Jérémy Létang
email: jeremy.letang@protonmail.com

Claudia De Melo

Charles Guillemard

Aymeric Vecchiola

Damien Rontani

Sébastien Petit-Watelot

Myoung-Woo Yoo

Thibaut Devolder

Karim Bouzehouane

Vincent Cros

Stéphane Andrieu

Joo- Kim
email: joo-von.kim@c2n.upsaclay.fr

Nanocontact vortex oscillators based on Co_2MnGe pseudo spin valves

I. INTRODUCTION

Spin-torque nano-oscillators (STNOs) are spintronic devices in which applied currents drive self-sustained magnetization oscillations, which provide a basis for electrical microwave oscillators on the nanoscale. They are promising for a number of different applications, such as rf communications [START_REF] Choi | Spin nano-oscillator-based wireless communication[END_REF][START_REF] Ruiz-Calaforra | Frequency shift keying by current modulation in a MTJ-based STNO with high data rate[END_REF], field sensing [START_REF] Tulapurkar | Spin-torque diode effect in magnetic tunnel junctions[END_REF][START_REF] Jenkins | Spin-torque resonant expulsion of the vortex core for an efficient radiofrequency detection scheme[END_REF] and neuro-inspired computing [START_REF] Torrejon | Neuromorphic computing with nanoscale spintronic oscillators[END_REF][START_REF] Romera | Vowel recognition with four coupled spin-torque nano-oscillators[END_REF][START_REF] Tsunegi | Physical reservoir computing based on spin torque oscillator with forced synchronization[END_REF][START_REF] Williame | Chaotic dynamics in a macrospin spin-torque nanooscillator with delayed feedback[END_REF]. An important challenge is to reduce the operating currents and improve signal-to-noise ratios. In this light, Heusler compounds [START_REF] Heusler | Über magnetische Manganlegierungen[END_REF][START_REF] Graf | Simple rules for the understanding of Heusler compounds[END_REF] are promising materials to meet these challenges. The near half-metallicity of these materials result in ultralow damping compared to standard ferromagnets [START_REF] Andrieu | Direct evidence for minority spin gap in the Co 2 MnSi Heusler compound[END_REF] and a large spin polarization [START_REF] Kübler | Understanding the trend in the Curie temperatures of Co 2 -based Heusler compounds: Ab initio calculations[END_REF], which reduces the threshold of spin-transfer torques and results in a larger magnetoresistive ratio in spin valve structures. This choice has been explored in vortex-based systems involving Co 2 Fe x Mn 1-x Si [START_REF] Yamamoto | Magnetic vortex in epitaxially-grown Co 2 (Fe,Mn)Si alloy[END_REF][START_REF] Yamamoto | Vortex spin-torque oscillator using Co 2 Fe x Mn 1-x Si Heusler alloys[END_REF][START_REF] Seki | Size dependence of vortex-type spin torque oscillation in a Co 2 Fe 0.4 Mn 0.6 Si Heusler alloy disk[END_REF], where a large output power of 10.2 nW [START_REF] Yamamoto | Vortex spin-torque oscillator using Co 2 Fe x Mn 1-x Si Heusler alloys[END_REF] and high quality factor of 5000 [START_REF] Seki | Size dependence of vortex-type spin torque oscillation in a Co 2 Fe 0.4 Mn 0.6 Si Heusler alloy disk[END_REF] have been reported.

In this paper, we present an experimental study of nanocontact vortex oscillators based on pseudo spin valves using Co 2 MnGe Heusler compounds. These are members of the broader class of materials with the composition Co 2 MnZ, where Z are different III B, IV B or V B elements, which have been studied in detail recently [START_REF] Andrieu | Direct evidence for minority spin gap in the Co 2 MnSi Heusler compound[END_REF][START_REF] Guillemard | Polycrystalline Co 2 Mn-based Heusler thin films with high spin polarization and low magnetic damping[END_REF][START_REF] Guillemard | Ultralow magnetic damping in Co 2 Mn-based Heusler compounds: Promising materials for spintronics[END_REF]. In particular, it has been demonstrated that Co 2 MnSi and Co 2 MnGe exhibit a combination of the lowest Gilbert damping, α, and highest spin polarization, P, among this class [START_REF] Guillemard | Ultralow magnetic damping in Co 2 Mn-based Heusler compounds: Promising materials for spintronics[END_REF]. In particular, ferromagnetic resonance experiments have shown that α = 4.6 × 10 -4 for Co 2 MnSi and α = 5.3 × 10 -4 for Co 2 MnGe at 290 K [START_REF] Guillemard | Ultralow magnetic damping in Co 2 Mn-based Heusler compounds: Promising materials for spintronics[END_REF], and both have close to the maximum polarization of P = 1, which are close to theoretical predictions [START_REF] De Groot | New class of materials: Half-metallic ferromagnets[END_REF][START_REF] Ishida | Search for halfmetallic compounds in Co 2 MnZ (Z=IIIb, IVb, Vb element)[END_REF][START_REF] Picozzi | Co 2 MnX (X=Si, Ge, Sn) Heusler compounds: An ab initio study of their structural, electronic, and magnetic properties at zero and elevated pressure[END_REF][START_REF] Kandpal | Calculated electronic and magnetic properties of the half-metallic, transition metal based Heusler compounds[END_REF][START_REF] Liu | Origin of low Gilbert damping in half metals[END_REF]. Lattice parameter mismatch is smaller between Co 2 MnGe and Au than between Co 2 MnSi and Au, which is used as spacer. We report here some results obtained on Co 2 MnGe-based spin valves.

A particular magnetic structure that underpins a class of spin-torque nano-ocillators is the magnetic vortex. In the nanocontact geometry, for example, the magnetic vortex represents a metastable state that is nucleated when the Zeeman potential associated with applied currents flowing through the nanocontact becomes sufficiently strong. As described by the Thiele equation [START_REF] Thiele | Steady-state motion of magnetic domains[END_REF], the vortex core gyrates around the nanocontact above a certain current threshold, whose motion is determined by the balance between the restoring force of the Zeeman confining potential, Gilbert damping, and spin torques. This steady-state gyration leads to a periodic variation of the magnetization in the nanocontact area, which can be detected electrically through resistance variations in a magnetoresistive stack such as spin valves. The sense of gyration is determined by the core polarity [START_REF] Thiele | Steady-state motion of magnetic domains[END_REF].

A unique feature of nanocontact vortex oscillators is the onset of periodic core reversal above a critical current [START_REF] Petit-Watelot | Commensurability and chaos in magnetic vortex oscillations[END_REF][START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF][START_REF] Létang | Modulation and phase-locking in nanocontact vortex oscillators[END_REF][START_REF] Yoo | Pattern generation and symbolic dynamics in a nanocontact vortex oscillator[END_REF]. This core reversal occurs concurrently with the steadystate gyration around the nanocontact, giving rise to self-phase locked and chaotic states. Core reversal involves the creation and annihilation of vortex-antivortex pairs [START_REF] Van Waeyenberge | Magnetic vortex core reversal by excitation with short bursts of an alternating field[END_REF], where the latter results in the emission of spin wave bursts [START_REF] Hertel | Exchange explosions: Magnetization dynamics during vortex-antivortex annihilation[END_REF]. This dynamics is driven primarily by currents flowing in the film plane (CIP), resulting in spin-transfer torques of the Zhang-Li form [START_REF] Zhang | Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets[END_REF] rather than the Slonczewski torques [START_REF] Slonczewski | Current-driven excitation of magnetic multilayers[END_REF] associated with currents perpendicular to the film plane in nanopillar geometries [START_REF] Pribiag | Magnetic vortex oscillator driven by d.c. spin-polarized current[END_REF][START_REF] Dussaux | Large microwave generation from current-driven magnetic vortex oscillators in magnetic tunnel junctions[END_REF][START_REF] Locatelli | Dynamics of two coupled vortices in a spin valve nanopillar excited by spin transfer torque[END_REF]. Therefore, it is interesting to enquire whether the low damping afforded by Heusler materials can affect the gyration process significantly as well as the synchronization process between gyrotropic motion and the core reversal since spin-wave relaxation times greatly exceed the typical timescale of the gyration and core-reversal periods. As we show here, the current-driven dynamics in Heusler-based pseudo spin valves can be quite rich and differ qualitatively from the behavior observed in permalloy-based systems. Some of the behaviors we observe might be inter-pretable as the result of coupled vortex dynamics in the free and reference layers.

This paper is organized as follows. In Sec. II, we describe the epitaxial film growth, along with the structural and magnetic characterization of the Co 2 MnGe films using x-ray diffractometry, in situ reflection high-energy electron diffraction (RHEED), vibrating sample magnetometry (VSM), and broadband ferromagnetic resonance (FMR). In Sec. III, we describe the frequency-domain analyses of the electrical power spectra. This is followed by a description of the timedomain analyses in Sec. IV, where mode stability is discussed. A discussion and concluding remarks are given in Sec. V.

