# Nonlinear magnetization dynamics driven by strong terahertz fields

###### Abstract

We present a comprehensive experimental and numerical study of magnetization dynamics triggered in a thin metallic film by single-cycle terahertz pulses of MV/m electric field amplitude and ps duration. The experimental dynamics is probed using the femtosecond magneto-optical Kerr effect (MOKE), and it is reproduced numerically using macrospin simulations. The magnetization dynamics can be decomposed in three distinct processes: a coherent precession of the magnetization around the terahertz magnetic field, an ultrafast demagnetization that suddenly changes the anisotropy of the film, and a uniform precession around the equilibrium effective field that is relaxed on the nanosecond time scale, consistent with a Gilbert damping process. Macrospin simulations quantitatively reproduce the observed dynamics, and allow us to predict that novel nonlinear magnetization dynamics regimes can be attained with existing table-top terahertz sources.

###### pacs:

Valid PACS appear here^{†}

^{†}preprint: APS/123-QED

Since Faraday’s original experiment Faraday (1846) and until two decades ago, the interaction between magnetism and light has been mostly considered in a unidirectional way, in which changes to the magnetic properties of a material cause a modification in some macroscopic observable of the electromagnetic radiation, such as polarization, ellipticity or intensity. However, the pioneering experiment of Beaurepaire et al. Beaurepaire et al. (1996), where femtosecond optical pulses were shown to quench the magnetization of a thin-film ferromagnet on the sub-picoseconds time scales, demonstrated that intense laser fields can be conversely used to control magnetic properties, and the field of ultrafast magnetism was born. Large research efforts are nowadays devoted to the attempt to achieve full and deterministic control of magnetism using ultrafast laser pulses Koopmans et al. (2000); van Kampen et al. (2002); Koopmans et al. (2005); Stanciu et al. (2007); Malinowski et al. (2008); Koopmans et al. (2010); Kampfrath et al. (2011); Kubacka et al. (2014), a fundamentally difficult problem that could greatly affect the speed and efficiency of data storage Walowski and Münzenberg (2016).

Recently, it has been shown that not only femtosecond laser pulses, but also intense single-cycle terahertz (THz) pulses Hoffmann et al. (2011) can be used to manipulate the magnetic order at ultrafast time scales in different classes of materials Nakajima et al. (2010); Yamaguchi et al. (2010); Kim et al. (2014); Bonetti et al. (2016); Shalaby et al. (2016). The main peculiarity of this type of radiation, compared with more conventional femtosecond infrared pulses, is that the interaction with the spins occurs not only through the overall energy deposited by the radiation in the electronic system, but also through the Zeeman torque caused by the magnetic field component of the intense THz pulse. This is a more direct and efficient way of controlling the magnetization, and to achieve the fastest possible reversal Tudosa et al. (2004); Gamble et al. (2009). However, an accurate description of the magnetization dynamics triggered by strong THz pulses is still missing.

In this Letter, we present a combined experimental and numerical study of the magnetization dynamics triggered by linearly polarized single-cycle THz pulses with peak electric (magnetic) fields up to 20 MV/m (66 mT). We investigate not only the fast time scales that are comparable to the THz pulse duration ( ps), but also the nanosecond regime, where ferromagnetic resonance oscillations are observed. Moreover, we write an explicit form of the LLG equation suitable to analyze terahertz-driven dynamics, that we use to predict yet-unexplored nonlinear magnetization dynamics regimes uniquely achievable with this type of excitation mechanism.

