Abstract
Nonlinear magnetization dynamics is essential for the operation of numerous spintronic devices ranging from magnetic memory to spin torque microwave generators. Examples are microwaveassisted switching of magnetic structures and the generation of spin currents at low bias fields by highamplitude ferromagnetic resonance. Here we use Xray magnetic circular dichroism to determine the number density of excited magnons in magnetically soft Ni_{80}Fe_{20} thin films. Our data show that the common model of nonlinear ferromagnetic resonance is not adequate for the description of the nonlinear behaviour in the low magnetic field limit. Here we derive a model of parametric spinwave excitation, which correctly predicts nonlinear threshold amplitudes and decay rates at high and at low magnetic bias fields. In fact, a series of critical spinwave modes with fast oscillations of the amplitude and phase is found, generalizing the theory of parametric spinwave excitation to large modulation amplitudes.
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Introduction
Nonlinear behaviour is observed in a vast range of physical systems. Although in some cases a transition from a wellbehaved and predictable linear system to a nonlinear or even chaotic system is detrimental, nonlinear phenomena are of high interest, owing to their fundamental richness and complexity. In addition, a number of technologically useful processes rely on nonlinear phenomena. Examples are solitonic wave propagation^{1}, high harmonic generation^{2}, rectification and frequency mixing^{3}. In many fields of physics, reaching from phonon dynamics to cosmology, anharmonic terms enrich the physical description but complicate the analysis. Often nonlinear effects can only be accounted for by performing cumbersome numerical threedimensional lattice calculations or the physical description relies on dramatic simplifications. An analytic theory describing nonlinear phenomena would thus be highly desired.
The transition between harmonic and anharmonic behaviour usually occurs when an external driving force exceeds a welldefined threshold^{4}. In the case of spinwave excitations at ferromagnetic resonance (FMR) discussed in this study, the nonlinear spinwave interaction depends on the amplitude of an external radio frequency (r.f.)driving field and can thus be easily controlled. At large excitation amplitudes, one observes an instability of nonuniform spinwave modes^{5}. Moreover, large excitation amplitudes and thus nonlinear behaviour is also essential in the switching process of the magnetization vector (for example, in memory devices). In spintronics, the reversal of the magnetization in nanostructures is one of the key prerequisites for functional magnetic random access memory cells. Equally important is the understanding of spin transfer torquedriven nanooscillators, which may function as radio frequency emitters or receivers. Both phenomena inherently involve large excitation amplitudes (and precession angles) of the magnetization vector deep in the nonlinear regime^{6,7,8,9,10,11,12}.
In this study, we combine measurements of longitudinal^{13} and transverse^{14} components of the dynamic motion of the magnetization vector by Xray magnetic circular dichroism (XMCD) as a function of r.f. power. At large driving amplitudes, our measurements clearly show that the lowfield nonlinear resonance behaviour cannot be described adequately using existing models for nonlinear magnetic resonance^{15,16}. To understand these data we develop a novel model that generalizes existing theories of spinwave turbulence. We show that the basic assumption of a timeindependent spinwave amplitude parameter is not justified at low magnetic bias fields. In fact, pronounced fast oscillations of the amplitude and phase occur and dominate the nonlinear response.
Results
Experimental configuration
Our experiments are performed using Permalloy (Ni_{80}Fe_{20}) films deposited on top of the signal line of coplanar waveguide structures. In all measurements, a magnetic bias field H_{B} forces the static magnetization to be oriented in the x direction. A magnetic r.f. field oriented along the y direction leads to a forced precession of M, as illustrated in Fig. 1. The precession of the magnetization vector is strongly elliptical due to the demagnetizing field. As indicated in Fig. 1, the Xray beam can be oriented at an angle θ=30° with respect to the film normal. In this geometry, the precession of the magnetization causes slight changes of the absorption of circularly polarized Xray photons detected by a photo diode in transmission. In a first set of measurements, the Xray beam is tilted in the y direction as illustrated in Fig. 1 (). A continuous wave r.f. excitation is synchronized to the Xray flashes. Owing to the large ellipticity of the magnetization precession, the detected signal is mostly given by the inplane magnetization M_{y} projected onto the Xray beam direction. When the phase of the magnetic r.f. driving field is set to 90° or 0° with respect to the Xray pulses, the measured signal represents either the real or the imaginary part (χ′ or χ′′) of the dynamic magnetic susceptibility ^{17} (cf. Fig. 2).
