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&lt;br/&gt;The relaxation of the magnetisation in magnetic materials is of great importance from both an applied and fundamental point of view. Conventional modelling of magnetisation dynamics employs the Landau-Lifshitz-Gilbert (LLG) equation [1], where the damping is included phenomenologically by the Gilbert damping term. This describes the coupling of the magnetic modes (given primarily by the atomic spin) to the non-magnetic modes (lattice vibrations and electron orbits) which are assumed to be in equilibrium. Recent studies of the dynamics induced by THz laser pulses has highlighted the necessity of understanding magnetisation relaxation beyond this assumption.&lt;br/&gt;In reality the spin and lattice dynamics mutually influence one another, hence it is necessary to employ a unified model of molecular and spin dynamics [2,3,4], Spin-Lattice dynamics (SLD). In the present work we employ a model where the transfer of energy and angular momentum between the lattice and the spin system (Fig. 1) is realised by the pseudo-dipolar coupling [3,4], which arises from the spin-orbit interaction and can be parameterised by magneto-elastic experiments. The spin system exchange parameters are taken from ab-initio parameterisation for BCC iron [2], while for the phonon system interactions potential we compare the Harmonic and Morse potentials. Our results in Fig. 1 show that equilibration of both sub-systems can be obtained on a sub-nanosecond timescale in both the microcanonical and canonical ensembles, the relaxation time being governed by the energy initially deposited in the system. We also observe that the equilibrium magnetisation is independent of the thermostat and by coupling the spin system only to the lattice vibrations the magnetisation temperature dependence can be reproduced without the need of a phenomenological spin damping [3].&lt;br/&gt;Our model allows the evaluation of the effective magnon-phonon damping (Fig. 2) which agrees well with the values measured in magnetic insulators where they depend predominantly on magnon-phonon coupling rather than on electronic effects. The functional form of the damping variation is quadratic, in accordance with the form of the pseudo-dipolar coupling term. The magnon-phonon damping has been also calculated for an alternate form of the coupling, where the pseudo-dipolar coupling is replaced by an on-site form, i.e a Néel-like anisotropy term. The latter leads to much smaller damping values, as shown in Ref. [3], suggesting that the magnon-phonon damping can clearly have complex behaviour depending on the properties of the system, especially the coupling term. Based on this remark, no universal behaviour of damping as a function of temperature can be deduced for spin lattice models.&lt;br/&gt;The model developed in this work [3] opens the possibility to describe the distinct dynamics of spins and phonons, necessary for the understanding of ultrafast magnetisation dynamics experiments and the subsequent angular momentum transfer between the two subsystems. We have demonstrated that the model works well in the absence of a phenomenological Gilbert damping, which consists mainly of electronic contributions, hence the SLD model can be employed to study magnetic insulators, such as YIG, where the principal contribution to damping is via magnon-phonon interactions.

**References:**

&lt;br/&gt;[1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).&lt;br/&gt;[2] P.-W. Ma et al., &lt;i&gt;Phys. Rev. B&lt;/i&gt;, vol. 78, no. 2, p. 024434, Jul. 2008.&lt;br/&gt;[3] M. Strungaru et al. &lt;i&gt;arXiv preprint arXiv:2010.00642&lt;/i&gt; (2020) (accepted in Phys. Rev. B)&lt;br/&gt;[4] M. Aßmann and U. Nowak, &lt;i&gt;J. Magn. Mag&lt;/i&gt;&lt;i&gt;n.&lt;/i&gt;&lt;i&gt; Mate&lt;/i&gt;&lt;i&gt;r 6&lt;/i&gt; 469 (2019): 217-223.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0415f635b3cb2e3e3855832e043401d0.jpg)
&lt;br /&gt;
&lt;i&gt;Fig.1: NVE (microcanonical – top panel) and NVT ( canonical – bottom panel) simulations for a 10 × 10 × 10 unit cell BCC Fe system. The spin system is initialised at a spin temperature T&lt;sub&gt;S&lt;/sub&gt; = 0K corresponding to a saturated magnetic state in the z direction, while the lattice velocities are initialised by a Boltzmann distribution at T &lt;sub&gt;L&lt;/sub&gt; = 1200 K (for NVE) and T &lt;sub&gt;L&lt;/sub&gt; = 600 K (for NVT). In both cases we obtain equilibration of the two subsystems under 1ns.&lt;/i&gt;

![](https://assets.underline.io/uploads/markdown_image/1/image/6755da1be71fc906a6b75c124d5d162a.jpg)
&lt;br /&gt;
&lt;i&gt;Fig.2 Damping parameter extracted from fitting the z component of the magnetisation for two different choices of potential: HP- Harmonic Potential (green open squares) and MP-Morse Potential (black open circles) as a function of temperature - Panel a) and as a function of the coupling strength – Panel b); The effective damping in Panel a) is calculated for a constant coupling strength of C = 0.5, while in Panel b) the damping is calculated for a constant temperature of T = 300K. The black and green lines represent the average damping parameter obtained from the simulations using the Morse and the Harmonic Potentials, respectively.&lt;/i&gt;

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 Magnon-phonon damping calculations in the spin-lattice dynamics model

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The relaxation of the magnetisation in magnetic materials is of great importance from both an applied and fundamental point of view. Conventional modelling of magnetisation dynamics employs the Landau-Lifshitz-Gilbert (LLG) equation [1], where the damping is included phenomenologically by the Gilbert damping term. This describes the coupling of the magnetic modes (given primarily by the atomic spin) to the non-magnetic modes (lattice vibrations and electron orbits) which are assumed to be in equilibrium. Recent studies of the dynamics induced by THz laser pulses has highlighted the necessity of understanding magnetisation relaxation beyond this assumption. In reality the spin and lattice dynamics mutually influence one another, hence it is necessary to employ a unified model of molecular and spin dynamics [2,3,4], Spin-Lattice dynamics (SLD). In the present work we employ a model where the transfer of energy and angular momentum between the lattice and the spin system (Fig. 1) is realised by the pseudo-dipolar coupling [3,4], which arises from the spin-orbit interaction and can be parameterised by magneto-elastic experiments. The spin system exchange parameters are taken from ab-initio parameterisation for BCC iron [2], while for the phonon system interactions potential we compare the Harmonic and Morse potentials. Our results in Fig. 1 show that equilibration of both sub-systems can be obtained on a sub-nanosecond timescale in both the microcanonical and canonical ensembles, the relaxation time being governed by the energy initially deposited in the system. We also observe that the equilibrium magnetisation is independent of the thermostat and by coupling the spin system only to the lattice vibrations the magnetisation temperature dependence can be reproduced without the need of a phenomenological spin damping [3]. Our model allows the evaluation of the effective magnon-phonon damping (Fig. 2) which agrees well with the values measured in magnetic insulators where they depend predominantly on magnon-phonon coupling rather than on electronic effects. The functional form of the damping variation is quadratic, in accordance with the form of the pseudo-dipolar coupling term. The magnon-phonon damping has been also calculated for an alternate form of the coupling, where the pseudo-dipolar coupling is replaced by an on-site form, i.e a Néel-like anisotropy term. The latter leads to much smaller damping values, as shown in Ref. [3], suggesting that the magnon-phonon damping can clearly have complex behaviour depending on the properties of the system, especially the coupling term. Based on this remark, no universal behaviour of damping as a function of temperature can be deduced for spin lattice models. The model developed in this work [3] opens the possibility to describe the distinct dynamics of spins and phonons, necessary for the understanding of ultrafast magnetisation dynamics experiments and the subsequent angular momentum transfer between the two subsystems. We have demonstrated that the model works well in the absence of a phenomenological Gilbert damping, which consists mainly of electronic contributions, hence the SLD model can be employed to study magnetic insulators, such as YIG, where the principal contribution to damping is via magnon-phonon interactions.

