Using rf voltage induced ferromagnetic resonance to study the spin-wave density of states and the Gilbert damping in perpendicularly magnetized disks

TL;DR

This paper introduces a method using rf-voltage induced ferromagnetic resonance to analyze spin-wave density of states and Gilbert damping in nanoscale magnetic disks, enabling detailed magnetic damping measurements.

Contribution

The study presents a novel approach to separate transverse and longitudinal susceptibility signals in rf-voltage FMR, allowing accurate determination of damping and spin-wave properties in nanoscale devices.

Findings

Resonance lines can be used to extract spin-wave density of states.

Device size affects the ability to resolve individual spin-waves.

Measured Gilbert damping in 90 nm disks is 0.011, higher than in unpatterned films.

Abstract

We study how the shape of the spinwave resonance lines in rf-voltage induced FMR can be used to extract the spinwave density of states and the damping within the precessing layer in nanoscale tunnel junctions that possess perpendicular anisotropy. We work with a field applied along the easy axis to preserve the uniaxial symmetry of the system. We describe the set-up to study the susceptibility contributions of the spin waves in the field-frequency space. We then identify the maximum device size above which the spinwaves can no longer be studied in isolation as the linewidths of their responses make them overlap. The rf-voltage induced signal is the sum of two voltages that have comparable magnitudes: a first voltage that originates from the transverse susceptibility and rectification by magnetoresistance and a second voltage that arises from the non-linear longitudinal susceptibility and the resultant time-averaged change of the micromagnetic configuration. The transverse and longitudinal susceptibility signals have different dc bias dependences such that they can be separated by measuring how the device rectifies the rf voltage at different dc bias voltages. The transverse and longitudinal susceptibility signals have different lineshapes; their joint studies can yield the Gilbert damping of the free layer of the device with a degree of confidence that compares well with standard FMR. Our method is illustrated on FeCoB-based free layers in which the individual spin-waves can be sufficiently resolved only for disk diameters below 200 nm. The resonance line shapes on devices with 90 nm diameters are consistent with a Gilbert damping of 0.011. This damping of 0.011 exceeds the value of 0.008 measured on the unpatterned films, which indicates that device-level measurements are needed for a correct evaluation of dissipation.

Figures

Tables

Table 1. Table 1: Summary of the expected lineshapes and linewidths for the different signals that can be encountered in rf -voltage-induced FMR experiments. An hyphen is inserted when the concept is not applicable.

Signal		Spectral	Peak-to-peak	Full Width	Zero crossings
		shape	separation	at Half Maximum	separation
Expected signals and their stimulus dependence:
$V_{1,ac}\propto V_{rf}^2$		$\Re e\left({\chi }_{xx}\right)$	$2\alpha \omega$	-	-
$V_{2,ac}\propto V_{rf}^2{V}_{dc}$		$\Delta M_z$	-	$2\alpha \omega$	-
Signal extraction procedure from experiments:
$\frac{d}{d{H}_z}{V}_{1,ac}^{exp}$ estimated from $\frac{d{V}_{ac}^{exp}}{d{H}_z}|{}_{V_{dc}=0}$		$\frac{d\Re e\left({\chi }_{xx}\right)}{d\omega }$	-	-	$2\alpha \omega$
$\frac{d}{d{H}_z}{V}_{2,ac}^{exp}$ estimated from ${{\left[\frac{d{V}_{ac}^{exp}}{d{H}_z}\right|}_{V_{dc}\ne 0}-\frac{d{V}_{ac}^{exp}}{d{H}_z}|}_{V_{dc}=0}]$		$\frac{d\Im m\left({\chi }_{xx}\right)}{d\omega }$	$\frac{2}{\sqrt{3}}\alpha \omega$	-	-

Equations

$$s(f_0)=\frac{1}{2{\gamma }_0{H}_z^{max}}{\int }_{\text{contour}}\frac{d{V}_{\text{ac}}}{d{H}_z}df,$$

$$V_{1,ac}=\frac{V_{rf}}{R+50}\frac{\delta R}{\delta M}\delta M_{rf}\text{ and}$$

$$V_{2,ac}=\frac{V_{dc}}{R+50}\frac{\delta R}{\delta M}\delta M_{ac}$$

$$V_{1,ac}\propto {V_{rf}}^2.$$

$$V_{2,ac}\propto {V_{rf}}^2{V}_{dc}.$$

$$a_{\text{crit}}^2=\frac{9A}{\alpha {\mu }_0{M}_S(H_z+H_k-M_s)}$$

$$\left\{\begin{array}[]{ccl}H^{\prime}(m_{y}+\alpha m_{x})+im_{x}\omega^{\prime}&=&h_{x}M_{s}\alpha\\ H^{\prime}(m_{x}-\alpha m_{y})-im_{y}\omega^{\prime}&=&h_{x}M_{s}\end{array}\right.$$

$${\chi }_{xx}=\frac{M_s{H}^{\prime }}{{H^{\prime }}^2-{\omega }^{\prime 2}+2\text{i}\alpha H^{\prime }{\omega }^{\prime }}$$

$${\chi }_{yx}=-\text{i}\frac{M_s{\omega }^{\prime }}{{H^{\prime }}^2-{\omega }^{\prime 2}+2\text{i}\alpha H^{\prime }{\omega }^{\prime }}$$

$$\Re e({\chi }_{xx})=\frac{M_s{H}^{\prime }({H^{\prime }}^2-{\omega }^{\prime 2})}{4{\alpha }^2{H^{\prime }}^2{\omega }^{\prime 2}+({H^{\prime }}^2-{\omega }^{\prime 2}{)}^2}$$

$$\Im m({\chi }_{xx})=-\frac{2\alpha M_s{\omega }^{\prime }{H^{\prime }}^2}{4{\alpha }^2{H^{\prime }}^2{\omega }^{\prime 2}+({H^{\prime }}^2-{\omega }^{\prime 2}{)}^2}$$

$$\Delta M_z\approx \frac{||h_x|{|}^2}{2M_S}\frac{M_s^2{H}^{\prime 2}}{[{H^{\prime }}^2-{\omega }^{\prime 2}{]}^2+4{\alpha }^2{H^{\prime }}^2{\omega }^{\prime 2}}$$

$$\Delta \omega =2\alpha \omega \text{ or equivalently, }\Delta H_z=2\alpha H^{\prime }$$

$$\Delta \omega =\alpha {\omega }_k\frac{\partial {\omega }_k}{{\gamma }_0\partial \tilde{H}}$$