II. FILM GROWTH AND SAMPLE FABRICATION

The pseudo spin valves based on Heusler compounds were grown by molecular beam epitaxy (MBE) on (001)-oriented MgO substrates. The multilayer stack studied is composed of Cr (10 nm) / Co (5 nm) / Co 2 MnGe (5 nm) / Au (1-10 nm) / Co 2 MnGe (15 nm) / Au-capping (2 nm), as schematically depicted in Fig. 1(a). The stoichiometry of the Co 2 MnGe films was accurately controlled by calibrating the atomic fluxes of each element through a quartz microbalance located at the sample position [START_REF] Guillemard | Issues in growing Heusler compounds in thin films for spintronic applications[END_REF]. The 15-nm-thick Co 2 MnGe corresponds to the free layer, while the Co(5 nm)/Co 2 MnGe (5 nm) composite bilayer plays the role of the reference layer in the pseudo spin valve as it possesses a larger coercive field compared with the Co 2 MnGe layer. In MBE-grown structures, it is often difficult to achieve a reference magnetic layer as a bottom electrode due to epitaxy constraints. Here, we succeeded in this respect by growing the 5-nm Co 2 MnGe film biased through exchange coupling with an epitaxial Co hard layer, as explained in more detail in the following.

First, a 10-nm-thick Cr buffer layer was grown on the MgO substrate to promote the epitaxial growth of hexagonal closest packed (hcp) Co which has a high magnetic anisotropy compared with Co 2 MnGe. X-ray diffractograms and RHEED patterns [Figs. 1(e The magnetic coupling between the Co and bottom Co 2 MnGe layers is confirmed by hysteresis loop measurements. In Fig. 2, the in-plane M-H hysteresis loops of the reference layer Co 2 MnGe (5 nm) / Co (5 nm) / Cr (10 nm) / MgO [Fig. 2(b)] is compared with the hysteresis loop of a single Co 2 MnGe soft magnetic layer (15 nm) / MgO [Fig. 2(a)]. The magnetic field is applied in plane along the main crystallographic axes of MgO (MgO [001] axis ⇒ 0 • ), as sketched in the inset of Fig. 2. The soft Co 2 MnGe magnetic layer [Fig. 2(a), coercivity of 1.6 mT] shows an in-plane four-fold magnetic anisotropy with easy axes at [100] and [010], which corresponds to the curves at 45 2(a)]. The coercivity enhancement in the reference layer (16.3 mT) is evidenced in Fig. 2(b), which confirms the strong exchange coupling between Co 2 MnGe and Co layers. It is well known that bulk hcp Co has a uniaxial magnetocrystalline anisotropy along its [0001] axis, denoted here as the c axis. According to the epitaxial relationship of the two Co domains here, the c axes are in plane and 90 • rotated from one domain to the other, leading to a fourfold magnetic anisotropy, with easy axes rotated 45 • from the Co [0001] direction [START_REF] Gu | Fourfold anisotropy and structural behavior of epitaxial hcp Co/GaAs(001) thin films[END_REF]. This finally leads to easy axes in the Co/Co 2 MnGe stack similar to the ones in Co 2 MnGe [see schematic at the inset of Fig. 2(b)]. This explains the fourfold magnetic anisotropy of the reference layer, with easy axes along [100] and [010] Co 2 MnGe directions, and the improved squareness of the hysteresis loop along these axes, compared with the single Co 2 MnGe layer.

The choice of the Au-spacer thickness represents a compromise between conduction electron scattering and interlayer exchange coupling (IEC) between the free and reference layers through Au. In this paper, Au-spacer layers of different thicknesses ranging from 1 to 10 nm were explored. A welldefined double-stepped hysteresis loop was obtained from 2 nm, which means that this thickness is enough to keep the free and reference layers magnetically decoupled [see Fig. 2(c)].

The stacks with 2-nm Au spacers were used to fabricate nanocontact oscillators. We defined 70 × 30 µm 2 mesas and an indentation back electrode wire in the stack down to substrate with argon ion beam etching. We first define a thin insulating resist (SU8) 40 nm thick, with a surface area of 40×40 µm 2 over the mesa. This layer represents the "indentation mask" through which the nano-indentation is performed. A thick resist (1-µm-thick SU8) was deposited to provide a good insulation between top and base electrodes over large areas. An aperture of 20×20 µm 2 is opened in the thick resist above the thin resist, defining the "indentation zone". A third resist is defined by optical lithography for top electrode lift-off. A nano-hole is thus created in the thin resist by using a conducting atomic force microscope tip as described in Ref. [START_REF] Bouzehouane | Nanolithography based on real-time electrically controlled indentation with an atomic force microscope for nanocontact elaboration[END_REF]. Samples were placed in a gentle reactive plasma (15 W) in 75% Ar / 25% O 2 gas mixture to clean up any remaining resist at the bottom of the nano-indented hole, i.e., at the surface of the stack, and to finely tune the nanocontact diameter. Following this, tantalum and gold were deposited to form the nanocontact and top electrodes. This process enabled us to realize nanocontacts with radii of 20 nm. The back electrode wires used for electrical control of the AFM tip is finally cut with a diamond spear.

We assume that these properties vary little between layers on the substrate and in the stack. Under a magnetic field loop, we measure at best for the full stack with nanocontact a magnetoresistance (MR) ratio of 0.12 ± 0.03%. A single 20nm-layer of Co 2 MnGe has an anisotropic magnetoresistance (AMR) ratio of 0.19%, which is comparable to the MR ratio of the full stack. However, the indentation process quality impacts the nanocontact quality, which can change the MR ratio.

III. FREQUENCY DOMAIN ANALYSIS

The microwave frequency measurement setup is illustrated in Fig. 3. A bias tee separates the DC current that is applied to the nanocontact and the high-frequency components that are generated from spin-torque driven dynamics of the magnetic vortex. The high-frequency branch is amplified by +50 dB before reaching a switch that directs the signal either to the spectrum analyzer or the oscilloscope. An in-plane magnetic field was applied on the sample.

As in previous studies of nanocontact vortex oscillators [START_REF] Petit-Watelot | Commensurability and chaos in magnetic vortex oscillations[END_REF][START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF][START_REF] Létang | Modulation and phase-locking in nanocontact vortex oscillators[END_REF][START_REF] Yoo | Pattern generation and symbolic dynamics in a nanocontact vortex oscillator[END_REF], it is necessary to nucleate a vortex (nominally in the free magnetic layer) as this state is not necessarily the ground state of the magnetization. This is achieved by applying a large DC current, here 9 or 10 mA which is toward the upper limit before sample degradation, while sweeping an inplane magnetic field. Simulations have shown previously in samples [START_REF] Petit-Watelot | Commensurability and chaos in magnetic vortex oscillations[END_REF][START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF][START_REF] Létang | Modulation and phase-locking in nanocontact vortex oscillators[END_REF][START_REF] Yoo | Pattern generation and symbolic dynamics in a nanocontact vortex oscillator[END_REF] with a different material that as the magnetization reverses, a domain wall sweeps through the free layer and as it crosses the nanocontact region, a vortex is nucleated through the competition between the current-induced Oersted field and spin-transfer torques. This nucleation process can result experimentally in slightly different micromagnetic configurations from one nucleation event to the next, which results in power spectra that can vary qualitatively. Examples of the different power spectra observed are presented in Fig. 4 for a nominal applied current of 8.5 mA, which establishes a common Zeeman energy potential associated with the current-induced Oersted-Ampère fields for the vortex gyration [START_REF] Mistral | Current-driven vortex oscillations in metallic nanocontacts[END_REF], but under different nucleation and applied field conditions. For these spectrum analyzer measurements, we used a resolution bandwidth of 500 kHz, a video bandwidth of [START_REF] Keatley | Imaging magnetisation dynamics in nano-contact spin-torque vortex oscillators exhibiting gyrotropic mode splitting[END_REF] 2000 points were acquired per spectrum. The power spectral density (PSD) of the measured MR oscillations shown is obtained after subtracting off a reference signal without spintorque driven oscillations; we give it in dB as here it represents a relative power above this reference signal. We can observe well-defined peaks in the sub-GHz regime, which is consistent with vortex gyration around the nanocontact with radii in the range of 100-200 nm. Figure 4(a) represents a case where the measurement was made under an in-plane applied field of 12.2 mT, with the frequency of the main peak at f 0 = 327.3 MHz and higher order harmonics up to 5 f 0 are visible in the frequency range displayed. Some parasitic noise is also observed, in particular, between the second and third harmonics. However, these peaks do not vary with applied current and can be attributed to artifacts related to external parasitic sources such as telecommunication networks. The spectral line of the fundamental frequency is in the range of 1 MHz, which is comparable to the resolution bandwidth used. As such, we find that the spectral line shape is best described by the pseudo-Voigt function

S (ν) = A η 1 σ √ 2π exp - (ν -ν 0 ) 2 2σ 2 + (1 -η) 1 π Γ/2 (ν -ν 0 ) 2 + (Γ/2) 2 , (1) with σ = Γ 2 √ 2 ln 2 , (2)
which represents a linear combination of a Gaussian line shape and a Lorentzian line shape, and approximates the Voigt function that describes the convolution between the Gaussian and Lorentzian functions. Here, A represents the amplitude, ν 0 is the central frequency, Γ is the full width at half maximum (which we refer to simply as the linewidth), and 0 ≤ η ≤ 1 describes the relative contribution of the (normalized) Gaussian line shape with respect to the (normalized) Lorentzian profile. The inset of Fig. 4(a) shows that the pseudo-Voigt function [Eq. ( 1)] accounts very well for the spectral line shape of the fundamental frequency peak, where we find a linewidth of Γ = 1.5 MHz, which is similar to characteristics seen in permalloy-based nanocontact vortex oscillators [START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF], and η = 0.77, which is consistent with the fact that the linewidth is close to the resolution bandwidth.