**References:**

 [1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [2] P.-W. Ma et al., Phys. Rev. B, vol. 78, no. 2, p. 024434, Jul. 2008. [3] M. Strungaru et al. arXiv preprint arXiv:2010.00642 (2020) (accepted in Phys. Rev. B) [4] M. Aßmann and U. Nowak, J. Magn. Magn. Mater 6 469 (2019): 217-223.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0415f635b3cb2e3e3855832e043401d0.jpg)
 
Fig.1: NVE (microcanonical – top panel) and NVT ( canonical – bottom panel) simulations for a 10 × 10 × 10 unit cell BCC Fe system. The spin system is initialised at a spin temperature TS = 0K corresponding to a saturated magnetic state in the z direction, while the lattice velocities are initialised by a Boltzmann distribution at T L = 1200 K (for NVE) and T L = 600 K (for NVT). In both cases we obtain equilibration of the two subsystems under 1ns.

![](https://assets.underline.io/uploads/markdown_image/1/image/6755da1be71fc906a6b75c124d5d162a.jpg)
 
Fig.2 Damping parameter extracted from fitting the z component of the magnetisation for two different choices of potential: HP- Harmonic Potential (green open squares) and MP-Morse Potential (black open circles) as a function of temperature - Panel a) and as a function of the coupling strength – Panel b); The effective damping in Panel a) is calculated for a constant coupling strength of C = 0.5, while in Panel b) the damping is calculated for a constant temperature of T = 300K. The black and green lines represent the average damping parameter obtained from the simulations using the Morse and the Harmonic Potentials, respectively.

technical paper

Voltage controlled magnetic anisotropy (VCMA) has attracted much attention, since it can dramatically reduce the power consumption during the writing in magnetic random access memory (MRAM) [1]. The voltage-induced anisotropy change triggers the magnetization precession, and a stable magnetization switching is confirmed by the application of appropriate voltage pulses [1]. It is also reported that the stable switching is significantly influenced by the relaxation time of the magnetization precession [2], and thus the magnetization dynamics under the application of the electric field is an important topic to be understood [2]. In this study, we report the electric-field modulation of perpendicular magnetic anisotropy (PMA) and the magnetization dynamics in MgO/Co/Pt trilayers by using time-resolved magneto-optical Kerr effect (TRMOKE). ITO(20) /MgO(10)/Co(1.2)/Pt(1.6)/Ta(10)/oxidized Si substrate (thickness in nanometer) was prepared by using a magnetron spattering system. The film was microfabricated into a Hall-bar structure and a gate electrode of ITO(15 nm) was deposited on the junction. Gate voltage was applied from -3 V to 3 V corresponding to the electric field from -0.3 V/nm to 0.3 V/nm. To investigate PMA, we measured anomalous Hall effect (AHE). The magnetization dynamics was measured by TRMOKE using high-power fiber laser with λ=1040 nm, a pulse width of 500 fs, and a repetition frequency of 100 kHz. During the measurement, an external magnetic field Hext in the range of 5-14 kOe was applied in the direction of 73 deg from the film normal. The measured precession was fitted with the damped oscillation function, exp(-t/τ)sin(ωt), where τ and ω denote relaxation time and angular frequency of the precession. By fitting the Hext dependence ofω, we estimated effective magnetic anisotropy field Hk and g-factor. Figure 1 shows the electric field E dependence of Hk estimated from AHE and TRMOKE. Hk decreases linearly with increasing the electric field in both measurements. The variation of PMA with the electric field η obtained from AHE and TRMOKE were -32 fJ/Vm and -47 fJ/Vm, respectively. Figure 2 (a) shows the electric field dependence of the inverse of relaxation time 1/τ obtained at Hext=14 kOe. 1/τ is known to consist of four contributions; effective Gilbert damping αeff, anisotropy axis distribution ΔθH, anisotropy distribution ΔHk and two-magnon scattering (TMS). The four contributions depend on Hext and E, since Hk depends on E, and we fitted Hext and E dependences of 1/τ by assuming material-dependent parameters which are independent of Hext and E. The extracted four contributions at Hext=14 kOe are also shown in Fig. 2 (a). The results indicate a large electric-field modification of 1/τ is caused by the two contributions: effective Gilbert damping and ΔθH, which have the same trend as the electric-field modification of 1/τ. Figure 2 (b) shows the electric field dependence of αeff. αeff is found to decrease with increasing electric field with a ratio of -13 % per 1 V/nm. Okada et al. reported a ferromagnetic resonance (FMR) study on MgO/CoFeB/Ta trilayers, and they showed PMA variation of η=-60 fJ/Vm and α modulation of -21 % per 1 V/nm [3]. These results are similar to our results, however their samples have MgO(001)/bcc-CoFe interface. Our study indicates similar PMA and α modulations are found in MgO/Co/Pt trilayers having MgO(111)/fcc-Co interface.

**References:**

[1] Y. Shiota, et al., Nat. Mater., 11, 39 (2011). [2] R. Matsumoto, et al., Appl. Phys. Express 12, 053003 (2019) [3] A. Okada, et al., Appl. Phys. Lett. 105, 052415 (2014).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/121663ae9205bffe840fa6e0e620c23c.jpg)
Fig.1 Electric field dependence of effective magnetic anisotropy field Hk of the MgO/Co/Pt trilayers.

![](https://assets.underline.io/uploads/markdown_image/1/image/03117393df9d4f663c79f64c0cda057f.jpg)
Fig.2 (a)Electric field dependence of the inverse of relaxation time 1/τ at an applied magnetic field Hext=14 kOe. Four contributions to 1/τ: effective Gilbert damping αeff, anisotropy axis distribution ΔθH, anisotropy distribution ΔHk and two-magnon scattering, are extracted by the fitting of Hext and E dependences of 1/τ, and are plotted in the same figure. (b)Electric field dependence of the effective damping constant αeff of the MgO/Co/Pt trilayers.