Full text

The frequencies of the magnetization eigenmodes of magnetic body reflect the energetics of the magnetization. As a result the frequency-based methods – the ferromagnetic resonances (FMR)Kittel (1948) and more generally the spin-wave spectroscopies– are particularly well designed for the metrology of the various magnetic interactions. In particular, measuring the Gilbert damping parameter $\alpha$ that describes the coupling of the magnetization dynamics to the thermal bath, specifically requires high frequency measurements. There are two main variants of these resonance techniques. The so-called conventional FMR and its modern version the vector network analyzerBilzer et al. (2007) (VNA)-FMR are established technique to harness the coupling of microwave photons to the magnetization eigenmodes to measure to anisotropy fieldsKittel (1948), demagnetizing fields, exchange stiffnessKubota et al. (2009), interlayer exchangeDevolder (2016) and spin-pumpingMizukami et al. (2001), most often at film level. More recent methods, like the increasingly popular spin-transfer-torque-(STT)-FMR, are developedTulapurkar et al. (2005) to characterize the magnetization dynamics of magnetic bodies embodied in electrical devices possessing a magneto-resistance of some kind.

In conventional FMR or VNA-FMR, the community is well aware that the line shape of a resonance is more complicated than simple arguments based on the Landau-Lifshitz-Gilbert equation would tell. There are for instance substantial contributions from microwave shielding effects Bailleul (2013) (”Eddy currents”) for conductive ferromagnetic films Lin and Kostylev (2015) or ferromagnetic films in contact with (or capacitively coupled to) a conductive layers. A hint to these effect is for instance to compare the lineshapes Lin and Kostylev (2015) for the quasi-uniform precession mode and the first perpendicular standing spin wave modes that occur in different resonance conditions. Note that the experimental lineshapes are already complex in VNA-FMR despite the fact that the dynamics is induced by simple magnetic fields supposedly well controlled.

In contrast, STT-FMR methods rely on torques [spin-orbit torques (SOT)Liu (2011) or STT] that have less hindsight that magnetic fields or that are the targeted measurements. These torques are related to the current across the device and the experimental analysis generally assumes that this current is in phase with the applied voltage. This implicitly assumes that the sample is free of capacitive and inductive responses, even at the microwave frequencies used for the measurement. A careful analysis is thus needed when the STT-FMR methods analyze the phase of the device response to separate the contribution of the different torquesTulapurkar et al. (2005); Sankey et al. (2008); Kubota et al. (2008). Besides, the quasi-uniform mode is often the sole to be analyzed despite that fact that the line shapes of the higher frequency modes can be very different Sankey et al. (2008). Finally, an external field is generally applied in a direction that is not a principal direction of the magnetization energy functional Cheng et al. (2013). While this maximizes the signal, this unfortunately makes numerical simulation unavoidable to model the experimental responses.

With the progress in MTJ technologies, much larger magneto-resistance are now availableIkeda et al. (2008), such that signals can be measured while maintaining sample symmetries, for instance with a static field applied collinearly to the magnetization. In addition, high anisotropy materials can now be incorporated in these MTJs. This leads to a priori much more uniform magnetic configurations in which analytical descriptions are more likely to apply. In this paper, we revisit rf-voltage induced FMR in a situation where the symmetry is chosen so that all torques should yield a priori the same canonical lineshape for all spinwaves excited in the system. We use PMA MTJ disks of sizes 500 nm, on which a quasi-continuum of more that 20 different spin-wave modes can be detected, down to sizes of 60 nm where only a few discrete spinwave modes can be detected. We discuss the lineshapes of the spin-wave signals with the modest objective of determining if at least the Gilbert damping of the dynamically active magnetic layer can be reliably extracted. We show that the linear transverse susceptibility and the non-linear longitudinal susceptibilities must both be considered when a finite dc voltage is applied through the device. We propose a methodology and implement it on a nanopillars made with a standard MgO/FeCoB/MgO free layer system in which we obtain a Gilbert damping of $0.011\pm 0.0003$. This exceeds the value of 0.008 measured on the unpatterned film, which indicates that device-level measurements are needed for a correct evaluation of dissipation.

The paper is organized as follows:

The first section lists the experimental considerations, including the main properties of the sample, the measurement set-up and the mathematical post-processing required for an increased sensitivity. The second section discusses the origins of the measured resonance signals and their main properties. The third section describes how the device diameter affects the spin-wave signals in rf-voltage-induced ferromagnetic resonance. The last section describes how the voltage bias dependence of the spinwave resonance signals can be manipulated to extract the Gilbert damping of the dynamically active magnetic layer. After the conclusion, an appendix details the main features of the spectral shapes expected in ideal perpendicularly magnetized systems.

I Experimental considerations

I.1 Magnetic tunnel junctions samples

We implement our characterization technique on the samples described in detail in ref. Devolder et al., 2016a. They are tunnel junctions with an FeCoB-based free layer and a hard reference system based on a well compensated [Co/Pt]-based synthetic antiferromagnet. All layers have perpendicular magnetic anisotropy (PMA). The perpendicular anisotropy of the thick ($t=2~{}\textrm{nm}$) free layer is ensured by a dual MgO encapsulation and an iron-rich composition. After annealing, the free layer has an areal moment of $M_{s}t\approx 1.8~{}\textrm{mA}$ and an effective perpendicular anisotropy field $\mu_{0}(H_{k}-M_{s})$ = 330 mT. Before pattering, standard ferromagnetic resonance measurements indicated a Gilbert damping parameter of the free layer being $\alpha=0.008$. Depending on the size of the patterned device, the tunnel magnetoresistance (TMR) is 220 to 250%, for a stack resistance-area product is $\textrm{RA}=12~{}\Omega.\mu\textrm{m}^{2}$. The devices are circular pillars with diameters varied from 60 to 500 nm. The materials, processing and device rf circuitry were optimized for fast switching Devolder et al. (2016a) spin-transfer-torque magnetic random access memories (STT-MRAMKhvalkovskiy et al. (2013)) ; the quasi-static dc switching voltage is $\approx 600$ mV. In the present report, the applied voltages shall never exceed 100 mV to minimize spin-transfer-torque effects. The fields will always be applied along (z) which is the easy magnetization axis. The sample will be maintained in the antiparallel (AP) state.