Indeed, qualitatively different behavior can be seen under other nucleation conditions, such those shown in Figs. 4(b)-4(d). In Fig. 4(b), we observe a clear doublepeak structure with high spectral power between the two maxima and a strong asymmetry across all the visible harmonics. Here, the power spectrum cannot be accounted for by a double Lorentzian fit as shown in the inset (we can neglect the Gaus-sian contribution because the features are much larger than the resolution bandwidth), where we can see the strongest discrepancy occurs in the region between the two maxima. In Fig. 4(c), we can observe a broad asymmetric triangularshaped spectral line with a comparatively large linewidth of 32 MHz, which is suggestive of a chaotic process [START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF]. An asymmetric double-peak structure can be seen Fig. 4(d), which, in contrast to the case shown in Fig. 4(b), can in fact be described by a double Lorentzian profile where the fitted central frequencies are 323.2 and 334.5 MHz, with linewidths of 2.5 and 2.4 MHz, respectively.

Finally in Fig. 4(e), we observe a similar case to Fig. 4(a) but with a lower in-plane applied field of 0.45 mT, where the fundamental peak appears at a lower frequency of 291.4 MHz with a broader spectral line. Again, this case can be well described by the pseudo-Voigt fit from which we obtain Γ = 2.5 MHz and η = 0.4. We note that the larger contribution from the Lorentzian profile is consistent with the larger linewidth, where convolution effects are less important.

In Fig. 5, we present color maps of the current dependence of these power spectra, which exhibit a richer behavior than permalloy standard nanocontact oscillators [START_REF] Létang | Modulation and phase-locking in nanocontact vortex oscillators[END_REF]. The color map represents a density plot in which the power spectra are shown as a function of applied current by assigning a color value to the amplitude of the peaks shown in in Fig. 4. Each of the five maps in Fig. 5 were acquired after a single nucleation event, where an example from each map is given in Fig. 4. Overall, we can observe an increase in the frequency of the main spectral line (along with its harmonics) as a function of applied current, which is consistent with behavior reported previously for nanocontact vortex oscillators. The spectra are acquired for descending current after successful nucleation. While gyration can be observed for currents down to around 4 mA in certain cases [Figs. 5(a These color maps highlight some of the different qualitative behaviors observed. Figure 5(a) represents the canonical case of nanocontact vortex oscillations with pure gyration without core reversal [START_REF] Mistral | Current-driven vortex oscillations in metallic nanocontacts[END_REF][START_REF] Pufall | Low-field current-hysteretic oscillations in spintransfer nanocontacts[END_REF][START_REF] Keatley | Direct observation of magnetization dynamics generated by nanocontact spin-torque vortex oscillators[END_REF], where the frequency of the spectral peaks vary monotonically as a function of applied current. Some steplike changes are seen in this variation, a feature that has been reported in other STNOs and attributed to the transition between different oscillatory mode profiles [START_REF] Krivorotov | Largeamplitude coherent spin waves excited by spin-polarized current in nanoscale spin valves[END_REF]. The disappearance of the signal below 4 mA can either result from the annihilation of the vortex or pinning, whereby the spintransfer torques do not allow the core to escape some local potential. The fact that this lower threshold varies between nucleation events suggests that this pinning is, in part, determined by the particular micromagnetic state present.

The current dependence of the double-peaked structure in Fig. 4(b) is given in Fig. 5(b), where a distinctive bandlike structure can be observed over the entire current interval over which the oscillations are present. As we indicate above, a double-Lorentzian fit does not adequately describe the observed spectra and provides a particularly poor description of the large levels of power observed in the frequency interval between the two peaks. This feature is particularly apparent in the color map in Fig. 5(b). The spectral profile is reminiscent of the power spectrum of a chirp signal, where the frequency varies in time between f 1 and f 2 , the positions of the two peaks. However, we have not identified a physical mechanism that would produce such a signal in our device.

Figure 5(c) provides an example of broad spectral lines, such as the PSD presented in Fig. 4(c), which might be interpreted as a signature of chaotic behavior. Indeed, the vortex gyration does not result in a sharp spectral line as in Fig. 5(a) that is typically seen for simple vortex gyration around the nanocontact. Chaos in nanocontact systems studied previously results from competing behaviors involving gyration and periodic reversals of the vortex core [START_REF] Petit-Watelot | Commensurability and chaos in magnetic vortex oscillations[END_REF][START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF][START_REF] Létang | Modulation and phase-locking in nanocontact vortex oscillators[END_REF][START_REF] Yoo | Pattern generation and symbolic dynamics in a nanocontact vortex oscillator[END_REF]. We do not see any evidence here of core reversal, which is typically characterized by the appearance of additional modulation sidebands in the power spectra. This suggests that another mech-anism involving transitions in the micromagnetic state might be present. We note that hints of a double-mode structure appear at lower currents (i.e., around 7 mA) before the onset of the broad excitation peak as the current is increased, which suggests that the dynamics here results from the interaction between two oscillatory modes.

In Fig. 5(d), we can see two distinct frequencies that are not related harmonically which appear to merge as the current is increased. In contrast to the case shown in Fig. 4(b), the double-peak structure here can be well described by a double Lorentzian fit, as illustrated in Fig. 4(d). Such a doublefrequency mode has been observed in other STNOs [START_REF] Krivorotov | Largeamplitude coherent spin waves excited by spin-polarized current in nanoscale spin valves[END_REF][START_REF] Pribiag | Long-timescale fluctuations in zero-field magnetic vortex oscillations driven by dc spin-polarized current[END_REF][START_REF] Kuepferling | Two modes behavior of vortex oscillations in spin-transfer nanocontacts subject to in-plane magnetic fields[END_REF][START_REF] Eggeling | Low spin current-driven dynamic excitations and metastability in spin-valve nanocontacts with un-pinned artificial antiferromagnet[END_REF][START_REF] Wang | Multiple-mode excitation in spin-transfer nanocontacts with dynamic polarizer[END_REF][START_REF] Keatley | Imaging magnetisation dynamics in nano-contact spin-torque vortex oscillators exhibiting gyrotropic mode splitting[END_REF][START_REF] Zhang | Tunable mode coupling in nanocontact spin-torque oscillators[END_REF] and can either result from the hopping between two frequencies or from the simultaneous emission of two frequencies. We cannot distinguish between these two scenarios from frequency domain measurements alone and further discussion on this point will be given in the following section.

Finally, in Fig 5(e), we observe a behavior consistent with the pure gyration regime but with the appearance of a microwave-quiet region within a small current interval between 8.9 and 9.3 mA. Such an extinction has been observed in other studies [START_REF] Martin | Tunability versus deviation sensitivity in a nonlinear vortex oscillator[END_REF][START_REF] Soucaille | Nanocontact based spin torque oscillators with two free layers[END_REF] but its mechanism has not been clearly identified. We offer an interpretation of this behavior in Sec. V.

IV. TIME DOMAIN ANALYSIS

To gain further insight into the origin of these multimode frequency spectra, we performed time domain measurements with a single-shot oscilloscope for a range of applied currents and in-plane magnetic fields. In Fig. 6, we present the PSD as a function of applied magnetic field under a fixed DC current of 7.5 mA. In Fig. 6(a), the Welch method [START_REF] Welch | The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms[END_REF] was employed to construct the average PSD from the time traces. At each value of the applied field, we recorded a 10-µs-long time trace which was then decomposed into half-overlapping 100ns windows. After applying a Hann filter to the time series data to minimize spectral leakage, a discrete Fourier transform was extracted from the windowed data. The power spectrum presented in Fig. 6(a) represents an average over these windows.