 Electric-field modulation of perpendicular magnetic anisotropy and damping constant in MgO/Co/Pt trilayers

Thin films with perpendicular magnetic anisotropy (PMA) have been widely studied for perpendicular recording media since the middle of the 1970s when Prof. Iwasaki suggested this approach [1]. Since them, a great effort has been done by the research community for the use of this kind of nanostructures with PMA in other technological applications such as in high-density spin-transfer torque magnetic random access memories (STT-MRAM) [2], skyrmion-based devices [3] or even in synthetic antiferromagnets for biomedical applications [4]. PMA in thin films can be usually achieved through different strategies such as by using materials with large magnetocrystalline anisotropy energy [5] as well as multilayer thin films with high surface anisotropy energy contributions [6]. Regarding the last alternative, Ikeda et al. [2] demonstrated in 2010 that amorphous CoFeB ultrathin films can show PMA and it is supported by the CoFeB/MgO interfacial anisotropy contribution. Since then, the development of non-magnetic/ferromagnetic superlattices with PMA, using CoFeB alloys as the ferromagnetic element, has received great attention. In particular, PMA in CoFeB/Pd was first demonstrated in bilayers by Fowley et al. [7] as well as in multilayers by Jung et al. [8]. Literature has reported that the effective anisotropy energy depends on the thickness of magnetic (tCoFeB) and non-magnetic (tPd) layers as well as on the number of bilayers (N). In this work, the dynamical response of [CoFeB (tCoFeB) / Pd (10 Å)]5 multilayered ultrathin films (1 Å ≤ tCoFeB ≤ 5 Å) were studied by using two complementary methods: broadband ferromagnetic resonance (Figure 1 a) and time-resolved magneto-optical Kerr effect (Figure 1 b). The perpendicular magnetization was confirmed for multilayers with tCoFeB ≤ 4 Å, while magnetization became in-plane oriented for tCoFeB ≥ 5 Å. This behaviour was explained by considering competing contributions from surface and magnetoelastic anisotropies. In addition, we have observed that the effective damping parameter is reduced with increasing tCoFeB (Figure 1 c), and the lower parameter, αeff ≈ (0.019 ± 0.001), was determined for the multilayer with 4 Å CoFeB thickness. We have suggested that for tCoFeB ≥ 4 Å the layer is continuous and the main contribution to effective damping is coming from spin-pumping when for lower thicknesses αeff is dominated by two-magnon scattering. Afterwards, we have studied the magnetic behaviour of ultrathin [CoFeB(tCoFeB)/Pd(10 Å)]N films. Intending to ensure that the system shows PMA, tCoFeB (3 or 4 Å) was fixed, while the number of bilayers (N) was ranged from 3 to 15. We observed that our samples show PMA and it is generally improved with N. In addition, we studied the dependence of αeff with the number of CoFeB/Pd repeats (N). Up today, literature in superlattices with PMA reported that αeff is proportional, inversely proportional or even, independent of N [9]. Therefore, the reported relationship between the damping parameter and N in superlattices with PMA is still unclear. We show that the evolution of αeff with N shows a complex behaviour that we have correlated with the magnetization process: larger αeff for samples with square hysteresis loops while αeff is independent of N for samples with bow-tie-shaped hysteresis loops.

**References:**

1. S. Iwasaki, Y. Nakamura and K. Ouchi, IEEE Trans. Magn., MAG-61, 1456 (1979). 2. S. Ikeda et al., Nat. Mater, 9, 721 (2010). 3. C. Moreau-Luchaire et al., Nature Nanotechnology, 11, 444 (2016). 4. R. Mansell et al., Sci. Rep,. 7, 4257 (2017). 5. N. Inaba , Y. Uesaka, and M. Futamoto , IEEE Trans. Magn., 36, 54 (2000). 6. C. J. Tatnall et al., J. Magn. Magn. Mater., 165, 391 (1997). 7. C. Fowley et al., IEEE Trans. Magn., 46, 2116 (2010). 8. J. H. Jung et al., Appl. Phys. Express, 3, 023001 (2010). 9. A. Barman et al., J. Appl. Phys., 101, 09D102 (2007); S. Mizukami et al., Appl. Phys. Express, 4, 013005 (2011); H.-S. Song et al., Appl. Phys. Lett., 102, 102401 (2013).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/dc5eba45a2ba94ed3193f118e071a51e.jpg)
Fig. 1. (a) VNA-FMR spectra of the [CoFeB (4 Å) / Pd (10 Å)]5 multilayer thin film with the external magnetic field applied perpendicularly to the sample plane. (b) Time-resolved magneto-optical Kerr effect (TR-MOKE) signals of a [CoFeB (2 Å) / Pd (10 Å)]5 multilayer thin film under different external applied fields (H = 5650, 5060, 4695, 4310 Oe and 3900 Oe) and when H was applied at θH = 78°. Theoretical curves (solid curves) are fits to the experimental data (open symbols). Data was moved along the vertical axis for clarity. (c) Evolution of the effective damping (αeff) vs tCoFeB and determined from the TR-MOKE measurements (Δ), the perpendicular VNA-FMR (●) and the in-plane VNA-FMR (○) measurements.

 Magnetization Dynamics of ultrathin [CoFeB (tCoFeB) / Pd]N films with perpendicular magnetic anisotropy

Tailoring and control of electromagnetic properties of synthetic magnetic structures is advantageous for high frequency applications, such as shielding materials in magnetic recording, where characterisation in the gigahertz frequency range is necessary to determine parameters for simulations and aid component design. In such applications it is desirable to have a high saturation magnetization, low coercivity, high permeability and close to zero magnetostriction. Meeting almost all of these requirements, NiFe is the material of choice for many high frequency applications. We present a systematic study of the effects of process settings, dopants and composition on the ferromagnetic resonance (FMR) and spin wave properties of NixFe100-x alloy thin films fabricated by magnetron sputtering and investigated using a coplanar waveguide, vector network analyser (VNA) [1], vibrating sample magnetometer (VSM) and x-ray diffraction (XRD), with the aim to determine any relation between microstructure and the static and dynamic magnetic properties. It has been found that changes in the NiFe crystal structure and deposition can have significant impact on the magnetic properties, examples of which can be seen in Fig. 1 and 2. This work extends into compositionally modulated superlattice structures to explore further possibilities with respect to material design and optimization [2]. VAMPIRE [3] atomistic simulations have been employed in conjunction with these experiments and to provide parameters for FMR simulations using the micromagnetic model, OOMMF [4].

