Measuring frequency fluctuations in nonlinear nanomechanical resonators
Olivier Maillet, Xin Zhou, Rasul R. Gazizulin, Bojan R. Ilic, Jeevak M. Parpia, Olivier Bourgeois, Andrew D. Fefferman, Eddy Collin
MMeasuring Frequency Fluctuations inNonlinear Nanomechanical Resonators
Olivier Maillet, † Xin Zhou, ‡ Rasul R. Gazizulin, † Rob Ilic, ¶ Jeevak M. Parpia, § Olivier Bourgeois, † Andrew D. Fefferman, † and Eddy Collin ∗ , † † Université Grenoble Alpes, CNRS Institut Néel, BP 166, 38042 Grenoble Cedex 9, France ‡ Université Grenoble Alpes, CNRS Institut Néel, BP 166, 38042 Grenoble Cedex 9, FranceNow at: Institut d’Electronique, de Microélectronique et de Nanotechnologie (IEMN), Univ.Lille & CNRS, 59000 Lille, France ¶ Center for Nanoscale Science and Technology, National Institute of Standards andTechnology, Gaithersburg, Maryland 20899, USA § Department of Physics, Cornell University, Ithaca, New York 14853, USA
E-mail: [email protected]
Phone: +33 4 76 88 78 31
Abstract
Advances in nanomechanics within recent years have demonstrated an always ex-panding range of devices, from top-down structures to appealing bottom-up MoS andgraphene membranes, used for both sensing and component-oriented applications. Oneof the main concerns in all of these devices is frequency noise, which ultimately limitstheir applicability. This issue has attracted a lot of attention recently, and the ori-gin of this noise remains elusive up to date. In this Letter we present a very simpletechnique to measure frequency noise in nonlinear mechanical devices, based on thepresence of bistability. It is illustrated on silicon-nitride high-stress doubly-clamped a r X i v : . [ c ond - m a t . m e s - h a ll ] J un eams, in a cryogenic environment. We report on the same T /f dependence of the fre-quency noise power spectra as reported in the literature. But we also find unexpected damping fluctuations , amplified in the vicinity of the bifurcation points; this effect isclearly distinct from already reported nonlinear dephasing, and poses a fundamentallimit on the measurement of bifurcation frequencies. The technique is further appliedto the measurement of frequency noise as a function of mode number, within the samedevice. The relative frequency noise for the fundamental flexure δf /f lies in the range . − . ppm (consistent with literature for cryogenic MHz devices), and decreaseswith mode number in the range studied. The technique can be applied to any types ofnano-mechanical structures, enabling progresses towards the understanding of intrinsicsources of noise in these devices. Keywords: nanomechanics, nonlinearity, bifurcation, frequency fluctuations, damp-ing fluctuations.
Within the past decade Nano-Electro-Mechanical-Systems (NEMS) have developed witha broad range of applications extending from physics to engineering. In the first place, theirsize makes them very sensitive transducers of force.
This had been demonstrated e.g. inthe seminal work of D. Rugar et al. in which a cantilever loaded by a magnetic tip reacheda detection sensitivity corresponding to the force exerted by a single electronic spin at adistance of about 100 nm. More recently, NEMS have been applied to the detection ofsmall quantities of matter (mass spectroscopy), with precision reaching the single proton. Nowadays, even the quantum nature of the mechanical degree of freedom is exploited forquantum information processing. In all applications, the quality of the device is intrinsically linked to its level of displayednoise. Specifically, frequency noise in NEMS appears to be a key limiting parameter whosephysical origin is still unknown. Besides, only few quantitative experimental studies areavailable in the literature, especially at low temperatures. The nonlinear frequencynoise reported for carbon-based systems is one of the most striking results, revealing thecomplex nature that the underlying mechanisms can possess.2requency noise (or phase noise ) can be understood in terms of pure dephasing , making an analogy with Nuclear Magnetic Resonance (NMR); and its impact on a mechanicalresonance can be modeled experimentally by means of engineered frequency fluctuations. The physical origin of intrinsic frequency noise is indeed still elusive, since all identified mech-anisms studied explicitly display much weaker contributions than the reported experimentalvalues: adsorption-desorption/mobility of surface atoms, experimentally modeled under aXe flow, or the nonlinear transduction of Brownian motion. These efforts in under-standing the microscopic mechanisms at work in mechanical dephasing are accompanied bytheoretical support. The nonlinear dephasing/damping has been proposed to originate innonlinear phononic interactions between the low frequency mechanical modes and thermalphonons. Finally, a common speculation reported in the literature is that frequency noiseis related to defects, which can be either extrinsic or constitutive of the material (likein a glass). The presence of these so-called Two-Level Systems (TLS) is also proposed toexplain damping mechanisms in NEMS, and have been shown recently to lead to peculiarfeatures (especially in the noise) for mesoscopic systems such as quantum bits and NEMS. Properly measuring frequency noise is not easy; a neat technique presented in the litera-ture relies on cross-correlations present in the two signals of a dual-tone scheme. Moreover,in the spectral domain dephasing and damping are mixed.
In order to separate thecontributions, one has to use both spectral-domain and time-domain measurements.