I.2 Measurement set-up

The pillars are characterized in a set-up (Fig. 1) inspired from spin-torque diode experiments Tulapurkar et al. (2005) but an electrical bandwidth increased to 70 GHz. The objective is to identify the regions in the {frequency, field} space in which the magnetization is responding in a resonant manner. The device is attacked with an rf voltage $V_{rf}$. A 10 dB attenuator is inserted at the output port of the synthesizer to improve its impedance matching so as to avoid standing waves in the circuit. This improves the frequency flatness of the amplitude of the stimulus arriving at the device. To ease the detection of the sample’s response, the rf voltage is pulse-modulated at an ac frequency $\omega_{ac}/(2\pi)=50~{}\textrm{kHz}$ (Fig. 1). The current passing through the MTJ has thus frequency components at the two sidebands $\omega_{rf}\pm\omega_{ac}$. The ac voltage which appears across the device is amplified and analyzed by a lock-in amplifier. We shall discuss the origin of this ac voltage in section II. Optionally, the device is biased using a dc sourcemeter supplying $V_{\textrm{dc}}$ and measuring $I_{dc}$.

Figure 2 shows a representative map of the $\frac{dV_{ac}}{dH_{z}}$ response obtained on a pillar of diameter 300 nm with $V_{dc}=10~{}\textrm{mV}$. As positive fields are parallel to the free layer magnetization, the spin waves of the free layer appear with a positive frequency versus field slope, expected to be the gyromagnetic ratio $\gamma_{0}$ of the free layer material (see appendix). Conversely, the reference layer eigenmodes appear with a negative slope, expectedly $-\gamma_{0}$, where this time $\gamma_{0}$ is gyromagnetic ratio of the reference layer material combination. Working in the AP state is thus a convenient way to easily distinguish between the spinwaves of the free layer and of the reference layers. Note that the gyromagnetic ratios $\gamma_{0}$ of the free layer mode and the reference layer modes differ slightly owing to their difference chemical nature. The free layer has a Landé factor $g=2.085\pm 0.015$ where the error bar is given by the precision of the field calibration; the reference layer modes are consistent with a 1.2% larger gyromagnetic ratio. The accuracy of this latter number is limited only by the signal-to-noise ratio in the measurement of the reference layer properties. Looking at Fig. 2, one immediately notices that the linewidths of the reference layer modes are much broader than that of the free layer. While the linewidh of the reference layer modes will not be analyzed here, we mention that this increased linewidth is to be expected for reference layers that contain heavy metals (Pt, Ru) with large spin-orbit couplings, hence larger damping factors Kambersky (2007).

I.3 Experimental settings

In practice, we choose an applied field interval of $[-110,110~{}\textrm{mT}]$ that is narrow enough to stay in a state whose resistance is very close to that of the remanent AP state. The frequency $\omega_{rf}/(2\pi)$ is varied from 1 to 70 GHz; we generally could not detect signals above 50 GHz. The practical frequency range $2\pi\times 50~{}\textrm{GHz}/\gamma_{0}\approx 1.6~{}\textrm{T}$ is much wider that our accessible field range. For wider views of the experimental signals (for instance when the spin-wave density of states is the studied thing), we shall thus prefer to plot them versus frequency than versus field. The response is recorded pixel by pixel in in the {frequency, field} space. The typical pixel size is $\{\delta H_{z}\times\delta f\}=$ {1 mT $\times$ 50 MHz}. The field and frequency resolutions are thus comparable (indeed $2\pi\times\delta f/\gamma_{0}=1.7$ mT).

I.4 Signal conditioning

I.4.1 Mathematical post-treatments

Finally, despite all our precautions to suppress the rectifying phenomena that do not originate from magnetization dynamics, we have to artificially suppress the remaining ones. This was done by mathematical differentiation, and we generally plot $\frac{dV_{ac}}{df}$ or $\frac{dV_{ac}}{dH_{z}}$ in the experimental figures (Figs. 2-5).

I.4.2 Dynamic range improvement by self-conformal averaging

A special procedure (Fig. 3) is applied when a better signal to noise ratio is desired while the exact signal lineshape and amplitudes are not to meant to be looked at. This procedure harnesses the fact that the normalized shape of the sample’s response is essentially self-conformal when moving across a line with $\frac{d\omega}{dH_{z}}=\gamma_{0}$ in the {frequency, field} parameter space (see appendix). The procedure consists in calculating the following primitive:

[TABLE]

in which the integration contour is the segment linking the points ($-H_{z}^{max},f_{0}-\gamma_{0}H_{z}^{max}$) and ($H_{z}^{max},f_{0}+\gamma_{0}H_{z}^{max}$) in the {field, frequency} parameter space. Such contours appear as pixel columns in Fig. 3(b). This primitive (eq. 1) is efficient to reveal the free layer spin-wave modes that yield an otherwise too small signal. For instance when only 7 modes can be detected in single field spectra [Fig. 3(a)], the averaging procedure can increase this number to typically above 25. The averaging procedure is also effective in suppressing the signals of the reference layer as these laters average out over a contour designed for the free layer mode when in the AP state. However as the linewidth of the free layer modes is proportional to the frequency, it is not constant across the contour; the higher signal to noise ratio is thus unfortunately obtained at the expense of a distorted (and unphysical) lineshape. Note also that this procedure can not be applied to the quasi-uniform precession mode as will be explained in section II.4.2).