**References:**

[1] Y. Ding et al. J. Appl. Phys. 96. (2004) [2] C.B. Hill et al. Meas. Sci. Technol. 24. (2013) [3] R.F.L. Evans et al. J. Phys.: Condens. Matter 26, 103202 (2014) [4] M. J. Donahue and D. G. Porter, “OOMMF User’s Guide, Version 1.0,” NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (Sept 1999).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ca3c03d9e435df9d90ed1b8a3bcadece.jpg)
Fig. 1 : FMR spectra at an applied field of 253Oe showing the broadening of the FMR and therefore increased damping with increasing Fe content in NixFe100-x thin films. Note particularly the most significant changes occur at x=40 which marks the fcc/bcc transition.

![](https://assets.underline.io/uploads/markdown_image/1/image/f564dafe40ca24a0b86a77ae748d9a43.jpg)
Fig. 2 : FMR spectra at an applied field of 253Oe illustrating the effect of increasing substrate temperature during deposition on Ni80Fe20, where a supression of the FMR response is seen at 300°C and above.

 Control of Static and Dynamic Magnetic Properties of NixFe100-x Alloy Thin Films

Unconventional computing utilizing the probabilistic nature of physical systems – probabilistic computing – has recently gathered much attention for its potential to effectively address computationally complex problems [1]. A stochastic magnetic tunnel junction (MTJ) is a promising element to construct a probabilistic bit (p-bit), a fundamental building block of the probabilistic computers, owing to fast fluctuation rate and high controllability as well as the manufacturability based on current spin-transfer torque (STT) magnetoresistive random access memory (MRAM) technology. Recently, a proof of concept of the probabilistic computer using stochastic MTJs was presented, where integer factorization was demonstrated as an illustrative example of a combinatorial optimization problem [2]. Performance of stochastic MTJs is characterized by a time-varying fluctuation of magnetization and time-averaged response to the external field and current, and in this regard, understanding the physical mechanism governing these properties is of high importance. In this research, we investigate the mechanism governing the sigmoid-like time-averaged response to external field and current in stochastic MTJs with various sizes. We use the same stack structure used in the demonstration of integer factorization [2], i.e., Ta(5)/ Pt(5)/ syntheic ferrimagnet/ Ta(0.3)/ Co19Fe56B25(1)/ MgO(1.1)/ Co19Fe56B25(t)/ Ta(5)/ Ru(5) (thickness in nm), which is deposited by dc/rf magnetron sputtering on a thermally oxidized Si substrate. Both top and bottom CoFeB layers have perpendicular easy axis. We design the thickness of the free layer t so that the thermal stability factor Δ0 is low enough to show superparamagnetic behavior. All films are processed into MTJs with various junction diameters D by electron beam lithography and Ar ion milling. We determine D of each device from their electrically measured resistance R, with a resistance area product RA determined from several MTJs with D ~ 80 nm. Typical TMR ratio is 90% and RA = 6.3 Ωμm2. The measurement circuit for the time-averaged resistance 〈R〉 is shown in Fig. 1(a). We monitor the magnetization fluctuation of the MTJ by measuring R vs. time under dc bias current I with various magnitudes using an oscilloscope connected in parallel to the MTJ. The relaxation time τ of the studied MTJ is found to range from 1 ms to 1 s, corresponding to Δ0 of 14 to 21, by assuming the Néel-Arrhenius law, τ = τ0expΔ0 with τ0 being attempt time 1 ns [3]. The 〈R〉 against external perpendicular magnetic field H and I for the MTJ with nominal D = 50 nm are shown in Figs. 1(b) and (c), respectively. Clear sigmoidal-like characteristics with respect to H and I are observed as a result of the change of the ratio of τ in parallel (P) and anti-parallel (AP) states τP/τAP. We normalize 〈R〉 using resistances at P and AP states as 〈r〉 ≡ τP/(τP + τAP). The normalized sigmoid curves 〈r〉 against H and I are shown in Figs. 1(d) and (e), respectively. In order to characterize the external field and current dependences of stochasticity of MTJs, we derive an analytical expression of the slope of the sigmoidal curve around the center (〈r〉 = 1/2) [see Figs. 1(d) and (e)]. We define the shift current IS and the shift field HS as I and H to take 〈r〉 = 1/2 (P state probability of 50%), respectively. When MTJs are in P (AP) state, τ is given by τP(AP) = τ0 exp{Δ0[1 ± (I - IS)/IC0]nI}, where nI is a switching exponent for I and IC0 the intrinsic critical current. Here, we assume nI = 2, which was derived by analyzing the Fokker-Planck equation [4]. Then, the partial derivative of 〈r〉 (= τP/(τP + τAP)) with respect to I at I = IS is derived to be ∂〈r〉/∂I|I=Is = Δ0/IC0, which has been used as a figure of merit of STT switching in nonvolatile MTJs [5-7]. We apply the same procedure for ∂〈r〉/∂H and obtain ∂〈r〉/∂H|H=Hs = Δ0/HKeff with HKeff being the perpendicular effective anisotropy field. Experimentally measured ∂〈r〉/μ0∂H|H=Hs and ∂〈r〉/∂I|I=Is for MTJs with different t are plotted as a function of D in Figs. 2(a) and (b), respectively. The H derivative of sigmoid curve ∂〈r〉/∂H|H=Hs monotonically increases with increasing D. Since the analytical expression of the H derivative, Δ0/HKeff, can be converted to MSAefft/2kBT (Aeff denotes an effective area for thermal activation), where the spontaneous magnetization MS, thermal energy kBT, and thickness t are independent of D, this result indicates that Aeff increases with the device size; in other words, relatively coherent magnetization switching takes place in this D regime. On the other hand, the I derivative, ∂〈r〉/∂I|I=Is, decreases with increasing D. The reason for this behavior is not clear but we note that a similar result is also observed in the size dependence of Δ0/IC0 in nonvolatile MTJs [5-7]. In conclusion, this work has studied the physical mechanism governing the sigmoid-like response of stochastic MTJs, which strongly affects the performance of probabilistic computing. We derived an analytical expression of the slope of sigmoid curve and experimentally show that the slope varies with the device size. This work is supported in part by JST-CREST JP1082016.

**References:**

[1] P. R. Feynman, Int. J. Theor. Phys. 21, 467 (1982). [2] W. A. Borders et al, Nature 573, 390 (2019). [3] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963). [4] T. Taniguchi et al., Phys. Rev. B 83, 054432 (2011). [5] J. Z. Sun et al, Phys. Rev. B 84, 064413 (2011). [6] H. Sato et al, Appl. Phys. Lett. 105, 062403 (2014). [7] B. Jinnai et al., IEEE IEDM 24.6 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/dbe0f712d916203b266d2d43fc513b3a.jpg)
Fig. 1 (a) Measurement circuit and (b),(c) ((d),(e)) time averaged resistance 〈R〉 (sigmoid curve 〈r〉) as a function of external perpendicular magnetic field µ0H and bias current I, respectively (μ0 denotes a permeability in vacuum).

