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Impact of intra-grain spin wave reflections on nano-contact spin torque oscillators

Anders J. Eklund,1, ∗ Mykola Dvornik,2 Fatjon Qejvanaj,3 Sheng

Jiang,4 Sunjae Chung,2, 5 Johan Akerman,2, 3, 4 and B. Gunnar Malm6

1Department of Physics, University of Oslo, Box 1048 Blindern, 0316 Oslo, Norway2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden

3NanOsc AB, Electrum 205, 164 40 Kista, Sweden4Department of Applied Physics, School of Engineering Sciences,

KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden5Department of Physics Education, Korea National University of Education, Cheongju 28173, Korea

6Division of Electronics and Embedded Systems, School of Electrical Engineering and Computer Science,KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden

(Dated: August 19, 2020)

We investigate the origin of the experimentally observed varying current-frequency nonlinearityof the propagating spin wave mode in nano-contact spin torque oscillators. Nominally identicaldevices with 100 nm diameter are characterized by electrical microwave measurements and show largevariation in the generated frequency as a function of drive current. This quantitative and qualitativedevice-to-device variation is described in terms of continuous and discontinuous nonlinear transitionsbetween linear current intervals. The thin film grain microstructure in our samples is determinedusing atomic force and scanning electron microscopy to be on the scale of 30 nm. Micromagneticsimulations show that the reflection of spin waves against the grain boundaries results in standingwave resonance configurations. For a simulated device with a single artificial grain, the frequencyincreases linearly with the drive current until the decreased wavelength eventually forces another spinwave anti-node to be formed. This transition results in a discontinuous step in the frequency versuscurrent relation. Simulations of complete, randomly generated grain microstructures additionallyshows continuous nonlinearity and a resulting device-to-device variation in frequency that is similarto the experimental levels. The impact of temperature from 4 K to 300 K on the resonance mode-transition nonlinearity and frequency noise is investigated using simulations and it is found that thepeak levels of the spectral linewidth as a function of drive current agrees quantitatively with typicallevels found in experiments at room temperature.

I. INTRODUCTION

The nano-contact spin torque oscillator1,2 (NC-STO)is a spintronic microwave oscillator in which the spintransfer torque3–5 (STT), induced by an electrical di-rect current, counteracts the Gilbert damping and en-ables a persistent precession of the magnetization in thefree layer. Through the giant magnetoresistance effect6,7,this precessing magnetization direction results in a cor-respondingly time-varying device resistance, which to-gether with the dc drive current produces an oscillat-ing voltage signal. The excitation of the magnetic freelayer can take place in the form of a circularly trajectingmagnetic vortex8,9 at lower frequencies (from hundredsof MHz to a couple of GHz) and, on the order of tensof GHz10, as various spin wave modes. These spin wavemodes include the solitonic ”bullet” mode11 and propa-gating spin wave mode12–14 for systems with an in-planemagnetic anisotropy for the free layer.

Out of these high-frequency modes, the propagatingspin wave mode possesses several features that makes itmore attractive for applications. First, it can be excitedexclusively while the bullet mode can only be excitedin conditions where also the propagating mode exists15.Simultaneous excitability with the possibility of mode-hopping between these modes accounts for the compar-atively high frequency-domain linewidths observed dur-

ing these conditions. Second, its propagating nature in-creases the size of the oscillating system and thereby in-creases the frequency stability. Third, it blue-shifts withthe application of increased drive current magnitude andthereby provides higher frequencies. Apart from the highfrequency range and comparatively high frequency stabil-ity most useful in microwave RF applications, the large-amplitude propagating spin waves are also attractive foruse in magnonic circuits16–18.

One general property of the propagating spin wavemode that has so far not been explained or modelled,is the rich variety of features in the frequency versuscurrent behavior f(IDC), which generally shows regionsof linear dependence joined by nonlinear transitions19,20.These nonlinear transitions can be either continuous ordiscontinuous, i.e., appear as f(IDC) bending or discretesteps in frequency. It may be considered natural to de-scribe the linear regions as sub-modes of the propagatingspin wave mode, but a physical mechanism behind sucha degeneracy has not yet been reported. The frequencysteps are of the order of 1 GHz and cannot be explainedas the much larger ∼10 GHz steps between the differ-ent higher-order Slonczewski modes21. We would liketo point out that the phenomenon under investigationis a higher order nonlinearity not to be confused withthe general auto-oscillator nonlinearity property22, whichconcerns the presence of coupling between the amplitudeand frequency of the magnetization precession. While

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the amplitude-frequency coupling is the mechanism thatmakes it possible to tune the STO frequency by changingthe magnitude of the drive current IDC, we see no phys-ical reason why this coupling by itself should give riseto the type of complexity found in the experimentallymeasured f(IDC) behavior.

The f(IDC) nonlinearity is of direct interest for anytehnological application of the NC-STO for two mainreasons: frequency stability (phase noise) and device-to-device variability. Within the nonlinear transitionintervals, the frequency stability is decreased whichis commonly observed as an increase in the spectrallinewidth19,23. More detailed measurements have shownthat the f(IDC) nonlinearity is associated with increasedlevels of both white and 1/f frequency noise23. Nomi-nally identical devices also differ significantly in the po-sition and type of the f(IDC) nonlinearity

20, which trans-lates to device-to-device variation in f(IDC). The sametype of qualitative and quantitative frequency variabil-ity that is found between devices can also be seen whenchanging the angle of the in-plane component20,24 orpolarity24 of the applied magnetic field. These stud-ies have concluded that the measured behavior is con-sistent with an oscillation that takes place in magnetic”hotspots”20 or ”subregions”24 defined by an inhomoge-neous effective magnetic field and/or spin polarization ra-tio originating from microstructural inhomogeneity. Thelinearity and nonlinearity of f(IDC) would in this con-text arise due to a complex interplay between these dif-ferent subregions. Although this is certainly a possiblescenario, we consider it less likely that the selection ofthe dominating subregion would be so sensitive to thedrive current IDC. We do not see a clear reason why thisselection would change and would rather expect that thefrequency would be set by the same subregion throughoutmost of the operating current range.

When setting up our simulations in an attempt torecreate the nonlinearity, we initially noted the sensitiv-ity to the boundary conditions of the simulation space. Inparticular, we found that periodic boundary conditionsin combination with a simulation space being about oneorder of magnitude larger than the nano-contact resultedin discontinuous steps in f(IDC). In this configuration,the spin wave propagated from the NC to any of the simu-lation space borders, re-entered from the opposing borderand still had a notable amplitude as it re-entered the NCarea. This led us into the conceptually simpler hypothe-sis that the f(IDC) nonlinearity originates from spin wavepropagation and STO self-interaction. Microstructuralinhomogeneity and spin wave reflection in combinationwith axial asymmetry in the oscillation mode would beconsistent with both the device-to-device variability andin-plane magnetic field dependence. Micromagnetic sim-ulations have shown that for applied magnetic fields hav-ing an in-plane component, the inclusion of the current-induced Oersted magnetic field indeed breaks the sym-metry of the propagating mode and propagation insteadtakes the form of a directed spin wave beam25,26.

In this work, we investigate the microstructure of thethin film in terms of the size of the metal grains andperform micromagnetic simulations with included grainboundaries. With reduced magnetic exchange couplingat the grain boundaries, the propagating spin wave be-comes reflected and travels back to the active region. Byself-locking, the spin wave reflections result in resonatingspin wave paths that each depends on the distance tothe reflecting grain boundary and the wavelength. Thisleads to multiple sets of resonance frequencies for the dif-ferent reflecting grain boundaries and provides a directphysical model for the f(IDC) sub-modes and their asso-ciated variability. As will be shown, the model is able torecreate both continuous and discontinuous f(IDC) non-linearity, the device-to-device variability (with reasonablequantitative agreement) and the correct variation of thespectral linewidth.

II. EXPERIMENTAL METHODS

The samples were fabricated by sputter deposition toform the stack Si/SiOx/Pd8/Cu30/Co8/Cu8/NiFe4.5/Cu3/Pd3 (thicknesses in nm). The film was then pat-terned into 16 × 8 µm2 mesas by optical lithographyand lift-off, followed by sputter deposition of a 30 nmSiO2 insulating layer. Circular nano-contacts with diam-eter dNC = 100 nm were patterned using electron beamlithography and etched using reactive ion etching. Thenano-contact vias were then metallized by Cu during thedeposition of the Cu1000/Au400 top contact, which wasalso defined by optical lithography and lift-off. The topcontact has a coplanar waveguide ground-signal-ground(GSG) configuration, where the S pad is connected to thenano-contact and the G pads are connected to the outerregions of the mesa through two 2× 4 µm2 vias.The electrical microwave measurements were per-

formed using a 40 GHz-rated GSG microwave probe fol-lowed by a bias-T and a 20–40 GHz low-noise amplifier(gain 28 dB, noise figure of 3.0 dB) before recorded ona spectrum analyzer. The bias current was supplied bya Keithley 6221 precision current source, with the pos-itive current direction defined as electrons flowing fromthe free NiFe layer to the fixed Co layer. The sampleand microwave circuit were mounted on an electricallycontrolled rotating holder with the sample positioned in-side the pole gap of an electromagnet. The current driv-ing the electromagnet was feedback-controlled using aPI controller with a calibrated Hall sensor positioned atthe center of one of the poles. In all measurements pre-sented in this work, the strength and angle of the appliedfield (away from the film plane) are Hext = 10.0 kOe andθext = 70◦. This field was selected based on Ref. 27 tooptimize the trade-off balance between oscillation powerand frequency stability (spectral linewidth), well abovethe critical angle of θext,c = 58◦ under which also thelocalized bullet mode is co-existingly excited.Atomic force and scanning electron microscopy (AFM,

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SEM) were performed on a separately prepared, unpat-terned Si/SiOx/Pd8/Cu30/Co8/Cu8/NiFe4.5/Ta3 filmto capture the structure of the NiFe free layer. AFMwas conducted with a JPK NanoWizard 3 NanoSciencemicroscope in the AC tapping mode using an AppNanoACTA tip with a (nominal) radius of curvature of 6 nm.The SEM measurement was performed using the in-lensdetector of a Zeiss Ultra 55 microscope at ∼3 mm work-ing distance, with a specified resolution of 1.6 nm at 1kV accelerating voltage.Ferromagnetic resonance (FMR) measurements of the

same unpatterned films, using a NanOsc InstrumentsPhaseFMR-40, gave the saturation magnetization andGilbert damping for the NiFe free layer of 4πMs,NiFe =10.1 kG, αG,NiFe = 0.0135 and Co fixed layer 4πMs,Co =19.8 kG, αG,Co = 0.0088.

III. SIMULATION METHODS

Simulations were performed using the open-source,GPU-accelerated micromagnetic simulation package28

mumax3 . For the homogeneous free layer simulations, a512× 512× 1 quadratic grid was used with a cell size of2.5× 2.5× 4 nm3 for the 1280× 1280× 4 nm3 free layerrepresentation. The fixed layer and spacer layer weremodelled as 1280× 1280× 8 nm3. An initial settling stepwas used to let the full stack relax into its static config-uration, taking into account the externally applied field,the dipolar field, the exchange field and the Oersted field.After settling, the fixed layer cells were kept static in or-der to reduce the computation time. By this approach,the dipolar field of the (static) fixed layer is automat-ically included in the simulation. Absorbing boundaryconditions were implemented similar to those in Ref. 29with three encapsulated frames, each with a width of 5% of the simulation space, with the damping parametersuccessively increasing to αG = 0.05, 0.15 and 0.45. Us-ing these settings and sweeping the current resulted in afrequency versus current behavior free from continuousor discontinuous nonlinearities for the oscillation regimeabove the threshold, while a linear 50 % reduction in thesimulation space resulted in slight frequency stepping dueto wave reflection against the simulation space borders.The majority of the material parameters for the fully

processed samples were taken from our previous workin Ref. 26 with slight adaptation to fit the thresh-old current and frequency of our experimental samplebatch. Selected values were: saturation magnetization4πMs,NiFe = 8.5 kG, 4πMs,Co = 17.0 kG; exchange stiff-ness Aex,NiFe = 1.1 · 10−11 J/m, Aex,Co = 2.1 · 10−11

J/m. For the spin torque, the polarization was 0.3 andthe Slonczewski parameter Λ = 1.0. The Gilbert damp-ing parameter was taken from our FMR measurements:αG,NiFe = 0.0135 and αG,Co = 0.0088.The Oersted field was calculated as that from an in-

finitely long conductor running down the nano-contact.The magnitude of the field increases linearly from the

center of the nano-contact out to the edge, outside whichit decays with the inverse distance from the center.For the simulations including the grain structure, the

grains were randomly generated using the Voronoi tes-selation extension to mumax3. Using another extension,the exchange coupling across the grain boundaries was re-duced by scaling. Tests using a constant scaling factor30

between all grains showed well-defined oscillation for 30–100 % coupling, while 20 % showed oscillation only forcurrents below 30 mA and 0–10 % resulted in broadbandnoise. Since the exchange coupling is highly sensitiveto the inter-atomic distance (it has been calculated31 todrop to 0 already at a distance of 1.5 times the crystallinedistance), we consider it more realistic to have a randominter-grain exchange distribution. Knowing little aboutthe grain-to-grain interface structure, we here make afirst approximation with a uniform distribution of 0–100% coupling. Both the grain tessellation and exchangescaling are set using the same specified seed number forthe random number generator, ensuring reproducibility.When simulating the grain structure, no significant dif-ference was observed between the full 1280 × 1280 nm2

and halved 640× 640 nm2 simulation spaces, indicatingthat only an insignificant amount of energy is propagatedto the simulation border and back to the active region inthis case. This is consistent with spin wave reflection oc-curring at the grain boundaries having a higher influencethan the simulation space border effects. Because of this,the simulations of the grainy free layer were performedusing the smaller simulation space in order to reduce thesimulation time.Majority of the simulations were performed at tem-

perature T = 300 K using an adaptive32 timestep whichusually settled at around 50 fs. The duration of the sim-ulations were 1 µs except for the homogeneous film pre-sented in Figure 3, where we used 100 ns. For the calcu-lation of the spectral linewidth plotted in Figure 7 the 1µs timetrace was split into two and the spectra averaged,resulting in an approximate spectral resolution of 2 MHz.

IV. RESULTS AND DISCUSSION

A. Experiment

We begin by exemplifying the diversity in the device-specific frequency versus current behavior. We do thisby measuring nine devices located in a line next to eachother, on the same chip. Any wafer-scale manufacturingvariability is hence kept at its minimum possible influ-ence. The spectral density as a function of the drivecurrent is shown in Figure 1 for the nine devices. Thefrequency as a function of current shows a blue-shiftingtrend for all the devices, consistent with the propagatingspin wave mode. At low currents, before the onset of theblue-shifting propagating mode, all the devices show aweaker pre-threshold mode with low, zero or even neg-ative tunability. The pre-threshold mode does in some

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FIG. 1. (a)–(i) Experimental power spectral density in dB over noise as a function of drive current for nine devices on the samechip, adjacent to each other.

cases, but not all, connect to the propagating mode.Apart from those general features, it can be said thatthe behavior differs qualitatively between the devices interms of the number of simultaneously excited frequen-cies, the position and height of the discontinuous fre-quency steps and the linearity or curvature. We refer toboth the discontinuous frequency steps and continuouslynonlinear current-dependence as nonlinearities, since thesimulated behavior for an ideal, homogeneous thin filmwas found to be highly linear above the threshold cur-rent. The devices in Figures 1(d,h,i) show a particularinstability in the low-current section of the propagatingmode, with worse defined frequencies.The devices in Figure 1 show resemblance to previ-

ously characterized NC-STOs in similar field configura-tions, in particular in terms of the presence of linearregions connected by nonlinearities that are either dis-continuous or continuous. The diversity among our de-

vices is large but we would like to point out that we herepresent completely non-selected data in its unprocessedform, without reducing it by extracting and displayingonly the dominant peak frequencies. The large sample-to-sample variation in the frequency versus current be-havior implies that the magnetization dynamics is highlydifferent between the devices. The differences can noteasily be explained by device-to-device variation in thenominal parameters such as the film thicknesses sincethis can not be expected to result in the qualitativelydifferent device characteristics. Nor does such reasoningexplain the emergence of the discontinuous and continu-ous nonlinearities. We identify these nonlinearities andtheir variation as a root cause of the sample-to-samplevariation. The (simultaneous) existence of several prop-agating modes with different frequencies is another openquestion.The large quantitative and qualitative differences be-

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FIG. 2. (a) Atomic force microscopy (phase contrast mode) and (b) scanning electron microscopy of the surface of the freelayer (Si/SiOx/Pd8/Cu30/Co8/Cu8/NiFe4.5/Ta3 stack).

tween the devices indicate that the origin of the differ-ences is a highly device-specific phenomenon. Figure 2(a)shows the AFM measurement of the structure of the freelayer film and Figure 2(b) shows the SEM measurementof the same film. Both the microscopes reveal a grainstructure with grains on the order of 30 nm size. Due tothe 4.5 nm small thickness of the free layer, it is reason-able to assume that all the grains are columnar.The grain structure constitutes a possible complex

source for the similarly complex device-to-device vari-ation and brings up the question of how reasonablethe homogeneous-film approximation is for modellingexchange-dominated propagating spin waves. It can beassumed that at the grain boundaries, the exchange in-teraction may be significantly reduced31 with the magni-tude of the reduction depending on the individual grain-to-grain interfaces. The random geometry of the grainstructure together with the likely random nature of theinter-grain exchange coupling reduction constitute a vastvariability space. We next turn to micromagnetic simu-lations in order to gain insights of the potential effectsthat the grain structure has on the propagation as wellas generation of the spin waves.

B. Simulation: homogeneous thin film

As a basis for the investigation of the impact of spinwave barriers in the free layer thin film, we first performsimulations of the nominal case of a perfect, homogeneousfilm.Figure 3(a,b) shows the simulated frequency versus

current behavior for the nominal case of a perfectly ho-mogeneous free layer film in the cases of exclusion (a)and inclusion (b) of the Oersted field. We find that inthe homogeneous film, there is a perfectly linear f(IDC)relation once above 20.4 GHz, i.e. above the onset thresh-

old region of the blue-shifting propagating mode. Abovethis threshold, the inclusion of the Oersted field doesnot result in any qualitative difference from the linearf(IDC) behavior. The effect of the Oersted field in thehomogeneous-film case is to shift the frequency up by∼0.6 GHz close to the threshold, increasing to ∼0.9 GHzat 40 mA.

The linear f(IDC) dependence is an important result,since it shows that there is no inherent mechanism forthe magnetization precession or the propagating spinwave mode that introduces nonlinear f(IDC) behavior.In other words, the amplitude-frequency coupling has aconstant nonlinearity coefficient within the operating fre-quency range. This shows that a more complex model ofthe system is required.

Before the onset of the propagating mode, the so calledpre-threshold behavior shows a more dramatic difference.When the Oersted field is included, there is a frequencyjump of 0.6 GHz from the pre-threshold mode to thepropagating mode whereas there is a continuous transi-tion without the Oersted field. This can be understoodby considering the local FMR landscapes. Without theOersted field, the local FMR frequency is homogeneousand there is no asymmetry that allows different oscilla-tion modes in different regions. This symmetry remainswhen the drive current is increased and the increasedspin torque eventually becomes strong enough to launchthe propagating wave. In the case of the Oersted field,there is an asymmetry between the center and the edgesof the NC that facilitates the generation of several modesin different volumes. This allows the propagating modeto be excited independently of the pre-threshold modeat a different, higher frequency. As the drive current isincreased and the mode volumes grow and start to over-lap, the propagating mode eventually extinguishes thepre-threshold mode. The intermodulation products vis-ible in the threshold current range 15–17 mA in Figure

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FIG. 3. Fourier amplitude of my (in dB) from simulations with a homogeneous free layer film at T = 4 K, (a) without and (b)with the Oersted field.

3(b) are a result of this coexistence26,33 in time and spaceof the two modes.Our result of a high degree of linearity in the frequency

as a function of IDC for the propagating mode is op-posite to the result in Ref. 34, where nonlinearity wasfound despite the effort of having implemented spin waveabsorbing boundary conditions. However, when we em-ployed identical boundary conditions (in a circle outsidethe nano-contact) and performed a spatial analysis wedid actually observe a certain amount of reflection fromthe ”absorbing” boundary that resulted in an artificiallyintroduced standing spin wave pattern. We we will seein the following sections that reflection and standing spinwaves in the physical system is a highly dominant mech-anism affecting the frequency selection in a way that in-troduces the f(IDC) nonlinearity.

C. Simulation: impact of a single barrier

As a first step towards investigating the possible effectsof the grain structure, we simulate the STO behavior inthe case where a single barrier is placed in the spin wavepath. Being a wave phenomenon, we expect that part ofthe incident wave is reflected back towards the source, i.e.the active, current-driven region directly below the NC.We set up a barrier in the free layer in form of an artificalrectangular grain with the exchange coupling between itand the surrounding film set to zero. The rectangulargrain has a width of 100 nm facing the NC and is 50 nmdeep. The NC is at the origin of the xy-plane of the sam-ple film; the externally applied field is aligned in the firstquadrant of the xz-plane and the barrier is positionedalong the positive y-direction. This is the direction intowhich the spin wave beam propagates in the case when

the Oersted field is included26; this is the side where thein-plane component of the external field and the Oerstedfield oppose each other, resulting in a decrease of the localFMR frequency. On the opposite side, the in-plane com-ponents add up and bring the local FMR frequency upto a level above the spin wave generation, hence blockingspin wave propagation in that direction.

The result from the simulations with the single, ar-tificial ”wall” grain is shown in Figure 4. Figure 4(a)displays the power spectral density as a function of theedge-to-edge separation distance dsep between the NCand the wall, for a fixed current of IDC = 27.75 mA. Atthis level of drive current, the nominal, ideal film oscilla-tion is stable at 23.1 GHz (see Figure 3(b)). Figure 4(a)shows a periodic pattern with a period of 45 nm. Thiscorresponds very well to half of the spin wave length of85−90 nm which we read off from the simulation space inthe unperturbed situation. At its peaks, the wall shiftsthe oscillation frequency away from the homogeneous-film case frequency up to 1.0 GHz positive and 0.3 GHznegative, where the magnitude of the frequency shift de-creases with dsep. The effect of the wall is visible upto dsep = 210 nm, where the STO once again attainsits ideal, homogeneous-film frequency. The downwardfrequency slope can be understood as the consequenceof a forced and gradually enlarged spin wave length asthe STO strives for spin wave resonance at a graduallylonger distance. The upward jumps in frequency occurwhen resonance eventually occurs for one additional nodeand anti-node, which rapidly forces a shorter wavelength.We note that the effect is asymmetric towards higher fre-quency, corresponding to a preferred situation of shorterwavelength (more nodes). This is a consequence of thegeneral coupling in STOs between the oscillation ampli-tude and frequency: the frequency increases with the

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FIG. 4. Simulations of a barrier with no exchange coupling tothe remaining free layer film. Fourier amplitude of my (in dB)for: (a) Varied NC-edge to obstacle separation distance dsep

for IDC = 27.75 mA. (b) Current sweep for dsep = 125 nm.

amplitude for the propagating spin wave mode in thismagnetic field. The strongest and dominating resonanceoccurs for the standing wave configuration that has thehighest amplitude and, as a consequence, also has thehighest frequency, shortest wavelength and highest num-ber of nodes. Larger amplitude and higher frequencyis obtained for resonance at small dsep due to the loweramount of spin wave damping along the shorter propa-gation path.

Figure 4(b) shows a current sweep for the case of fixeddsep = 125 nm. The artificially introduced spin wavebarrier introduces nonlinearity in f(IDC) similar to thetype that is characteristic for the experimental devices;we notice the appearance of a discontinuous frequencystep of 0.65 GHz at 27 mA and a small degree of contin-uous nonlinear behavior both below and above the step.

Compared to the homogeneous film case in Figure 3(b),the introduction of the barrier generally pushes the fre-quency higher. This is consistent with the general prefer-ence of selecting a higher frequency (i.e. squeezing in anadditional standing wave node), as found in Figure 4(a).This examplifies how a reflected spin wave can alter the

generation frequency, and can be considered as STO self-interaction. This is made possible by the ability of theSTO to be pulled towards the frequency of an injectedsignal and phase-lock to it, which has previously beenshown to occur both for injected electrical RF signals35–40

and incoming spin waves41–46. We find that in the case ofspin wave reflection and self-interaction, the wavelengthbecomes tuned to form a standing wave between the NCand the barrier in a positive feedback loop. The STOwill stabilize its oscillation by tuning its frequency suchthat an anti-phase reflected wave is avoided.The degree to which the STO adapts the frequency to

the available standing wave-frequencies depends on theamplitude of the incoming reflected spin wave in relationto the STO amplitude. If an incoming anti-phase wave isweak enough, the STO frequency will be unaffected. Insuch a case, the reflected wave merely constitutes a per-turbation that can be expected to introduce phase noisebut not shift the frequency. In Figure 4(a) we find thatthe distance where frequency-shifting becomes negligiblefor our simulated devices is around dsep = 210 nm awayfrom the NC edge.Similar standing spin wave effects were obtained when

the same barrier was left with full exchange couplingacross its boundaries and instead the NiFe Gilbert damp-ing parameter αG was increased to 1.0.

D. Simulation: impact of grain structure

Figures 5(a,b) show the simulated behavior of twograiny free layers with random inter-grain exchange re-duction within 0–100 %. These two ”devices” exem-plify both continuous and discontinuous nonlinearities:the sample in Figure 5(a) show a highly continuous be-havior while in Figure 5(b) there is a discontinuous fre-quency transition at 29 mA. The inclusion of grains inthe model thus provides a direct explanation of the ori-gin of the f(IDC) nonlinearity in NC-STOs. The grainboundaries form spin wave barriers with varying degree ofreflectance and orientation with respect to the propagat-ing spin waves and thus have varying degrees of impacton the STO.We next investigate the simulated device statistics of

f(IDC) for randomly generated grainy films. Figure 5(c)shows the extracted frequency versus current for ninesimulated devices together with the mean value for boththe simulated and experimental (Figure 1) cases. Thehomogeneous-film case is included for reference. The sim-ulated grainy films all have their frequencies shifted up-wards compared to the nominal homogeneous-film case.The shift of the mean frequency increases gradually with

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FIG. 5. Simulations with grain microstructure. Fourier amplitude ofmy (in dB) for: (a) Seed #1 and (b) Seed #5. (c) Extractedfrequency versus current for nine simulated devices and a homogeneous film. Mean values for simulated and experimental deviceseries. Inset: The standard deviation of the frequency σf of the simulated and experimental device series.

the drive current from 500 MHz at 20.5 mA to 1400MHz at 32.5 mA. In fact, we do not observe a singlefrequency below the homogeneous-film case for any sim-ulated device at any current level. This shows that theSTO also for the grain microstructure case always se-lects a standing spin wave pattern that results in an up-ward rather than downward frequency shift and is in linewith the asymmetry towards higher frequency (and am-plitude) found in the single-barrier case, Figure 4(a). Inthe grain case, the standing spin wave pattern can bealtered both by changing the number of nodes towards agiven grain boundary, or by changing to another domi-nant grain. The large number of possible mode configu-rations gives a high probability of always finding a modewith the preferred positive frequency shift.

We note that the experimental mean f(IDC) actuallyevolves with a factor of two lower slope df/dIDC than the

simulated grain and homogeneous cases. Our free simu-lation parameters were initially tuned to give largely cor-rect threshold current and frequency for the propagatingspin wave mode, but were not adjusted to fit the exper-imental f(IDC) relation for higher drive currents. Webelieve that this discrepancy in the simulations might bedue to the true spin torque efficiency being lower thanthe value used for the simulations. It can also be re-lated to the real Oersted field, which is probably lowerthan what is calculated using the inifite-wire approxi-mation. There is also a lateral spread in current dueto the device design47, where the current is intended toflow at the bottom of the spin valve mesa from the NCregion out to the ground contacts. The lateral currentspread decreases the current density in the NC regionand further modifies the Oersted field. We have also nottaken into account any possible temperature dependence

9

for the magnetization (i.e., the FMR frequency), whichwould decrease at higher IDC due to the higher electricalpower dissipation. Exploring this parameter space whilefine-tuning the distribution function for the inter-grainexchange coupling to achieve even better correspondenceis beyond the scope of the present work. Exact corre-spondence is also not necessary for discussing the mech-anisms of spin wave generation, reflection and interactionand their consequences.

The inset of Figure 5(c) shows the standard deviation(between devices) of the frequency σf for the simulatedand experimental devices. The standard deviation of thesimulated devices increases linearly over the simulatedcurrent interval with a factor of three, from 0.2 GHz to0.6 GHz. This factor of three coincides with the increaseof the grain-induced frequency shift as a function of drivecurrent in Figure 5(c). It is a natural behavior that thedevice-to-device variation that is due to grain-inducedfrequency shifting is directly proportional to the meanmagnitude of the frequency shift.

In the upper half of the simulated current range thestandard deviation of the frequency reaches the levelsfound in the experiment, i.e. 400 − 600 MHz. Thisquantitative correspondence strengthens the hypothesisof grain-induced spin wave reflection as a main sourceof the device-to-device variation. Since the trends arenot the same (constant versus linearly increasing), thereremains modelling aspects in particular for lower drivecurrents. The most straight-forward approach wouldbe to further decrease the inter-grain exchange couplingto force the grain effect down to lower oscillation am-plitudes. For the experimental devices at lower drivecurrent, there is also a more prominent appearance ofthe pre-threshold mode which perturbs the propagatingmode in the devices in Figure 1(d,e,h,i). It is beyond thescope of this paper to fully reproduce the pre-thresholdbehavior.

Figure 6(a) shows the mode structure for seed #5 atIDC = 23.5 mA. The structure is shown as the time-averaged oscillation amplitude, as opposed to an instan-taneous snapshot of the propagating waves. The oscil-lation amplitude falls off non-monotonously outside thenano-contact and forms nodes and anti-nodes in a com-plex interference pattern. Points of spin wave reflectioncan be identified where there is a discontinuous drop inthe oscillation amplitude. Four clear reflectors are indi-cated at points A, B, C and D in Figure 6(a). Lookingat the exchange coupling at the same points in Figure6(b) reveals that the reflection occurs at grain bound-aries where the exchange coupling has been strongly re-duced. Grain boundaries A and B are oriented so thattheir normal direction is pointing approximately towardsthe NC. At point C there are two possible strong re-flectors oriented at approximately 45◦ angle relative tothe NC direction and it is not clear that they reflect spinwaves back directly to the NC. However, the proximity tothe NC still results in interference effects at strong am-plitudes close to the main oscillation. Around point D

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there are multiple grain boundaries that are too close tothe NC to form a node and anti-node, but that nonethe-less act to confine the oscillation. The STO stabilizes atthe stationary oscillation state that results in the leastamount of conflict between the reflected spin waves re-turning from points A, B, C and D.

We finally investigate the impact of the grain struc-ture on the frequency stability of the oscillation. For thisstudy we select seed #5 since it contains the cases of bothcontinuous and discontinuous nonlinearity in f(IDC) andcompare its spectral linewidth to the homogeneous filmcase. Simulations were carried out for sample temper-atures T of 300, 150, 77 and 4 K. Figure 7 shows thefrequency and spectral linewidth (full width at half max-

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FIG. 7. Simulation of the seed #5 grain microstructure andthe homogeneous film case at sample temperatures T = 300,150, 77 and 4 K. (a) Frequency versus drive current. (b)Spectral linewidth (FWHM) versus drive current. Linewidthvalues below our effective resolution of 0.5 MHz have beenplotted at this value. The homogeneous film at T = 4 K isbelow the linewidth resolution for the entire current rangeand has been omitted.

imum) as functions of the drive current IDC.

Figure 7(a) shows that as the temperature is decreased,the frequency for a given current also decreases. Thissimulation result agrees with our previous experimentalresults48 for similar devices and magnetic field. More in-terestingly, the temperature decrease induces a shift ofthe nonlinear operating points to higher drive currents.

The nonlinearity is primarily a function of the frequency,which can best be seen at the operating point around 30mA, where there is a discontinuous step which goes from25 GHz up to 25.5 GHz for all temperatures. This is anatural consequence of the standing spin wave landscape,where the operating frequency is set by the optimum spinwave length. The nonlinearity around 26 mA is continu-ous for all temperatures.

The nonlinearity at 23 mA changes character as thetemperature changes: at 300 K it is continuous butbreaks up into a discontinuous transition at 150, 77 and 4K. This illustrates that there is no fundamental differencein the origin of the continuous and discontinuous nonlin-earity. In the discontinuous case there are two resonancestates where the oscillator selects one at a time. Aroundthe transition, thermal energy may be able to kick thetrajectory back and forth between the states which can beobserved as mode-jumping49,50. In the continuous case,the oscillator enters a trajectory that is intermediate tothe two underlying resonances. In this continuous tran-sition case, the nano-contact spin torque oscillator canreadily be analyzed within the framework of the generalnonlinear auto-oscillator theory.

Figure 7(b) shows that the nonlinear operating pointsare associated with a destabilization of the frequency.For homogenous films, the linewidth is largely indepen-dent of the drive current and reaches a maximum value of6.7 MHz for T = 300 K. With the grain microstructure,the linewith varies from the homogenous-film values ofsingle MHz inside the linear regions up to 85-90 MHz atthe nonlinear operating points. As the temperature isdecreased, the maximum linewidth points shift to highercurrents. This occurs since the entire f(IDC) relationis moved to higher currents, as previously discussed forFigure 7(a). The two nonlinearities at higher current (26and 30 mA) both show a decreasing value for the max-imum linewidth as the temperature is decreased. Thisis the generally expected behavior for single modes de-scribed by the nonlinear auto-oscillator theory22, wherethe nonlinar amplification factor ν along with the tem-perature determines the linewidth. The same nonlin-ear amplification factor has also been shown to be ap-plicable in the multi-mode case51 with thermally acti-vated mode-jumping. The situation of mode-hopping be-tween multiple excitable modes creates an increased sen-sitivity to thermal fluctuations of the oscillation powerthrough a decrease in the power restoration rate Γp.Since ν ∼ 1/Γp, this theory can be used to at least qual-itatively explain the substantially increased linewidth atthe nonlinear operating points. Conversely, our work ex-plains the origin of the different modes and gives a phys-ical justification of the applicability of the multi-modetheory for the analysis of the nano-contact STO as car-ried out in Ref. 51.

The low-current nonlinearity at 23 mA again shows adifferent behavior. Here the maximum linewidth insteadincreases when the temperature is decreased from 300 Kto 77 K. At 4 K the linewidth has decreased to single-

11

MHz values, similar to the case of a well-defined discon-tinuous nonlinearity. Since this nonlinearity changes itsnature from continuous to discontinuous when changingT from 300 K to 77 K we cannot expect neither single-nor multi-mode theory to accurately describe the tem-perature dependence of the frequency stability across thetransition. There may also be additional instability in-duced by the simultaneous availability of both single- andmulti-mode solutions. More detailed study of the tran-sition from continuous into discontinuous nonlinearity isbeyond the scope of this work.

V. CONCLUSIONS

Experimental spectra from nine nominally identical de-vices has been presented and their qualitative behaviorhave been described. The sample-to-sample variation interms of the frequency as a function of current is signif-icant quantitatively as well as qualitatively, with a com-mon feature being linear regions that are connected bynonlinearities that can be either continuous or discontin-uous (in the form of a frequency step). This qualitativebehavior has been reproduced in simulations incorporat-ing the ∼30 nm grain structure measured using AFMand SEM, with randomly reduced inter-grain exchangecoupling. The reduction of the inter-grain exchange cou-pling results in spin wave reflection, which in turn facil-itates self-locking of the STO to geometry-defined res-onant frequencies. The spin wave resonance preferablyacts to increase the oscillation frequency compared tothe homogeneous film case. Each of the strongly reflect-ing grain boundaries constitutes one resonance conditionand the final frequency selection for a given current isdetermined by the inherent STO frequency and the rel-ative strengths of the different reflections. The differentstanding spin wave modes act to increase the spectrallinewidth by more than one order of magnitude at op-erating points where several of them are simultaneouslyexcitable.In all, the investigation shows that spin wave reflec-

tion and resonance against grain boundaries constitute aphysically reasonable model that is able to explain the

origin of the continuous and discontinuous nonlinearitiesin the frequency versus current. This model also explainsa large part of the device-to-device variation as stemmingfrom the random grain structure, with partially quanti-tative agreement with the experimental device variation.For improved agreement, we suggest future modellingwork to further reduce the inter-grain exchange couplingand increase the effects of spin wave reflection to its max-imum. Other possible sources of variability are inhomo-geneity in the magnetic parameters such as the satura-tion magnetization (or film thickness) and spin polariza-tion ratio. Given the grain microstructure and assumingcolumnar growth throughout the thin film stack, it wouldbe natural to assign these varying properties at the grainlevel.The different grain-induced resonance conditions can

be viewed as separate spin wave sub-modes which are si-multaneously excitable at the nonlinear operating points.This instability explains the elevated level of the spec-tral linewidth at the nonlinearities, provides a physicalmotivation for multi-mode oscillator theory and explainsthe origin of the apparent nonlinearity as it is treatedin single-mode theory. Since improving the crystallinequality of sputtered metallic films is considered to bedifficult, further work on controlling and improving thedevice performance is suggested to focus also on the ma-chining of artificial spin wave reflection boundaries thatcan dominate over the random grain boundaries.

ACKNOWLEDGMENTS

The authors gratefully acknowledge Ahmad Abedinfor assistance with the SEM measurements and FedericoPevere and Faraz Khavari for assistance with the AFMmeasurements. We thank the PDC Center for High Per-formance Computing, KTH Royal Institute of Technol-ogy, Sweden, for providing access to part of the comput-ing resources used in this research. The computationalresults presented have in part been achieved using theVienna Scientific Cluster (VSC). This work was finan-cially supported by the Swedish Research Council (VR)through the projects 2017-04196 and 2012-05372.

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