II Origin and nature of the rectified signal

Let us now discuss the origin of the demodulated ac voltage. In this section, we assume that the reference layer magnetization is static but not necessarily uniformly magnetized. We can thus express any change of the resistance by writing $\delta R=\frac{\delta R}{\delta M}\delta M$ where $\delta$ has to be understood as a functional derivative with respect to the free layer magnetization distribution.

II.1 The two origins of the rectified signals

The ac signal can contain two components $V_{1,~{}ac}$ and $V_{2,~{}ac}$ of different physical origins Devolder et al. (2016b). The first component is the ’standard’ STT-FMR signal: the pulse-modulated rf current is at the frequency sidebands $\omega_{rf}\pm\omega_{ac}$ and it rectifies to ac any oscillation of the resistance $\delta R_{rf}$ occurring at the frequency $\omega_{rf}$. We simply have $V_{1,~{}ac}=\delta R_{{rf}}\times i_{\omega_{rf}\pm\omega_{ac}}$.

The second ac signal ($V_{2,~{}ac}$) is related to the change of the time-averaged resistance due to the population of spinwaves created when the rf current is applied Cheng et al. (2013). Indeed the time-averaged magnetization distribution is not the same when the rf is on or off. This change of resistance $\delta R_{ac}$ can revealed by the (optional) dc current $I_{dc}$ passing through the sample, i.e. $V_{2,~{}ac}=\delta R_{ac}\times I_{dc}$.

Note that a third rectification channel Kupferschmidt et al. (2006) can be obtained by a combination of spin pumping and inverse spin Hall effect in in-plane magnetized systems Mihalceanu et al. (2017). This third rectification channel yields symmetric lorentzian lines when applied to PMA systems in out-of-plane applied fields (see eq. 23 in ref. Kupferschmidt et al., 2006). Besides, the spin-pumping is known to be largely suppressed by the MgO tunnel barrier Baker et al. (2016), such that we will consider that we can neglect this third rectification channel from now on. In summary, we have:

[TABLE]

[TABLE]

This has important consequences.

II.2 Compared signal amplitudes in the P and AP states

The first important consequence of Eq. 2 and 3 is that the signal amplitude depends on the nature of the micromagnetic configuration. As intuitive, both $V_{1,~{}ac}$ and $V_{2,~{}ac}$ scale with how much the instantaneous device resistance depends on its instantaneous micromagnetic configuration. This is expressed by the sensitivity factor $\frac{\delta R}{\delta M}$ which is essentially a magneto-resistance. We expect no signal when the resistance is insensitive to the magnetization distribution at first order (i.e. when $\frac{\delta R}{\delta M}\equiv 0$).

In our samples, the shape of the hysteresis loop (Fig. 1) seems to indicate that the free layer magnetization is very uniform when in the Parallel state. Consistently, the experimental rectified signal were found to be weak signals when in the P state. Conversely, there is a pronounced curvature in the AP branch of the $R(H_{z})$ hysteresis loop (see one example in the inset of Fig. 1). This indicates that the resistance is much dependent on the exact magnetization configuration when in the AP state. Consistently, this larger $\frac{\delta R}{\delta M}$ in the AP state is probably the reason why the rectified signal is much easier to detect in the AP state for our samples.

II.3 Bias dependence of the rectified signals

The second important consequence of Eqs. 2-3 concerns the dependence of the rectified ac signals $V_{1,~{}ac}$ and $V_{2,~{}ac}$ on the dc and rf stimuli. As $\delta M_{rf}$ scales with the applied rf torque according to a linear transverse susceptibility ($\Re e(\chi_{xx})$, see appendix), $V_{1,~{}ac}$ is expected to scale with the rf power ${V_{rf}}^{2}$ (see Eq. 2) independently from the dc bias, i.e. we have

[TABLE]

In contrast, $\delta M_{ac}$ is related to a longitudinal susceptibility and is thus quadratic with the rf torques (see appendix). Using Eq. 3, we thus expect the following bias dependence

[TABLE]

II.4 Peculiarities of the quasi-uniform precession (QUP) mode

The last important consequence of Eqs. 2-3 concerns specifically the quasi-uniform precession (QUP) mode that shows a peculiar ac signal.

II.4.1 Quasi-absence of STT-FMR like signal for the quasi-uniform precession mode

In the idealized macrospin case (see appendix) the uniform precession is perfectly circular with no rf variation of $M_{z}$ at any order. If the fixed layer was uniformly magnetized along exactly (z), this would lead $V_{1,~{}ac}=0$ such that the signal of the QUP mode would be given by purely $V_{2,~{}ac}$. This qualitatively ’pure $V_{2,~{}ac}$ character’ is confirmed experimentally by the fact that the signal of the QUP mode systematically changes sign with $V_{dc}$ in our sample series (not shown).

II.4.2 Strong dependence of the QUP signal amplitude with the applied field

In addition, the experimental signal of the quasi-uniform mode is found to disappear at low fields [see Figs. 2, 3(b) and 4(a, b)] exactly at the apex of the AP branch of the hysteresis loop (Fig. 1), i.e. for the field leading specifically to $\frac{dR}{dH_{z}}=0$. Moreover, the amplitude of the ac signal of the QUP mode appears to be essentially linearly correlated with the loop slope $\frac{dR}{dH_{z}}$ (not shown). For instance, the $V_{2,~{}ac}$ of the QUP mode changes sign when the applied field crosses the apex of the $R(H_{z})$ loops (Fig. 1).

The reason stems probably from the sensitivity factor $\frac{\delta R}{\delta M}$ and its correlation with the loop slope $\frac{dR}{dH_{z}}$; in some sense, a large loop slope should translate in a large sensitivity factor. While a numerical evaluation of this correlation goes beyond the scope of this paper, we stress that if the magnetization was perfectly uniform there would be a one-to-one correlation between loop slope $\frac{dR}{dH_{z}}$ and magnetoresistance sensitivity factor $\frac{\delta R}{\delta M}$. This trend remains qualitatively true for the QUP mode. Indeed as the hysteresis loop is monitoring the spatial average of the magnetization, it is more insightful for the uniform mode than for any other (higher order) modes whose dynamic profiles spatially average to essentially zero Klein et al. (2008); the correlation between $\frac{dR}{dH_{z}}$ and $\frac{\delta R}{\delta M}$ is thus expected to be maximal for quasi-uniform changes of the magnetization configuration.