![](https://assets.underline.io/uploads/markdown_image/1/image/f482273c059d8cc39adbc62d2afcd307.jpg)
Fig. 2 (a)(b) Slopes of sigmoid curve 〈r〉 against μ0H and I, respectively, as a function of junction diameter D measured for the devices with different CoFeB free layer thicknesses t.

 Physical Mechanism Governing Sigmoid Curves of Stochastic Magnetic Tunnel Junctions

The magnetization reversal of a ferromagnetic layer using a single ultrashort optical pulse in a GdFeCo/Cu/[Co/Pt] spin-valve structure was first demonstrated in 2018 [1]. This work has attracted attention for future ultrafast and energy-efficient magnetic storage or memory devices. However, the mechanism and the role of the magnetic properties of the ferromagnet as well as the time scale of the magnetization switching are not fully understood. The single GdFeCo layer demonstrate all-optical helicity-independent switching (AO-HIS). In that case, the magnetization of this Rare-earth – Transition Metal alloy can be switched with one single femto-second pulse whatever light helicity is used. AO_HIS was attributed to the presence of two exchange coupled magnetic sublattices (the Rare Earth and the Transition Metal one) showing different relaxation times after the ultrafast laser excitation [2] in the GdFeCo/Cu/[Co/Pt] spin-valve studied in ref [1]. the switching of the [Co/Pt] ferromagnetic layer is observed after a single femto-second laser pulse. It has been carefully checked that no exchange coupling between the two magnetic layers exist. The [Co/Pt] switching would then result from the combined effect of the spin current generated by the AO-HIS of the GdFeCo layer [3] and the laser-induced heating of the ferromagnet. More recently single-shot all-optical magnetization switching was investigated in a GdFeCo/Cu/[CoxNi1-x/Pt] spin-valve structure. It was demonstrated that the threshold fluence for switching both the GdFeCo and the ferromagnetic layer depends on the laser pulse duration and the thickness and the Curie temperature of the ferromagnetic layer. We were able to explain most of the experimental results using a phenomenological model. This work provides a way to engineer ferromagnetic materials for energy efficient single-shot all-optical magnetization switching [4,5].

**References:**

[1] S. Iihama, Y. Xu,M. Deb, G. Malinowski, M. Hehn, J. Gorchon, E.E. Fullerton, S. Mangin, Single-Shot Multi-Level All-Optical Magnetization Switching Mediated by Spin Transport. Adv. Mater. 2018, 30 (51), 1804004. https://doi.org/10.1002/adma.201804004. [2] Radu, I.; Vahaplar, K.; Stamm, C.; Kachel, T.; Pontius, N.; Dürr, H. A.; Ostler, T. A.; Barker, J.; Evans, R. F. L.; Chantrell, R. W.; Tsukamoto, A.; Itoh, A.; Kirilyuk, A.; Rasing, T.; Kimel, A. V. Transient Ferromagnetic-like State Mediating Ultrafast Reversal of Antiferromagnetically Coupled Spins. Nature 2011, 472 (7342), 205–208. https://doi.org/10.1038/nature09901. [3] Choi, G.-M.; Min, B.-C. Laser-Driven Spin Generation in the Conduction Bands of Ferrimagnetic Metals. Phys. Rev. B2018, 97 (1), 014410. https://doi.org/10.1103/PhysRevB.97.014410. [4] Q. Remy, J. Igarashi, S. Iihama, G. Malinowski, M. Hehn, J. Gorchon, J. Hohlfeld, S. Fukami, H. Ohno and S. Mangin Energy Efficient Control of Ultrafast Spin Polarized Current to Induce Single Femtosecond Pulse Switching of a Ferromagnet Adv. Sci. 2001996 (2020) [5] J. Iigarasi, Q. Remy, S. Iihama, G. Malinowski, M. Hehn, J. Gorchon , J. Hohlfeld, S. Fukami, H. Ohno, S. Mangin, Engineering single-shot all-optical switching of ferromagnetic materials Nano Lett 20, 12, 8654–8660 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9fea65b7d901ecef44256e9537a2f645.jpg)
By combining a single femtosecond laser pulse with an ultra-short spin current, generated by the demagnetization of a ferrimagnetic GdFeCo layer, it is possible to reverse the magnetization of a ferromagnetic Co / Pt multilayer. The energy required for this reversal can be controlled by modifying the properties of the different layers

 Engineering Spintronic devices to observe ferromagnetic layer switching induced by a single- femto-second current pulse INVITED

Ferrimagnetic materials (FiM) represent a promising direction for the realization of spin-based devices since they can combine the THz ultrafast dynamics typical of antiferromagnets (AFMs) with an easier way to control the magnetic state via well-established optical and electrical methods already applied to FMs, such as the magneto-optical Kerr effect (MOKE)1 or anomalous Hall effect (AHE)2. FiMs can be modeled as AFMs by considering two antiferromagnetically-coupled sublattices3-5 but, unlike AFMs, the two magnetic sublattices can have a different magnetic moment and angular momentum. Consequently, the net magnetization and angular momentum of FiMs can be varied by changing either the FiM’s chemical composition6 or temperature7 leading to two different compensation points: the magnetic compensation point (MCP), i.e. a zero net magnetization state, and an angular momentum compensation point (AMCP), i.e., a vanishing net angular momentum compensation state. In this work, we micromagnetically analyze, by means of Petaspin micromagnetic solver5, the magnetization dynamics of a current-driving transition metal/rare earth ferrimagnet in a spin Hall geometry as a function of the uncompensation parameter of the angular moments of the two sublattices8. This approach allows us to show the crucial effect of non-uniform dynamics in the system which cannot be caught by the macrospin approximation. Particularly, we have reported the switching time as a function of the uncompensation parameter (FIG.1) for a current |J| = 1.3 TA/m2. We have shown the switching time diverges at the AMCP, where the only dynamical solution is a self-oscillation state. More interestingly, a finite discontinuity of the switching time appears near the MCP. We show that this discontinuity is related to a change of the switching mechanism, from uniform to non-uniform dynamics with a corresponding increase of the switching time in the latter case. We have additionally analyzed the effect of the interfacial Dzyaloshinskii-Moriya interaction (IDMI) on the switching time and oscillations amplitude. These are relevant characteristics for the design of magnetic memories (switching time) and spin torque nano-oscillators (oscillation amplitude) since they will determine their performances. In FIG.1, we can see the effect on the switching time of the IDMI and how it is beneficial for the use of FiMs as storage devices since it promotes shorter switching times as it advantageously assists the nucleation of new domains. In this work, we also study the self-oscillation frequency and amplitude as a function of the current density. We show that the current thresholds between the changes of dynamical state are well described by the analytical macrospin formula8. We observe a linear dependence of the frequency on current independently of the IDMI and compensation point in agreement with the analytical formula8. For example, by considering the MCP case (FIG.2), the amplitude of the oscillations decreases non-monotonically as the current increases. For the self-oscillation case, we observe that the effect of IDMI is detrimental for the applicability of FiM as THz sources because it induces a significant reduction of the self-oscillations amplitude at low currents (see Fig. 2). Besides, as we can see in Fig.2, the peak-to-peak amplitude of the oscillations non-monotonically decreases as the current increases. This is due to the non-uniform oscillation mediated by domain walls. For low currents, the domain wall periodicity is larger than the device size and, hence, the whole device oscillates with roughly the same phase As we increase the current, the periodicity decreases thus averaging the out-of-plane component of the Néel vector over the sample size reduces the value of the peak-to-peak amplitude. Consequently, we obtain the minimum amplitude when the periodicity is close to the device size. By further increasing the current, the periodicity becomes smaller and the average magnetization value increases Our results give a better insight into the dynamics excited in FiMs thus paving the way for the design of more performant ferrimagnetic memory and THz nano-oscillator devices.