In a similar way to how the spectral characteristics vary under applied perpendicular magnetic fields [START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF], we observe a variety of behaviors as the in-plane applied field is changed at fixed current. We can identify a number of qualitatively distinct features, indicated by the dashed lines, such as narrow spectral lines, line broadening, spectral power decrease, and the presence of several modes. To better understand whether these features, in particular, spectral line broadening, are associated with chaotic dynamics, we use the noise titration technique to quantify whether highly nonlinear dynamics are present in physical processes [START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF][START_REF] Poon | Titration of chaos with added noise[END_REF][START_REF] Hu | Chaos in atomic force microscopy[END_REF]. This technique compares the accuracy of linear and nonlinear autoregressive models based on Volterra-Wiener series of degree d and memory depth κ as one-step predictors of the data. Truncated optimal linear and nonlinear models are selected to minimize the Akaike criterion and then statistically tested against each other Noise limit (%) (using an F test or a Whitney-Mann test) to decide on the best model fitting the data. To detect the presence of chaos, the aforementioned comparison between the two optimal models is performed on the experimental time series corrupted by numerical additive white Gaussian noise with increasing variance. The variance level leading for the first time to a lower normalized residual sum of squares from the linear model is called the noise limit (NL). A sufficient condition for the detection of chaos is then NL> 0 and its magnitude (expressed in %) represents the strength of chaos, similar to the value of the largest Lyapunov exponent characterizing the average rate of exponential divergence between nearby trajectories in phase space [START_REF] Kantz | Nonlinear Time Series Analysis[END_REF]. For our analysis, we have considered a degree d = 2 with a memory depth κ = 10 and used a time-delay embedding of τ e = 100 ∆t with ∆t = 12.5 ps, the sampling period of the experimental time series. We have also concentrated our analysis on data that was band-pass filtered in the frequency band [100 MHz, 1 GHz], where the relevant dynamical features of the nanocontact vortex oscillator are concentrated. The time-delay embedding allows us to sub-sample the time series and remove the sample-to-sample correlations, which would otherwise lead our experimental data to always be better described by linear models [START_REF] Barahona | Detection of nonlinear dynamics in short, noisy time series[END_REF]. In Fig. 6(b), we have plotted the NL as a function of the external magnetic field µ 0 H x and we can observe that significant, nonzero values of NL correlate with regions in the PSD map in which spectral line broadening appears or where several modes are present. A vanishing NL is observed for negative in-plane fields where a strong spectral line is seen, and in the range of 7 to 10 mT where a loss in the output occurs.

To shed light on the nature of the different dynamical regimes observed, indicated by the dashed lines in Fig. 6(a), we provide a deeper analysis of the time series data with a number of different methods as shown in Fig. 7. The case at µ 0 H x = -4.1 mT at which a single mode excitation is observed [Figs. 7(a)-7(d)] serves to illustrates these methods. Figure 7(a) shows the computed PSD using the Welch method, where we can observe narrow spectral lines consistent with the stable mode gyration, similar to the case shown in Fig. 5(a). Figure 7(b) illustrates the spectrogram constructed from Fourier transforms of the 100-ns halfoverlapping windows, which shows that the oscillation mode frequency remains stable over the entire duration of the time trace. In Fig. 7(c), we give an example of the measured oscillation signal over a 100-ns window, where both the raw and bandpass filtered signal (with a rectangular filter between 200 and 600 MHz) are shown. The filtered signal represents a well-defined oscillatory signal with small fluctuations in the amplitude. To quantify the frequency stability of the mode with a higher temporal resolution than that afforded by the spectrogram, we obtain a crude estimate of the instantaneous frequency by computing the interval between successive zerocrossings of the filtered signal in Fig. 7(c) and attributing this to a half oscillation period, T 0 /2. The evolution of T 0 with time is given in Fig. 7(d), where both the raw data and fivepoint moving average is shown over an interval of 300 ns. This behavior further confirms the stability of this mode, where only small fluctuations about the mean value occur.

Let us now discuss the case of small positive magnetic fields, µ 0 H x = 5.2 mT, at which broadening of power spectrum can be inferred from Fig. 6(a). The corresponding power spectrum, shown in Fig. 7(e), exhibits a broad peak at the fundamental frequency and is reminiscent of the case in Fig. 5(c). In contrast to the stable gyration mode [Fig. 7(b)], the spectrogram for this case in Fig. 7(f) shows a strong jitter in the mode frequency, which remains localized within a range of ∼100 MHz but fluctuates strongly in time. The noise level at this field value is about 40% [Fig. 6(b)], which is suggestive of a chaotic process. The time trace in Fig. 7(g) does not reveal any qualitatively different features as the previous case, but an analysis of the instantaneous period reveals large fluctuations over a range exceeding 1 ns, which appear to be randomly distributed in time [Fig. 7(h)].

As we noted previously, there is a range of applied fields from approximately 7 to 9 mT in which a microwave-quiet region is observed. As the in-plane field is further increased beyond this range, the reappearance of the microwave signal involves an intermittent signal. An example of this behavior occurs at µ 0 H x = 9.5 mT for which the power spectrum is shown in Fig. 7(i). In contrast to the other cases discussed until now, the power spectrum exhibits additional sidebands below the fundamental frequency peak. The spectrogram in Fig. 7(j) shows a largely quiet spectrum that is punctuated by intermittent oscillations with the peak structure in Fig. 7(i). While statistics on such events are limited, the intermittence appears to be aperiodic and is not accompanied by any discernible transient dynamics -at least within the time scale given by the 100-ns window used to compute the spectrogram. This can be seen in Fig. 7(k) where the oscillations emerge from the noise (at the center of the time window) without any additional transients. When present, the oscillatory signal exhibits additional oscillations in the amplitude, leading to the sidebands below the fundamental frequency, which is suggestive of a beat pattern. This is further corroborated by the analysis of the instantaneous period in Fig. 7(l), where a regular variation over a range of ≈ 0.8 ns can be seen. The proportion of the intermittent signal that occupies the entire 10-µs window studied increases as the field is increased; there is 0% of signal at 7.9 mT, 7% at 8.8 mT, 22% at 9.5 mT, 81% at 10.2 mT, and 100% at 13.8 mT.

Finally, under a higher in-plane applied field of µ 0 H x = 14.7 mT, we observe a double-peak structure with a broad base [Fig. 7(m)], with some discernible modulation side bands that are a few dB above the noise floor. The spectrogram in Fig. 7(n) reveals that spectral power remains present at both frequency branches of ∼ 375 MHz and ∼ 425 MHz across the entire time trace, which is suggestive of mode coexistence rather than mode hopping. In the time trace shown in Fig. 7(o), we can discern some variations in the period of the filtered signal. This is quantified in Fig. 7(p) where oscillations in the instantaneous period can be seen. Like the example in Fig. 7(l), this behavior is reminiscent of frequency modulation in which the oscillator frequency varies periodically in time.

Clear evidence of mode hopping was observed in another set of experiments in which the applied current was varied in a small interval, as shown in Fig. 8(a). Here, the range of applied currents in which microwave power is observed is less than 1 mA, where for decreasing currents a double-mode spectrum can be seen followed by a microwave-quiet region with the exception at 8.3 mA, which is reminiscent of the case in Fig. 5

V. DISCUSSION AND CONCLUDING REMARKS

Clear signatures of vortex gyration were detected in a similar current and frequency range to behavior previously reported in permalloy-based nanocontact systems [START_REF] Petit-Watelot | Commensurability and chaos in magnetic vortex oscillations[END_REF][START_REF] Devolder | Chaos in magnetic nanocontact vortex oscillators[END_REF][START_REF] Létang | Modulation and phase-locking in nanocontact vortex oscillators[END_REF]. There are, however, a few notable differences. First, core reversal does not appear to be triggered in the Heusler-based systems studied here, despite similar gyration frequencies being reached at the upper limit of the applied current. The onset of periodic core reversal results in the appearance of modulation sidebands, a feature that has not been observed in the Heusler-based nanocontact discussed here. This absence may be due to the fact that sufficiently large current densities have not been reached, since the onset of core reversal typically occurs above 10 mA in permalloy-based systems. Another possibility is that the larger exchange constant (Co 2 MnGe value being half to full Co value, i.e., 15 to 30 pJ/m, which is higher than permalloy or Ni value, which is 10 pJ/m) means that the critical deformation is not reached for core reversal under the spin torques applied [START_REF] Tretiakov | Vortices in thin ferromagnetic films and the skyrmion number[END_REF][START_REF] Guslienko | Dynamic origin of vortex core switching in soft magnetic nanodots[END_REF][START_REF] Yoo | Analytical model of the deformation-induced inertial dynamics of a magnetic vortex[END_REF]. Alternatively, core reversal is, in fact, present, but the low damping values result in a long relaxation time between each core reversal event. As a result, the modulation frequency is comparatively low and the sideband is indiscernible from the main oscillation peak.