Normalization of the XMCD signal
This measurement of the susceptibility is normalized to static hysteresis loops also measured by XMCD, as shown in Supplementary Figs 1 and 2 (setup and spectra, respectively). Thus, only nonthermal excitation of the magnetization is detected in units of M_{s}. For the measurements shown in Fig. 2 the microwave phase and frequency (ω_{p}=2π·2.5 GHz) are kept fixed, whereas for the resulting resonance curves the magnetic bias field H_{B} is swept for different amplitudes of the excitation field h_{rf}. When the excitation field is increased above a critical amplitude of ∼0.2 mT, the main absorption shifts to lower fields. This effect is a consequence of the shift of the phase φ of the uniform mode above the threshold r.f. field. We find phase shifts of up to 35° at the smallangle resonance field H_{FMR} when the excitation amplitude is increased (Fig. 2c).
Longitudinal component of the magnetization
Any magnetic excitation (coherent or incoherent) leads to a decrease of M_{x} of the order of gμ_{B}, where g is the gfactor and μ_{B} the Bohr magneton. Therefore, to determine and separate coherent and incoherent components of the excitation we perform an additional measurement that is sensitive only to the longitudinal component of the magnetization vector M_{x}. For this, the sample is tilted in the x direction (), the frequency of the r.f. signal is detuned by a few kHz from a multiple of the 500 MHz synchrotron repetition rate. In this way the phase information is averaged out and the experiment is only sensitive to the average longitudinal magnetization component. Lockin detection in this case is achieved by amplitude modulation of the r.f. excitation. The corresponding signal is normalized again to static XMCD hysteresis loops.
On the one hand the measured decrease of the longitudinal magnetization component 〈ΔM_{x}〉 is proportional to the density of nonthermal magnons n_{k} excited in the sample^{18}. On the other hand, the population of the uniform mode n_{0}=n_{k=0} can also be calculated from the timeresolved (coherent) measurement of M_{y} in the linear excitation regime. For this, the energy stored in the coherent magnon excitation ΔE can be written as:
where V is the volume of the sample. The reduction of the longitudinal magnetization per magnon is found to be ∼5.5 gμ_{B}. This large value is due to the highly elliptical precession and in agreement with expected from linear spinwave theory.
In Fig. 3, the driving field dependence of 〈ΔM_{x}〉 from the timeaveraged longitudinal XMCD–FMR experiment is shown and compared with the calculated 〈ΔM_{x}〉 values obtained from the timeresolved transverse measurement of M_{y}(t). It is noteworthy that the transverse component is only sensitive to n_{0} magnons. In the linear regime, both curves coincide and the excited magnon population only contains uniform k=0 magnons. Above a critical r.f. field of ∼0.2 mT, the two curves separate, owing to saturation of the uniform magnon occupation density n_{0} and the parametric excitation of higher k magnons in the nonlinear regime^{5,19,20}, that is, the difference between the curves shown in Fig. 3 corresponds to the parametric excitation of additional k≠0 spin waves.
Discussion
The saturation of the homogeneous mode population as a function of r.f. field amplitude (Fig. 3) is the consequence of an increased relaxation rate for this mode: the nonlinear coupling of the uniform mode to nonuniform spin waves opens additional relaxation channels. In this regime, the energy pumped into the homogeneous mode by the r.f. excitation is only partly relaxed by intrinsic uniform mode damping. In fact, a significant portion of the energy is distributed to nonuniform modes by additional magnon–magnon scattering processes and subsequently relaxed by intrinsic damping as illustrated in Supplementary Fig. 3. Conservation of energy requires that the energy pumped into the magnetic system is equal to the energy relaxed to the lattice by intrinsic Gilbert damping of the dynamic modes^{21}:
where n_{0,k} are the magnon densities, and ħω_{0,k} and η_{0,k} are the magnon energies and relaxation rates, respectively. At the FMR condition (ω_{p}=ω_{0}), only uniform magnons are directly pumped and all other magnons (with density n_{k≠0}) are indirectly excited via nonlinear magnon–magnon processes. When we assume the latter to be secondorder Suhl instability processes (ω_{k}=ω_{0}), we can calculate the expected magnon relaxation rate. The result of this is shown in the inset of Fig. 3 (dashed line). In the experiment, however, we find an increase of ∼50% for the average relaxation rate η compared with the relaxation rate of the uniform mode η_{0}≈0.8 ns^{−1} at ω_{p}=2π·2.5 GHz. This increased relaxation rate is unexpected, as microscopic theory of magnetic damping^{22} does not predict a wave vector dependence of the Gilbert damping constant α for the relatively small wave vectors that are relevant.