Macrospin simulations for small demagnetization effects Magnetization dynamics of a uniform ferromagnetic material can be modeled and understood from macrospin simulations where the description of the magnetic state is simplified by a single-domain approximation. The phenomenological Landau-Lifshitz-Gilbert (LLG) equation Landau and Lifshitz (1935); Gilbert (2004) can be used to obtain a first approximate description of the magnetization continuum precession and relaxation (damping). A basic feature of the LLG equation is that it preserves the length of the magnetization vector ( const.) Horley et al. (2011) and hence does not account for laser-induced demagnetization effects. Nevertheless, the importance of optical-induced demagnetization phenomena cannot be neglected and a limitation to constant magnetization in the description of magnetization dynamics ignores several physical phenomena such as spin waves or scattering effects Walowski and Münzenberg (2016). One possibility to account for optically- or thermal-induced demagnetization is using the Landau-Lifshitz-Bloch (LLB) equation Garanin et al. (1990), which has been shown to describe ultrafast demagnetization correctly Atxitia et al. (2017). In contrary to the indirect (thermal) coupling present in visible- and near-infrared light-matter interaction, THz radiation can directly couple to the spin system via magnetic dipole interaction (Zeeman interaction) Hirori and Tanaka (2016). Furthermore, femtosecond optical laser excitation and THz excitation deposit very different energy densities on the sample, i.e. the corresponding energy for THz excitation amounts to 0.01 meV/atom which is 4 orders of magnitude smaller than for optical laser excitation Bonetti et al. (2016). The THz excitation as used in our experimental data is thus nonthermal and the H component of the electromagnetic field is driving the magnetization dynamics.

In a first approximation, we utilize the phenomenological LLG equation with nonconstant magnitude and solve it numerically. The phenomenological LLG equation reads (see Supplementary Material):

with , where the gyromagnetic ratio is 28.025 GHz/T and the Gilbert damping , magnetization ( being the amplitude and the unit vector), saturation magnetization , and effective magnetic field (including the external field , THz field and demagnetization field ).

Here, the first and second terms on the right-hand side of the equation describe coherent spin precession (magnetic torque) and the macroscopic spin relaxation (Gilbert damping), respectively, whereas the third term describes the fast demagnetization along the direction controlled by the function . In the following, we propose a semi-empirical approach to identify and, consequently, the time evolution of magnetization amplitude from experimental demagnetization data. Simulation parameters for our study of the \ceCoFeB magnetization dynamics and experimental data are taken from time-resolved magneto-optical Kerr effect (TR-MOKE) measurements as described in Experimental details as well as from Ref. Bonetti et al. (2016). Throughout all macrospin simulations the Gilbert damping was set to 0.01, which is in accordance with independent measurements of for \ceCoFeB from conventional FMR spectroscopy (not shown here).

The LLG equation shown above is numerically solved in spherical coordinates (see the Supplementary Material for details) using a custom-made Python solver that is utilizing a 4th-order Runge-Kutta method and provides a simple implementation and good accuracy of the numerical solution Horley et al. (2011). In this semi-empirical approach, the demagnetization is modeled by using experimental data, where the change in magnetization modulus as a function of time is calculated from the cumulative integral of the incident THz pulse, and scaled with experimentally obtained demagnetization values (see Ref. Bonetti et al. (2016) for more details) which determines the function . The shape of the incident THz pulse (in the time domain) is experimentally obtained through electro-optical sampling in GaP.

From the described integration, we obtain the value of the total magnetization as a function of time, . Following a fast demagnetization step the magnetization recovers on a timescale of 100 ps, which is characterized by an exponential function. The obtained values of are then used at each time step in the solver in order to compute the dynamics of the magnetization orientation, i.e. , and , in terms of the spherical angles . In a similar way, the THz pulse as function of time is obtained from measurements of . The externally applied magnetic field bias is constant over time and consists of two components a main component, , along the -axis direction and a small component, , in the sample plane parallel to the -axis direction. The anisotropy field is parallel to the -axis, i.e. oop, and applied in opposite direction to the magnetization.