Mode hopping [START_REF] Pribiag | Long-timescale fluctuations in zero-field magnetic vortex oscillations driven by dc spin-polarized current[END_REF][START_REF] Kuepferling | Two modes behavior of vortex oscillations in spin-transfer nanocontacts subject to in-plane magnetic fields[END_REF][START_REF] Keatley | Imaging magnetisation dynamics in nano-contact spin-torque vortex oscillators exhibiting gyrotropic mode splitting[END_REF][START_REF] Zhang | Tunable mode coupling in nanocontact spin-torque oscillators[END_REF][START_REF] Muduli | Decoherence and mode hopping in a magnetic tunnel junction based spin torque oscillator[END_REF][START_REF] Heinonen | Decoherence, mode hopping, and mode coupling in spin torque oscillators[END_REF][START_REF] Dumas | Spinwave-mode coexistence on the nanoscale: A consequence of the oersted-field-induced asymmetric energy landscape[END_REF], mode coexistence [START_REF] Krivorotov | Largeamplitude coherent spin waves excited by spin-polarized current in nanoscale spin valves[END_REF][START_REF] Wang | Multiple-mode excitation in spin-transfer nanocontacts with dynamic polarizer[END_REF][START_REF] Zhang | Tunable mode coupling in nanocontact spin-torque oscillators[END_REF][START_REF] Iacocca | Mode-coupling mechanisms in nanocontact spintorque oscillators[END_REF], and mode variations with I DC [START_REF] Wang | Threemode behavior of spin-transfer vortex oscillators with dynamic polarizer[END_REF] have already been reported in the literature for other oscillator systems. Mode coexistence can be attributed to two oscillation modes, for example, where an oscillatory mode is present in each of the magnetic layers of a coupled bilayer system. On the other hand, mode hopping describes the transition, quite likely thermally activated, between two distinct oscillatory regimes within a given system, which possess distinct frequencies. Our observations in the time domain appear to be consistent with both of these scenarios, where the variation of the in-plane field allows us to transit from one behavior to the other.

Preliminary micromagnetics simulations of the nucleation process indicate that a vortex can be nucleated in both the free and reference layers. While dipolar coupling between the layers is minimized in the uniformly in-plane magnetized state, it can become significant when vortices are present in both layers. In nanopillar structures, dipolar coupling between vortices in the free and reference layers can result in coupled dynamics [START_REF] Locatelli | Dynamics of two coupled vortices in a spin valve nanopillar excited by spin transfer torque[END_REF][START_REF] Lebrun | Nonlinear behavior and mode coupling in spin-transfer nano-oscillators[END_REF], which leads to narrow spectral lines. Here, we can offer an interpretation of the observed behavior in Figs. 6-8 in terms of coupled vortex dynamics, where the vortices in the free and reference layers both undergo steadystate gyrations as a result of the CIP torques. A single stable frequency signal, such as in Fig. 7 Full micromagnetics simulations of the coupled bilayer system prove to be particularly challenging to implement primarily because of the low values of the Gilbert damping constant. In systems with self-sustained oscillations, the damping constant enters as a prefactor in the relaxation rate of amplitude fluctuations toward the limit cycle [START_REF] Kim | Spin-torque oscillators[END_REF]. Because the value of α in the Co 2 MnGe films studied is about 20 times smaller than its counterpart in permalloy (α ∼ 0.01), it means that transient dynamics associated with changes in applied currents or fields, which might last 25 ns in a permalloy-based system, would take up to 500 ns to damp out in the Heusler system. In order words, the transient dynamics toward the limit cycle would take much longer to simulate than several periods of gyration itself, which is needed to compute simulated maps of the PSD. Further simulations will be pursued but a detailed numerical study remains beyond the scope of the present paper.

Finally, we note that the signal-to-noise ratio of the pseudo spin valve is not significantly higher than that of permalloybased nanocontact vortex oscillators we studied previously. This is unexpected since we anticipated a strong MR signal based on the higher spin polarization of the Heusler compound used. We speculate that this is a consequence of the nanocontact geometry, where the large current densities flow perpendicular to the film plane and also within the film plane [START_REF] Petit-Watelot | Electrical properties of magnetic nanocontact devices computed using finiteelement simulations[END_REF]. The overall electrical signal is therefore based on a combination of CPP-GMR and CIP-GMR, with possible contributions also from AMR related to the CIP flow through the vortex structure. As such, the half-metallic nature of the Heusler compound is not fully exploited by the device geometry, which is not the case, for example, for oscillators based on nanopillars.

In summary, we present an experimental study of the current-driven magnetization dynamics in nanocontact oscillators based on epitaxially-grown pseudo spin valve structures with the Heusler compound Co 2 MnGe. The hard ferromagnetic layer was successfully fabricated as a bottom electrode by growing the Co 2 MnGe film biased through exchange coupling to an epitaxial Co layer. The absence of interlayer RKKY coupling between the free and reference layers was shown for spacer thicknesses of 2 nm with low values of the Gilbert damping constant in the range of 6 × 10 -4 preserved at room temperature. We have demonstrated that vortex dynamics in nanocontact oscillators based on Heusler compounds can lead to rich power spectra, which comprise phenomena such as mode coexistence and mode hopping. Such features are possibly linked to the dipolar-coupled magnetization dynamics of the vortex cores in the free and reference layers, and accentuated by the low values of the Gilbert damping constant.

  ) and 1(i)] confirm the epitaxial relationship: [110] (001) Cr [100] (001) MgO, where Cr [100] axis is 45 • rotated from MgO [100] axis [see schematic in Fig. 1(b)]. The Co layer starts to grow on Cr with the body-centred cubic (bcc) crystal structure and relaxes to its hcp structure after a critical thickness around four atomic planes [36], as indicated by the appearance of new streaks in the RHEED pattern of Fig. 1(g). RHEED patterns [Fig. 1(g)] and xray diffractograms [Fig. 1(i)] are consistent with the epitaxial relationship of [0001] (1120) Co || [110] (001) bcc Cr. The same RHEED pattern was obtained by turning the sample 90 • , which means that Co [0001] direction is a fourfold rotational symmetry axis. This is consistent with considering two domains 90 • rotated in-plane, that is, the Co [0001] azimuth parallel to either the [110] or [110] Cr azimuths, as sketched in Fig. 1(c). A similar bicrystal epitaxial relation has been reported for several Co/Fe and Co/Cr layers deposited by MBE [37-39]. Co 2 MnGe was observed to epitaxially grow on the relaxed Co layer with the epitaxial relationship [110] (001) Co 2 MnGe [0001] (1120) hcp Co [Fig. 1(d), left], as confirmed by the RHEED pattern performed at the film surface [Fig. 1(h)]. Gold was chosen for the spacer layer first because epitaxial and continuous Au films were obtained on Co 2 MnGe [17] and, second, because Co 2 MnGe was also observed to grow epitaxially on Au. Both Co 2 MnGe films are consistent with a chemically ordered L2 1 structure, as indicated by x-ray diffraction measurements, by the presence of half streaks in the RHEED patterns taken along the Co 2 MnGe [110] azimuth [see, for example, Fig. 1(h)] and confirmed by high-resolution transmission electron microscopy [Fig. 1(j)].

FIG. 1 .

 1 FIG. 1. (a) Schematic of the pseudo spin valve stack. (b) Epitaxial relationships between the Cr layer and the MgO substrate. (c) Epitaxial relationship between the Co and Cr layers (left), and schematic of the HCP Co (1120) planes (right). (d) Epitaxial relationship between the Heusler Co 2 MnGe film (noted CMG) and HCP Co (left), and schematic of CMG (001) planes (right). (e)-(h) RHEED patterns obtained for each layer of the stack along the [100] azimuth of the MgO substrate. (i) X-ray diffractogram of the pseudo spin valve. (j) High-resolution TEM micrograph of the pseudo spin valve stack. Fourier transform (FT) pattern performed to the Co 2 MnGe free layer along the [110] axis is displayed at the inset.

FIG. 2 .

 2 FIG.2. In-plane hysteresis loop at different angles for (a) the free layer: CMG / MgO, (b) the reference layer: CMG / Co / Cr / MgO, and (c) the pseudo spin valve with Au-spacer layer of 2 nm (EA meaning easy axis). Horizontal scales are different between figures.

FIG. 4 .

 4 FIG.[START_REF] Jenkins | Spin-torque resonant expulsion of the vortex core for an efficient radiofrequency detection scheme[END_REF]. Power spectral density (PSD) of magnetoresistance oscillations at I DC = 8.5 mA for nucleation events under different applied in-plane fields. (a) Single mode with narrow spectral peaks and higher order harmonics at µ 0 H x = 12.2 mT, (b) double frequency mode at µ 0 H x = 0.23 mT, (c) broad asymmetric peaks at µ 0 H x = -2.0 mT, (d) double frequency mode at µ 0 H x = -12.4 mT, and (e) single mode at µ 0 H x = 0.45 mT. The PSD is given in relative power with respect to a reference signal that has been subtracted from the curves displayed. The insets show fits to the fundamental frequency peak using a pseudo-Voigt function for (a) and (e), a double Lorentzian for (b) and (d), and single Lorentzian for (c).