**References:**

[1] K. Fleischer, N. Thiyagarajah, Y.-C. Lau, et. al., Phys. Rev. B, Vol. 98, 134445 (2018). [2] K. Ueda, M. Mann, C.-F. Pai, A.-J. Tan, and G.S.D. Beach, Appl. Phys. Lett., Vol. 109, 232403 (2016). [3] V. Puliafito, R. Khymyn, M. Carpentieri, et. al., Phys. Rev. B, Vol. 99, 024405 (2019). [4] L. Sánchez-Tejerina, V. Puliafito, P. Khalili Amiri, et. al., Phys. Rev. B, Vol. 101, 14433 (2020). [5] L. Sánchez-Tejerina, R. Tomasello, V. Puliafito, et. al., IEEE Magn. Lett. , Vol. 11, 1 (2020). [6] S. Woo, K.M. Song, X. Zhang, et. al., Nat. Commun. , Vol. 9, 959 (2018). [7] L. Caretta, M. Mann, F. Büttner, et. al., Nat. Nanotechnol. , Vol. 13, 1154 (2018). [8] I. Lisenkov, R. Khymyn, J. Akerman, et. al., Phys. Rev. B, Vol. 100, 100409 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e1f65e0922be4090a862a18c147e0267.jpg)
Fig. 1 Switching time as a function of the effective angular momentum uncompensation factor (top axis) of the two sublattices for a current density of |J| = 1.3 TA/m2. The black line corresponds to a dynamics without DMI, while the red line corresponds to a system with non-vanishing DMI. Orange line stands for simulations without demagnetizing field neither DMI. In blue the values of switching time are highlighted for MCP, in magenta there is the line that refers to AMCP.

![](https://assets.underline.io/uploads/markdown_image/1/image/8bed697f0af4114545008dbfb8097492.jpg)
Fig. 2 Amplitude of self-oscillations at the MCP vs applied current for vanishing and non-vanishing IDMI. The snapshots represent the spatial distribution of the 1st-sublattice magnetization at different current densities as indicated by the arrows and arbitrary times. The colors refer to the z-component of the magnetization.

 Micromagnetic understanding of switching and self-oscillations in ferrimagnetic materials

In the field of magnetism and spintronics chiral magnetic structures, such as spin spirals, chiral domain walls and skyrmions [1-2], are intensively investigated due to their fascinating properties such as potentially enhanced stability and efficient spin-orbit torque driven dynamics [3-4]. It has been shown that these structures are stabilized by the Dzyaloshinskii-Moriya interaction (DMI) that favours a chiral winding of the magnetisation. While the investigation of static structures and slow (ns) dynamics of chiral magnetic structures has been intensified recently, experimental studies addressing the ultimate fs-ps dynamics of the chirality have been elusive so far as ultrafast pump-probe experiments have so far concentrated on the collinear order in magnetic systems [5-7]. In this work we employ circularly polarized light pulses from an XUV free-electron laser and investigate time-resolved the evolution of the chiral order of domain walls in magnetic thin film samples by an IR pump - X-ray magnetic scattering (XRMS) probe experiment (Fig. 1a) [8]. Using samples with interfacial DMI and perpendicular magnetic anisotropy exhibiting labyrinth-like domain patterns (Fig. 1b), we measure in the same experiment both the (CL+CR) sum signal (Fig. 1d) corresponding to the ferromagnetic order in the domains (average collinear domain magnetisation mz2+mx2) and the difference signal (CL-CR) (Fig. 1e) corresponding to the average chiral order in the domain walls (chiral order correlator mz mx). In contrast to the (CL+CR) signal, phase information is preserved in the dichroic signal (CL-CR) allowing to determine the domain wall chirality (left/right-handed) and character (Bloch (helical), Néel (cycloidal)) [9-10]. Fig. 1f shows the orthoradial profile of the dichroic signal, which indeed confirms the prevalence of fully right-handed Néel-type domain walls [9-10]. This provides a tool to individually probe the time resolved dynamics of the chiral magnetic order in the domain walls [8]. In a next step we investigated the time evolution of the scattering signals upon IR laser excitation. The key step is the comparison of the collinear ferromagnetic order dynamics and the chiral order dynamics. To ascertain both, we calculated the average intensity for the sum and the difference signal as a function of the wavevector Qr for different time delays. With this the numerical integration of the radial profiles for each time delay leads to the data shown in Fig. 2a, which demonstrates the evolution of the total intensity of the sum and the difference signal as a function of delay time normalized to the unpumped total intensity. The data shown in Fig. 2b is the result of a second experimental run. We find an ultrafast intensity decrease of both signals in the sub-ps regime with similar time constants. However, a significantly faster recovery of the chiral signal in the sub-ns timescale is observed [8]. In order to explain the key finding of the experimentally observed faster recovery of the chiral signal after laser excitation we can envisage two different mechanisms: (1) a change in the size ratio between domain walls and domains caused by an increase of the domain wall width during the whole investigated time frame or (2) a faster recovery dynamics of the chiral order within the domain walls compared to the ferromagnetic order in the domains leading to a faster build-up of the chiral magnetisation [8]. We subsequently investigate the origin of the faster recovery of the chiral signal by performing numerical simulations of the scattering signal, which reproduce the experimental findings. We conclude from these that the main driver behind our experimental findings is likely mechanism (2) leading to a faster recovery of the chiral order in the domain walls in comparison to the ferromagnetic order in the domains. In the future further fundamental aspects will be studied in detail e.g. the dependence of the timescales of the chiral order build-up on the absolute strength of the DMI by varying the heavy metal layers. The better control of the DMI and the chirality of spin structures on the ultrafast timescale can finally allow for the controlled ultrafast manipulation of chiral magnetism, e.g. ultrafast writing of chiral topological objects such as skyrmions and pave the path to applications in the field of ultrafast chiral spintronics.