From extensive micromagnetic simulations, we found that the experimental data can in fact be reproduced quite well using a wave vectorindependent damping parameter (red points in the inset of Fig. 3). This in turn implies that the physical explanation for the deviation from the Suhl model is to be found within the framework of the Landau–Lifshitz–Gilbert (LLG) equation. To unravel the physical origin for this behaviour, we develop a model based on the LLG equation that allows computing the properties of the critical spin waves in kspace in an efficient manner. This partially analytic approach can provide insight into the physics and reduces the computational effort drastically. We start from the LLG equation. Below the threshold excitation amplitude, only the uniform mode has a considerable amplitude. Therefore, we can restrict our considerations to linear terms in the k≠0 spinwave amplitude m^{k}. By algebraic transformations, one can show that the time evolution of m^{k} is then governed by a harmonic oscillator equation, which is parametrically driven by the uniform mode. In our case, this driving mostly occurs due to the dipolar fields h^{k}, which strongly depend on the angle between the timedependent uniform magnetization and the kvector. A numerical time integration of the differential equation as a function of kvector easily allows extracting the nonlinear dispersion and the decay rates for the spin waves in kspace. An example of this is shown in Fig. 4a. The calculation is performed for the parameters that correspond to the XMCDFMR experiments and the excitation amplitude was chosen close to the instability threshold. The fundamentally new finding from our model is that the critical spin waves (inverse life times approaching zero) do not precess at the driving frequency as expected for the fourmagnon scattering processes that usually lead to the secondorder Suhl instability^{23,24,25}. Instead, we find that the spin waves precess nonmonochromatically at frequencies that are halfinteger multiples of the driving frequency with additional oscillations of their amplitude and phase at the driving frequency (see Fig. 5a). In addition to the nonlinear shift of the spinwave dispersion, we also observe a pronounced frequency locking effect to halfinteger multiples of the driving frequency in the vicinity of the wave vectors for the critical spin waves (Fig. 4b).
To demonstrate more clearly that this nonlinear behaviour is fundamentally different from Suhl instabilities, we further simplify the numerical model. We assume that the time dependence of the potential term in the parametric oscillator equation can be written as Ω^{2}(τ)=a−2q cos(2τ), where the transformation of the time t→τ must be chosen to represent the instability process of interest (see further details in the Methods section). Although this assumption cuts off higherfrequency components, the simplified model is still able to reproduce our instability process as well as Suhl instability processes with the use of only two parameters (a and q). With these simplifications, the parametric oscillator equation assumes the form of the Mathieu equation (see Methods). Here with ω_{k} equal to the mean frequency of the spin wave, ω_{mod} is the frequency of the modulation and the parameter q is the modulation strength. The instability diagram for this twoparameter equation is well known, owing to its significance for fundamental quantum mechanical problems^{26} and shown in Fig. 5b. By mapping the spinwave instability processes onto this diagram, we find that although Suhl instabilities belong to the first instability region, the instabilities that we observe at low bias fields belong to the third instability region.
What distinguishes the first instability region from the others is that the modulation parameter is small (q/a<<1) and a perturbative approach can be used. In this case, the unperturbed states are harmonic oscillations with an amplitude that may only vary slowly in time. It is this assumption of a slowly varying envelope that breaks down for the higher instability regions and prevents the description of the instability processes that we observe by standard spinwave instability theory. When the modulation amplitude (q/a) can no longer be considered small, the spinwave precession also becomes considerably anharmonic, that is, higher Fourier components, separated by the frequency of the modulation, become important. This situation is mathematically identical to the motion of a quantum mechanical particle in a onedimensional periodic potential when the kinetic energy becomes comparable to the potential height. Although for a quantum particle the amplitude and phase of a wave function oscillate with the spatial period of the potential, the parametric spin wave behaves in a similar manner in the time domain. Both situations can be mathematically described in terms of the Hill equation.