  ) and 5(b)], the signal can disappear at a higher threshold current of 7 mA, as shown in Figs. 5(c)-5(e). Thin horizontal lines in the color maps represent artifacts arising from parasitic electromagnetic signals, such as cellular telephone networks or WiFi signals, which are independent of the current and persist in the absence of the oscillator signal.

FIG. 5 .

 5 FIG. 5. Power spectral density (PSD) maps as a function of DC current for given magnetic field and different nucleation events. Spectra are measured every 0.1 mA. (a) Narrow signal at µ 0 H x = 12.2 mT, (b) two-frequency mode at µ 0 H x = 230 µT, (c) chaotic signal at µ 0 H x = -2.0 mT, (d) two-frequency mode at µ 0 H x = -12.4 mT, (e) narrow signal at µ 0 H x = 450 µT. The red arrows indicate the corresponding spectra shown in Fig. 4.

FIG. 6 .

 6 FIG. 6. (a) Color map of the power spectral density (PSD) computed from time-domain measurements as a function of applied in-plane magnetic field, µ 0 H x , at a current of 7.5 mA. The dashed lines indicate field values at which time-averaged power spectral densities and spectrograms are shown in Fig. 7. (b) Noise limit computed with the noise titration technique where nonzero values are consistent with chaotic dynamics. The points in light orange correspond to the field values indicated by the dashed lines in (a).

FIG. 7 .

 7 FIG. 7. Analysis of time series data at 7.5 mA for the four values of the in-plane magnetic field µ 0 H x indicated by dashed lines in Fig. 6(a): (a)-(d) -4.1 mT, (e)-(h) 5.2 mT, (i)-(l) 9.5 mT, and (m)-(p) 14.7 mT. The power spectrum computed with half-overlapping 100-ns windows are shown in (a), (e), (i), (m), with the corresponding spectrograms presented in (b), (f), (j), (n). Examples of raw (gray) and band pass (200-600 MHz) filtered (blue) time traces over an interval of 100 ns are given in (c), (g), (k), (o). The instantaneous period computed from successive zero crossings of the filtered time traces over an interval of 300 ns is given in (d), (h), (l), (p) where both the single (gray dots) and five-point moving averaged values (blue lines) are shown.

  (e). The spectrograms for three values of the applied current are shown in Figs. 8(b), 8(d), and 8(f), where clear transitions can be seen between two well-defined modes. While the statistics are limited, the dwell times appear to fluctuate randomly between the two states, with transitions being more frequent as the current is increased. The corresponding power spectra are shown in Figs. 8(c), 8(e), and 8(g), which exhibit a clear double-peak structure that more closely resembles Fig. 4(d) than Fig. 7(m).

FIG. 8 .

 8 FIG. 8. (a) Power spectral density computed from time-domain measurements as a function of applied current, I, under an applied field of µ 0 H x = -0.7 mT. The dashed lines labeled with symbols indicate current values at which spectrograms constructed with halfoverlapping 100-ns windows (b), (d), (f) and the corresponding PSD computed with the Welch method are shown: (b), (c) 8.8 mA, (d),(e) 8.9 mA, and (f), (g) 9.0 mA.

  (a)-7(d), results from the synchronized motion of the two vortices, leading to improved phase coherence and narrower spectral lines. Conditions for synchronization degrade as the in-plane field is increased, leading to unsynchronized motion along independent trajectories with close frequencies, as shown in Figs. 7(m) and 7(n) [and possibly also in Figs. 7(e) and 7(f)]. Through the dipolar coupling, the resulting dynamics shows signs of chaos as detected from noise titration, which is not unexpected for coupled oscillators. The intermittent signal in Figs. 7(i) and 7(j) does not exhibit any signs of transient dynamics, for example, associated with oscillator death, so we can interpret the loss in microwave power as resulting from a synchronized regime in which the oscillators are in anti-phase, which leads to a drastic reduction in the overall time-varying MR of the pseudo spin valve. As the in-plane field is varied outside of this range of intermittence, we observe a clearer separation between the frequency of the two gyration modes, which is particularly evident in Figs. 7(m) and 7(n) with the presence of two distinct branches. In this light, we can then interpret the mode hopping in Figs. 8(b), 8(d), and 8(f) as synchronized motion in which the master/slave relationship alternates between the free and reference layer oscillators. This is consistent with the narrower spectral lines seen as compared with the two-mode behavior in Figs. 7(m) and 7(n).

  • and 135 • since the [100] axis of Co 2 MnGe is rotated by 45 • about the [100] axis of MgO [see inset in Fig.

  kHz, a sweep time of 10 s per spectrum, and

			Bias tee
		ac+dc	dc
	Au		ac	DC generator
	Resist
	Au Co	Heusler Cr	+50 dB amplification
			Switch
		Oscilloscope		Spectrum analyzer
	FIG. 3. Electrical circuit for frequency-and time-domain measure-
	ments.

ACKNOWLEDGMENTS

The work was supported by the Agence Nationale de la Recherche under Contract No. ANR-17-CE24-0008 (CHIP-MuNCS). The authors thank S. Migot for FIB preparation and J. Ghanbaja for TEM imaging. J.L. acknowledges financial support from the EOBE Doctoral School of Université Paris-Saclay. D.R. gratefully thanks C.-S. Poon for sharing information on the implementation of the titration technique. D.R. acknowledges financial support from the Chair in Photonics.


L Desplat

J.-V Kim

Entropy-reduced retention times in magnetic memory elements: A case of the Meyer-Neldel Compensation Rule

We compute mean waiting times between thermally-activated magnetization reversals in a nanodisk with parameters similar to a free CoFeB layer used in magnetic random access memories. By combining Langer's theory and forward flux sampling simulations, we show that the Arrhenius prefactor can take values up to 10 21 Hz, orders of magnitude beyond the value of 10 9 Hz typically assumed, and varies drastically as a function of material parameters. We show that the prefactor behaves like an exponential of the activation energy, which highlights a case of the Meyer-Neldel compensation rule. This suggests that modeling information retention times with a barrier-independent prefactor in such magnetic storage elements is not justified.

The Meyer-Neldel (MN) rule, also known as entropyenthalpy compensation or the isokinetic rule, has been known empirically to chemists since the 1920s [1] and to physicists since 1937 [2]. It describes many processes across the natural sciences [3], such as biological death rates [4], transport in semiconductors [2,5], chemical reaction rates [6], and geological diffusion profiles [7]. For thermally activated processes whose rate k follows an Arrhenius-type law [8],

k = f 0 e -β∆E , (1)
in which ∆E is the internal energy barrier, β = (k B T ) -1 is Boltzmann's factor, and f 0 is the Arrhenius prefactor, then if the Meyer-Neldel rule applies [3,9,10],

ln f 0 = ln f 00 + ∆E E 0 + b, (2)
where E 0 is a characteristic energy of the family of transitions, f 00 is a positive factor and b is a constant. A general result is that processes described by Eq. ( 2) should possess a large activation energy compared to the thermal energy and the fundamental excitations in the system. The quanta of excitation of the heat bath are typically bosons, such as phonons in solid state physics, photons in the case of electronic transitions, or magnons in magnetism. Since ∆F = ∆E -T ∆S, where ∆F is the Helmholtz free energy, Eq. ( 1) may be reformulated as

k = f 00 e -β∆F = f 00 e ∆S/k B e -β∆E , (3)
and compensation follows if ∆S/k B = ∆E/E 0 + b. This type of relation seems to naturally arise in the case of large activation energies, by counting the number of ways the heat bath can furnish the energy needed to overcome the barrier [11]. Refs. 9 and 10 find that compensation should be associated to multi-excitation processes. Within the magnetism community, it is common practice to consider the attempt frequency as a characteristic timescale of the dynamics, i.e., f 0 ∼ 1 GHz. The knowledge of the internal energy barrier then naturally leads to a direct estimation of the rate of thermally activated magnetic transitions via Eq. ( 1) [12][13][14][15]. In particular, this is the case for the mean waiting time between magnetization reversals in nanostructures, such as nm-thick disks used in magnetoresistive random access memories (MRAM). Their stability can be evaluated as ∆ = β 300 ∆E, where β 300 is Boltzmann's factor at T = 300 K [16]. A typical metric for information storage is a 10-year retention time, which requires a minimum of ∆ ≈ 50. Recently, efforts to estimate the thermal stability of magnetic skyrmions have hinted that such an approach does not hold. In systems with a multidimensional phase space, the Arrhenius prefactor cannot be interpreted as a literal "attempt frequency", because it also carries an important activation entropy [17][18][19][20][21]. Nevertheless, this effect has often been attributed to the nontrivial topology of magnetic skyrmions, rather than a general result [17,20]. Meanwhile, more recent developments in MRAM have focused on storage elements with perpendicular magnetic anisotropy (PMA). The introduction, in the free layer, of elements with a strong spin orbit coupling, in order to enhance the PMA, has been shown to also induce a large interfacial Dzyaloshinkii-Moriya interaction (DMI) [22]. The typical configuration at the saddle point (SP) for the switching of the magnetization in such systems is two oppositely magnetized domains separated by a domain wall [23]. The DMI selects a preferred chirality of the wall and lowers its energy, thus leading to lower activation energies and, from the assumption f 0 ∼ 1 GHz, dramatically reduced retention times [23,24].