**References:**

[1] Jiang, W. J. et al. Phys. Rev. 704, 1–49 (2017). [2] Everschor-Sitte, K., Masell, J., Reeve, R. M. & Kläui, M. J. Appl. Phys. 124, 240901 (2018). [3] Fert, A., Cros, V. & Sampaio, J. Nat. Nanotech. 8, 152–156 (2013). [4] Woo, S. et al. Nat. Mater. 15, 501–506 (2016). [5] Koopmans, B. et al. Nat. Mater. 9, 259–265 (2010). [6] Battiato, M., Carva, K., & Oppeneer, P. M. Phys. Rev. Lett. 105, 027203 (2010). [7] Pfau, B. et al. Nat. Commun. 3, 1100 (2012). [8] Kerber, N. et al. Nat. Commun. 11, 6304 (2020). [9] Zhang, S. L. et al. Phys. Rev. B 96, 094401 (2017). [10] Chauleau, J.-Y. et al. Phys. Rev. Lett. 120, 037202 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0a6311ce5b2275e4d9718738f6370d87.jpg)
Figure 1: Experimental setup and diffraction images. (a) Measurement geometry: A magnetic thin film sample is pumped by an optical infrared laser pulse and probed by a circularly polarized X-ray FEL pulse. Afterwards an IR-protected CCD detector records the magnetic SAXS pattern. (b) MFM image of a typical labyrinth domain pattern of the [Ta(5.3 nm)/Co20Fe60B20(0.93 nm)/Ta(0.08 nm)/MgO(2.0 nm)]x20/Ta(1.6 nm) sample. (d) The resulting sum = CL+CR of the diffraction pattern confirms that the diffraction corresponds to the magnetic domains observed by MFM. (e) The dichroic scattering signal = CL-CR and its azimuthal dependence (f) confirms the presence of (c) right-handed chiral Néel (cycloidal) domain walls [8].

![](https://assets.underline.io/uploads/markdown_image/1/image/6eaf4721a65ec4df5813cb74ec270cf9.jpg)
Figure 2: Time dependence of total scattering intensities and second moments. Time evolution of the (a,b) radially integrated intensity from two different experimental runs and (c,d) the 2nd moments of the sum signal (ferromagnetic order) and the difference signal (chiral magnetic order) from the scan shown in (a). The solid lines in (c,d) provide a guide to the eye [8].

 Faster chiral versus collinear magnetic order recovery after optical excitation revealed by femtosecond XUV scattering INVITED

Ultrafast optical generation of chiral magnetic structure is one of the most promising and highly sought-after technology for next generation magnetic memory devices. Recent experimental demonstration of generation of skyrmions with ultrafast laser pulse [1] has revealed a new horizon in this field. However, a proper theoretical understanding of the underlying interactions taking place at different timescales as well as main stimulants of the process are still in darkness. Most of the existing studies focus on the magnetic interactions only and completely neglect the interaction between the electric field of the laser and the electronic wavefunction of the material and thus create a huge gap in the complete understanding of the laser assisted magnetization dynamics. We bridge this gap by employing a quantum-classical hybrid method [2] to study the evolution of quantum states as well as the magnetic moments on equal footing and demonstrate that how a stable spin spiral can be generated from a collinear anti-ferromagnetic spin chain with an ultrashort laser pulse [3] (Fig.1,2). We use a minimal tight binding model to define our antiferromagnetic chain with a scalar hopping and onsite magnetic exchange. The laser electric field is modelled as a Gaussian pulse which couples to the hopping. On being hit by the pulse several different mechanisms take place at different timescale starting with an instantaneous change in occupation of states resulting in a gradual buildup of effective torque which leads to the magnetization dynamics. The magnetization dynamics being a slower process compared to the quantum evolution lags by approximately 50fs behind the incident of laser. To initiate this process the initial state must deviate from an ideal collinear state. A stronger deviation shows a faster convergence to the steady state configuration. The mixing of quantum states gives rise to different emergent spin mixing interactions which drives the subsequent dynamics. The system finds its chiral state within a timescale of 1ps. The timeframe of the different steps of the spiralization process is of the same order of the timescales associated with the ultrafast demagnetization which indicates towards the similar underlying physics. After 1ps the dynamics becomes slower and the chain gradually attains a uniform steady chiral structure. We characterize this evolution in terms of the relative angle between adjacent magnetic moment and vector chirality which clearly distinguish different time frames. The relative angle and the induced chirality can be tuned by the strength of the amplitude field. We further show that these effects are fairly robust against thermal fluctuations. Our results thus reveal several salient features of optical manipulation of chirality which has not been revealed in existing literature and thus would be instrumental in their experimental realization.

**References:**

[1] F. Büttner et.al. Nat. Matter. 20, 30 (2021). [2] A. Ono and S. Ishihara, Phys. Rev. Lett. 119, 207202 (2017). M. D. Petrović, B. S. Popescu, U. Bajpai, P. Plecháč, and B. K. Nikolić, Phys. Rev. Applied 10, 054038 (2018). [3] S. Ghosh, F. Freimuth, O. Gomonay, S. Blügel and Y. Mokrousov, arXiv:2011.01670

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/87c5fef65fb042eaab838c75b45a688d.jpg)
Fig. 1: Formation of a spin spiral (solid arrows) from an collinear antiferromagnetic (transparent arrows) chain with 32 sites.

![](https://assets.underline.io/uploads/markdown_image/1/image/ed03b2ecc70a85f0fdc71272a0a68860.jpg)
Fig. 2: Time evolution of the components of magnetization over time for 1st (solid) and 17th (transparent) magnetic moment. The yellow line shows the laser pulse.