We would like to note that this type of behaviour has not been considered so far, and that it is worthwhile to examine previous experiments in the light of these findings. In particular, we believe that these nonlinear processes can explain the apparent wavevectordependent Gilbert damping reported by two groups under similar experimental conditions^{27,28,29}. Reviewing the frequencydependent measurements of nonlinear ferromagnetic resonance performed by Gerrits et al.^{27}, we are able to accurately reproduce their experimentally found threshold fields with our model. In Fig. 6, corresponding calculations are shown for our sample thickness. Our simulations indicate that for frequencies below 5 GHz, spin waves oscillating at are parametrically excited before the secondorder Suhl instability can set in (spin waves oscillating at f_{p}). We therefore conclude that the observed threshold fields correspond to the type of instability processes described in the present work.
To verify the applicability of our model also for Suhl instability processes, the r.f. threshold amplitude fields are calculated as a function of magnetic bias field (socalled butterfly curves) and compared with published experimental results^{30} for subsidiary absorption (firstorder Suhl instability) and for resonance saturation (secondorder Suhl instability)^{24}. We find very good agreement with the measurements in both cases using a wavenumberindependent intrinsic Gilbert damping parameter. Furthermore, the nonlinear frequency shift is also observable in the butterfly curves of firstorder Suhl instability thresholds^{30}. According to our calculations, the frequency locking effect can explain why the experimental threshold fields in^{30} increased less abruptly above the resonance field than the authors expected.
We would like to point out that the validity of the above calculations is verified by extensive micromagnetic simulations^{31}. As shown in the inset of Fig. 3 and by the points in Fig. 5a, the results of the micromagnetic simulation are in excellent agreement with our experimental results and our model description of the nonlinear dynamics. We also find that the micromagnetic simulations agree very well in terms of the threshold excitation field, the wave vector of the critical mode and the nonlinear frequency shift. An example of the spinwave spectral density in kspace obtained from micromagnetic calculations is shown in Supplementary Fig. 4. In agreement with predictions from our model (see Fig. 4a)), the critical spinwave modes actually oscillate nonharmonically at for low magnetic bias fields.
In conclusion, we investigate experimentally and theoretically the nonlinear magnetization dynamics in magnetic films at low magnetic bias fields. Our analysis leads to a new and more general description of parametric excitation not limited to small amplitudes of the modulation parameter. Using this method, we find a new class of spinwave instabilities that dominate the nonlinear response at low magnetic fields. For these modes, we also find pronounced frequency locking effects that may be used for synchronization purposes in magnonic devices. By using this effect, effective spinwave sources based on parametric spinwave excitation may be realized. Our results also show that it is not required to invoke a wave vectordependent damping parameter in the interpretation of nonlinear magnetic resonance experiments performed at low bias fields. The recipe that has been developed here should prove very valuable for the general description of nonlinear magnetization dynamics. Specifically, the model allows a fast prediction of the critical spinwave modes.
Methods
Sample preparation
The samples are composed of a metallic film stack grown on top of a 100nmthick Si_{3}N_{4} membrane supported by a silicon frame, to allow transmission of Xrays. The film system is patterned into a coplanar waveguide by lithography and liftoff processes. The nominal 40nmthick Ni_{80}Fe_{20} layer is isolated from the 160nmthick copper layer by a 5nmthick film of Al_{2}O_{3}. The isolation layer and the low conductivity of Ni_{80}Fe_{20} ensure that 95% of the r.f. current flow in the Cu layer, leading to a welldefined inplane r.f. excitation of the sample. The driving field is oriented along the y direction perpendicular to the external d.c. field (transverse pumping).
Xray magnetic circular dichroism–FMR
XMCD is measured at the Fe L_{3} absorption edge. For circularly polarized Xrays, the dichroic component of the signal is proportional to the magnetization component along the Xray beam direction. The size of the probed spot on the sample is defined by the width of the signal line of the coplanar waveguide structure (80 μm) and by the lateral dimension of the Xray beam (900 μm). The transmitted Xray intensity is detected by a photodiode^{32}. All measurements are performed at the PM3 beamline of the synchrotron at the Helmholtz Zentrum Berlin in a dedicated XMCD chamber. The magnet configuration is shown in Supplementary Fig. 1. The microwave excitation in the XMCD–FMR experiment is phase synchronized with the bunch timing structure of the storage ring, so that stroboscopic measurements are sensitive to the phase of the magnetization precession. Synchronization is ensured by a synthesized microwave generator, which uses the ring frequency of 500 MHz as a reference, to generate the required r.f. frequency in the GHz range. The phase of the microwave excitation with respect to the Xray pulses is adjusted by the signal generator. To allow for lockin detection of the XMCD signal, the phase of the microwaves is modulated by 180° at a frequency of a few kHz. Owing to the synchronization of the microwave signal and the Xray bunches, the magnetization is sampled at a given constant phase. The amplitude of the modulated intensity is proportional to the dynamic magnetization component projected onto the Xray beam, as illustrated in Fig. 1. This signal is normalized by static XMCD hysteresis loops. The normalized signal is an absolute measure of the amplitude of the magnetization dynamics, evaluated in units of the saturation magnetization or as cone angle of the precession of the magnetization vector. Supplementary Fig. 2 shows typical XMCD measurements.