Allof these techniques may not be well suited for large amplitude signals (especially when thesystem becomes bistable), preventing the exploration of the nonlinear range where nonlineardamping/dephasing may dominate.In this Letter we present a method based on bifurcation enabling a very simple mea-surement of frequency noise in nonlinear bistable resonators. Building on this method, wecharacterize the intrinsic frequency noise of high-stress Silicon Nitride (SiN) doubly-clampedbeams in cryogenic environment (form 1.4 K to 30 K). In particular, we study the three firstsymmetric modes ( n = 1 , , ) of one of our devices, and demonstrate the compatibility of3ur results with existing literature. The temperature-dependence is indeed similar to Ref., but we find an unexpected damping noise which is amplified through the bifurcation mea-surement. This result is distinct from the reported nonlinear phase noise of Ref. in whichthe device was not bistable. The phenomenon seems to be generic, and we discuss it in theframework of the TLS model. Note that our results demonstrate the existence of an ultimatelimit to the frequency resolution of bifurcation points in nonlinear mechanical systems. -5 -1.0x10 -5 -5.0x10 -6 -6 -5 -5 -5 -5 -5 -5 X , Y ( V ) f (Hz) V(t)I(t)
Figure 1:
Device and setup. SEM image of the 250 µ m device measured in this work. The gateelectrode (brown) is not used here. The actual NEMS device is the red-colored string in betweenthe two (light blue) electrodes. The lock-in detector (violet), magnetic field and drive generator (ingreen) are also depicted in a schematic fashion to illustrate the magnetomotive technique. esults and discussion A typical doubly-clamped NEMS device used in our work is shown in Fig. 1. It consistsof a 100 nm thick SiN device covered with 30 nm of Al. The width of the beam is 300 nmand the length L = 250 µ m. Another similar sample of L = 15 µ m has been characterized.The beams store about 1 GPa of tensile stress, and we define A to be their rectangularcross-section. For fabrication details see Methods below. The device is placed in a Hecryostat with temperature T regulated between 1.4 K and 30 K, under cryogenic vacuum ≤ − mbar. The motion of the beam is driven and detected by means of the magnetomotivescheme. For experimental details see Methods below.
A ) f d o w n X q u a d r a t u r e Y q u a d r a t u r e
Amplitude (nm)
F r e q u e n c y ( H z ) f u p B ) R u p R Amplitude (nm)
F r e q u e n c y ( H z ) R d o w n Figure 2:
Nonlinear (Duffing) resonance. A) Duffing resonance line (X and Y quadratures) mea-sured on the 250 µ m device at 4.2 K in vacuum, for a drive force of 81 pN, in a 1 T field. Thedirections of frequency sweeps are depicted by arrows. Vertical dashed lines indicate the two bifur-cation points f up,down . B) Amplitude R = √ X + Y corresponding to A). Bifurcation points areindicated with their amplitudes R up,down . A Laplace force F ( t ) = F cos(2 πf t ) with F ∝ I LB is created with a static in-planemagnetic field B and an AC current I fed into the metallic layer (Fig. 1). Fields B ofthe order of 1 T, and currents I up to 0.5 µ A have been used. The detected signal is theinduced voltage V ( t ) proportional to velocity. It is measured with a lock-in from which wecan obtain the two quadratures X, Y of the motion. We call R = √ X + Y the amplitude ofthe motion (at a given frequency), defined in meters peak. For all the T , B settings used in5he present work, the Al layer was not superconducting. A key feature of the magnetomotivescheme is that it enables the ability to tune the Q factor of the detected resonances: thisis the so-called loading effect.At low drives, in the linear regime, the quality factor of the resonance Q = f / ∆ f isdefined from the linewidth ∆ f and the resonance frequency f of the mode under study. Weconsider here only high- Q resonances Q (cid:29) . In this limit, the X peak is a simple Lorentzian,whose full-width-at-half-height gives ∆ f . For large excitation forces, our doubly-clampedbeams’ mechanical modes behave as almost ideal Duffing resonators. A typical Duffingresonance is shown in Fig. 2. The maximum of the resonance shifts with motion amplitudeas f max = f + βR max . β is the so-called Duffing parameter. We assume β > , but thecase β < is straightforward to adapt. R max is the maximum amplitude of motion; italways satisfies R max = F Q/k with very good accuracy. k is the mode’s spring constantwith f = π (cid:112) k /m and m the mode mass. In the nonlinear Duffing regime, a dampingparameter ∆ f can still be defined from the Q factor deduced from the peak height R max .When frequency noise is negligible, the so-called decoherence time T = 1 / ( π ∆ f ) definedfrom such frequency-domain measurements is just equal to T , the relaxation time of theamplitude R in time-domain. When R max ≥ √ R s a response hysteresis opens (see Fig. 2). The point in the ( R, f ) space at which this begins is called the spinodal point , with R s = / (cid:113) ∆ fβ . Beyond thispoint, two stable states coexist in a range of frequencies, with the system jumping from oneto the other at f up (in an upward frequency sweep) and f down (for downward). These arethe two bifurcation points . The maximum amplitude of the resonance is reached only on the6 Frequency (Hz)