In this Letter, we show by applying Langer's theory [25] and forward flux sampling (FFS) [26,27] that the prefactor for magnetization reversals in nm-thick disks can take extreme, seemingly non-physical values and vary drastically as a function of material parameters. We find that it behaves like an exponential of the activation en- ergy, which stems from a linear variation of the activation entropy with the energy barrier, and thereby demonstrates a case of the Meyer-Neldel compensation rule.

To illustrate these compensation effects for magnetic memory elements, we follow the example in Ref. 23 and study a perpendicularly-magnetized CoFeB ultrathin film in a nanopillar within the micromagnetic approximation. The geometry comprises a disk of 32 nm in diameter with a thickness of d = 1 nm, which we model using finite difference cells 1 × 1 × 1 nm 3 in size. We take an exchange constant of A = 10 pJ/m, a saturation magnetization of M s = 1.03 MA/m, and a variable interfacial DMI constant D > 0. Dipolar interactions are treated in the local approximation through the use of an effective perpendicular anisotropy,

K = K u -(N z -N x )µ 0 M 2 s /2
, where N i are the demagnetizing factors of the disk [28] and K u = 0.77 MJ/m 3 is the base value. Below a critical DMI strength of D c = 4π -1 √ AK, the ground state is degenerate between uniformly-magnetized ferromagnetic (FM) states along the +z ("up") and -z ("down") directions. Above D c , noncollinear states are favored, and the domain wall energy, σ w = 4 √ AK -πD, becomes lower than that of the FM state. Our goal here is to directly calculate the information retention time of the disk -i.e., the mean waiting time, τ = k -1 , between magnetization reversals.

We begin with the search for minimum energy paths (MEPs) connecting the two stable FM states -'A' and 'B' -through the energy surface, and the precise identification of the first-order saddle point, 'S', along the MEP. This is carried out with the geodesic nudged elastic band method [29]. The MEP is shown in Fig. 1a. The reversal takes place via the nucleation of a domain wall at the boundary, which then propagates through the disk.

The SP is found as the domain wall reaches the center of the disk, in agreement with previous studies [16,23,30]. Note that because of the staircase boundary resulting from the discretization, the wall at the SP is not free to rotate around the z-axis, and the lowest energy realizations at 'S' exist as two perpendicular orientations of the wall, along the x-or the y-axis. In the absence of dipolar interactions or DMI, there is however a degeneracy between Bloch and Néel walls, which manifests in the appearance of a Goldstone eigenmode of zero-energy fluctuations. In reality, dipolar couplings favor Bloch walls in perpendicularly magnetized thin films, and this Goldstone mode has no physical relevance. Therefore, we consistently apply a minimum DMI of 0.25 mJ/m 3 , which naturally selects Néel walls [31,32], in agreement with Ref. [23]. We set the effective anisotropy at K = 187 kJ/m 3 [23] and we vary D up to 3.5 mJ/m 2 . Fig. 1b shows the energy profile along the reaction coordinate for different values of D. For D 2.5 mJ/m 2 , the configuration with the domain wall in the center becomes metastable. In Fig. 1d, we show the internal energy barrier as a function of D. As one can expect, the energy of the wall varies linearly with D [23,24,30]. We then set D = 0.25 mJ/m 2 and vary K from 0.1 to 0.26 MJ/m 3 . Below this lower bound, the characteristic wall width, δ w = A/K, becomes comparable with the size of the disk and the behavior of the system begins to change. The energy profiles are shown in Fig. 1c, and the corresponding energy barriers are shown in Fig. 1d. Once more, the result matches analytical predictions, as the energy of the wall varies like √ K. Next, retention times are calculated as a function of D and K through two distinct approaches. The first relies on the Kramers method [8] following Langer's approach [25] for which the rate constant is expressed as

k = λ + 2π Ω 0 e -β∆E = λ + 2π i λ A i j |λ S j | e -β∆E . (4)
λ + carries the dynamical contribution and corresponds to the rate of growth of the instability at the saddle point. We provide details on its derivation in the Supplemental Material (SM) [33]. The λ A,S are the curvatures of the energy surface at 'A' and 'S'. Ω 0 therefore depends on the details of the fluctuations in the initial state and at the SP and carries the entropic contribution. The theory constitutes the most complete extension of Kramers' method to a multidimensional energy surface, but it is in principle restricted to intermediate to high dampings. In that limit, it can be applied to magnetic spin systems [34,35] and has been successfully used to calculate the lifetime of magnetic skyrmions [19,21,36]. At low damping, the density of states deviates more significantly from the equilibrium Maxwell-Boltzmann distribution, so the assumptions of the theory become less valid. Since the typical damping factor for CoFeB thin films is α = 0.01 [37],

we compute rates in a range of α from 0.5 to 0.01, while bearing in mind that their accuracy decreases as α decreases. Another important assumption in Kramers' framework is sink boundary conditions past the barrier, i.e., barrier recrossings are not considered. In that sense, the theory should be limited to metastable states decaying to a lower energy minimum. In practice, it can be applied to a symmetric potential [38], but yields only a forward rate and neglects the recrossings that will necessarily take place. Therefore, the waiting times between reversals are under-evaluated.

Our second approach to rate calculations is forward flux sampling (FFS). This path sampling method, initially developed in the field of biochemistry [26,27] for simulating rare events and computing activation rates, has also been successfully adapted to magnetic systems to treat problems such as reversal rates in graded media grains [39,40] or collapse rates of skyrmions [21]. In the latter, it yields good agreement with Langer's theory. The method consists in sampling the stochastic magnetization dynamics at finite temperature [41] but is significantly more efficient than direct Langevin simulations [39] and, as opposed to Langer's theory, does not require any additional assumptions to hold. It is therefore a convenient way to check whether the rate of a particular transition can reasonably fall within the framework of Langer's theory. It relies on a set of interfaces in configuration space, {Λ A , Λ 0 , . . . Λ n = Λ B }, defined as iso-surfaces of a monotonically varying order parameter, ζ. A flux of trajectories of the system through each interface is computed by Langevin trial runs, which involves integrating the stochastic Landau-Lifshitz equation [42]. The rate constant of the transition can be decomposed into a product of the partial fluxes as

k = Φ A,0 n-1 i=0 P (Λ i+1 |Λ i ), (5)
where Φ A,0 is the rate at which trajectories starting from region 'A' cross the first interface Λ 0 , and the conditional probabilities, P (Λ i+1 |Λ i ), correspond to the probability that a trajectory coming from 'A' that crossed Λ i for the first time will cross Λ i+1 before returning to 'A'. For best efficiency, FFS requires a pertinent choice of the order parameter. Here we choose the z component of the magnetization averaged over all sites, ζ = V -1 dV m z . In Fig. 2, we show the mean waiting time between reversals at T = 300 K as a function of DMI [Fig. 2a ] and anisotropy [Fig. 2b] calculated with a finitedifference implementation of Langer's theory, and FFS simulations performed with a homemade code [21] and MuMax3 [43,44]. With the latter, we performed a single run with full dipole-dipole interactions, which is labeled in Fig. 2(a) as 'DDI'. We find a good qualitative agreement between Langer's method and FFS. As expected, FFS yields larger values of τ than those of Langer's the- ory, because barrier recrossings are quite frequent. As a result, the flux of trajectories past the SP does not quickly approach unity, like it would if recrossings were negligible. When we decrease the damping in FFS, deviations from Langer's theory increase as the assumptions of Langer's become less valid. Micromagnetics simulations are also performed with full dipolar interactions and α = 0.01. This case is closer to physical systems and is the furthest one from Langer's framework, but the lifetimes are still comparable with Langer's theory. Fig. 2a shows that a D of 1 mJ/m 2 leads to a reduction of τ by about one or two orders of magnitude, and not five-six orders of magnitude like previously anticipated from the constant f 0 approximation [23]. In both cases, the largest stability factors of 32 and 40 [Fig. 1d] respectively yield a retention time of the order of 10 -1 -10 -2 s, and a few seconds.

We now examine the variation in the Arrhenius prefactor, which we show in Figs. 3a and3b as a function of D and K, respectively. We find values up to 10 17 Hz, which is orders of magnitude greater than the value of 10 9 Hz typically used for estimating thermal stability. These may appear unphysical, but the dynamical contribution, λ + , does fall in the GHz range, as we show in the SM [33]. The large values of f 0 stem from the entropic contribution. Following previous works [8,19,21,45], we define the change in configurational entropy as

e ∆S/k B ≡ β 2π i λ A i j λ S j , (6)
where is defined for positive energy curvatures corresponding to stable modes of fluctuations. It follows that Eq. ( 4) can be expressed as

k = λ + 2πβ|λ S 1 | e ∆S/k B e -β∆E , (7)
in which λ S 1 is the negative curvature at the barrier top and is associated with the unstable mode. The system overcomes the barrier by following this mode -in our case, it corresponds to the translation of the wall. If ∆S exhibits a linear dependence on ∆E, we have a case of compensation.