Theoretical model
We start with the LLG equation in the following form:
with , where we assume M_{x}>m^{0}>>m^{k}, that is, the uniform precession m^{0} is smaller than the static uniform magnetization and the nonuniform dynamic magnetization m^{k} is much smaller than m^{0}. H_{eff} is the effective field consisting of the external field, the exchange field (h_{exch}=2Ak^{2}/(μ_{0}M_{s})) and the dipolar field (). Here we use a thin film approach^{33} for the dipolar tensor instead of the more complicated expressions that we use for the analysis of parametric instability in thicker films^{34}. To first order in the nonuniform spinwave amplitudes, we find an equation of the form:
with timedependent components D_{ij}. The coupled coordinates can be separated by applying a time derivative. The result for looks as follows (where we drop the superscript):
with and . This form corresponds to the differential equation of a parametric oscillator. By substituting
with , we can eliminate the damping term:
with . Now we introduce the dimensionless parameter x=ω_{mod}t/2 and assume that varies periodically with the frequency ω_{mod}. One thus obtains:
where we use , to find approximate solutions. This assumption only implies that the timedependent modulation of the parametric oscillator is sinusoidal with a single frequency (for example, the driving frequency ω_{p}). Equation (8) can then be written in the form of the Mathieu equation:
According to Floquet’s theorem^{35}, the solutions of this equation are of the form:
where the complex number ν=ν(a, q) is called the Mathieu exponent and P is a periodic function in x (with period π). The parameters a and q depend on the properties of the spin wave. From the knowledge of ν for a given kvector, one can predict the behaviour of the corresponding spin wave as a function of time: For example, as soon as the imaginary part of exceeds the exponent in equation (6) the spin wave becomes critical. Furthermore, the real part of corresponds to the frequency of the spin wave. This method can be used to quickly find the dispersion and the lifetimes of all possible spinwave modes when the parameter ν is evaluated as a function of k_{x} and k_{y}.
Micromagnetic calculations
Micromagnetic simulations that confirm our conclusions are performed using an open source graphic processor unitbased code Mumax^{31}. The simulated sample volume is 80 μm × 20 μm × 30 nm. Time traces of 500 ns are computed to extract the numerical values. Standard parameters for Ni_{80}Fe_{20} are used in the simulations: saturation magnetization M_{s}=8 × 10^{5} A m^{−1}, damping constant α=0.009 and exchange constant A=13 × 10^{−12} J m^{−1}. We find the best agreement between the simulations and the experiments for a Ni_{80}Fe_{20} thickness of 30 nm, whereas in the experiments the nominal thickness of the Ni_{80}Fe_{20} layer is 40 nm. We attribute this discrepancy mostly to the surface roughness of the Ni_{80}Fe_{20} layer in the experiments.
Additional Information
How to cite this article: Bauer, H. G. et al. Nonlinear spinwave excitations at low magnetic bias fields. Nat. Commun. 6:8274 doi: 10.1038/ncomms9274 (2015).
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Acknowledgements
Financial support by the German Ministry of Education and Research (BMBF) through project number W05ES3XBA (VEKMAG), by the German research foundation (DFG) through project WO1231 and by the he European Research Council through ERC grant number 280048 (ECOMAGICS) are gratefully acknowledged.
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G.W. and C.H.B. designed and supervised the experiments. P.M., T.K. and G.W. prepared the experimental setup. H.G.B., P.M. and G.W. performed the experiments. The samples were prepared by G.W. and H.G.B. H.G.B. analysed the data, performed numerical simulations and developed the theoretical model. H.G.B., C.H.B. and G.W. wrote the paper. All authors analysed the data, discussed the results and commented on the manuscript.
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Bauer, H., Majchrak, P., Kachel, T. et al. Nonlinear spinwave excitations at low magnetic bias fields. Nat Commun 6, 8274 (2015). https://doi.org/10.1038/ncomms9274
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DOI: https://doi.org/10.1038/ncomms9274
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