A m p l i t u d e ( n m ) f d o w n - f f u p - f f s - f R s H y s t e r e t i c r a n g e
Figure 3:
Bifurcation branches. Calculated bifurcation frequencies from Eqs. (1-2) for the 250 µ mdevice and magnetomotively-loaded Q of (1 T field). upper branch. These frequencies write explicitly: f up = f + 2 βR up − (cid:114)(cid:0) βR up (cid:1) −
14 ∆ f for R up ≥ R s , (1)and: f down = f + 2 βR down − (cid:114) ( βR down ) −
14 ∆ f for / √ R s < R down ≤ R s (Case 1) ,f down = f + 2 βR down + (cid:114) ( βR down ) −
14 ∆ f for R down ≥ / √ R s (Case 2) . (2)These functions are displayed in Fig. 3. As we increase the driving force F , the maxi-mum amplitude R max linearly increases while the peak position shifts quadratically. Theupper bifurcation point f up then shifts monotonically towards higher frequencies, with amonotonically increasing amplitude R up when F is increased. On the other hand, the lowerbifurcation point f down has first an amplitude R down that decreases ( f down being given by Case 1 above), and then it increases again ( f down being then defined through Case 2 ). Atthe spinodal point R s , f up = f down = f s = f + √ ∆ f .The method we present builds on the work by Aldridge and Cleland: the bifurcation7 s f = 0 . 4 8 H z Count f d o w n - < f d o w n > ( H z ) fdown-
F r e q u e n c y ( H z )
A ) - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 005 01 0 01 5 0 s d f = 0 . 2 6 H z Count d f ( H z ) d f (Hz) T i m e ( s ) 1 0 - 4 - 3 - 2 - 4 - 2 B ) S d f (a.u.) F r e q u e n c y ( H z )
Figure 4:
Statistics on frequency at low amplitudes. A): histogram obtained on the f down relaxationpoint of Fig. 2 (left inset: actual frequency time-trace; right inset: power spectrum of the frequencyfluctuations, displaying /f -type structure). B): histogram obtained on the frequency jumps δf computed from the f down time-trace (left inset: δf time-trace; right inset: power spectrum). Thelines are Gaussian fits, and the power-law dependencies of the spectra are discussed in the text. positions are essentially arbitrarily well defined (in the sense that in an ideal system only thethermal activation of the bifurcation process will limit the stability), and can be used forsensing/amplifying. We thus devise a technique enabling the characterization of frequencyfluctuations themselves; indeed, the imprint of frequency fluctuations had been reportedearlier in noise-induced bifurcation relaxation. We show in Fig. 4 histograms obtained onthe f down frequency position of the resonance of Fig. 2. They are measured by rampingthe frequency down from above f up at constant speed, and measuring the switching to thehigher branch through a threshold detection. We repeat this protocol typically 1000 timesto acquire enough statistics (see Fig. 4 graph A left inset). The histogram obtained directlyfrom the frequency time-trace is then fit to a Gaussian (of standard deviation σ f , graph Ain Fig. 4), while the power spectrum S f ( f ) of the fluctuations is also computed (defined asthe FFT of the auto-correlation function). It displays a /f -type structure (see Fig. 4 Aright inset).Some precautions have to be taken in order to ensure that the acquired data is unbiased:we first make sure that the bifurcation jump occurs within a single point of the acquisi-tion trace (we thus have to lower the filtering time constant of the lock-in compared to8ig. 2). Typically, we take one point every 40 ms with a frequency resolution typically 10times smaller than the measured Gaussian spread. Second, we verify that we do not sufferfrom Brownian-type motion amplitude noise (at the mode frequency) that would activaterelaxation of states when we are close enough to the bifurcation points. Such activatedbifurcation generates non-Gaussian and asymmetric statistics, which is ramping-speed de-pendent. No such characteristics have been seen in our experiments: we first check thatthe ramping speed (of order . − Hz/sec) does not change the measured histogram; andsecond, we add a controlled amount of force noise (at the mechanical resonance) in orderto see when relaxation is indeed noise-activated. We see that a force noise equivalent to abath temperature of about K has to be reached in order to affect the frequency statistics.Note that K is also the range of effective temperatures that are needed in order to see(asymmetric) frequency fluctuations transduced from Brownian motion through the Duffingnonlinearity. Clearly, at 4.2 K with no added noise no such phenomena can occur. In thefollowing, we make sure that no extra force noise is injected in the setup while measuringfrequency fluctuations. Finally, the frequency drifts of our generators are characterized: wetake two of them of the same brand, and measure the frequency stability of one against theclock of the other. Slow frequency fluctuations occur at the level of 1 mHz for 1 MHz signalsover minutes, and 10 mHz for 10 MHz. This is at least two orders of magnitudes smallerthan what is seen here over the same periods of time, and can be safely discarded.We see that frequency fluctuations display a typical /f -type behaviour (right inset inFig. 4 A), as reported by others. Indeed, the time-trace has clearly some slowly driftingcomponent (left inset in the same graph). This means that the statistics obtained dependson the acquisition bandwidth . For pure /f noise, the standard deviation σ f (which is thesquare root of the power spectrum integral) depends on (cid:112) ln ( f high /f low ) , with f high thefastest frequency probed (defined from the time needed to acquire 1 bifurcation trace ∆ t min ,about 10 seconds) and f low the lower frequency cutoff (set by the total acquisition time, about3 hours). In order to be as quantitative as possible, we look for an estimate of frequency9oise which is as much independent from the protocol as possible. We therefore study the frequency jumps δf ( t ) = f down ( t i +1 ) − f down ( t i ) instead of f down ( t ) , see Fig. 4 B left inset.The variance of this quantity σ δf is the well known Allan variance, computed for acquisitiontime ∆ t min = t i +1 − t i . Note that this quantity safely suppresses equipment low frequencydrifts (like, besides the one characterized for the generator, e.g. temperature drifts due tothe He bath).For a perfectly /f noise, the Allan variance at ∆ t min → is independent of f high , f low (see Methods). However, our power spectrum fits S /f (cid:15) with (cid:15) ≈ . ± . . We calculatethat over the most extreme settings that have been used, our Allan variance should not havechanged by more than 50 % (and in the data presented here by much less). Moreover, toprove that the ∆ t min → assumption can be applied we display in the right inset of Fig. 4B the power spectrum S δf ( f ) of δf fluctuations: the data match the spectrum ∝ f directlycomputed from the fit of S f ( f ) .We can then compare the values of σ f and σ δf that have been obtained. According totheory (see Methods) the first one should be about twice the second one in our conditions.This is indeed what we see in Fig. 4, confirming that σ δf is a good quantitative measurementof frequency fluctuations. Besides, the frequency stability defined as σ δf /f ≈ . ppm fallswithin the expected range according to reported measured devices. We thus demonstratethat our simple technique is functional: the key being that since the spectrum is /f -type,we do not need to be especially fast to characterize frequency noise. With a Phase LockedLoop (PLL) setup one could measure fluctuations on much shorter timescales, but our 10 srepetition rate is perfectly well adapted.We now build on our method to thoroughly characterize frequency noise in SiN string de-vices. Let us apply this technique to the upper bifurcation point f up . The method explainedabove is easily reversed in sweep direction and threshold detection. Similar time-trace, spec-trum and histogram to Fig. 4 obtained this way are shown in Fig. 5. We see that thepower spectrum displays the same /f (cid:15) law with (cid:15) ≈ . . This is true for the complete10 Count d f ( H z ) s d f = 8 H z d f (Hz) T i m e ( s ) 1 0 - 4 - 3 - 2 - 2 Sf (a.u.)
F r e q u e n c y ( H z )
Figure 5:
Large amplitude statistics. Similar graphs as in Fig. 4 obtained on the upper bifurcationpoint f up (frequency jumps time-trace in the left inset, power spectrum of frequency in the rightinset and histogram of δf in the main of the graph). The line is a Gaussian fit, while the powerspectrum follows /f . (see text). study we realized on the same device, and proves that different data sets can be consistentlycompared. The histogram is again Gaussian. But surprisingly, σ δf is now much bigger on theupper branch than on the lower one. We therefore make a complete study as a function ofdriving force ( i.e. motion amplitude). We discover that the standard deviation σ δf dependsquadratically on motion amplitude R . Measuring at another magnetic field B , we find thatit also depends linearly on the Q of the mechanical mode. However, measuring at differenttemperatures T in the 1.4 K - 30 K range, we realize that the small amplitude value obtainedis temperature-dependent, while the large amplitude one is not.This suggests the normalized plot of Fig. 6, where the increase σ δf ( T, R up,down ) from theextrapolated σ δf ( T, R s ) is plotted against the normalized variable R up,down /R s . The notation R s ∼ means that the value is obtained from the linear fit, extrapolating at R → . Inorder to verify the robustness of the result, we realize the same analysis with a similar deviceof 15 µ m length. Some typical data is displayed in the inset of Fig. 6. The noise propertiesobtained for this other device are very similar to the initial 250 µ m long beam (but thequadratic dependence is different). However the spectra better fit with (cid:15) ≈ . ± . .11 , (cid:9)(cid:12)(cid:7)(cid:1)$(cid:26)(cid:1)(cid:22)(cid:23)"(cid:25)(cid:21)(cid:23)(cid:1)!(cid:29)(cid:1)(cid:16)(cid:14)(cid:9)(cid:7)(cid:1)(cid:10)(cid:11)(cid:7)(cid:1)(cid:2)(cid:7)(cid:5)(cid:12)(cid:1)(cid:18)(cid:3)(cid:13)(cid:1)(cid:11)(cid:5)(cid:9)(cid:1)(cid:15)(cid:1)!(cid:29)(cid:1)(cid:16)(cid:14)(cid:12)(cid:1)(cid:7)(cid:7)(cid:7)(cid:1)(cid:2)(cid:8)(cid:1)(cid:18)(cid:3)(cid:13)(cid:1)(cid:11)(cid:5)(cid:9)(cid:1)(cid:15)(cid:4)(cid:1) (cid:1)(cid:22)(cid:28) (cid:1) sd f[T,Rup down] - sd f[T,Rs~ 0] (Hz) (cid:17) (cid:9)!(cid:29)(cid:1)(cid:1)(cid:22)(cid:28) (cid:6)(cid:17) (cid:9)(cid:31) (cid:1)(cid:1)(cid:2)(cid:27)(cid:28)(cid:1)!(cid:27)(cid:25) (cid:31)(cid:3) (cid:4) (cid:8)(cid:12)(cid:1)$(cid:26)(cid:1)(cid:22)(cid:23)"(cid:25)(cid:21)(cid:23) Figure 6:
Universal plot for mode 1. Frequency noise increase vs squared amplitude normalized tothe spinodal value R s , for the first mode n = 1 of the 250 µ m device. Various T and B (hence Q )have been used (see legend). Squares stand for upper branch bifurcation, triangles for lower. Inset:same result obtained with a 15 µ m beam having β = 1 . × Hz/m (4.2 K and Q = 17 000 redsquares; 800 mK and Q = 31 000 blue dots; the magnetomotive field broadening was negligible).The green line is a linear fit (see text). We proceed with similar measurements performed on modes n = 3 and n = 5 of the250 µ m beam sample. All spectra display the same /f . dependence as the n = 1 mode.The same normalized plots are displayed in Fig. 7. However, this time the inferred quadraticdependencies are much weaker.The nonlinear dependence of the frequency noise is rather unexpected; indeed, the non-linear dephasing features observed for carbon-based devices have not been reported fornitride structures. A possible source for such an effect could be a purely intrinsic prop-erty of the bifurcation effect itself. However, since our statistics could not be altered byreasonable changes in effective temperatures and frequency-sweep ramping speeds, such anexplanation is improbable. If we then suppose the bifurcation process to be perfectly ideal,the nonlinear frequency noise observed should originate in one of the parameters defining thebifurcation frequencies. When the experiment is performed reasonably far from the spinodal12 , (cid:4) (cid:7)(cid:9)(cid:6)(cid:1)(cid:25)(cid:16)(cid:1)(cid:13)(cid:14)(cid:23)(cid:15)(cid:12)(cid:14) (cid:1) sd f[T,Rup down] - sd f[T,Rs~ 0] (Hz) (cid:11) (cid:7)(cid:22)(cid:19)(cid:1)(cid:1)(cid:13)(cid:18)(cid:24)(cid:17) (cid:5)(cid:11) (cid:7)(cid:20) (cid:1)(cid:1)(cid:2)(cid:17)(cid:18)(cid:1)(cid:22)(cid:17)(cid:15)(cid:21)(cid:20)(cid:3) (cid:10)(cid:18)(cid:13)(cid:14)(cid:1)(cid:8) , (cid:4) (cid:1) sd f[T,Rup down] - sd f[T,Rs~ 0] (Hz) (cid:11) (cid:7)(cid:22)(cid:19)(cid:1)(cid:1)(cid:13)(cid:18)(cid:24)(cid:17) (cid:5)(cid:11) (cid:7)(cid:20) (cid:1)(cid:1)(cid:2)(cid:17)(cid:18)(cid:1)(cid:22)(cid:17)(cid:15)(cid:21)(cid:20)(cid:3) (cid:7)(cid:9)(cid:6)(cid:1)(cid:25)(cid:16)(cid:1)(cid:13)(cid:14)(cid:23)(cid:15)(cid:12)(cid:14) (cid:10)(cid:18)(cid:13)(cid:14)(cid:1)(cid:9) Figure 7:
Universal plot for modes 3 and 5. Normalized frequency noise plot for the same µ mdevice, for modes n = 3 (left) and n = 5 (right). The lines are linear fits (see text). point (which is always our case), we have: f up ≈ f + δφ ( t ) + βR up , (3) f down ≈ f + δφ ( t ) + 3 βR down , (4) R up ≈ R max , (5) R down ≈ / (cid:18) R max ∆ fβ (cid:19) / , (6)adapted from Eqs. (1-2), where we explicitly introduced the stochastic frequency variable δφ ( t ) . For strings β ∝ / ( m f ) (cid:0) E Y AL (cid:1) , the nonlinear frequency noise could be caused by(Gaussian) fluctuations of the Young’s modulus E Y = E + δE ( t ) . However, to have themeasured characteristics, this noise would have to be δE/E ∝ / ∆ f and mode-dependent,which is difficult to justify: this explanation seems again unphysical.The only possibility left is an internal motional noise with R max = R max + δR , leadingto fluctuations ∝ (cid:16) δRR max (cid:17) βR max . The proper scalings, as reported in Figs. 6 and 7,are thus only achieved by assuming damping noise δ Γ( t ) with δRR max = − δ Γ∆ f . As a result,13t follows from Eqs. (3-6): f up ( t ) ≈ f + δφ ( t ) + βR max (cid:16) − δ Γ( t )∆ f (cid:17) , f down ( t ) ≈ f + δφ ( t ) + / (∆ f βR max ) / together with f s ( t ) = f + δφ ( t ) + √ ∆ f (cid:16) δ Γ( t )∆ f (cid:17) . Thismeans that both bifurcation frequencies suffer from frequency noise δφ , while only the upperbranch experiences damping fluctuations δ Γ : they are amplified by the measurement methodthrough a factor βR max / ∆ f . Note that the frequency noise extrapolated at R → on theupper branch matches the one of the lower branch, but does not equal the one obtained atthe spinodal point, simply because the expressions Eqs. (3-6) do not apply near R s ; this isemphasized through the writing R s ∼ in our graphs.Table 1: Mode parameters, frequency and damping fluctuations for modes n = 1 , and (250 µ m long beam, 4.2 K).Mode number n Freq. f Unloaded Q Duffing β σ δf at R s ∼ σ δ ∆ f from R up (cid:29) R s n = 1 0 . MHz
600 000 ±
10 % 8 . ± . × Hz/m . ± . Hz . Hz ±
10 % n = 3 2 . MHz
450 000 ±
10 % 1 . ± . × Hz/m . ± . Hz . Hz ±
15 % n = 5 4 . MHz
400 000 ±
20 % 5 . ± . × Hz/m . ± . Hz . Hz ±
30 %
The Allan deviation σ δf extrapolated to R → is thus characteristic of the frequencynoise δφ , while the slopes of the graphs in Figs. 6 and 7 are √ times the Allan deviation σ δ ∆ f of the damping fluctuations. To our knowledge, the latter has not been reportedin the literature so far. The mode parameters together with these 4.2 K frequency anddamping noise figures are summarized in Table 1. σ δ ∆ f is temperature-independent in therange studied, while σ δf is linear in T ; this is displayed in Fig. 8. The same temperature-dependence of frequency noise has been reported in Ref. (within an overall scaling factor)for a very similar device. In order to compare the various results, values from the literatureare presented in Table 2. We give the Allan deviation when it is reported, otherwise welist the direct frequency noise; the damping noise in the third line is recalculated from Fig.IV.19 in Ref. We can only speculate on the microscopic mechanisms behind the reported features. Theentities generating such noise are thought to be atomic-scale two level systems (TLS), which14 (cid:1) sd f[T,Rs~ 0]2 (Hz2) (cid:10)(cid:2)(cid:9)(cid:3) (cid:7)(cid:8)(cid:5)(cid:1)(cid:25)(cid:17)(cid:1)(cid:13)(cid:14)(cid:23)(cid:15)(cid:12)(cid:14)(cid:4)(cid:1)(cid:17)(cid:18)(cid:13)(cid:14)(cid:1)(cid:6) (cid:5)(cid:1)(cid:9)(cid:1)(cid:14)(cid:24)(cid:21)(cid:20)(cid:11)(cid:19)(cid:18)(cid:16)(cid:11)(cid:21)(cid:14)(cid:13)(cid:1)(cid:23)(cid:11)(cid:16)(cid:22)(cid:14) Figure 8:
Temperature dependence of frequency fluctuations. Allan variance σ δf as a function oftemperature (first mode of 250 µ m device), as obtained for small motion R → R s ∼ . The line is alinear fit, with the T = 0 K extrapolated value emphasized by the arrow (see text). The varianceof damping fluctuations σ δ ∆ f is constant in the same range of temperatures. Table 2: Mode parameters, frequency and damping fluctuations for different devices (fun-damental flexure n = 1 ). The damping noise figure in the third line is recalculated from Fig.IV.19 in Ref. Device Freq. f Q σ δf or σ f σ δ ∆ f µ m SiN/Al d.c. beam 4.2 K (this work) . MHz
600 000 ±
10 % 0 . ± . Hz . Hz ±
10 %15 µ m SiN/Al d.c. beam 4.2 K(this work) . MHz
18 000 ±
10 % 1 . ± . Hz . Hz ±
10 %2 × µ m Si/Al goalpost 4.2 K Refs. . MHz ±
10 % 1 ± . Hz on f . Hz ±
10 %380 µ m SiN d.c. beam 5 K Ref. . MHz . Hz on f X . µ m Si cantilever Room Temp. Ref. . MHz
Hz Xcould be defects or intrinsic to the materials in use.
A signature that also supportsthis view is the presence of telegraph frequency noise in many NEMS experiments (see e.g.
Ref. ). Ref. analyzed frequency noise in beams, i.e. structures with no built-in stress.These Authors assumed that thermally activated motion of a defect in a double well potentialfrom one minimum to the other caused a shift in the local Young’s modulus. The motion ofmany such defects (following the mathematical arguments of Ref. ) then causes a change inthe average Young’s modulus and consequently a change in the resonance frequency of the15eam with power spectrum ∝ T /f . The same argument was applied in Ref. to analyzethe frequency noise of string structures, where the built-in stress is large, even though theresonance frequency is nearly independent of the Young’s modulus in the high-stress limit.We thus believe that it is more appropriate, in the interpretation of the present measurementson strings, to consider frequency fluctuations due to stress fluctuations . Indeed, point defectsin crystals are characterized as elastic dipoles that cause an orientation dependent changein the strain (and consequently the stress) of a crystal. In our highest Q device, for the first flexure n = 1 frequency fluctuations at 4.2 K repre-sent about 20 % of the linewidth, and damping fluctuations about 5 %. These parametersfall with mode number n (see Tab. 1), while both frequency f and linewidth ∆ f increase.This means that the effect of fluctuations is the strongest on the first mode, but is usuallydifficult to visualize on standard frequency-sweeps or time-decay data; it is for instance ex-pected that for a device rather equivalent to our 250 µ m beam, the measurements performedin Ref. did not report any such features (see Methods). Conclusion
In this Letter we present a very simple and reliable method to measure and characterizefrequency noise in bistable resonators. The technique has been employed to describe thor-oughly the intrinsic frequency noise of high-stress silicon-nitride doubly-clamped beams. Themeasurements have been performed at low temperatures in cryogenic vacuum, on two devicesof very different lengths/fundamental resonance frequencies. The 3 first symmetric flexuralmodes of the longest beam have been studied.We report on the Allan deviations of the frequency noise, presenting the same basicfeatures as in Refs.: spectra of /f -type, scaling linearly with temperature. The reportedmagnitudes of the noise δf /f fall in the . − . ppm range, as expected for MHz devices.We have also found unexpectedly damping fluctuations , which are amplified in the vicinity16f bifurcation points. Our technique seems the most adapted for the detection and thequantification of such a noise process to date. We find that damping noise can be as largeas about 5 % of the total width of the resonance peak in our highest Q devices. It also setsa finite resolution attainable for the measurement of the frequency position of bifurcationpoints.These features seem ubiquitous to all NEMS devices, and we do believe that dampingnoise and frequency noise originate in the same microscopic mechanism. But the latterremains elusive, and the most discussed candidate is based on Two-Level Systems (TLS). Because of the strength of the frequency noises reported here for high-stress devices, wepropose that TLS are responsible for noise on the stored stress in the structure instead ofthe Young’s modulus, as it was proposed in earlier papers discussing stress-free beams ( e.g.
Ref. ).Our technique can be easily adapted to any types of devices, including MoS and carbon-based systems in which nonlinear frequency noise has been reported. We think that itshould help advance the understanding of the underlying fundamental microscopic mecha-nisms that also significantly degrade the properties of existing NEMS devices, and hindertheir applicability.
Methods
Device Fabrication
The structure was fabricated using e-beam lithography on a silicon substrate covered with a
100 nm silicon nitride (SiN) layer. The stochiometric nitride was grown using low pressure chemical vapordeposition (LPCVD) at the Cornell NanoScale Science & Technology Facility (CNF). It stores abiaxial stress of about 1 GPa. A 30 nm Aluminum coating has been evaporated onto the samplein a Plassys e-gun machine. Its resistance at low temperature is about 1 k Ω for the 250 µ m longdevice, and 100 Ω for the 15 µ m one. It served as a mask for the structure during the SF Reactive on Etching (RIE) step used to pattern the SiN. The structure was then released using a final XeF etching of the underlying silicon. Measurements
The voltage drive was delivered by a Tektronix AFG3252 generator, through a bias resistancewhich created the drive current. The motion was actuated with the magnetomotive techniquethrough a force F ( t ) = I ζLB cos(2 πf t ) , which also leads to the detection of the velocity ˙ x ( t ) of the oscillation through a voltage V ( t ) = ζLB ˙ x ( t ) . ζ is a mode-dependent shape factor. Inthe high Q limit, the velocity in frequency domain is iω x ( ω ) with ω = 2 πf , hence an inverteddefinition for the signal quadratures X and Y with respect to displacement x ( t ) . Due to thesymmetry of the scheme, only symmetric modes ( n = 1 , , · · · ) can be addressed ( ζ = 0 otherwise).The magnetic field was generated with a small superconducting coil fed with a 10 A Kepco currentsource. The detected signal was processed with a Stanford SR 844 RF lock-in amplifier. Due tothe finite impedance of the electric circuit seen by the NEMS (whose own characteristic impedancevaries as B ), the mechanical resonances are loaded by an additional damping ∝ B . This providesthe ability to tune the quality factors in situ . Our calibration procedure is described in Ref. It enables us to give all mechanical parameters in S.I. units (we thus quote X , Y in meters), whileminimizing the loading effect. Loading is negligible for the 15 µ m beam, but still large in the 250 µ mdevice. Spectra Mathematical Properties
Let us consider a frequency power spectrum for the stochastic resonance frequency ω = 2 πf of type S ω ( ω ) = A / | ω | (cid:15) (defined from −∞ to + ∞ ). The variance can be defined from the integral of thespectrum, leading to σ ω = π (cid:82) S ω dω = A π (cid:20) ω − (cid:15)low − ω − (cid:15)high (cid:15) (cid:21) with ω low and ω high the lower and higherfrequency cutoffs imposed by the experiment. For (cid:15) → we have σ ω ∝ (cid:112) ln [ ω high /ω low ] . Since δω = ω ( t i +1 ) − ω ( t i ) ≈ ∂ω ( t ) /∂t × ∆ t min when ∆ t min → , we have S δω ( ω ) ≈ ∆ t min ω S ω ( ω ) . Thus σ δω ≈ A π ( ∆ t min ω high ) (cid:20) ω − (cid:15)high − ( ω low /ω high ) − (cid:15) − (cid:15)/ (cid:21) . The Fourier Transform imposes ∆ t min ω high ≈ π ,and in the case (cid:15) → we obtain σ δω ∝ (cid:113) − ( ω low /ω high ) , which is essentially independent of the utoffs. For (cid:15) (cid:54) = 0 , a small dependence to the bandwidth appears in the Allan variance σ δω . Forour acquisition bandwidths, this does not result in a too large scatter in data (within error bars). Impact on Frequency-domain & Time-domain Measures
With a noise of type S ω ( ω ) = A / | ω | (cid:15) , we can take as an estimate of the relevant fluctu-ations timescale τ − c ∼ ω low /π : the weight is at the lowest accessible frequencies. We thusalways verify σ ω τ c (cid:29) , which means that the phase diffusion of the mechanical mode is inthe Inhomogeneous Broadening limit (IB), in analogy with Nuclear Magnetic Resonance. In frequency-domain, the response χ meas ( ω ) is the convolution of the standard (complex-valued,defining the two quadratures) susceptibility χ ( ω ) with the (Gaussian) distribution of frequencies p ( δϕ ) = 1 / (cid:112) πσ ω exp (cid:0) − δϕ /σ ω (cid:1) , with δϕ = 2 πδφ in Rad/s. This means that at each scannedfrequency ω , the measurement is performed over a long enough timescale such that all fluctua-tions are explored. On the other hand, the small damping fluctuations are simply filtered out bythe acquisition setup (here, a lock-in amplifier): they have no relevant impact on the resonancepeak measured, even at very large motion amplitudes. We conclude that only frequency noisewill contribute to the definition of a T , the decoherence time involving relaxation T and dephas-ing σ ω . In time-domain, in the linear regime the complex susceptibility ¯ χ meas ( t ) ( i.e. decayof the two quadratures) is simply the Fourier Transform (FT) of χ meas ( ω ) . It can also be writ-ten ¯ χ meas ( t ) = (cid:104) exp ( iδϕt ) (cid:105) ¯ χ ( t ) with ¯ χ ( t ) the FT of χ ( ω ) and (cid:104) exp ( iδϕt ) (cid:105) = exp (cid:0) − σ ω t (cid:1) theaverage over frequency fluctuations. In the nonlinear regime, the averaged decay of the two quadra-tures writes ¯ χ meas ( t ) = (cid:104) exp ( iδϕt ) (cid:105) (cid:28) exp (cid:18) − i βR max ∆ ω δγ t κ [ t ] (cid:19)(cid:29) ¯ χ ( t ) with κ [ t ] = − exp[ − t ∆ ω ] t ∆ ω and ¯ χ ( t ) defined in Ref., the second average being over damping fluctuations δγ = 2 πδ Γ ; we wrote ∆ ω = 2 π ∆ f in Rad/s. The function κ is characteristic of the decay of the nonlinear frequencypulling due to the Duffing term, ∝ R max . In practice, this assumes that both quadratures aremeasured independently, averaging many decay traces starting from the same (noisy) t = 0 ampli-tude R max , imprinted by the slow fluctuations of damping δγ . The second average can be explicitlycalculated: (cid:28) exp (cid:18) − i βR max ∆ ω δγ t κ [ t ] (cid:19)(cid:29) = exp (cid:34) − (cid:18) βR max ∆ ω (cid:19) σ δ ∆ ω ( t κ [ t ]) (cid:35) . In time-domain,the impact of damping fluctuations (of variance σ δ ∆ ω ) is thus amplified by the same term as in thebifurcation measurement: βR max ∆ ω . However, to have a measurable effect (within experimental error ars) both amplitude R max and fluctuations σ δ ∆ ω have to be very large; in experiments of the typeof Ref. based on devices similar to our 250 µ m, no such effect has been reported. Measuring thedecay of the two quadratures leads then to the definition of ¯ T , roughly equivalent to T . Notethat the decay of the motion amplitude | ¯ χ ( t ) | remains unaffected by frequency and damping noise,leading to the proper T definition. Acknowledgement
We thank J. Minet and C. Guttin for help in setting up the experiment, as well as J.-F. Motte, S.Dufresnes and T. Crozes from facility Nanofab for help in the device fabrication. We acknowledgesupport from the ANR grant MajoranaPRO No. ANR-13-BS04-0009-01 and the ERC CoG grantULT-NEMS No. 647917. This work has been performed in the framework of the European Mi-crokelvin Platform (EMP) collaboration. At Cornell we acknowledge support from the NSF underDMR 1708341.
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Anelastic Relaxation in Crystalline Solids ; Academic Press: NewYork, 1972. f down X quadrature Y quadrature A m p li t ude ( n m ) Frequency (Hz) f up -20 0 20 050100 C oun t f (Hz) f =8 Hz f ( H z ) Time (s) -3 -2 -1 0 1 2 3050100150 f =0.26 Hz C oun t f (Hz) f ( H z ) Time (s) -5 -1.0x10 -5 -5.0x10 -6 -6 -5 -5 -5 -5 -5 -5 X , Y ( V ) f (Hz) V(t)I(t)
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