 Ultrafast electronic manipulation of antiferromagnetic spin spiral states

The observation of ultrafast and coherent control of spins in magnetic materials at room temperature [1-2] has prompted the experimental and theoretical efforts to understand the underlying mechanisms driving such magnetic phenomena. In order to understand the driving mechanisms, in this non-equilibrium regime, it is necessary to disentangle the relevant interactions (exchange, spin-lattice, electron-phonon, coulomb etc.). Stoichiometric B2 ordered epitaxial (001) FeRh undergoes a first order magnetic phase transition from antiferromagnetic (AFM) to ferromagnetic (FM) at ≈ 380K. In the AFM phase of FeRh, only Fe has net magnetic moment, whereas in the FM phase both Fe and Rh carry net moments. The phase transition is also accompanied by a ≈ 1% isotropic lattice expansion in the bulk BCC structure, and changes in electronic structure, see [3-5] and refs. therein. Since magnetic and structural dynamics can occur at different timescales, it makes FeRh an ideal candidate for disentangling relevant interaction mechanisms at different timescales. The phase transition has been extensively studied theoretically and experimentally [6-10], in thermal equilibrium (static heating) and at ultrafast time scales (fs- optical excitation), but a precise knowledge of the roles of the electronic, phononic and spin sub-systems remains elusive. We have studied the laser-driven AFM to FM phase transition in FeRh with X-ray absorption spectroscopy (tr-XAS) around the Fe L3 edge. The experiments were performed at the SCS Instrument of the European XFEL. We also performed the density functional theory (DFT) calculations to simulate the XAS spectra at different time delays including magnetic, lattice and temperature changes. Temperature modelling was done using two temperature model (2TM). In this contribution, we will discuss the changes in the electronic band structure of FeRh in the femto- and picosecond timescales, derived from the X-ray absorption spectroscopy measurements at the Fe L3 absorption and compare them with the density functional theory simulations.

**References:**

[1] E. Beaurepaire et al., Phys. Rev. Lett. 76, 4250 (1996) [2] J.-Y. Bigot et al., Nature Physics 5, 515 (2009) [3] J.-U. Thiele et al. Appl. Phys. Lett. 82, 2859 (2003) [4] A.X. Gray et al., Phys. Rev. Lett. 108, 257208 (2012). [5] C. Baldasseroni et al., Appl. Phys. Lett. 100, 262401 (2012). [6] J.-U. Thiele, M. Buess, C.H. Back, Appl. Phys. Lett 85, 2857 (2004). [7] G. Ju et al., Phys. Rev. Lett. 93, 197403 (2004) [8] I. Radu et al., PRB 81, 104415 (2010) [9] S. Mariager et al., Phys. Rev. Lett. 108, 87201 (2012) [10] F. Quirin et al., Phys. Rev. B 85, 020103(R) (2012)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/92b1687e04e78b6abfbee5f8574ccb81.jpg)
Figure-1: a: X-ray absorption (XAS) changes: (dashed line: experiment) near the Fe L3 edge before and after laser excitation at a fixed time delay of 70 fs, (solid lines: DFT theory) XAS changes corresponding to temperature of 2000K 
 
b: a: X-ray absorption (XAS) changes: (dashed line: experiment) near the Fe L3 edge before and after laser excitation at a fixed time delay of 11.5 ps and 70 us, (solid lines: DFT theory) XAS changes corresponding to net changes including temperature (400K), lattice (0.5 % expansion) and magnetic changes (AFM->FM)

 Ultrafast dynamics in electronic band structure of optically excited epitaxial FeRh: X-ray absorption spectroscopy and density functional theory approach

Recently, the effects of inertia in the spin dynamics of ferromagnets were reported to cause nutation resonance [1-8] at the frequencies higher than the conventional ferromagnetic resonance. It was shown that spin-orbit induced inertia is responsible for nutation, which should be considered together with magnetization precession in a magnetic field. Recently nutation in ferromagnets was confirmed experimentally [1], since nutation and precession operate at substantially different time scales, and conventional microwave ferromagnetic resonance (FMR) spectroscopy techniques do not easily reach the high-frequency (sub-Terahertz) regime, required to observe the inertia effect which in addition yields much weaker signal. The interest in the ultrafast spin dynamics has inspired various theoretical studies. For instance, a relation between the Gilbert damping constant and the inertial regime characteristic time was elaborated in ref. [2]. The exchange interaction, damping, and moment of inertia can be calculated from the first principles as shown in [3]. In [4], the inertial regime was introduced in the framework of the mesoscopic nonequilibrium thermodynamics theory, and it was shown to be responsible for the nutation superimposed on the precession of magnetization. Wegrowe and Ciornei [5] discussed the equivalence between the inertia in the dynamics of uniform precession and a spinning top within the framework of the Landau–Lifshitz–Gilbert equation generalized to the inertial regime. This equation was studied analytically and numerically in ref. [6, 7]. In this digest we present new results on the frequency dependence of high-frequency magnetic susceptibility in ferromagnets taking inertia effects into account [8]. We performed calculations based on the Landau-Lifshitz-Gilbert equation with the additional inertia term and obtained an analytical solution. These results provide a basis for further studies of magnetocrystalline anisotropy and damping in magnetic materials in the sub-Terahertz range and furthermore a guideline for experimental research. Using circular variables of magnetization and magnetic field we obtained the dependence of the susceptibility on frequency for the right-hand precession, i.e. positive polarization (Fig. 1(a)). Due to the contribution of nutation, the frequency and linewidth of this resonance are slightly different from the ones of usual FMR. In Fig. 1(b) the nutation resonance possessing negative polarization is presented. Note that the polarizations of ferromagnetic and nutation resonances are reversed. In addition, we derived a general analytical expression for the linewidth of nutation resonance in ferromagnets, and will show its dependence on magnetization, the Gilbert damping, the inertial relaxation time and applied magnetic field. We find that the nutation linewidth can be tuned by the applied magnetic field, and this tunability breaks the direct relation between losses and the linewidth.

**References:**

[1] K. Neeraj, N. Awari, S. Kovalev et al., Nat. Phys. (2020), doi: 10.1038/s41567-020-01040-y [2] M. Fähnle, D. Steiauf, and C. Illg, Phys. Rev. B, Vol. 84, 172403 (2011). [3] S. Bhattacharjee, L. Nordström, and J. Fransson, Phys. Rev. Lett., Vol. 108, 057204 (2012). [4] M.-C. Ciornei, J. M. Rubí, and J.-E. Wegrowe, Phys. Rev. B, Vol. 83, 020410 (2011). [5] J.-E. Wegrowe and M.-C. Ciornei, Am. J. Phys., Vol. 80, p. 607 (2012). [6] E. Olive, Y. Lansac, and J.-E. Wegrowe, Appl. Phys. Lett., Vol. 100, 192407 (2012). [7] E. Olive, Y. Lansac, M. Meyer et al., J. Appl. Phys. 117, 213904 (2015). [8] M. Cherkasskii, M. Farle, and A. Semisalova, Phys. Rev. B 102, 184432 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/06be23fe2d97187eb18d84ec0746a005.jpg)
Fig. 1. (a) The FMR peak with nutation. (b) The nutation resonance. The calculation was performed for γ/(2π)=28 GHz T-1, μ0M0=1 T, μ0H0=100 mT, α=0.0065 and τ = 10-11 s.

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Nutation resonance in ferromagnets

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