While this property – the disappearance of the QUP mode signal when $\frac{dR}{dH_{z}}=0$ – can be used to distinguish the QUP mode from the higher order spin waves, the pronounced field dependence of the QUP signal complicates the analysis, as it prevents to conveniently analyze the field derivative of the ac signal ( I.4.1). In the remainder of this paper we shall focus on only higher order modes to avoid such difficulties.

II.5 Signals for non-uniform spin-waves

Before analyzing the spin-wave density of states (section III), let us comment on the amplitude of the STT-FMR-like signal $V_{1,~{}ac}$ for the non-uniform spin-waves. In the perpendicular magnetization state, these spin-waves have a circular precession Kalinikos and Slavin (1986). By symmetry, the resistance is not expected to change during a period of circular precession when in the perfect collinear cases and for radial spin waves maintaining the cylindrical symmetry of the system. In other words, when the dynamical magnetization of the eigenmode maintains the cylindrical symmetry and when the free and reference layers equilibrium magnetizations follow $\vec{M}_{\textrm{free}}\times\vec{M}_{\textrm{ref}}=\vec{0}$ everywhere in the (xy) plane, with $\times$ being the conventional vector product) the device resistance is not expected to oscillate. While we can not identify to what extent we depart from this ideal situation, we speculate that this perfect collinearity does not happen in practice at least because of finite thermal fluctuations. The effect of thermal fluctuations on the device resistance is not averaged out for non-uniform spin-waves, while it could be essentially averaged out for the QUP mode analyzed earlier. In practice a finite variation of the resistance $\delta R_{rf}\neq 0$ is always present during a precession period for a non-uniform spin-wave. This provides a finite sensitivity to any spin-wave mode. This resistance variation at $\omega_{rf}$ has the spectral shape of a transverse susceptibility term $\Re e(\chi_{xx})$ (see appendix).

III Spin wave density of states against lateral confinement

Any reliable analysis of a spectral lineshape or linewidth requires to determine priorly how many spin-waves contribute to the lineshape under study. Therefore, before discussing the lineshapes of the individual spin-wave modes, let us determine how the lateral confinement influences the measured rectified signal. The impact of the device diameter on the spectral signals is reported in Fig. 4.

III.1 Spin-waves within the references layers

Fig. 4 indicates that the modes of the reference layers have frequencies that are almost not affected by the device diameter. This fact is related to the well compensated character of the synthetic antiferromagnet that composes the reference layers. Indeed the internal demagnetizing fields compensate to some extent, such that they do not influence the frequency of the acoustical mode of a SAF as much as the anisotropies and the interlayer exchange couplings do.

III.2 Spin-waves within the free layer

Conversely, the frequencies of the modes of the free layer are strongly affected by the device diameter (Fig. 4). First, the modes are pushed to higher frequencies as the device is shrunk. At remanence, the lowest frequency mode is at $f_{\textrm{QUP}}=12.3~{}\textrm{GHz}$ for a diameter of 500 nm; it reaches 19.5 GHz for 60 nm devices (not shown). Second, the frequency spacing between the free layer modes increases substantially when downsizing the device.

The first effect – increase of $f_{\textrm{QUP}}$ at downscaling – is indicative of a dependence of some effective fields with the device diameter. Among the effective fields, the only ones that vary with the diameter are the exchange fields (positive contribution to the frequency $f_{\textrm{QUP}}$ if magnetization is non-uniform), the demagnetizing fields (positive contribution to the frequency $f_{\textrm{QUP}}$ at downscaling) and the local effective anisotropy fields in case some process damages alter locally the interface anisotropy at the perimeter of the free layer (negative contribution to the frequency $f_{\textrm{QUP}}$ at downscaling) or alter the local magnetization of the rim of the free layer (positive contribution to the frequency $f_{\textrm{QUP}}$ at downscaling). The exchange fields are related to the non uniformities of either the static configuration –the fact that the AP state is not perfectly uniform as inferred previously from the loop in Fig. 1– or non uniformities of the dynamic magnetization –i.e. the fact that the quasi-uniform mode is not a strictly uniform mode–. If the frequency increase was due to the sole demagnetizing effects, it could be estimated from the demagnetizing factors of disks Mizunuma et al. (2013) which are $N_{z}\approx 1-(3\pi/8)t/a$, where $t$ and $a$ are the thickness and radius of the free layer. However a $f_{\textrm{QUP}}$ against $1/a$ plot (not shown) has a perceivable curvature near all sizes; an unwise linear fit through $f_{\textrm{QUP}}$ against the expected $\gamma N_{z}$ would give a slope of $\approx 2.5~{}\textrm{T}$, which is obviously too large for the magnetization of the free layer. This indicates that the sole change of the global shape anisotropy with the device diameter is insufficient to account for the increase of $f_{\textrm{QUP}}$ at downscaling: exchange contributions or non uniformities induced by process damages also contribute to the frequencies. Exchange contributions should not contribute for the largest devices, however even for those devices the experimental frequencies are larger than the ones expected from global shape anisotropy only, which argues for some process damages. Since the TMR is almost independent of the device size Devolder et al. (2016a), we can reasonably assume that the MgO/FeCoB interface is not substantially affected by the patterning and that consequently the interface anisotropy is essentially preserved at the rim of the free layer. We conclude that part of the increase of $f_{\textrm{QUP}}$ at downscaling is due to a reduced magnetization (magnetically ”dead” or weak zone) near the edges of the free layer. This interpretation is probably very much stack and process technology dependent, hence it should not be considered as general.