Experimental details Room temperature experimental data is obtained from a time-resolved pump-probe method utilizing the magneto-optical Kerr effect (TR-MOKE), Refs. Freeman and Smyth (1996); Hiebert et al. (1997). A sketch of the experimental setup is presented in Fig. 1 (a). Strong THz radiation is generated via optical rectification of 4 mJ, 800 nm, 100 fs pulses from a 1 kHz regenerative amplifier in a lithium niobate (\ceLiNbO3) crystal, utilizing the tilted-pulse-front method Hoffmann and Fülöp (2011). The orientation of the THz polarization, i.e. the orientation of electric and magnetic field component of the THz pulse, is controlled using a set of two wire grid polarizers, one variably oriented at 45 and a second fixed to 90 (with respect to the original polarization direction of ). The magnetic field component of the THz pulse is fixed along the -axis direction and is therefore flipped by 180 by rotating the first polarizer. An amorphous \ceCoFeB sample (\ceAl2O3 (1.8nm)/\ceCo40Fe40B20 (5nm)/\ceAl2O3 (10nm)/Si substrate) is placed in the gap of a 200 mT electromagnet as well as on top of a 0.5 T permanent magnet. The orientation of the externally applied field is along the -axis direction, i.e. out-of-plane (oop) with respect to the sample surface. The THz pump beam, with a spot size ⌀ mm (FWHM), and the 800 nm probe beam, with a spot size ⌀ m (FWHM), overlap spatially on the sample surface in the center of the electromagnet gap. The FMR response is probed as a function of both the magnetic bias field as well as the THz pump field . Since the incident light orientation is close to the surface normal of the film sample, the magneto-optical Kerr effect (MOKE) signal is proportional to the oop component of the magnetization, i.e. polar MOKE geometry. Time overlap of the pump and probe beams is attained by varying the optical path length of the probe beam. The probe beam reflected from the sample surface is analyzed using a Wollaston prism and two balanced photo-diodes, following an all-optical detection scheme van Kampen et al. (2002).

Results and discussion The experimental data demonstrating THz-induced demagnetization and the magnetic field response of the spin dynamics is shown in Fig. 1 (c-f). For short timescales on the order of the THz pump pulse, 1 ps, the polar MOKE is sensitive to the coherent response of the magnetization, i.e. its precession around the THz magnetic field, shown in Fig. 1 (c)+(e). Within 100 fs after time zero ( 5 ps) a sudden demagnetization step of the order of 0.1-0.2% is observed. The demagnetization step is followed by a ’fast’ relaxation process, 1 ps and subsequent a ’slow’ recovery of the magnetization on a timescale, 100 ps. During and after magnetization recovery, a relaxation precession, i.e. ferromagnetic resonance (FMR), is superimposed, see Fig. 1 (d)+(f). The FMR response is mainly determined by the saturation magnetization , the effective anisotropy field (i.e. demagnetization field ) and the external applied magnetic field . The effect of reversing the externally applied magnetic field, i.e. changing the equilibrium state of the magnetization defined by the balance between external and anisotropy field, on the MOKE measurements is illustrated in Fig. 1 (c)+(d). This data shows that all the different processes just identified (demagnetization, coherent magnetization response ( ps) and FMR response) are indeed magnetic, as they all reverse their sign upon reversal of the sign of the magnetic field. Fig. 1 (e)+(f) instead depict the effect of reversing the THz field polarity on the magneto-optical measurements, while keeping the externally applied field constant. This data illustrate the symmetry of the magnetic response with respect to the terahertz magnetic field. The demagnetization and FMR response, whose amplitude scales quadratically with the terahertz magnetic field, remain unchanged upon reversal of the terahertz magnetic field, while the coherent magnetization response, linear in , reverses its sign.

The dependence of the THz-induced FMR response on the magnetic bias field () is shown in Fig. 2 (a). Oscillation amplitude and resonance frequency are summarized in Fig. 2 (b). The oscillation amplitude is obtained from fitting a damped sinusoidal function to the data shown in Fig. 2 (a) and the resonance frequency follows from subsequent Fourier transformation of that fitting function. The relationship between magnetic resonance field and FMR frequency can be described by the phenomenological Kittel equation Kittel (1948)

(1) |

with gyromagnetic ratio and effective magnetization , where is the perpendicular anisotropy field. From magnetization measurements at 300 K (not shown) the saturation magnetization 1.84 T and an anisotropy field 0.76 T are obtained. The observed experimental period of the ferromagnetic resonance is 360 10 ps (141 5 ps) for external fields 180 mT (450 mT), respectively. The resulting macroscopic spin relaxation time of the FMR is . The FMR oscillation decays on a comparable long timescale, 10 ns.

A detailed picture of the THz-induced demagnetization and FMR response for an external magnetic field from a 0.5 Tesla permanent magnet and a THz pulse of = 18 MV/m ( 60 mT) is presented in Fig. 3. The applied magnetic field can be deduced from modeling of the experimental data with our semi-empirical macrospin simulations aiming for best agreement between data and model. For the the applied magnetic field for the experimental data in Fig. 3, one finds = 448 mT (out-of-plane component) and = 56 mT (in-plane component). A good agreement between experimental data and macrospin simulations is found both on a short timescale after arrival of the pump pulse showing coherent precession and demagnetization, Fig. 3 (a), as well as on long timescales showing magnetization recovery and FMR at a frequency of about 7.2 GHz, Fig. 3 (b). It can be noticed that amplitude of the measured FMR oscillations is slightly recovering between the first and second FMR maximum. Such a behavior can be expected considering the fact that the magnetization is recovering on a similar timescales.

