Tunability versus deviation sensitivity in a non-linear vortex oscillator
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Tunability versus deviation sensitivity in a non-linear vortex oscillator
S.Y. Martin, C. Thirion, C. Hoarau, C. Baraduc, and B. Diény SPINTEC, UMR-8191,CEA-INAC/CNRS/UJF-Grenoble 1/Grenoble-INP,17 rue des martyrs, 38054 Grenoble Cedex 9, France Institut Néel, CNRS et Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France (Dated: October 18, 2018)Frequency modulation experiments were performed on a spin torque vortex oscillator for a widerange of modulation frequency, up to 10 % of the oscillator frequency. A thorough analysis of theintermodulation products shows that the key parameter that describes these experiments is thedeviation sensitivity, which is the dynamical frequency-current dependence. It differs significantlyfrom the oscillator tunability discussed so far in the context of spin-transfer oscillators. The essentialdifference between these two concepts is related to the response time of the vortex oscillator, driveneither in quasi-steady state or in a transient regime.
PACS numbers: 75.76.+j, 42.60.Fc, 75.78.FgKeywords: Spin Transfer Torque, Magnetic tunnel junction, Magnetic vortex dynamics, Tunable Oscillators,Modulation and Tunability
I. INTRODUCTION
Spin-transfer oscillators are based on the excitationof magnetization precession by a large dc current den-sity. These oscillators combine many interesting features,such as broad-band frequency operation, small size, easyintegration and scalability. In this context, vortex os-cillators are promising devices, with intrinsically highoutput power and narrow linewidth . The oscillationis due to the gyrotropic motion of a magnetic vortexset into motion by the spin-transfer torque. Such sys-tems were initially studied with nanocontacts on spinvalve structures , and, later, in magnetic tunnel junc-tion nanopillars under an out-of-plane applied field.When characterizing an oscillator for telecommunica-tion applications, studying the modulation of the output( carrier ) by the information channel ( modulation wave )is essential to evaluate its potential. Moreover, suchfrequency modulation experiments lead to a better in-sight into the magnetization dynamics. Up to now, suchstudies were devoted only to macrospin oscillators. Herewe performed low-noise frequency modulation measure-ments on a vortex oscillator, based on a magnetic tunneljunction nanopillar in an in-plane field . We also de-velop an analysis based on modulation theory to treatthe whole set of data at once. It appears that the de-scription previously used to describe modulation exper-iments where the modulation frequency was much lessthan the natural frequency, does not apply when themodulation frequency is a significant fraction of the nat-ural frequency. More precisely, the vortex response timeappears to play a significant role, so that the concept ofdeviation sensitivity has to be introduced to explainthe observations. Until now, deviation sensitivity wasoverlooked in the case of spintronic oscillators: it cor-responds to the dynamical dependence of the oscillatorfrequency with an applied current that varies with time.We show that this frequency dependence differs stronglyfrom that of a quasi-static experiment. We emphasize that the concept of deviation sensitivity differs signifi-cantly from the tunability discussed so far in the contextof macrospin oscillators. II. EXPERIMENT
Our samples are magnetic tunnel junc-tions with an ultra-low resistance area prod-uct ( . µ m ) of the following composition: IrM n /CoF e /Ru . /CoF e . /AlOx/CoF e /N iF e .The subscripts represent the layers’ thickness in nm .They are etched as pillars of
300 nm diameter. In aprevious paper we have shown that injecting a largedc current through the sample induces the formationof a magnetic vortex in the free layer due to the largeOersted field. At some critical current, spin transfertorque induces gyrotropic motion of the vortex, thusleading to a large rf response. Its signature is seen in thepower spectral density of the junction, that shows a largepeak around
400 MHz and up to − harmonics. Thedynamical properties of this oscillator were thoroughlystudied with respect to synchronization, leading us todeduce it behaves as a parametric oscillator . Here weexamine frequency modulation on the same samples,in the same experimental conditions. In contrast toother modulation experiments performed on spintronicoscillators , we sweep the modulation frequency andnot the modulation power. The tunnel junction is biasedwith a large dc current and is simultaneously excited bya small “low” frequency current ( ω m / π = 3 −
40 MHz )provided by a microwave source. The experiment isrealized as follows: the sample is polarized by a dccurrent through the dc-port of a bias-tee and the ac-portis connected to a power-splitter. The ac-current isdelivered by a microwave source to one port of thispower-splitter, whereas the power spectral density of thesample is measured on a spectrum analyzer connectedto the other port. The amplitude of the dc current used( I dc ) is typically about
20 mA whereas the amplitude ofthe ac current ( i ac ) is about , corresponding to afrequency deviation of approximatively
15 MHz . In thefollowing, the modulation power is corrected using theexperimental attenuation factors.Before performing the frequency modulation experi-ment, the oscillator natural frequency dependence withbias current is investigated by measuring the power spec-tral density for various dc currents (Fig. 1). We observethat the dependence of oscillator frequency on bias cur-rent can be fitted by a third order polynomial: ω ( I dc + i ) = ω + a i + b i + c i (1)where the higher order terms, with coefficients b and c ,provide small corrections to linearity. This direct relation (cid:1)(cid:2) (cid:1)(cid:3) (cid:4) (cid:3) (cid:2)(cid:2)(cid:4)(cid:4)(cid:2)(cid:5)(cid:4)(cid:2)(cid:3)(cid:4)(cid:2)(cid:6)(cid:4)(cid:2)(cid:2)(cid:4)(cid:2)(cid:7)(cid:4)(cid:2)(cid:8)(cid:4)(cid:2)(cid:9)(cid:4) (cid:10)(cid:11)(cid:12)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20) (cid:21) (cid:12) (cid:13)(cid:22)(cid:11)(cid:13)(cid:14) (cid:23)(cid:24) (cid:16) (cid:17) (cid:25) (cid:26) (cid:27) (cid:20) (cid:10)(cid:11)(cid:12)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20) (cid:1)(cid:2) (cid:1)(cid:3) (cid:4) (cid:3) (cid:2) (cid:5)(cid:7)(cid:4)(cid:3)(cid:3)(cid:7)(cid:6)(cid:4)(cid:4)(cid:6)(cid:9)(cid:7)(cid:2)(cid:7)(cid:4)(cid:7)(cid:3)(cid:7) (cid:16) (cid:19) (cid:18) (cid:28) (cid:29)(cid:30) (cid:15) (cid:11)(cid:31)(cid:13) (cid:16)(cid:16) (cid:14) ! (cid:26) (cid:27) (cid:1) (cid:5) " (cid:3) (cid:20)(cid:17) Figure 1: (Color on line) Dependence of the oscillator fre-quency with bias current around the working point at I dc =20 mA . The solid line corresponds to the polynomial fit. Thevertical dashed lines show the maximum current variation im-posed by the modulation experiment. Inset: amplitude of theoscillator signal with bias current around the same workingpoint. Fig. 1, 4 and 5 correspond to the same sample. between the oscillator frequency and the current flow-ing through the device implies that modulating the mi-crowave current induces a periodic change of the instan-taneous oscillator frequency. On long time scales, suchfrequency modulation results in a spectral power densityas in Fig. 2. We observe several peaks: the carrier, at fre-quency ω c , close to the natural frequency ω of the oscil-lator, and several sidebands at frequencies ω l = ω c + l ω m , l being a positive or negative integer. In the following,peaks will be labeled by the corresponding l : the carrier( l = 0 ), the first order sidebands ( l = ± ), the secondorder sidebands ( l = ± ),... Similarly, sidebands are seenaround the carrier harmonics. In this paper, however, wefocus on the frequency range around the fundamental fre-quency, where the peaks are the largest. The measuredspectrum evolves as we sweep up the modulation fre-quency. Sidebands shift further apart and the peak am-plitudes vary: some peaks become larger, others smaller.
430 440 450 460 470 4800.00.20.40.60.8
Spectrum frequency (MHz) PS D ( µ V / H z ) m f m Figure 2: (Color on line) Power spectral density (PSD) ofthe vortex oscillator subjected to a frequency modulation per-formed at f m = ω m / π = 13 . , with a modulating powerof − . corresponding to a current i ac = 0 . . Thecentral peak is the carrier and on both sides, first and secondorder sidebands are clearly observed. The total power estimated from the peak amplitudes isobserved to be conserved, as expected from frequencymodulation theory . This evolution can be observed inthe color map (Fig. 3) obtained by measuring many spec-tra at different modulation frequencies: a cross-section ofthis map is therefore a spectrum like Fig. 2. In this colormap, peak frequencies are represented as a function ofthe modulation frequency. The carrier is the horizontalline at
454 MHz . The l -order sidebands shift apart fromthe carrier with a slope ± l . Peak extinctions are easilyseen: at a given modulation frequency, a peak disappearscompletely, eventually the carrier or the ± l side-bands.Extinctions follow specific pattern, as shown in the inset.We will show that this pattern gives valuable indicationsabout the parameters that govern the oscillatory behav-ior. III. DATA ANALYSIS
Our experimental results can be analyzed by using theanalytical formulation of frequency modulation . Bytaking into account the equation of the frequency depen-dence with bias current (Eq. 1), and replacing i with i ac cos( ω m t ) in this equation, we obtain the expressionof the instantaneous frequency ω ( t ) . Then the phase φ = R ω ( t ) dt can easily be calculated: φ = ω c t + B ω m sin( ω m t ) + B ω m sin(2 ω m t ) + B ω m sin(3 ω m t ) S pe c t r u m f r equen cy ( M H z ) Modulation frequency (MHz) 10 -6 -4 -2 Figure 3: (Color on line) Frequencies of the different peaks,carrier and sidebands, as a function of the modulation fre-quency. The color scale codes the power spectral density in alogarithmic scale. Inset: zoom on the region of low modula-tion frequency where the extinctions of the peaks are clearlyseen. with ω c = ω + 1 / b i ac (2) B = a i ac + 3 / c i ac (3) B = b i ac / (4) B = c i ac / (5)Knowing that e iz sin θ = P n J n ( z ) e inθ , the oscillator re-sponse in frequency space V max e iφ can be written as : V max e iω c t X n,m,p J n ( β ) J m ( β ) J p ( β ) e i ( n +2 m +3 p ) ω m t (6)where β i = B i /ω m , J k are Bessel functions and V max isthe signal amplitude of the oscillator without modula-tion. Since ω c = ω + 1 / b i ac , the carrier frequency isusually not exactly the natural oscillator frequency, un-less the oscillator frequency depends linearly on bias cur-rent. In our case, the deviation of the carrier frequencyfrom the natural frequency is almost below experimentalaccuracy, thus giving another proof of the quasi-linearityof frequency with current. From Eq. 6, we see that eachpeak labeled by the number l has therefore an amplitudeequal to a sum of products of Bessel functions, wherethe indices n, m, p must verify n + 2 m + 3 p = l . Thisinfinite sum may be truncated since J n ( x ) is negligiblewhen n > x . A very good approximation is, however,obtained with a much less drastic criterion. With ourexperimental values of B j , keeping only the terms with n, m, p ≤ gives a close approximation to the exact so-lution.At this stage, it is worth re-examining the mathemat-ical formula. In particular, suppose that B and B are less than 5% of B , which is the case for our vortex oscil-lator. In that case, Eq. 6 has the following properties: i)The extinctions of the carrier, i.e. the zeros of the carrieramplitude function, are controlled almost exclusively byparameter B . The impact of the two other parameterson the position of the zeros is below the experimentalaccuracy. ii) B controls the asymmetry between rightand left sidebands. If B = 0 , the right and left side-bands are equal, whereas when B > (resp. B < ),the left (resp. right) sideband becomes larger. The effectof B on the dissymmetry is linked to the fact that B is proportional to the only odd power term in the poly-nomial expression of ∂ω/∂i . iii) B mostly modifies theamplitude: for example B > amplifies both first-ordersidebands and reduces the second-order. The effect isopposite when B < .From these properties, the three parameters B , B , B , or equivalently a, b, c , can be extractedfrom the whole set of data, without fitting individualcurves. Once a, b, c are determined, it is possibleto reproduce the amplitude dependence of each peak(carrier, 1st and 2nd order sidebands) with modulationfrequency, for each microwave power used, with noadjustable parameter. By contrast, fitting single curvesindependently cannot lead to reproducible parameters.In particular, b, c and V max vary from fit to fit, sincethey all act mostly on the curve amplitude. In this case, V max is even observed to change by 20-30%, which is notconsistent with power conservation. Now let us describeour method in more detail. We know that the carrierextinctions are only controlled by B . Thus the zerosare necessarily the same as the zeros of J ( B /ω m ) ,since the higher order Bessel functions have a very weakimpact on the zeros position. In Fig. 4, the last carrierextinction ω ∗ m, is plotted as a function of the appliedmicrowave current. Mathematically ω ∗ m, must verify B /ω ∗ m, = x ∗ , where x ∗ is the first zero of the Besselfunction J . Equivalently the extinctions of the l ordersidebands correspond to B /ω ∗ m,l = x ∗ l where x ∗ l is thefirst zero of the Bessel function J l . Since the carrierextinctions are obtained on a larger scale of modulationcurrent, it is more appropriate to fit the carrier extinc-tions to obtain B with a reasonable accuracy. Thusthe parameters a and c are perfectly determined and B and B are fixed for each modulation current. b is thendetermined from the sideband asymmetry. To do this,the ratio between the maximum values of the right andleft first order sidebands is calculated numerically as afunction of B . Comparison to the experimental value ofthe ratio gives a specific value of B . By repeating thisoperation for different applied microwave currents, it ispossible to extract b with reasonable accuracy. In ourcase, we found: a = 13 . / mA , b = − . / mA , c = 0 .
23 MHz / mA and V max = 300 nV / √ Hz . Withthese four parameters, we can reproduce the whole set ofdata i.e. the carrier ( l = 0 ) and sidebands ( l = ± , ± )amplitudes obtained at different values of the mod-ulation current, i. e. × curves. The !!!!!!!!! """""" Carrier ! st sideband " nd sideband Modulation amplitude (mA) E x t i n c t i on s ( M H z ) Figure 4: (Color on line) Values of the modulation frequencyat which the last extinction is observed, when increasing mod-ulation frequency. Blue dots: extinctions of the carrier; greendiamonds: average value of the extinctions of the right andleft first order sidebands; red triangles: average value of theextinctions of the second order sidebands. Solid lines corre-spond to the fit ω ∗ m,l = B ( i ac ) /x ∗ l , for l = 0 , , . comparison between experimental data and calculatedcurves is quite satisfactory (see Fig. 5). Such a successlends confidence to the extracted parameters. A morecomplex approach including amplitude modulation is not necessary here. The reason is that the oscillationamplitude depends weakly on the current in thesesystems (cf Fig. 1). We also verify a posteriori our initialassumption: for the highest modulation current used, B and B are smaller than 5% of B . IV. DISCUSSION
Finally, let us compare the parameter a extractedfrom frequency modulation experiment with the valueof a obtained from the fit of Fig. 1. Surprisingly, itappears that the two values are quite different. Forexample, for the sample considered here, the fit of Fig. 1gives a ≈ / mA whereas the frequency modulationexperiment gives a ≈
13 MHz / mA . The origin of thisdiscrepancy, observed in all samples, must come fromthe dynamics of the experiments. One experiment isperformed in a quasi-static regime, whereas the other isperformed at a few MHz . In the frequency modulationexperiment, the instantaneous frequency is tuned ratherquickly in comparison to the natural oscillator frequency:a period of the modulation cycle corresponds to the timenecessary to perform 10 to 100 orbits. It was alreadyshown experimentally and numerically that a vortexcannot immediately jump from one frequency to another.Thus the agility is not infinite and the typical transitiontime is of the order of to
80 ns . In our frequencymodulation experiment, the frequency is continuouslyvaried with a cycle period corresponding to this typical
I = 1.31739droite
Modulation frequency (MHz) st sidebandCarrier A m p li t ude ( n V . H z - ½ ) nd sideband Figure 5: (Color on line) Amplitude of the carrier and of theright sidebands as a function of the modulation frequency, fora modulation current i ac = 1 . . The line represents thecalculated amplitude using Eq. 6 with the values of the pa-rameters a, b, c, V max determined by the procedure explainedin the text. Similar results are obtained for the left sidebands. time. The vortex dynamics is therefore expected to bein a transient regime and not in a stationary state. It isthen reasonable that the deviation sensitivity ∂ω/∂i ac appears significantly different from the tunability ∂ω/∂I dc . This difference has never been pointed out sofar in spin transfer oscillators : frequency modulationdata were collected at much lower modulation frequencyrelatively to the carrier frequency and analyzed usingthe frequency-current dependence determined with aquasi-static experiment. So, up to now, the conceptof deviation sensitivity has been ignored in the fieldof spin transfer oscillators. In previous studies, itwas reasonable to consider tunability as the relevantparameter since the modulation period was much longerthan the transition time between stationary dynamicalstates.In conclusion, we have shown that four parametersare enough to account for all our frequency modulationexperiments. One of those parameters is the deviationsensitivity, which appears to differ significantly fromthe tunability measured in a quasi-static regime. Sincea modulation experiment consists of a fast continuouschange of states, the characteristic time of the vortexdynamics must be taken into account. In our case, themodulation period approaches the transient time, so thedynamic experiment can no longer be considered as aquasi-static experiment. Hitherto in spintronic oscil-lators, the difference between tunability and deviationsensitivity had not been observed. Here we show theessential difference between these two concepts: a change from a quasi-steady state to forced transient dynamics. M.R. Pufall, W.H. Rippard, M.L Schneider, andS.E. Russek, Physical Review B , 140404 (2007). Q. Mistral, M. van Kampen, G. Hrkac, J.-V. Kim, T. De-volder, P. Crozat, C. Chappert, L. Lagae, and T. Schrefl,Physical Review Letters , 257201 (2008). A. Dussaux, B. Georges, J. Grollier, V. Cros, A. V.Khvalkovskiy, A. Fukushima, M. Konoto, H. Kubota,K. Yakushiji, and S. Yuasa, et al., Nature Communica-tions , 8 (2010). S.Y. Martin, N. Mestier, C. Thirion, C. Hoarau, Y. Con-raux, C. Baraduc, and B. Diény, Physical Review B ,14, 144434 (2011). M. Golio,
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