The second effect – the increased frequency spacing between the modes at small diameters – is the expected effect of the confinement of the spin waves and the resulting increase of the exchange contribution to the mode frequenciesDevolder et al. (2016b). The eigenmodes of perpendicularly magnetized circular disks are well understood and can be described analytically in a semi-quantitative manner Kakazei et al. (2004); Klein et al. (2008); Arias and Mills (2009); Naletov et al. (2011); Nedukh et al. (2013); Munira and Visscher (2015). The frequency spacing between the lowest frequency modes scales with $\gamma_{0}H_{J}$ where $H_{J}=\frac{2Ak^{2}}{\mu_{0}M_{S}}$ is a generalized exchange field with $A$ the exchange stiffness. The effective wavevector $k$ is reminiscent of the lateral confinement and reads $k^{2}=({u_{2}^{2}-u_{1}^{2}})/a^{2}\approx 9/a^{2}$ where $u_{1}$ and $u_{2}$ are the first zeros of the first and second Bessel functions Klein et al. (2008). The lowest frequency spinwave modes can be resolved only if their frequency spacing is comparable or greater than their linewidth $2\alpha\gamma_{0}(H_{z}+H_{k}-M_{s}+H_{J})$ (see appendix).

This condition can be used to define a critical device diameter:

[TABLE]

For large devices with $a\gg a_{\textrm{crit}}$ we expect to observe a quasi-continuum of overlapping modes above $f_{\textrm{QUP}}$, while discrete non-overlapping modes are anticipated in the opposite limit. Typical parameters of an FeCoB-based free layer include a magnetization of $\mu_{0}M_{s}=1.2$ T and a Gilbert damping ofDevolder et al. (2016c) $\alpha=0.01$. From the quasi-uniform mode frequency, we can get our effective anisotropy which is $H_{k}-M_{s}=330$ kA/m. If the exchange stiffness of the free layer was bulk-like (i.e. $A=22$ pJ/m) like in ref. Devolder et al., 2016b, the critical diameter would be $2a_{\textrm{crit}}=444~{}\textrm{nm}$. In practice the small frequency spacings between the modes of our samples indicates that the exchange stiffness of our free layer is in the range of 6-7 pJ/m i.e. well below the bulk value. This estimate of the exchange stiffness was deduced assuming perfectly pinned boundary conditions for the spin-waves at the device edge, which is a questionable Guslienko et al. (2002) assumption. However the exchange stiffness is anyway weak in the free layer and this can be also qualitatively seen directly from the spin wave spectroscopy: indeed the frequency spacing of the lowest modes of the reference layer system is typically twice larger that the frequency spacing of the lowest modes of the free layer [see for instance Fig. 3(b)]. While the reason for this small value of the free layer exchange stiffness is not entirely clear, we emphasize that having such a small exchange stiffness is not uncommon in magnetic systems that comprise only a small number of atomic layers, starting for instance from 2 pJ/m for a single layer of iron Romming et al. (2015). Anyway with these parameters, we expect a clear separation of the lowest frequency modes at remanence provided that the device diameter is much smaller than $2a_{\textrm{crit}}=250~{}\textrm{nm}$.

In practice for 300 and 500 nm devices a fine structure can still be detected in the spin-wave density of states [see Fig. 4(c)] but it is hard to count the modes and guess their frequencies out of this fine structure. In the remainder of this paper, we shall thus only consider devices of diameter less than 200 nm, in which the different spin-wave modes can be unambiguously resolved [see Fig. 4(e-f)].

IV Lineshape evolutions with bias and extraction of the Gilbert damping

Let us now compare the shapes of the experimental rectified signal with those expected (see appendix). For that purpose we harness the different bias dependences (Table 1) of the rectified signals $V_{1,~{}ac}$ and $V_{2,~{}ac}$ to isolate each of them in the experimental signal $V_{ac}$. We identify $V_{1,~{}ac}$ to the experimental curve $V_{ac}$ measured at ${V_{dc}=0}$, and we construct an estimate of $V_{2,~{}ac}$ by subtracting $V_{ac}$ measured at ${V_{dc}=0}$ from that measured at ${V_{dc}\neq 0}$ (see Table 1). Note because of this subtraction, any dependence of the spin-wave frequency with the dc voltage will prevent the measurement of the voltage dependence of the damping factor or of the voltage dependence of the exciting torques.

We illustrate this procedure in Fig. 5 in which we plot the field derivatives of the so-calculated $V_{1,~{}ac}$ and $V_{2,~{}ac}$ rectified signals in both fixed field or fixed frequency experimental conditions for a device of diameter 90 nm. We center the curves on the second lowest frequency eigenmode since it provides the largest signals and it is reasonably separated from both the quasi-uniform precession mode and from the other high order modes. The obtained experimental $V_{1,~{}ac}$ curves [see Fig. 5 (a) and (d)] have the expected line shapes (see appendix) with a negative peak surrounded by two tiny positive halos (areas shaded in red). The separation between the two zero crossings is $9.5\pm 0.5~{}\textrm{mT}$ or $285\pm 10~{}\textrm{MHz}$. The obtained experimental $V_{2,~{}ac}$ curves also have the expected line shape of the derivative of a Lorentzian distribution (see appendix). As expected, the sign of the response changes with the dc bias voltage. The separation between the positive and negative maxima of the distribution are $6.1\pm 0.5~{}\textrm{mT}$ and $170\pm 10~{}\textrm{MHz}$.

These four different ways of measuring the linewidths [Fig. 5 (a, d, b, e)] are consistent with a free layer damping of $\alpha=0.011\pm 0.0003$. Indeed this value of damping would predict linewidths of $2\alpha f=295~{}\textrm{MHz}$ and $2\mu_{0}\alpha\omega/\gamma_{0}=9.54~{}\textrm{mT}$ (materialized as black bars in [Fig. 5 (a, d) and (d)]) and $1.15\alpha f=171~{}\textrm{MHz}$ or $1.15\alpha\omega/\gamma_{0}=6.21~{}\textrm{mT}$ (materialized as blue bars in [Fig. 5 (b, c, e, f)]).

This proposed value of damping is also consistent with the linewidths of higher order spin waves that appear at larger frequencies but with a lower signal. This is illustrated in fig. 6 where a comparison is drawn between the values of the damping estimated for each applied field from the second (and most intense) mode and from the third mode for a device of diameter 80 nm. The used procedure is a direct fit of the experimental lineshapes to the derivative of Eq. 7 with the damping, the resonance frequency and the signal amplitude as free parameters. For this specific device, the estimates of the damping parameter are subjected to a random error of standard deviation 0.0018 around a mean value of $\alpha=0.0119$. It is interesting to note that the non-local contributions Nembach et al. (2013); Wang et al. (2015) to the damping expected for the relatively large wavevectors of the second and third spin-wave modes seem to be too small to be observed in our samples. Note that as mentioned earlier, the same procedure can in principle not be applied to the quasi-uniform mode since it exhibits a strong dependence of the mode amplitude with the field which invalidates the procedure to some extent. For the sake of completeness of this paper, we have anyway fitted the experimental QUP lineshapes with the field derivative of Eq. 8; this is not possible near zero field, as the corresponding signal vanishes. The value of the Gilbert damping that would be illegitimately deduced would be $0.01$, i.e. 20% lower than the correct value. Besides, the estimates from the QUP mode would exhibit a substantially larger spread in the fit results [compare the histograms in in fig. 6(b)]. For these two reasons, we consider that the reliable estimate of the damping is the one extracted from the non-uniform modes.

Above 100 mV of dc bias, the amplitude of the constructed experimental $V_{2,~{}ac}$ start to depart from proportionality with $V_{dc}$ and a frequency shift is observed, as expected when dc field-like spin torques are applied. This comes with by a distortion of the line shape, probably linked to the modification of the spin waves lifetimes by spin-transfer torque as commonly observed in in-plane magnetized MTJs Petit et al. (2007).

V Summary and conclusions

In this paper, we have studied how to use rf-voltage-induced ferromagnetic resonance to study the spin-wave density of states and the Gilbert damping in perpendicularly magnetized disks embodied in magnetic tunnel junctions. We have applied the field along the easy axis to preserve the cylindrical symmetry of the magnetization energy functional. The interest of this configuration is that all the current-induced torques that potentially excite the dynamics yield the same type of susceptibility spectral shape. Additionally, this configuration is the sole in which the applied field and the frequency play similar roles near FMR so that consistency crosschecks between variable-field and variable-frequency experiments can be performed to reveal and suppress potential experimental artefacts.

Working in a situation in which the fixed layer and the free layer are oppositely magnetized in a convenient way to classify the spin-waves according to their hosting layer, as the two sub-systems have opposite eigenmode frequency-versus-field slopes. The dc bias dependence of the signal of the quasi-uniform mode is peculiar and can be used to ambiguously identify the free layer quasi-uniform mode in the manifold of spin-waves. The non-uniform (higher order) spin-waves are easier to analyze, as their amplitudes weakly depend on the applied field so that field differentiation can be used safely for background subtraction. Optionally, the dynamic range of the experiment can be improved by self-conformal averaging of the resonance spectra.

The unambiguous identification of the spin-wave frequencies requires devices that are sufficiently small to avoid that the spinwave modes overlap into a quasi-continuous density of states. The critical device size is set by the exchange stiffness, the damping, the magnetization and the effective anisotropy field. In practice, device diameters below 200 nm are needed in our low-damped FeCoB-based PMA system.

For each spin-wave mode, the rf-voltage-induced spin-wave spectra contain contributions from two different physical mechanisms. The first one is the standard STT-FMR-like signal, whose spectral shape is a linear transverse susceptibility term. It is independent from the dc voltage applied across the MTJ. The second one is a variation of the time-averaged magnetic configuration when the rf voltage is applied. It is proportional to the dc voltage applied across the MTJ and it has the spectral shape of a non-linear longitudinal susceptibility. The bias dependence can be used to separate these two signals. The analysis of their spectral shape yields the Gilbert damping within the precessing layer. A single value of the damping factor is found to account for the lineshapes of all studied spin-waves.

The spectra of rf-voltage-induced rectified voltages for a vanishing dc voltage bias are in principle sufficient to get the Gilbert damping of the dynamically active layer. However as microwave methods are prone to artefacts, a consistency check exploiting the bias dependence of the resonance spectra is useful for a consolidation of the numerical estimation of the damping.

This work was supported in part by the Samsung Global MRAM Innovation Program, who provided also the samples. Critical discussions with Vladimir Nikitin, Jean-Paul Adam, Joo-Von Kim and Paul Bouquin are acknowledged.

VI Appendix: susceptibilities in an idealized PMA film

In this appendix, our aim is to determine the transverse and longitudinal microwave susceptibility versus frequency $f$ and static field $H_{z}$ for a PMA film as a response to an harmonic transverse field $h_{x}\cos(\omega t)\vec{e_{x}}$. We shall write the equations with this transverse field $h_{x}$ but any other effect that yields a torque possessing a component transverse to the static magnetization will yield similar lineshapes. This includes current-induced Oersted-Ampere fields but also Slonczewski STT and field-like STT as soon as the reference layer magnetization $\vec{M}_{\textrm{ref}}$ is not strictly collinear with that of the free layer $\vec{M}_{\textrm{free}}$. The susceptibility tensor will be used to deduce the shape of the line expected in rf-voltage-induced FMR, as summarized in Table 1. Throughout this appendix, we assume a dc field $H_{z}$ perfectly perpendicular to the plane and a free layer magnetization $\vec{M}=M_{z}\vec{e}_{z}+m_{x}\vec{e}_{x}+m_{y}\vec{e}_{y}$, where the transverse terms are assumed small and written as complex numbers in the frequency space. For conveniency, we will use the notation $H^{\prime}=H_{z}+H_{k}-M_{s}$. We shall also write the frequencies in field units and define $\omega^{\prime}=\omega/\gamma_{0}$. This is meant to emphasize the fact that the generalized field $H^{\prime}$ and the generalized frequency $\omega^{\prime}$ play very similar roles in the FMR of perpendicularly magnetized macrospin when near resonnance. We shall systematically assume that $\alpha\ll 1$ and only keep the lowest order of the damping terms in the equations. In sections VI.1 -VI.3 we make the macrospin approximation. This approximation is discussed in section VI.4.

VI.1 Transverse linear susceptibility

Following the usual procedure, we project the linearized Landau-Lifshitz-Gilbert equation along $\vec{e}_{x}$ et $\vec{e}_{y}$ :

[TABLE]

We then invert this system of equations to get the susceptibilities $m_{x}=\chi_{xx}h_{x}$ and $m_{y}=\chi_{yx}h_{x}$. They are:

[TABLE]

and

[TABLE]

Several points are worth to remind:

(i) The dc transverse susceptibilities are $\chi_{xx}^{dc}=M_{s}/H^{\prime}$ and $\chi_{yx}^{dc}=0$.

(ii) The in-phase transverse susceptibility $\chi_{xx}$ is peaked at the FMR condition $\omega^{\prime}=H^{\prime}$. It reaches $\chi_{xx}^{FMR}=-\frac{1}{2i\alpha}\chi_{xx}^{dc}$. (iii) When near the FMR condition, we have $\chi_{yx}\approx-i\chi_{xx}$. Hence the two transverse components of the magnetization are in quadrature and the forced precession is essentially circular. Note that this holds true despite the fact that the pumping field is linearly polarized (i.e. along $(x)$ only). It would also remain true for other (e.g STT) pumping torques.

From Eq. 5 we deduce the classical expressions for the real and imaginary parts of the transverse susceptibility:

[TABLE]

[TABLE]

The lineshapes given by the above expressions are shown in Fig. 7. Their main properties are summarized in Table 1.

VI.2 Longitudinal non-linear susceptibility

Let us now express the non-linear change of the longitudinal magnetization $\Delta M_{z}$ that occurs due to the precession. This can be viewed as an rf-induced reduction of the remanence. Using the circularity of the precession near the FMR resonance and the conservation of the magnetization norm to second order in $m_{x,~{}y}$, one gets:

[TABLE]

where $||h_{x}||$ is the (constant) amplitude of the applied rf field. It is worth noticing that the longitudinal loss of the magnetization is stationary (constant in time) despite the fact that the magnetization precesses continuously.

VI.3 Lineshapes and linewidths of the susceptibiliies and their derivatives

The different susceptibility expressions are plotted in Fig. 7. The lineshape of the functions $\Delta M_{z}$ and $\chi_{xx}$ are essentially determined by their denominator which are the fast varying functions of eqs. 8 and 9. As $\chi_{xx}$ and $\Delta M_{z}$ are two signatures of the same resonance process, their denominators are equal (see eqs. 8 and 9) and a simple algebra confirms that $\Im m(\chi_{xx})$ and $\Delta M_{z}$ lead to the same frequency or field linewidths (Half Width at Half Maximum) which are:

[TABLE]

Note that $\Delta\omega$ (resp. $\Delta H_{z}$) is also the frequency (resp. field) spacing between the positive maximum and negative maximum of $\Re e(\chi_{xx})$ about the FMR condition.

We stress that this linewidth is different from that obtained by conventional FMR in which people examine the derivative of the absorption signal $\frac{d\Im m(\chi_{xx})}{dH_{z}}$ versus $H_{z}$. The peak-to-peak separation of the conventional FMR signal is $\Delta H_{z}=\frac{2}{\sqrt{3}}\alpha H^{\prime}=\frac{2}{\sqrt{3}}\alpha\omega^{\prime}$. The factor $2/\sqrt{3}$ is 1.1547. The spectral shapes of $\Re e(\chi_{xx})$ and $\Delta M_{z}$ as deduced above are to be used in the main part of the paper to describe respectively $V_{1,~{}ac}$ and $V_{2,~{}ac}$ (Table 1).

VI.4 Linewidth beyond the macrospin approximation

Some words of caution are needed as the calculations done so far the appendix are for the uniform precession mode of a macrospin, while most of our experimental results were obtained on the higher order (non-uniform) spin-waves. When modeling non-uniform spin-waves, one needs to take into account additional exchange and dipole-dipole terms.

For non-uniform spin-waves in perpendicularly magnetized system, the exchange fields related to the non uniformity of the dynamical magnetizations $m_{x}$ and $m_{y}$ can be added to $H^{\prime}$ to form a new generalized field $\tilde{H}=H+H_{k}-M_{s}+\frac{2Ak^{2}}{\mu_{0}M_{S}}$, where $k$ is a generalized wavevectorKlein et al. (2008); Naletov et al. (2011); Munira and Visscher (2015). The additional exchange contributions act on $m_{x}$ and $m_{y}$ on equal footing, hence they maintain the circularity of the precession. The Gilbert linewidth for a non-uniform spin-wave is simplySlavin and Kabos (2005); Kim (2006):

[TABLE]

Where this equation holds as $\tilde{H}$ is a circular term. As a result of Eq. 11 the proportionality of an eigemode linewidth to its frequency (Eq. 10) is not broken by the exchange contributions in non-uniform spin-waves.

Conversely, the directional nature of the dipole-dipole interaction is such that the related effective fields act differently on the two dynamical magnetizations $m_{x}$ and $m_{y}$ for spin-waves having a non-radial character. As a result the dynamic demagnetizing fields can not be simply added to the circular precession $H^{\prime}$ term and they induce some ellipticity of the precession of the non-uniform spin-waves. Dipole-dipole interactions make the eigenmode frequency non-linear with the field (see Eq. 52 in ref. Kalinikos and Slavin, 1986). For the lowest lying non-uniform spin-wave the dipole-dipole stiffness field is $M_{S}kt/2$ with $k\approx\pi/a$. It is negligible against the generalized field $\tilde{H}$ for the thickness used in practice in PMA systems meant for spin-torque applications. The precession stays therefore essentially circular for all the modes observed experimentally here; we will thus consider that it is legitimate to use Eq. 10 and deduce the damping from the ratio of the half frequency linewidth to the eigenmode frequency.

Finally, we would like to mention that our method is not restricted to the PMA materials only: it should hold when the considered spin-waves are quasi-circular. In particular, this is the case of exchange-dominated spin waves in in-plane magnetized systems Dvornik and Kruglyak (2011).