It is interesting to explore the expected response of the magnetization to THz fields with strength larger than the ones considered in this work, but nowadays accessible with table-top sources. For THz fields 100 the demagnetization follows the square of the amplitude of the terahertz field, see Ref. Bonetti et al. (2016). Lacking detailed experimental data, it is reasonable to assume that THz-induced demagnetization for higher THz peak fields ( 100 ) can be described by the positive section of an error function, allowing for a quadratic behavior for small demagnetization and a saturation for large demagnetization approaching 100%. From our experimental data we derive a functional description of the demagnetization as a function of the THz field such as Demag = , with THz peak field and fitting parameter . (See the Supplementary Material for further details.)

With this assumption, the macrospin simulation results for THz fields 20 and 200 are presented in Fig. 4 (a-b) and Fig. 4 (c-d), respectively. For 200 a clear nonlinear response of the magnetization to the THz field is found, illustrated by the phase shift of the FMR oscillation and a second harmonic oscillation in the component of the magnetization. The simulated THz-induced demagnetization for 200 is on the order of 20%. In Fig. 4 (b)+(d) the Fourier spectrum of the FMR oscillation for the and components of the magnetization at THz pump peak fields of 20 MV/m and 200 MV/m are depicted. The Fourier data of the 200 MV/m simulation shown in Fig. 4 (c) clearly shows a second harmonic peak at 14 GHz, present for but not for . A similar behavior, i.e. a phase shift of the FMR response and high harmonic generation was observed recently by performing FMR spectroscopy of thin films irradiated with femtosecond optical pulses inducing either ultrafast demagnetization Capua et al. (2016) or by exciting acoustic waves Chang et al. (2017). In our case, the high-harmonic generation process is solely driven by the large amplitude of the terahertz magnetic fields that are completely off-resonant with the uniform precession mode. This would allow for exploring purely magnetic dynamics in regimes that are not accessible with conventional FMR spectroscopic techniques, where high amplitude dynamics are prevented by the occurrence of so-called Suhl’s instabilities, i.e. non-uniform excitations degenerate in energy with the uniform mode. Such non-resonant, high THz magnetic fields are within the capabilities of recently developed table-top THz sources Shalaby et al. (2018), and can also be generated in the near-field using using metamaterial structures as described by Refs. Polley et al. (2018a, b).

In summary, we investigated magnetization dynamics induced by moderate THz electromagnetic fields in amorphous \ceCoFeB, in particular the ferromagnetic resonance response as a function of applied bias and terahertz magnetic fields. We demonstrate that semi-empirical macrospin simulations, i.e. solving the Landau-Lifshitz-Gilbert equation with a non-constant magnitude of the magnetization vector to incorporate THz-induced demagnetization effect, are able to describe all the details of the experimental results to a good accuracy. Existing models of terahertz spin dynamics and spin pumping would need to be extended to include the evidence presented here Bocklage (2016, 2017). Starting from simulations describing experimental data for THz-induced demagnetization we extrapolate that THz fields one order of magnitude larger drive the magnetization into a nonlinear regime. Macrospin simulations with THz fields on the order 200 , introducing a significant demagnetization of 20%, and a marked nonlinear behavior, apparent from a phase shift of the FMR oscillation and second harmonic generation of the uniform precessional mode. We anticipate that our results will stimulate further theoretical and experimental investigations of nonlinear spin dynamics in the ultrafast regime.

M.H. gratefully acknowledges support from the Swedish Research Council grant E0635001, and the Marie Skłodowska Curie Actions, Cofund, Project INCA 600398s. The work of M.d’A. was carried out within the Program for the Support of Individual Research 2017 by University of Naples Parthenope. M.C.H. is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. 2015-SLAC-100238-Funding. M.P. and S.B. acknowledge support from the European Research Council, Starting Grant 715452 “MAGNETIC-SPEED-LIMIT”.

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