [4]. A notional ‘spin memory loss’ term is often used to represent an interfacial contribution, while for Pt in particular, a magnetic polarisation arises in proximity (PIM) to a ferromagnetic material, opens a question about the role that PIM has upon interfacial spin phenomena, with contradictory reports that it has either a profound effect [5] or no effect [6] on spin transport. Here we present the results of a detailed systematic spin transport and PIM study in the same FM/HM samples. Structures with either a constant Pt thickness and a spacer layer thickness wedge, or a constant spacer layer and a Pt wedge were deposited. A combination of dynamic and static magnetic measurements, along with element specific resonant x-ray measurements at the APS, Argonne Lab, were used to address both the interfacial scattering contribution and the role of PIM in Pt on spin transport and to determine their relative contributions. Figs. 1 and 2 show damping and PIM data, with a persisting Pt moment with a 3nm Au layer, and a clear trend observed between the PIM and dynamic behaviour. These results highlight the role of the d-d orbital hybridisation and the induced moment for efficient interfacial transport, which is key for many spintronic applications.

![](https://assets.underline.io/uploads/markdown_image/1/image/ca60624dc0cead773db990e20080f85b.jpg)

Fig 1: Measured XMCD as a function of Au spacer thickness at both the Pt (blue) and Au (orange) L3 edges for NiFe (7 nm)/ Au / Pt (4 nm)

![](https://assets.underline.io/uploads/markdown_image/1/image/f3928c773ef3592e934476655f51d0ce.jpg)

Fig 2: Damping and Pt XMCD for CoFe/Au/Pt and NiFe/Au/Pt. Dotted lines indicate bulk damping contributions.

The role of proximity induced magnetism on spin transport in multilayered systems

Spin-Hall nano oscillators (SHNOs) [1-4] have recently emerged as an alternative for microwave signal generation and bio-inspired oscillatory computing [3,4]. However, the ferromagnetic layer of the SHNO comes with drawbacks: (a) need for strong external magnetic fields, and (b) limited operational frequencies. As a solution, compensated ferrimagnets provide an alternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like spin-orbit-torque (SOT) behavior. For use in SHNOs, it is however important to ensure that such films retain their ferrimagnetic behavior also in ultrathin films. In this work, ferrimagnetic Gdx(FeCo)1-x thin films were grown using co-sputtering of Gd and Fe87.5Co12.5 on high resistance (HR) Si (100) substrates, and their magneto-dynamics was studied using ferromagnetic resonance measurements at room temperature. By tuning the stoichiometry of Gdx(FeCo)1-x films, a nearly compensated behavior is for the first time demonstrated in 2nm thin film (Fig. 1). Values for the effective magnetization and effective Gilbert damping constant (α) of 0.02 Tesla and 0.0078±0.0002 are obtained for 2nm Gd24.4(FeCo)75.6 film, where α is comparable to the lowest value obtained in thick films [5]. The vertical red and yellow lines in Fig. 1(c) show literature values [5,6] for the spin angular momentum (xa) and magnetic momentum (xm) compensation points in thick films. The stoichiometry for the magnetic moment compensation in this work is very close to those [Fig. 1(c)]. In search of high-frequency auto-oscillations in SHNOs, constrictions with 50-300 nm widths were also prepared using Gdx(FeCo)1-x (2-10nm)/Pt(5nm) based stacks and promising results using Brillouin Light Scattering show SOT controllable ferromagnetic modes in these devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/9abf7a526aa9ca9c52175f16a9ed7bff.png)

Fig. 1. (a) Frequency (f) vs. resonance field and (b) Resonance linewidth vs f, of 2 nm thick Gdx(FeCo)1−x films, with x the Gd atomic %. (c) Effective magnetization and effective Gilbert damping constant vs. composition of 2 nm Gdx(FeCo)1−x films; solid symbols represent values obtained by fitting the experimental data to the Kittel’s equation [7] and equation (10) in reference [8], while solid lines are a guide to the eye.

Ultrathin ferrimagnetic GdFeCo films with very low damping

We present an overview of new techniques that can probe the properties of magnetic materials on very short time scales. We present a pump-probe spectroscopy method utilizing tabletop extreme UV (EUV) light produced by high harmonic generation. This EUV source allows us to spectroscopically probe ferromagnetism with multi-element specificity and a time resolution on the order of a few-femtoseconds. This made it possible to uncover that surprisingly, it is possible to excite the spin system in pure ferromagnets and alloys on few-femtosecond time scales [1,2]. With the implementation of a flux monitoring reference arm, we are able to remove source instability and improve data quality while reducing acquisition times.

New Techniques for Probing the Dynamic Properties of Magnetic Materials

Recently, it has been theoretically and experimentally demonstrated that the effects of inertia of magnetization should be considered in the full description of spin dynamics at pico- and femtosecond timescales [1-8]. The nutation motion of magnetization is a manifestation of inertia of the magnetic moments. A rigorous derivation including inertia in the Landau-Lifshitz-Gilbert equation was carried out by Mondal et al. in the Dirac-Kohn-Sham framework [6,7]. Here, we show that inertia effect in magnetization dynamics results in a new type of spin waves, i.e. nutation surface spin waves, which propagate at terahertz frequencies in in-plane magnetized ferromagnetic thin films. Considering the magnetostatic limit, i.e. neglecting exchange coupling, we calculate dispersion relation. In addition, we find that the nutation surface spin waves are backward spin waves [1]. The phase shift between precessing magnetic moments propagates as a spin wave through the ferromagnet because of dipole-dipole or exchange coupling (Fig. 1(a)). Magnetic inertia effects, which are expected to contribute to dynamics of spin waves, originate from spin-orbit coupling. We find that when taking inertia into account one finds that the deviation of alignment of localized moments will propagate through the spin system in the form of both precession and nutation motions, i.e. in ferromagnetic materials one needs to add to all “conventional” spin wave modes a high frequency wave-like motion with small amplitude caused by inertia. Moreover, a different type of waves having predominantly inertial nature appears in ferromagnetic thin films, which we call here nutation surface spin waves. Since these waves have terahertz frequencies (compared to typically GHz frequencies of other spin wave modes), they can be plotted as a small deviation on top of a “frozen” precession motion (Fig. 1(b)).

![](https://assets.underline.io/uploads/markdown_image/1/image/e702e12d86e64bbc071819e5db941fcf.jpg)

Fig. 1. (a) The precession spin wave without inertia (red curve). The blue arrow indicates the motion of the magnetization M. (b) The nutation surface spin wave (purple curve) with a frequency considerably higher than in (a) plotted with small blue circles on top of the “frozen” precession motion.

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