In Fig. 4 we plot ∆S/k B , as defined in Eq. ( 6), as a function of ∆E for different values of D and K. The inset shows the corresponding behavior of the prefactor on a logarithmic scale. We present values of K up to 1 MJ/m 3 , for which f 0 reaches 10 21 Hz. We reiterate that such values include large entropic contributions, while relaxation processes captured in λ + remain governed by the Landau-Lifshitz equation and lie in the GHz range. In both cases, we find a linear relation between them. The inverse slope is E 0 [Eq. ( 2)], the characteristic Meyer-Neldel energy of the family of transitions. Note that β does not impact the slope of the graph, and we find compensation regardless of the temperature, as long as Langer's framework remains valid (β∆E ≥ 5 [34]). When varying D, we find E 0 = 0.92Ad, while when varying K, we have E 0 = 2.06Ad, in which Ad sets the energy scale of magnons. Ultimately, the activation entropy is found to be more detrimental to the retention time than the DMI.

We offer two ways to interpret this result. First, configurational entropy characterizes the change of volume available to thermal fluctuations on the energy surface induced when the system reaches the SP. Compensation implies that the volume at the saddle point is greater than that of the stable state, i.e., there are more accessible states around the SP, which makes it more likely FIG. 4. Activation entropy at T =300 K, ∆S300/kB, computed with Eq. ( 6), as a function of the activation energy normalized by the characteristic exchange energy, ∆E/(Ad), for variations in D and K. Solid lines represent linear fits to the data. The inset shows the corresponding variation of f0 for α = 0.5 with the same horizontal scale. to be visited. This change of volume increases with the barrier height. This explains the frequent recrossings observed in FFS simulations, and the fact that the system seems to spend a long time in the vicinity of the SP. The magnon dispersion relation in the long wavelength limit behaves like (M s /2γ)ω A (q) = Aq 2 + K in the FM state, where q 2 = q 2

x + q 2 y and γ is the gyromagnetic constant. For magnons propagating parallel to the wall at the SP, i.e, along x, gapless modes appear with the dispersion (M s /2γ)ω S (q x ) = Aq 2

x [46,47]. The presence of these gapless, low energy modes at the SP is in line with a largest entropy at the barrier top. In the SM [33], we show that the logarithm of the ratio of products of these eigenfrequencies along x, i.e., ln i ω A i /ω S i , can be linked back to Ω 0 in Eq. ( 4) and tends to behave like ∆E if ∆E Ad. A second interpretation lies in the dynamics of the transition. With increasing barrier height, a larger number of small excitations is required to overcome the barrier, and the number of ways to combine these excitations increases [3,[9][10][11]. Compensation therefore results from a multi-excitation process, akin to a "dynamical entropy" of the bath [11]. The two interpretations are not incompatible, because a large volume around the SP also implies that there must be many pathways on the energy surface that lead to it. Besides the MEP which involves the reaction coordinate, the other paths must necessarily mix the eigencoordinates, which results in the excitation of many magnon modes. It has been suggested that compensation should occur for magnon-driven transitions [11], with a characteristic energy determined by the exchange (A) and the energy dependence of the density of states of magnons (ρ(E) ∼ √ E). When varying A at constant D and K, we do not find compensation, which seems to confirm this idea. The difference by a factor of two in the slope between D-and K-driven com-pensation may stem from the lifting of the degeneracy between counter-propagating azimuthal modes in confined structures with D [48]. The authors of Ref. 11 suggest that compensation should appear for ensembles of transitions for which ∆E varies between members, but the effective coupling to the heat bath, remains the same. Compensation may therefore be a general feature of magnetic systems with large activation energies.

- 1 FIG. 1 .

 11 FIG. 1. (a) Minimum energy path for the magnetization reversal in the disk from the 'A' ("up") to the 'B' ("down") state through the saddle point 'S'. (b, c) Energy profile along the reaction coordinate as a function of (b) DMI, D, and (c) anisotropy, K, with β300 ≡ (kBT300) -1 at T = 300 K. (d) Internal energy barriers as a function of D and K.

FIG. 2 .

 2 FIG. 2. Mean waiting time between reversals at 300 K, τ , as a function of the (a) DMI, D, and (b) anisotropy, K, computed with Langer's theory and FFS for different values of the Gilbert damping, α. The insets give a comparison between Langer's theory and the estimate based on f0 = 1 GHz. In (a), a single run with full dipole-dipole interactions (DDI) is also shown.

FIG. 3 .

 3 FIG. 3. Arrhenius prefactor, f0, as a function of the (a) DMI, D, (b) anisotropy, K, computed with Langer's theory and FFS for different values of the Gilbert damping, α. Points involving full dipole-dipole interactions (DDI) were obtained with energy barriers from Ref. 23.

The authors graciously thank Nicolas Reyren for providing published data that we used to compute f 0 in Figure 3. This work was supported by the Agence Nationale de la Recherche under Contract No. ANR-17-CE24-0025 (TOPSKY) and the University of Strasbourg Institute for Advanced Study (USIAS) for a Fellowship, within the French national programme "Investment for the Future" (IdEx-Unistra).


Marie Amalric

Liping Wang

Pierre Pica

Santiago Figueira

Mariano Sigman

Stanislas Dehaene

Fast comprehension of embedded geometrical primitives and rules in human adults and preschoolers

 Studies of sequence learning have outlined one possible mechanism by which complex mental representations are constructed out of simpler primitives: the human ability to extract complex nested structures from sequential inputs.

 Experiments in infants, preschoolers, and adults without access to education have demonstrated the existence of innate "core knowledge" for space, endowing humans with spontaneous intuitions of geometry.

 The question therefore arises whether a capacity for the internal representation and manipulation of nested sequences also underlies the acquisition of mathematics.

 We propose to formalize the human sensitivity to mathematical rules through a "language of thought" that allows the formation of complex representations from a small repertoire of primitives.

d'

CONCLUSIONS

 Simple rotations and axial symmetries were all detected and quickly used by human adults and 5-years-old children.

Point symmetry was more challenging for French preschoolers or Munduruku adults than for French adults.

 Human subjects were able to detect most of the embedded expressions we used to define our visuospatial sequences such as simple repetition, concatenation, and some repetition with variation.  The analysis of error patterns provided direct evidence for hierarchical embedding: superficial rules were acquired more quickly and induced fewer errors than deeper rules.

 With age, a geometrical language endowed with nested rules seems to arise even in the absence of formal schooling, as Munduruku adults who lacked school-based education, performed better than 5-years-old kids.  In children, the failure with complex sequences could arise from limitations in working memory and not necessarily to a lack of understanding nested structures.

 The theoretical complexity of a sequence was an excellent predictor of its mean error rate, and we confirm that minimal description length is a reasonable approach of adult sequence learning capacity.  Additional primitives, both geometrical and non-geometrical still need to be added to our model to complete its description of "core geometry".

RESULTS

Subjects saw the first locations of a given sequence and had to point to the next ones. If they were mistaken, the sequence restarted.

Participants:

23 French adults and 47 5-years-old children, and 14 Munduruku teenagers and adults.

Repeat +1 01234567 K=5 Repeat +2 02460246 K=5 [+0, [+1] 7 ] [+0, [+2] 7 ] Alternate 0213243546 K=8 2arcs 01237654 K=8 [+0, +2] 8 {+1}] [[+0, [+1] 3 ] 2 <V>] 2squares 02467135 K=8 4segments 01726354 K=7 4diagonals 04152637 K=7 2points 02020202 K=7 [[+0, [+2] 3 ] 2 <-1>] [[+0, H] 4 {-1}] [[+0, P] 4 {+1}] [[+0, +2] 4 {+0}] 2rectangles 05416327 K = 10 2crosses 04512673 K=7 Irregular 04715236 K = 16 4points 02360236 K = 11 [[[+0, -3] 2 {P}] 2 <-2>] [[+0, P ] 4 {-3}] [ +0,P,B,+2,P,B,+1,H ] [[+0,+2,+1,+3] 2 <+0>] +1 or H +2 -1 or A -2 0 B V P

Knowledge of geometrical primitive rules

Learning of embedded rules

•

  If K(P) ≤ Kmax: the next location is defined by P:• If K(P) > Kmax:the next location is chosen randomly:prf = [prf, rand([1..8])] [+0, [+1] 7 ] … [+0,[[+1]²]²<A>,+2,[+1]²] In comparison to irregular baseline, most of the regular sequences were well learnt.  Error pattern directly reflects the hierarchical internal representation of sequences.  Error rate at specific data point indicates how well a given rule is understood: e.g. in "4segments", even data points reveal that all axial symmetries are detected by all groups of participants.Complexity threshold: