Stabilization of phase noise in spin torque nano oscillators by a phase locked loop

Steffen Wittrock,1, ∗ Martin Kreißig,2 Bertrand Lacoste,3 Artem Litvinenko,4 Philippe

Talatchian,4 Florian Protze,2 Frank Ellinger,2 Ricardo Ferreira,3 Romain Lebrun,5

Paolo Bortolotti,5 Liliana Buda-Prejbeanu,4 Ursula Ebels,4, † and Vincent Cros5, ‡

1Unite Mixte de Physique CNRS, Thales, Univ. Paris-Saclay,1 Avenue Augustin Fresnel, 91767, Palaiseau, France

2Chair for Circuit Design and Network Theory, Technische Universitat Dresden, 01062 Dresden, Germany3International Iberian Nanotechnology Laboratory (INL), 471531 Braga, Portugal

4Univ. Grenoble Alpes, CEA, INAC-SPINTEC, CNRS, SPINTEC, 38000 Grenoble, France5Unite Mixte de Physique CNRS, Thales, Univ. Paris-Saclay,

1 Avenue Augustin Fresnel, 91767 Palaiseau, France(Dated: October 26, 2021)

The main limitation in order to exploit spin torque nano-oscillators (STNOs) in various potentialapplications is their large phase noise. In this work, we demonstrate its efficient reduction bya highly reconfigurable, compact, specifically on-chip designed PLL based on custom integratedcircuits. First, we thoroughly study the parameter space of the PLL+STNO system experimentally.Second, we present a theory which describes the locking of a STNO to an external signal in a generalsense. In our developed theory, we do not restrict ourselves to the case of a perfect phase lockingbut also consider phase slips and the corresponding low offset frequency 1/f2 noise, so far the maindrawback in such systems. Combining experiment and theory allows us to reveal complex parameterdependences of the system’s phase noise. The results provide an important step for the optimizationof noise properties and thus leverage the exploitation of STNOs in prospective real applications.

I. INTRODUCTION

Spin torque nano oscillators (STNOs) are nano-sizedoscillators, which convert a supplying dc current intoa rf electrical signal through magnetization dynam-ics and basic spintronic phenomena1. Within thelast decade of intensive research on spintronics, theyhave been identified as promising candidates for next-generation multifunctional microwave spin-electronics1,2.In addition to their nanometric size (∼ 100 nm) andlow power consumption, STNOs in general benefitfrom a high frequency tunability along with compat-ibility with standard CMOS technology3,4 and semi-conductor manufacturing processes. Potential appli-cations are manifold and go beyond the integrationinto future wide-band high-frequency communicationsystems2,5–9: From high data transfer rate hard diskreading10, spin wave generation11,12 for e.g. magnonicdevices13,14, broadband microwave energy harvesting15

or frequency detection16–18, to bio-inspired neuromorphiccomputing19,20. Facilitating many of the mentioned po-tential applications, an intrinsic effect of the underlyingmagnetization dynamics and noteworthy the importantparticularity of STNOs is their non-isochronicity, i.e. thecoupling between the oscillator’s amplitude and phase21.This specifically allows tuning of the STNO frequencyvia the dc current. However, the STNO’s nanoscalesize and nonlinearity come at the cost of a relativelypoor phase noise performance, identified as the majordrawback in order to exploit STNOs in real practicalapplications22,23, especially in the field of rf communi-cations. In order to substantially improve the phase andfrequency stability, a standard means is the integrationof the oscillator into a phase locked loop (PLL) system,

which on a circuit level continuously corrects the oscil-lator’s phase fluctuations through comparison with anexternal reference. This approach applied to STNOs re-cently gained attention in theory24,25 as well as in termsof practical realizations3,4,26–30. It represents an impor-tant step towards STNO applicability and system inte-gration. Indeed, PLL systems, usually implemented witha VCO (”Voltage controlled oscillator”), are widely usedin microelectronics and rf communications for variousrf applications, such as clock recovery, frequency (de-)modulation, stable frequency generation, or frequencysynthesis, etc.31,32. However, the design of such a systemcompatible to STNOs is challenging, particularly due totheir strong nonlinearity, which leads to a coupling ofamplitude and phase and consequently an enhancementof the phase noise.

The PLL system used in this study has been specifi-cally designed based on custom integrated circuits pro-viding a highly reconfigurable and compact system.Thanks to a specially designed programmable amplifier33

and a wide range frequency divider34 with high sensitiv-ity, the PLL can be operated in a large frequency rangeof 0.1-10 GHz. In the following, we summarize the basicprinciple of this PLL and refer to Refs.3,4,33,34 for furthertechnical details. We demonstrate the PLL performanceon two types of STNOs in different frequency ranges, i.e.magnetic tunnel junctions whose free layer adopts eitherthe vortex state or is uniformly magnetized. Further-more, a general theoretical model is developed to describethe noise characteristics of STNOs submitted to an ar-bitrary external signal. We not only consider the phasedynamics in the perfectly phase locked state, but alsotake phase slips into account that are a main drawbackin such systems35,36 but that have so far not yet been

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considered in theoretical evaluations to correctly predictand analyze phase noise properties. Based on the theo-retical approach, the PLL operation and noise propertiesof the PLL-locked STNO are described and determinis-tic expressions are provided. The results from the theo-retical model reproduce well the different experimentallyobserved features. Moreover, it provides a fundamentalinsight onto the complex dependencies of the PLL oper-ation on the STNO and PLL parameters and herewithshows the routes to efficiently reduce the system’s phasenoise. Our analysis underlines the necessity to appropri-ately adapt the PLL to the STNO and vice versa andconsequently provides the adequate parameter space toachieve that.

II. THE PLL SYSTEM

(a) (b)

Figure 1: Basic concept of a PLL: Stabilization of theSTNO by locking the phase of the divided oscillatorsignal to a fixed reference. The phase difference is

continuously analyzed. (a) Scheme of the PLLcomponents. (b) Periodic signals in the PLL showing

the correction mechanism.

The basic concept of a PLL is depicted in fig. 1. Theoscillator’s output signal frequency is divided by the di-vider ratio N and subsequently compared with an ex-ternal reference of frequency close to fref ∼ fosc/N . Aphase frequency detector (PFD) evaluates the phase dif-ference between these two signals whereafter the loopfilter outputs a correction to the dc current Idc pro-portional to the phase difference. Due to its frequencytuning capability df/dIdc, the STNO frequency is thenpulled/pushed to operate at the stabilized PLL outputfrequency fosc = Nfref. Its phase noise is reduced withinthe PLL operation bandwidth.

The challenge in the circuit development of the STNO-PLL (see fig. 2) lies in the adaptation of the PLLconcept37 to the specifications of STNOs. This particu-larly includes the requirement for a large PLL bandwidth,which mainly determines the ability to significantly re-duce the noise characteristics and phase diffusion in thecorresponding oscillator. Usually, PLL bandwidths incommercial rf applications range from a few kHz to afew 100 kHz. However, due to the typical noise ampli-tude in STNOs, bandwidths in the range of a few MHz(vortex STNOs), or up to a few tens of MHz (uniformSTNOs) are necessary, although very large bandwidthsoften lead to detrimental spurious characteristics. The

Figure 2: Picture of the developed PLL-chip with itscomponents.

developed PLL system exploited in this work exhibits abandwidth of ∼ 2.5 MHz. Furthermore, it is highly dy-namic and reconfigurable. With its large range of oper-ation frequencies of 0.1-10 GHz, it can be operated withdifferent STNO configurations, namely in particular vor-tex based (STVOs) as well as uniform STNOs. The in-tegrated circuit design3,4,33,34 is shown in fig. 2.

III. EXPERIMENT: PLL OPERATION UPONVORTEX AND UNIFORM STNOS

We present here the PLL operation for both vortexbased STNOs (STVOs, operating at 300-500 MHz) andfor uniformly magnetized STNOs (operating in the 5 GHzrange). More experimental details are given in the ap-pendix A.

A. Vortex based STNOs

Vortex based STNOs (STVOs) rely on the spin transferinduced dynamics of a noncollinear magnetization dis-tribution, namely the gyrotropic motion of a magneticvortex core38,39. Typical frequencies of these devices liein the 100 MHz to 1 GHz range. They exhibit large-amplitude oscillations with output powers up to the µWrange40 and a good phase coherence compared to otherSTNO realizations.

An example for the frequency characteristics of aSTVO (see appendix A for detailed sample informa-tion) vs. the applied dc current Idc is presented in fig.3. It shows a frequency tunability of about df/dIdc ≈10 MHz/mA, which is exploited in order to stabilize thephase noise of the sample by directly injecting the PLLfeedback current into the STVO.

In fig. 4, we show the emitted power spectrum (fig.4a) and phase noise41 (fig. 4b) for the operation of theSTVO device when the PLL feedback loop is off (black)and on (red). The radiofrequency PSD peak amplitude(fig. 4a) is increased by 25 dB up to −19 dBm/MHzand the corresponding FWHM decreased from ∼ 1 MHzdown to < 1 Hz , nominally the resolution limit of

3

Power [dBm/MHz]

Figure 3: Frequency vs. injected current Idc, measuredfor a STVO at room temperature under a perpendicular

field of µ0H⊥ = 455 mT. The color scale indicates theemitted power.

3 3 2 3 3 4 3 3 6 3 3 8 3 4 0- 9 0- 8 0- 7 0- 6 0- 5 0- 4 0- 3 0- 2 0

P L L b a n d w i d t h

P L L o f f P L L o n

Powe

r [dBm

/MHz

]

F r e q u e n c y [ M H z ]

0 H = 4 5 5 m T

(a)

1 0 4 1 0 5 1 0 6 1 0 7 1 0 8

- 1 0 0

- 8 0

- 6 0

- 4 0

- 2 0 0 H = 4 5 5 m T

Phas

e Nois

e PSD

[dBc

/Hz]

O f f s e t f r e q u e n c y [ H z ]

P L L o f f P L L o n

P L L b a n d w i d t h

(b)

Figure 4: Comparison for operation of the STVO whenthe PLL is off (black curve) and on (red curve) for (a)

the emission radiofrequency power spectral density(PSD) and (b) the phase noise PSD. The PLL

parameters are: frequency divider ratio N = 10, chargepump CP = 8 and coupling strength ε ∼ CP/N = 0.8.

Details of the experiment are provided in the appendix.

our spectrum analyzer. The corresponding phase noise(fig. 4b) of the free running state has the typical 1/f2

noise dependence22,23,42 in the full offset-frequency range,while the phase noise upon PLL operation is constantwithin a given bandwidth ∆ωBW and hence is efficientlyreduced by more than 50 dB at a 10 kHz offset from thecarrier frequency.

The PLL operation bandwidth ∆ωBW is an importantparameter, because it defines the frequency band aroundthe carrier for which the phase deviations are detectedand stabilized. The feedback response corrects the phaseand ”kicks” it periodically on a time scale given by theinverse of ∆ωBW (see fig. 5b). Hence, ∆ωBW directlycorrelates with the phase noise level: the higher it is,the more the phase noise can be reduced at low offsetfrequencies. For the STNO-PLL system presented here,it is around ∆ωBW /(2π) ≈ 2.5 MHz (see fig. 4) and is,as derived in section IV, related to the loop filter and thePLL parameters.

In fig. 5, we present the measured phase deviationfrom the mean phase for the cases when the PLL is off(black) and on (red, blue) at different operation con-ditions. For the unlocked, free running STVO (black

(a) (b)

Figure 5: Phase deviation from the mean phase. (a)With and without PLL at µ0H⊥ = 455 mT,

corresponding to the datasets shown in fig. 4. (b)Perfect phase locking upon PLL operation (red curve,µ0H⊥ = 455 mT ) and occurrence of phase slips (blue,

µ0H⊥ = 310 mT).

curve in fig. 5a), the deviation to the mean phase islarge and varies significantly, i.e. more than ±100π rad.The improvement of the STVO’s coherence upon PLLoperation is highlighted by the red curve in fig. 5, forwhich good phase stabilization within at least 10 ms (to-tal measurement time, see fig. 5a) is achieved. Fromthe phase deviations over time we calculate the stan-dard deviation σψ of the stabilized phase for the phaselocked state. For PLL on and an applied field of 455 mT,we find σψ = 0.25 rad which is in good agreement withvalues obtained using commercial PLLs, lying betweenσψ = 0.237 rad and σψ = 0.923 rad28–30. However, atsome operation conditions, the PLL does not achieve acomplete phase locking but instead, phase slips by 2π atsome instances in time occur in the phase deviation, asshown by the blue curve in fig. 5b. They occur when thefrequency divider raises a counting error due to too largefluctuations above ∆ωBW (see Ref.28). The phase slipsreflect a diffusive phase noise process and hence, mani-fest themselves as a 1/f2 contribution in the low offsetfrequency range of the phase noise PSD, as observed forinstance in fig. 6a for the blue-colored curves (note, thecyan-colored curve is based on the same measurementdata as the blue curve in fig. 5b).

In order to better assess how the STNO and PLL pa-rameters and the phase jumps affect the PLL perfor-mance, we present in fig. 6a, a series of phase noise (PN)PSDs for different PLL parameter settings, such as thefrequency divider ratio N , reference signal frequency fref

and coupling strength ε ∼ CP/N , with CP the chargepump of the PLL feedback as a product of correctioncurrent and driver gain4. It can clearly be seen thatthe locking and desynchronization of the STNO insidethe PLL circuit strongly depend on the PLL parame-ter settings along with the parameters that determinethe free running STNO performance (dc current and ap-plied magnetic field). In fig. 6a, phase noise PSDs areshown for zero frequency mismatch between free running(f = 330 MHz) and PLL frequency and for the two caseswhen the PLL is turned off (red) as well as for the PLL

4

104 105 106 107

-90

-80

-70

-60

-50

-40

-30

-20

N

free, 330 MHz, μ0H=105 mT, ν=3

Ref 8.25, N 40, CP 9, 330, 0.225 Ref 16.5, N 20, 11, 330, 0.55

free, 480 MHz, μ0H=525 mT, ν=5

Ref 5, N 98, CP 15, 490, 0.153 Ref 4.94, N 99, CP 15, 489, 0.15 Ref. 24.33, N 20, CP 6, 486.6, 0.

Pha

se N

ois

e P

SD

[dB

c/H

z]

Frequency Offset [Hz]

Ref [MHz] N CP f [MHz] ε8.25 40 9 330 0.22516.5 20 11 330 0.55

5 98 15 490 0.1534.94 99 15 489 0.152

24.33 20 6 486.6 0.3

Ref [MHz] N CP f [MHz] ε [a.u.]8.25 40 9 330 0.22516.5 20 11 330 0.55

(a)

4 8 0 4 8 2 4 8 4 4 8 6 4 8 8 4 9 0 4 9 2 4 9 4 4 9 6- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

Powe

r [dBm

]

F r e q u e n c y [ M H z ]

R e f . : 4 , 9 4 M H z , N = 9 8 R e f . : 4 , 9 4 M H z , N = 9 9 R e f . : 2 4 , 3 3 M H z , N = 2 0

(b)

Figure 6: (a) Phase noise PSDs for two free runningconditions (red & dark blue) and different parameter

sets of the combined system STVO + PLL as indicatedin the inset table. (b) Power emission spectra of the

locked STVO at different PLL parameters N and fref

resulting in different central frequencies. Same colors in(a) and (b) match; cyan dataset with ε = 0.3

corresponds to the blue phase deviation curve in fig. 5b.

turned on (pink, orange). For this set of PLL parame-ters, the STVO is well locked to the PLL in the frequencyrange below the bandwidth ∆fBW = ∆ωBW /(2π), wherethe phase noise is flat and well reduced. Furthermore,it is observed that upon increasing of the PLL couplingε ∼ CP/N the phase noise is more efficiently decreased.For another set of data (blue-colored curves in fig. 6a)with non-zero frequency mismatch, we find that phaseslips become important. For instance for the cyan curvewith N = 20 (fref = 24.33 MHz), ε = 0.3 and a fre-quency mismatch of 6.6 MHz, a significant 1/f2 noisecontribution appears for offset frequencies below the cor-ner frequency (transition between 1/f0 and 1/f2 curveshapes) fcorner ≈ 105 Hz, while the phase noise is ef-ficiently reduced in the frequency range between thePLL bandwidth ∆fBW and fcorner. For lower couplingε = 0.153, ε = 0.152 combined with a larger frequencymismatch of 10 MHz and 9 MHz respectively, the phasenoise reduction is less efficient, i.e. phase jumps becomemore frequent. In consequence, the 1/f2 noise levels arehigher and the corner frequency fcorner approaches thePLL bandwidth ∆fBW . Furthermore, comparing the twocurves of similar coupling strength ε = 0.153 (blue) and

ε = 0.152 (green-blue), the blue one with larger frequencymismatch shows a larger phase noise level. This demon-strates also the importance of the frequency mismatch forthe occurrence rate of the phase slips. A larger nonlinearamplitude-phase coupling, quantified by the parameter ν(see ν = 5 vs. ν = 3 in fig. 6a), favours the occurrenceof phase slips and also leads to a generally larger noiselevel at higher frequency offsets. Comparing the datasetsin fig. 6a for zero and non-zero frequency mismatch, onemore important feature needs to be pointed out. Forthe latter set (blue-colored curves), a small resonancepeak is visible around the bandwidth frequency. Fig. 6asuggests that its amplitude, and as well the bandwidthitself, scale with the PLL coupling parameter ε and thenon-linear coupling parameter ν.As a final point, we demonstrate in fig. 6b, that the de-veloped PLL is very versatile and can also be used forfrequency synthesis. Notably, its output frequency canbe easily shifted by several MHz through either the PLLdivider ratio N and/or the reference frequency fref.

To conclude, the PLL measurements on vortex STNOsreveal several characteristic features, such as the PLLbandwidth ∆ωBW , the phase noise reduction in thelocked state, the phase slips leading to a 1/f2 contri-bution and a resonance around ∆ωBW . They depend onthe free running STNO performance and parameters aswell as on the PLL settings. These complex dependencieswill be further discussed in sections IV-V where theoret-ical expressions for the phase noise of the PLL-STNOsystem are derived considering stochastic white noise.

B. Uniform STNOs

To demonstrate the developed PLL chip’s large opera-tional frequency range of 0.1–10 GHz, we also operate thePLL with other available STNO devices. These are in-plane quasi-uniformly magnetized magnetic tunnel junc-tions (MTJs), composed of an in-plane magnetized ref-erence layer and an in-plane magnetized free layer. Themagnetic stack is very close to the vortex MTJs but witha much thinner free layer to stabilize the uniform in-planemagnetization. The applied in-plane magnetic field isadjusted for operation around 4-5 GHz, leading to a freerunning emission power of 10-50 nW and a linewidth of10-20 MHz.

Time domain characterization of the free running out-put voltage signal of this in-plane (IP) STNO reveals thatthe oscillations are not stable in time, with extinctions(strong amplitude reductions and possible loss of phasecoherence) over short time scales (1 ns). The averagetime of stable oscillations was ∼ 0.1 - 1µs. Large amountof extinctions result in an increased phase noise of the freerunning state as compared to the vortex and other uni-form STNO devices28,29. Furthermore, the free runninglinewidth is larger than the PLL bandwidth. This alto-gether makes the operation of the PLL with the availableuniform magnetized STNOs more difficult and lowers the

5

Figure 7: Measurement of a uniform STNO in the PLLcircuit: Phase noise spectral density for the free runningand the locked oscillator at different PLL parameters.

locking capability. Notably we find that the PLL waslocked only over finite time scales, in the best cases thiswas 20-60µs. Therefore, when evaluating the phase noiseover the full measured time trace for devices with veryhigh number of locking failures (i.e. locking times muchbelow 100µs), we observe a phase noise reduction of only12 dB at 100 kHz offset frequency for a PLL bandwidthof 2 MHz. However, when evaluating the phase noise forthe case of low number of locking failures and lockingtimes of 20-60µs (see fig. 7), the phase noise reductionat 100 kHz offset frequency increases to 20 dB, the noiseplateau is well pronounced and the resonance around thePLL bandwidth frequency is well visible at large cou-pling. Nevertheless, when increasing the PLL couplingε = CP/N in this case, similar features can be observedas for the PLL operation with a STVO: occurrence ofa plateau, corner frequency below which the 1/f2 phasediffusion sets in and a more pronounced ”bump” aroundthe PLL bandwidth. To conclude, the phase noise of thevortex and uniform STNOs reveals very similar featuresin the PLL phase-locked state and the designed PLL sys-tem is operational for both device configurations, i.e. fora large frequency range.

IV. THEORY: PLL PHASE DYNAMICSINCLUDING STOCHASTIC WHITE NOISE AND

PHASE SLIPS

In addition to the experimental results, we develophere a complete theoretical approach in order to assessthe noise properties of STNOs, especially when they aresubject to an external signal. This includes, i.a., theSTNO synchronization to fractional or harmonic frequen-cies, frequency modulation, or finally the exploitation ofa PLL onto the STNO. We do not restrict to the assump-tion of perfect phase locking but, importantly, also takethe occurence of phase slips into account, which have notyet been treated in this context. However, as we presentexperimentally in sec. III, they are the main drawbackin the synchronization of a STNO. With our theoretical

approach, we are able to give deterministic expressionsthat fully determine the characteristics and noise prop-erties of STNOs upon an external forcing. In section V,we discuss the results with a focus on the operation ofa PLL but stress, that the discussion remains generaland valid also for other forcing signals. A comparisonbetween the experimental results of section III and thedeveloped theoretical expressions allows us to fully ex-plore the complex parameter space of the PLL (but notlimited to that) upon a STNO in detail.

We first present the phase equations for non-isochronous auto-oscillators such as STNOs, includingexternal forcing and noise. From this, we derive an ex-pression for the noise power spectral density (PSD) forperfect phase locking. Subsequently, we consider the oc-currence of phase slips based on the Brownian motioninside the synchronization potential of the locked STNO.

A. Power and phase equations for theauto-oscillator under external forcing

The nonlinear auto-oscillator theory21,43 is used to de-scribe the STNO phase equations under external forcing.It describes the oscillations through the complex ampli-tude c(t) =

√p(t)e−iφ(t), with p(t) = |c|2 being the os-

cillation power and φ(t) the phase:

c+ iω(|c|2)c+ Γeff(|c|2)c = f(t) . (1)

The oscillation angular frequency is denoted by ω andΓeff = Γ+ − Γ− is the effective damping rate with posi-tive rate Γ+ representing the losses of the system, andnegative one Γ− representing the system’s gain. f(t)is a function, which allows a description of the sys-tem’s interaction with the environment, e.g. includingnoise processes and an injected external signal. Dueto the dependence of the damping terms on the am-

plitude (dΓ−(p)dp < 0 and dΓ+(p)

dp > 0), the oscillation is

described by a limit cycle with stable oscillation powerp0, which is obtained when the positive and negativedamping terms equal: Γ−(p0) = Γ+(p0). Assuminga small perturbation δp of the stable oscillation powerp(t) = p0 + δp due to noise yields a characteristic damp-

ing rate Γp = πfp =[dΓ+

dp (p0)− dΓ−dp (p0)

]p0 of small

power deviations back to the stable limit cycle21,43. Theparameter ν = dω(p)/dp · p0/(πfp) is the normalized di-mensionless nonlinear frequency shift and quantifies thecoupling between phase and amplitude due to nonlinear-ity.

In order to study the system for an injected exter-nal signal, we add a periodic interaction term f(t) =Ωaexte

−i(β+φext) to eq. (1). At first, the external sourceis assumed spectrally pure and noise-free; Ω denotes thecoupling strength to the oscillator, aext the injected signalamplitude, β the coupling phase, which might be differentfor amplitude and phase (βp and βφ, resp.) and which canalso account for a time delay, and φext the phase of the

6

external signal. Explicitly writing eq. (1) for the phaseand power separately21,22, the system can be linearizedaround p0. Upon renormalization ρ = (p − p0)/(2p0) ofsmall deviations from p0 due to the external signal, oneobtains:

ρ = −2Γpρ+ εpaext cos(nψ − βp) (2a)

ψ = ∆ω − εφaext sin(nψ − βφ) + 2νΓpρ . (2b)

We introduce here the phase difference ψ = φ − φext/nand frequency mismatch ∆ω = ω − ωext/n betweenSTNO and external signal and the normalized couplingεp,φ = Ωp,φ/

√p0. We assume the nonlinear damping

rate larger than the coupling Γp εφ. The param-eter n ∈ Q accounts for either a possible fractionalsynchronization35,44,45 or for the frequency division n =1/N of the PLL system.

From the stationary solutions to eqs. (2), the equilib-rium phase difference ψeq and power deviation ρeq canbe derived. For the equilibrium state to be stable, all theeigenvalues of the Jacobian J of the dynamical system(2) must be negative. The latter is given by:

J =

(−2Γp −nεpaext sin(βp + nψeq)2νΓp −nεφaext cos(βφ + nψeq)

). (3)

and fundamentally characterizes the system’s stabil-ity. Note that the theoretical approach in this section israther general and indeed valid for any external sourceapplied to the STNO.

B. General form of the noise PSD at efficientphase locking and white noise

As elaborated in the experiments (sec. III), the forcedoscillator can respond in two ways to stochastic noise pro-cesses. First, when it is locked to a stable phase differ-ence ψeq, the fluctuations occur inside a synchronizationpotential valley around this value. The second mecha-nism describes discrete phase jumps between two 2π/nperiodic equilibrium states, separated through a poten-tial barrier. In this section, we first consider only thelocked case while in section IV C, also the occurrence ofphase slips will be taken into account.

In the presence of thermal noise, ψ and ρ are subject tosmall variations around their stationary values ψeq andρeq. The variation dynamics can be described in linearalgebra and response theory by the time-independent ma-trices A = −J and Σ = σ01 through a linear stochasticdifferential equation with additive noise46:

Xt = −AXt + ΣHt . (4)

Xt = (δρ, δψ)ᵀ

is a Gaussian stationary Ornstein-Uhlenbeck process, Σ describes the correlation matrixof the normalized amplitude and phase noise processesHt = (ηp, ηφ)

ᵀwith diffusion constant σ0 (with the dif-

fusion coefficient Dn of the Gaussian noise process, it is

σ20 = 2Dn(p0)/p0 = 2∆ω0 with ∆ω0 the STNOs linear

half-linewidth43,47). By evaluating the eigenvalues of A,a classical stability analysis can be performed. Trans-formation into the frequency space and exploiting thecharacteristics of the Gaussian noise process yields thephase noise PSD (for details see appendix B 1):

Sδψ2(ω) =σ2

0

4Γ2p

1 + ν2 + v2(λ2

1 + v2)(

λ22 + v2

) , (5)

with λ1,2 = λ1,2/(2Γp) and v = ω/(2Γp) normalizedquantities and the eigenvalues of the system matrix Agiven by:

λ1,2 = Γp (1 + u)(

1±√

1− 4γ)

.

We have defined the following variables:

u =nεφaext

2Γpcos(nψeq − βφ)

γ =det(A)

(tr(A))2 =

aextn

2Γp(1 + u)2

· (εφ cos(nψeq − βφ) + νεp sin(nψeq − βp)) . (6)

It can be seen that γ determines the eigenvalues to bereal or complex valued. A discussion of the result (5) isgiven in section V.

C. Phase slip dynamics

phase slip

Figure 8: Representation of the phase slip dynamics inthe periodic synchronization potential V (ψ).

As seen in section III, phase slips by ±2π/n are one ofthe major drawbacks in terms of STNO performance ingeneral and of STNO performance within a PLL systemin particular. They add an additional diffusive contribu-tion to the noise PSD. Hence in the following, we studythe physics of these phase jumps similar to a Brownianmotion in a periodic potential field V (ψ)48,49. Withinsuch a model, the dynamics of the phase difference ψ canbe characterized by an effective drift velocity ve and dif-fusion constant De, defined asymptotically through48,49:

ve = limt→∞

〈ψ(t)〉t

; De = limt→∞

〈ψ2(t)〉 − 〈ψ(t)〉22t

. (7)

7

In terms of phase noise, this diffusive mechanism trans-lates to the described 1/f2 shaped noise characteristics:

Sδψ2(ω) =De

ω2. (8)

The phase slip mechanism within the synchronization po-tential V (ψ) is shown in fig. 8, where thermal fluctua-tions induce a phase slip to another minimum. The driftand diffusion due to the phase slips are determined bythe potential barrier between two minima in V (ψ), thefrequency mismatch ∆ω between STNO and external sig-nal, whose sign sets the favored drift direction, and theintrinsic diffusion D of the phase difference. We deriveexpressions for these parameters and describe the fluctu-ation dynamics within the Fokker-Planck formalism46,50.

Under the assumption that phase slips occur at timescales larger than the power relaxation rate τ 1/Γp,the derivative ρ of the amplitude deviation in the cou-pled eqs. (2) is small compared to Γpρ and hence ρ canbe expressed as ρ(ψ(t)). Inserting this into the stochas-tic differential equation for the dephasing ψ (eqs. (2)including noise), one obtains:

ψ =∆ω − εφaext sin(nψ − βφ) + νεpaext cos(nψ − βp)+ σ0ηφ + σ0νηp︸ ︷︷ ︸

σ0

√1+ν2η

. (9)

The term under the curly bracket renormalizes the noiseterms and defines the diffusion D = (1 + ν2)σ2

0/2 of thephase dynamics. Eq. (9) corresponds to the stochasticAdler equation51,52 and defines the periodic synchroniza-tion potential V (ψ) (see appendix C) with local minimaat ψ = ψeq + 2πk/n, k ∈ Z.

In the Fokker-Planck formalism46,50 upon the poten-tial V (ψ), the 2π/n periodic probability density P(ψ, t)of the dephasing value ψ at time t describes the fluctu-ation dynamics (see appendix C for details). Analyti-cal expressions for the parameters ve and De can thenbe found from the stationary probability density P0(ψ),evaluated through the corresponding continuity equationwith probability current jP (see appendix C for details).They can be given as:

ve =nD

πsinh

(πF

2

) ∣∣IiF/2 (E/2)∣∣−2

, (10)

and

De = D cosh (πF/2)∣∣IiF/2 (E/2)

∣∣−2. (11)

with Iia(x) the modified Bessel function53, (a, x) ∈R, E = [2aext(νεp + εφ)] /(nD) a normalized coupling,and the normalized frequency mismatch F = 2∆ω/(nD).We treat the diffusion constant De using the approachpresented by Stratonovich48 assuming a small drift (fre-quency detuning ∆ω much smaller than the periodic cou-pling terms in (9)). Note that in general, the evaluation

of De is rather complex. However, the derived relationis a good approximation for small normalized frequencydetuning F = 2∆ω/(nD) 148, typically aimed at PLLoperation. More details of the calculus can be found inthe appendix C and for a more detailed study of thediffusion in a tilted periodic potential it is referred toRefs.54,55. As mentioned, both drift and diffusion pa-rameters classify the phase slips. Hence, they are furtherdiscussed in section V.

V. PHASE NOISE PSD OF A PLLCORRECTED STNO

The given results in section IV are quite generally ap-plicable to describe the phase dynamics under noise forany external periodic forcing. For PLL operation, the ex-ternal signal corresponds to a modulation feedback withn = 1/N . The feedback signal in the form of a spinpolarized current acts on the STNO amplitude21 andhence the coupling parameters can be set to εp 6= 0 andεφ = 0. Note that this situation simplifies the discussionbut however, the results remain qualitatively valid evenwhen εφ 6= 0 (see appendix B 2). From eqs. (2), the im-portant PLL bandwidth can be calculated. It describesthe locking capability defined by the difference betweenthe external frequency ωe and the multiple of the naturalfrequency nω:

∆ωBW = nνεpaext = nνΩpaext√p0

. (12)

Interestingly, the PLL bandwidth (eq. (12)) scales withthe nonlinearity factor ν. The latter consequently favorsan efficient phase locking through the PLL. The couplingstrength Ωp depends on the interaction mechanism andthe STNO’s intrinsic parameters. For STNOs, the PLLfeedback mechanism is usually a direct correction cur-rent, for which Ωpaext ∼ dΓ(I)/dI|Idc IPLL, or a mag-netic correction field generated by a field line.

A. Phase noise PSD in the phase locked statewithout phase slips

In fig. 9, we display the evaluated phase noise PSD(eq. (5)) for different values of γ corresponding to theratios of amplitude coupling εp, PLL divider rate n andnonlinearity ν (eq. (6)). Evaluating eq. (6) with thebandwidth, eq. (12), the parameter γ describes thePLL bandwidth relative to the nonlinear damping rate:γ ∼ ∆ωBW /(2Γp). To compare with the experimental re-sults from section III, we set the parameters fp = Γp/π =10 MHz, ν = 10 and FWHM = 2∆ω0

(1 + ν2

)= 450 kHz

fixed. The results reproduce well the different featuresobserved in the experiments (sec. III), such as the flat-tening of phase noise at low offset frequencies and a res-onance peak around the cutoff frequency. From eq. (5),

8

103 104 105 106 107 108 109

Offset Frequency [Hz]

−150

−125

−100

−75

−50

−25

Noi

seP

SD

[dB

c/H

z]

fp νfp

γ = 0

γ = 0.01

γ = 0.1

γ = 1

γ = 10

Figure 9: Phase noise PSD for different parameters ofγ. The black curve represents the free-running STNO.

the offset frequency limit values in the graph 9 can bedetermined:

limω→∞

Sδψ2(ω) =σ2

0

ω2(13a)

limω→0

Sδψ2(ω)

=σ2

0(1 + ν2)

a2extn

2ε2pν2 −∆ω2

=σ2

0(1 + ν2)

∆ω2BW −∆ω2

. (13b)

In the high offset frequency limit, eq. (13a), the phasenoise PSD of the PLL-corrected signal joins the one of afree running oscillator (black curve in fig. 9).In the low offset frequency limit, eq. (13b), Sδψ2 is inde-pendent of ω and thus adopts a constant value. Its levelis inversely proportional to the PLL bandwidth ∆ωBW .Through ∆ωBW (see eq. (12)), the parameters n andεp can lower the low offset frequency noise plateau forincreasing values; the dependence on ν is more compli-cated and further discussed in sec. V C. Eq. 13b alsoshows that the phase noise level depends on the frequencydetuning ∆ω. Large detuning increases the phase noiselevel until at ∆ω = ∆ωBW , the detuning reaches its max-imum in the synchronization bandwidth and the phasenoise diverges.

Frequency cutoff and resonance in the phase noise PSD

The noise PSD representation, eq. (5), can be inter-preted as a filter with transfer function H(f), correlated

with the PSD through: SY (f) = |H(f)|2 SX(f), with Xthe input and Y the output signal (here, this is the ther-mal white noise process and the phase noise, resp.). Thenumerator in (5) yields a cutting frequency of

ωcut = 2Γp√

1 + ν2 ,

which arises from the inverse first order lowpass filter ofthe numerator in eq. (5). For large nonlinear parametersν > 1, it can be written in terms of the real frequencyfcut ≈ νfp, also plotted in fig. 9.

The denominator in eq. (5) represents a low pass filterof 2nd order56 with a natural frequency ωn = 2Γp(1 +u)√γ and quality factor Q =

√γ. For frequencies higher

than ωn, the system decreases by 1/f4. Subsequentlydetermining the filter frequency taking the damping D =1/(2Q) into account gives:

ωd = ωn√

1− 2D2 = 2Γp(1 + u)

√γ − 1

2. (14)

It allows evaluating the filter stability, similar to a stan-dard harmonic oscillator as can be also determined bythe eigenvalues in eq. (5). For values γ < 1/2, the filteris overdamped and strictly decreases with the frequency.For γ > 1/2, the filter is underdamped and peaks at ωd,also represented in fig. 9 for higher values of γ. Com-paring the maximum filter value at ωd with the low fre-quency limit (13b), the amplitude of the overshoot peakis expressed as:

S(ωd) =4γ2

4γ − 1S(ω → 0) .

Thus, the overshoot becomes more pronounced at highervalues of γ, which is related to the coupling εp, divider ra-tio n and nonlinearity parameter ν (eq. (6)), or expresseddifferently as γ ∼ ∆ωBW /(2Γp). This is discussed exper-imentally (sec. III) and theoretically shown in fig. 9.

B. Parameters of the phase slip dynamics

In general, the occurrence of phase slips should be re-duced for efficient PLL operation and hence, the effectivedrift ve and diffusion De in the synchronization potentialV (ψ) minimized.

Thus, we characterize in fig. 10 their dependence onthe PLL and STNO parameters. Therefore, the normal-ized coupling E = 2aextνεp/(nD), defining the relativepotential barrier height between phase slips, and fre-quency mismatch F = 2∆ω/(nD) are introduced. Fig.10a reveals a strong influence on the frequency mismatch:For small normalized detuning F , we observe that forhigher coupling E the effective diffusion De and also theeffective drift coefficient ve can be minimized, hence thephase slips suppressed resulting in a locking of the STNO.However, the situation changes for higher detuning val-ues F : The drift coefficient converges to the detuning∆ω and the effective diffusion De significantly increases.Even for increasing coupling E, the effective diffusionconcurrently increases further if the target frequency istoo far from the free-running frequency. This behaviourof ve and De with detuning F and coupling E enablesus to understand the difference in the two measured bluecurves of coupling value ε = 0.152 and ε = 0.153 in fig.6a: We see that despite a larger coupling, the noise leveland as well the phase slip dynamics can be less favorable.This is because of the larger frequency mismatch of thespecific curve, which, in the picture of fig. 10a increases

9

0 0.2 0.4 0.6 0.8 1 1.2

0

0.5

1

1.5

Drift

0 0.2 0.4 0.6 0.8 1 1.2

0

1

2

3

4

Diffusion

(a)

10-2

10-1

100

101

102

10-15

10-10

10-5

100

(b)

Figure 10: Coefficients determining the phase slipdynamics in the periodic potential V : (a) Drift anddiffusion coefficients ve and De as a function of the

normalized frequency mismatch F at different couplingstrength values E. (b) Normalized effective diffusion for

F = 0 as a function of the coupling E.

the effective diffusion constant De. It is to be noted thatDe can even exceed the intrinsic phase diffusion D atlarge detuning (fig. 10a). In case of a sufficiently lowdetuning, a higher coupling results in a decrease of theeffective diffusion as it is depicted in graph 10b, in excel-lent agreement with the presented measurement data insection III.

Phase slip limit considerations

On long time scales (see eq. (7)) it is: 〈ψ〉 = ve and thediffusion describes a Gaussian noise process with variance∆ψ2 = 〈ψ2〉 − 〈ψ〉2 = 2Det, i.e. a 1/f2 phase noise PSDcharacteristics23. If phase slips occur, the STNO is neverperfectly locked in terms of a noise plateau at low offsetfrequencies. Thus, the phase noise PSD always readoptsa 1/f2 shape with Sδψ2(ω) = De/ω

2 at low offset fre-quencies below a characteristic frequency at which phaseslips occur.

At high diffusion D →∞ (high temperature), the nor-malized magnitudes F & E tend to zero implying themodified Bessel function to equal one: D → ∞ ⇒ F ,

E → 0 ⇒ IiF/2 (E/2) → 1. Consequently, drift and dif-fusion (eqs. (10) & (11)) correspond to the free-runningoscillation: ve → ∆ω & De → D and the periodic poten-tial is negligible.

For large coupling E 1, the situation depends on thedetuning F , as discussed above. At simultaneously small

detuning F 1, it follows∣∣IiF/2 (E/2)

∣∣−2 → πEe−E .As a consequence, it means that at small detuning, phasedrift ve and as well diffusion De exponentially decreasewith the coupling strength E, likewise decreasing thephase slip probability. This again agrees well with theperformed measurements presented in section III.

C. Discussion: Phase noise PSD including phaseslips: PLL performance and parameter space

In fig. 11, we illustrate more in detail the depen-dence of the phase noise PSD on the system parame-ters εp, ν and n. The contribution due to phase slipsSδψ2(ω) = De/ω

2 can be summed to the phase noisePSD at efficient locking (eq. (5)). In fig. 11a, the low-ered noise plateau at low offsets with increasing couplingis observed for constant values of ν and n. For the highestshown coupling value, a resonance bump is seen slightlyabove fp, as discussed in sec. V A. Furthermore, the dis-cussed exponential decrease ∼ exp(−E) of the phase slipeffective diffusion De with the coupling strength can beobserved (here, F = 0), decreasing the phase slip phasenoise contribution for increasing coupling. Fig. 11b sum-marizes the complex dependence of the noise PSD on theSTNO nonlinearity parameter ν for constant couplingεp/(2Γp) = 0.025 and n = 1. As it is derived in sec. IV(see eq. (12)), we observe that the PLL bandwidth is en-hanced by the oscillator’s nonlinearity ν. However, ν alsostrongly broadens the STNO linewidth and enhances thediffusion by (1+ν2)21,57, which can in fig. 11b mainly berecognized at higher offsets. Moreover, the noise overshotat the resonance frequency of the 2nd order filter trans-fer function (see eq. (14)) is likewise proportional to νand narrows the PLL performance. Thus, the filter over-shoot at ωd close to the PLL bandwidth becomes morepronounced with the nonlinearity, also recognized in fig.11b. Furthermore taking the phase jump dynamics intoaccount, the normalized coupling strength E is propor-tional to ν and antiproportional to nD. On the contrary,it is D ∼ (1 + ν2) and thus the nonlinearity lowers thePLL performance in terms of phase slip dynamics, whatis perfectly retrieved in the curves of fig. 11b in whichthe best noise characteristics are found for ν = 1.

As a last parameter, the divider ratio n is to be consid-ered. Its influence on the phase noise is shown in fig. 11cfor ν = const = 10: A high value leads to a lower noiseplateau at low frequencies but likewise also increases theresonant 2nd order filter response (see sec. V). Againtaking the phase slip dynamics into account, n decreasesthe coupling strength E ∼ (nD)−1 and phase slips aremore likely to occur at high n, especially recognized for

10

102 103 104 105 106 107 108 109

Offset Frequency [Hz]

−150

−125

−100

−75

−50

−25

0N

oise

PS

D[d

Bc/

Hz]

fp νfp

εp/(2Γp) = 0

εp/(2Γp) = 0.025

εp/(2Γp) = 0.05

εp/(2Γp) = 0.1

εp/(2Γp) = 0.5

(a)

102 103 104 105 106 107 108

Offset Frequency [Hz]

−140

−120

−100

−80

−60

−40

−20

0

20

Noi

seP

SD

[dB

c/H

z]

fp

n = 1

ν = 0.1

ν = 1

ν = 10

ν = 16

n = 1

ν = 0.1

ν = 1

ν = 10

ν = 16

(b)

102 103 104 105 106 107 108

Offset Frequency [Hz]

−140

−120

−100

−80

−60

−40

−20

0

Noi

seP

SD

[dB

c/H

z]

fp

ν = 10

n = 0.01

n = 0.1

n = 1

(c)

Figure 11: Phase noise (PN) PSDs covering the parameter space of ν, n and εp. (a) PN for different couplingcoefficients εp. (b) PN for different parameter values ν at n = const = 1. (c) PN for different parameter values n atν = const = 10. The green curve in (b) and (c) is the same one. The following constant values are chosen: F = 0,fp = 1 · 107 Hz, FWHM = 450 kHz = 4π(1 + ν2)∆f0, and, if not explicitely indicated differently, ν = 10, εp = 0.025,

n = 1.

n = 1 in fig. 11c.The discussed dependences are in complete agreement

with the presented measurements in section III. The dis-cussion highlights that an adequate trade-off between thedescribed mechanisms and parameter dependences mustalways be found for an efficient noise suppression by thePLL.

VI. CONCLUSION

We present the experimental implementation of twodifferent STNOs – a STVO and a uniform STNO (sec-tion III) – into an on-chip integrated PLL developed forthis purpose. We find an efficient phase noise reductionof 50 dB @ 10 kHz for the vortex and of 20 dB @ 100 kHzoffset for the uniform STNO with fc ≈ 340 MHz andfc ≈ 4.8 GHz, respectively. Furthermore, we thoroughlyanalyze the PLL performance for different control param-eters and find rather complex dependences due to theintrinsic large nonlinearity of our oscillators. The occur-rence of phase slips, mainly caused by the large intrinsicnoise in STNOs, is identified as the main drawback forthe exploitation of the PLL system.

In complement to the experimental results, we providea complete theoretical framework to analyze the perfor-mance of the PLL system upon operation with a nonlin-

ear spin torque nano oscillator. This model excellentlydescribes the system’s parameter characteristics and re-veals the physics of the low offset frequency noise due tophase slip dynamics. Note that in general, this theoret-ical development is not restricted to only the case of aPLL but the approach can easily be employed for anyexternal signal applied to the STNO. It is found thatthe dependence of the noise PSD on the PLL and STNOparameters is complex and its optimization requires atrade-off between all of those, implying a prior analysisof the parameter space. Our approach of a highly config-urable PLL chip is a perfect basis to handle the systemcomplexity since the important parameters can be easilyadjusted3,4. Thus, it illustrates a first step towards flexi-ble, integrated and hybrid systems for accurate frequencygeneration utilizing STNOs.

ACKNOWLEDGMENT

S.W. acknowledges financial support from LabexFIRST-TF under contract number ANR-10-LABX-48-01. The work is supported by the French ANR project”SPINNET” ANR-18-CE24-0012. We acknowledgethe Plateforme Technologique Amont (PTA), Grenoble,France for access and support for the nanofabrication

∗ Present address: Max Born Institute For Nonlinear Op-tics & Short Pulse Spectroscopy, Max-Born-Str. 2A, 12489Berlin, Germany; Email: steffen.wittrock@mbi-berlin.de

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56 System known as PT2 system, whose parameters can beidentified through the amplitude representation |H(iω)| ∼1/

√(1− (ω/ω0)2

)2+ (2D · ω/ω0)2 with filter parameters

damping D, quality factor Q = 1/(2D) and natural, un-damped frequency ω0

60.57 V. S. Tiberkevich, A. N. Slavin, and J.-V. Kim, Physical

Review B 78 (2008), 10.1103/physrevb.78.092401.58 M. Quinsat, D. Gusakova, J. F. Sierra, J. P. Michel,

D. Houssameddine, B. Delaet, M.-C. Cyrille, U. Ebels,B. Dieny, L. D. Buda-Prejbeanu, J. A. Katine, D. Mauri,A. Zeltser, M. Prigent, J.-C. Nallatamby, andR. Sommet, Applied Physics Letters 97, 182507 (2010),http://dx.doi.org/10.1063/1.3506901.

59 L. Bianchini, S. Cornelissen, J.-V. Kim, T. De-volder, W. van Roy, L. Lagae, and C. Chap-pert, Applied Physics Letters 97, 032502 (2010),http://dx.doi.org/10.1063/1.3467043.

60 G. Schulz and K. Graf, Regelungstechnik 1: Lineare undnichtlineare Regelung, rechnergestutzter Reglerentwurf, DeGruyter Studium (De Gruyter, 2015).

13

Appendix A: Experimental details

1. Samples

The measured vortex based and uniform STNOs havethe following structure: BE/AF/SAF/MgO/FL/cap.SAF denotes the polarizing layer and consists of asynthetic antiferromagnet (SAF) pinned by an antifer-romagnet (AF). It has the same structure for bothtypes of devices with PtMn(20)/Co70Fe30(2)/Ru(0.85)/Co40Fe40B20(2.2)/Co70Fe30(0.5) and the nm layer thick-ness in brackets. MgO is the tunnel barrier whose thick-ness and oxidation time is adjusted to yield a nomi-nal resistance area product of 1.45 Ωµm2 (1.5 Ωµm2) forthe vortex (uniform) MTJs. The free layer (FL) is abilayer of Co40Fe40B20(t1)/Ta(0.2)/Ni80Fe20(t2) withthicknesses t1 = 1.5 nm (2 nm) and t2 = 7 nm (2 nm)for the vortex (uniform) MTJs, respectively. The Taserves as a Boron pump and decouples the CoFeB andNiFe layers to assure a high tunneling magnetoresistance(TMR). BE denotes the bottom electrode material withTa(3)/CuN(30)/Ta(5) and cap is the capping materi-

al with Ta(2(3))/Ru(7) for the vortex (uniform) de-vices. All MTJ stacks were sputter deposited by INLusing a Singulus-Timaris deposition tool onto high re-sistivity Si substrates with an additional 500 nm SiO2

layer. The MTJ stacks are subsequently annealed for 2 hat T = 330 C at an applied magnetic field of 1 T. Theywere then patterned at the PTA facilities (Grenoble) bySPINTEC using standard optical and e-beam lithogra-phy processes as well as Ar ion etching. The magne-toresistance values (at room temperature) of the STNOnanopillars were around 50%. Data are presented for vor-tex (uniform) STNOs with diameter of D = 300(80) nm,leading to an emission frequency range of 200-400 MHz(under an out-of-plane field) for the vortex STNOs and3-6 GHz (under in-plane field) for the uniform STNOs.

2. Measurements

The different STNO devices are directly employed onthe presented PLL chip. It delivers the dc current in or-der to operate the STNO at self-oscillations, which aresustained by the resulting spin transfer torque. The PLLchip with STNO is subjected to an applied magnetic fieldthat is specified in the related sections. The emitted rftime signals are recorded by a single-shot oscilloscopemeasurement and the emission spectra are gathered bysimultaneous employment of a spectrum analyzer. In or-der to obtain noise data, the measured time signal isprocessed via the Hilbert transform method22,58,59.

Appendix B: Theory – efficient PLL phase locking

1. Power spectral density

The formal solution of eq. (4) is:

Xt = e−AtX0 +

∫ t

0

e−A(t−s)ΣHsds ,

and by evaluating the eigenvalues of A, a classical sta-bility analysis can be performed. However, to determinethe noise PSD of the system, a Fourier transform of sys-tem (4) is conducted. Assuming Ht deterministic andcontinuous for the Fourier transform to be defined, oneobtains:

X(ω) = (A+ 1iω)−1

ΣHt(ω) .

With SH(ω) the PSD of Ht, the noise PSD of Xt is inconsequence described by:

SX(ω) = (A+ 1iω)−1

ΣSH(ω)Σᵀ (Aᵀ − 1iω)−1

.

Here, the white noise processes inHt are independent andthus its PSD is diagonal with each process’ covariance asthe entries of SH(ω) = 1. Thus, one obtains:

SX(ω) = σ20(A+ 1iω)−1(Aᵀ − 1iω)−1 .

Taking the matrix A in its general form:

A =

(a11 a12

a21 a22

)the inverse matrix can be computed:

S−1X σ2

0 = (Aᵀ − 1iω) (A+ 1iω)

=

(a2

11 + a221 + ω2 Λ∗

Λ a212 + a2

22 + ω2

),

with Λ = a11a12 + a21a22 + iω(a12− a21). Its determi-nant is given through the systems’ eigenvalues λ by:

D = det(S−1X σ2

0

)= det (Aᵀ − 1iω) det (A+ 1iω)

= (λ1 − iω)(λ2 − iω)(λ1 + iω)(λ2 + iω)

= (λ21 + ω2)(λ2

2 + ω2) .

Now inverting S−1X σ2

0 , one obtains for the general formof the noise power spectral density matrix:

SX(ω) =σ2

0

D adj(S−1X σ2

0

)=σ2

0

D

(a2

12 + a222 + ω2 −Λ∗

−Λ a211 + a2

21 + ω2

).

(A.1)

14

2. Generalization through renormalization of thecoupling terms

In a general case, the theoretical calculations startingfrom eqs. (2) might be simplified by the introduction of acomplex notation, allowing for the generalization of someof the discussed results when εp, εφ 6= 0:

εp = εpeiβp , εφ = εφe

iβφ , ε = εpeiβ .

Here, the last variable is the new general coupling anddefined by:

ε = iεp + εφ/ν ,

with the norm ε and argument β given by:

ε =√ε2p + ε2φ/ν

2 + 2εpεφ sin(βφ − βp)/νcos(β) = −εp/ε · sin(βp) + εφ/(νε) cos(βφ)

sin(β) = εp/ε · cos(βp) + εφ/(νε) sin(βφ) .

This gives the following substitutions for a real variablex:

ε cos(x− β) = εp sin(x− βp) + εφ/ν cos(x− βφ)

ε sin(x− β) = −εp cos(x− βp) + εφ/ν sin(x− βφ) .

This approach can be used in order to generalize the re-sults that are specifically discussed for the situation ofa PLL, such as eqs. (12) or (13b). It allows for a morequalitative discussion, not only limited to the case of aPLL, but indeed, for any applied external synchroniza-tion signal. For instance, the stochastic Adler equation(eq. (9)) would become:

ψ = ∆ω − νε sin(nψ − β) + σ0

√1 + ν2η ,

which is more easily recognized as the common stochas-tic Adler equation due to the renormalized coupling term.

Appendix C: Theory – Phase slip dynamics

The dephasing dynamics is represented in fig. 8. It isdescribed as a Brownian motion in a periodic potentialfield, given by:

V (ψ) = −[ψ∆ω +

νεpaext

nsin(nψ − βp)

+εφaext

ncos(nψ − βφ)

], (A.2)

with diffusion D = (1 + ν2)σ20/2. Local minima of V

are found at ψ = ψeq + 2πk/n, k ∈ Z. Phase slips areinduced by thermal fluctuations and described by jumpsbetween the minima in V . The drift is determined by∆ω with favored drift direction set by the sign of ∆ω.

The fluctuation dynamics can now be describedthrough the probability density P(ψ, t) of the de-phasing value ψ at time t within the Fokker-Planckformalism46,50:

∂P

∂t=

∂

∂ψ[V ′P] +D

∂2P

∂ψ2, (A.3)

with V ′ = dV/dψ. Subsequently, the continuity equa-tion is described by:

∂P

∂t(ψ, t) = −∂jP

∂ψ(ψ, t) ,

with the probability current

jP(ψ, t) = −V ′P−D ∂P/∂ψ . (A.4)

For P0 the stationary probability, jP is constant andequals the drift ve. It is in a limit value examination:

limt→∞

P(ψ, t) = P0(ψ)Z√

4πDete−

(ψ−vet)24Det ,

with Z a normalization constant. The solution forthe probability density P0 is 2π/n periodic and can beevaluated solving eq. (A.4):

P0(ψ) = −

ψ∫−∞

jP eV/Ddψ′

e−V/D

= −veD

eπF

1− eπF

ψ+ 2πn∫

ψ

eV (ψ′)−V (ψ)

D dψ′

︸ ︷︷ ︸=:I−(ψ)

. (A.5)

We introduce the normalized frequency mismatch F =2∆ω/(nD) and define the integral I∓(ψ) with index ∓denoting the equivalent integration variables [ψ → ψ +2π/n], corresponding to a ”-” sign in the argument of theexponential, and [ψ−2π/n→ ψ], corresponding to a ”+”sign in the argument of the exponential: V (ψ′)∓ V (ψ) .We took advantage of the periodicity of V , disrespectingthe linear drift, i.e. V (ψ− 2πk/n) = V (ψ) + 2πk/n ·∆ω.Thus, the integral on the left-hand side of eq. (A.5) canbe evaluated in a geometrical series48,49:

ψ∫−∞

eV (ψ)/Ddψ =

∞∑l=0

ψ+ 2πn∫

ψ

eV (ψ− 2π

n)

D dψ

=1

1− eπF∫ ψ+ 2π

n

ψ

eV (ψ)/Ddψ .

From eq. (A.5), drift ve and diffusion De coefficients,which basically determine the phase slip dynamics, canbe evaluated.

15

a. Drift coefficient ve

The drift coefficient ve is identified through the nor-malization of P0

48,49 in eq. (A.5):

n

2π

∫ 2πn

0

P0(ψ)dψ = 1

⇒ ve =D(1− e−πF

)n2π

2πn∫0

I−(ψ)dψ

.

The integral in the denominator was evaluated byStratonovich48, who finds:

ve =nD

πsinh

(πF

2

) ∣∣IiF/2 (E/2)∣∣−2

, (A.6)

with Iia(x) the modified Bessel function53, (a, x) ∈ R,and E = [2aext(νεp + εφ)] /(nD) a normalized coupling.

b. Diffusion coefficient De

We treat the diffusion constant De using the approachpresented by Stratonovich48 assuming a small drift (fre-

quency detuning ∆ω much smaller than the periodic cou-pling terms in (A.2)). Then, the probability current vefor the stationary solution P0 can be treated as a positiveand a negative current jP,±:

ve = jP,+ − jP,− withjP,+jP,−

= e−πF .

The effective diffusion is then determined by the relativediffusion through the energy barriers of potential differ-ence πF on both sides:

De =π

n(jP,+ + jP,−) =

π

n

e−πF + 1

e−πF − 1ve ,

and is therefore calculated to:

De = D cosh (πF/2)∣∣IiF/2 (E/2)

∣∣−2. (A.7)

Note that in general, the evaluation of De is rather com-plex. However, the derived relation is a good approx-imation for small normalized frequency detuning F =2∆ω/(nD) 148, typically aimed at PLL operation. Amore detailed study of the diffusion in a tilted periodicpotential can be found in Refs.54,55.