Symmetry breaking in a mechanical resonator made from a carbon nanotube
SSymmetry breaking in a mechanical resonator made from acarbon nanotube
A. Eichler , , J. Moser , , M. I. Dykman , and A. Bachtold , ICFO - Institut de Ciencies Fotoniques,Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain, Institut Catal`a de Nanotecnologia, Campus de la UAB, E-08193 Bellaterra, Spain, and Department of Physics and Astronomy,Michigan State University, East Lansing, Michigan 48824, USA (Dated: November 9, 2018)
Abstract
Nanotubes behave as semi-flexible polymers in that they can bend by a sizeable amount. Whenintegrating a nanotube in a mechanical resonator, the bending is expected to break the symmetry ofthe restoring potential. Here, we report on a new detection method that allows us to demonstratesuch symmetry breaking. The method probes the motion of the nanotube resonator at (nearly)zero-frequency; this motion is the low-frequency counterpart of the second overtone of resonantlyexcited vibrations. We find that symmetry breaking leads to the spectral broadening of mechanicalresonances, and to an apparent quality factor Q that drops below 100 at room temperature. Thislow Q at room temperature is a striking feature of nanotube resonators whose origin has remainedelusive for many years. Our results shed light on the pivotal role played by symmetry breaking inthe mechanics of nanotube resonators. PACS numbers:Keywords: a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov carbon nanotube is a unique system that can be seen both as a crystal and as a polymer.Its crystallinity confers excellent mechanical properties to nanotube-based resonators [1–6],such as high resonant frequencies [7, 8] and low dissipation at low temperature [9, 10].As a result, these resonators are well suited for ultra-sensitive detection of mass [11, 12],charge [2, 3] and force [13]. A nanotube has also much in common with a polymer as bothcan bend by a large amount. In a resonator, the bending can be generated by the mechanicaltension that builds in during the fabrication process, as well as by the electrostatic force usedin most studies. This curvature is expected to have profound consequences on the dynamicsof nanotube resonators, since the transverse vibrational modes lack inversion symmetry.In a bent nanotube, if one thinks of a vibrational mode as an oscillator, its potential is notsymmetric with respect to the displacement from the equilibrium position (Fig. 1a). Thisleads to a nonlinear term in the restoring force that depends quadratically on the displace-ment, F = mβ · δz ( t ) with m the effective mass of the resonator, β a constant quantifyingthe strength of the symmetry breaking effect, and δz ( t ) the transverse displacement of theresonator for the given mode. The mechanism underlying the effect can be understood asfollows. For a resonator that is curved and clamped at both ends, the length is different for+ δz and − δz and, therefore, the tension induced by the motion is asymmetric with respectto δz (Fig. 1b).In a potential with broken symmetry, the equilibrium position of the mode depends onits vibrational amplitude z vibra (see Fig. 1a). Indeed, if the resonator vibrates as δz ( t ) = z vibra · cos( ωt ), the quadratic term in the restoring force becomes F = mβ z vibra ωt )] . (1)The second term in the bracket leads to the second overtone, that is, motion at 2 ω . Thefirst term corresponds to a time-independent force and, therefore, generates a shift of theequilibrium position, δz eq . In other words, it is possible to move the equilibrium position byvarying z vibra . This motion is slow, since it is limited by the ring-down time associated to z vibra . When considering the thermal motion of such a resonator, the power spectrum of thedisplacement is expected to feature a peak at zero-frequency, its width being roughly the in-verse of the ring-down time [14, 15]. To the best of our knowledge, this low-frequency motionof the equilibrium position of high- Q oscillators has not been observed in nanomechanicalresonators or other condensed-matter systems.2he device consists of a single carbon nanotube that is clamped by two metal electrodesand is suspended over a trench. A gate electrode is defined at the bottom of the trench(Fig. 1c). The fabrication is described elsewhere [10]. Briefly, we pattern the three elec-trodes and the trench using standard electron-beam lithography techniques. We grow thenanotube by chemical vapor deposition (CVD) in the last fabrication step in order to avoidcontamination [9]. All measurements are performed at 65 K to avoid Coulomb blockade [2, 3].We have studied 5 nanotube devices in total. We discuss in the following the data for onedevice. Data for a second device yielding similar results are shown in supplementary sectionXI.We employ a new technique to detect the motion of nanotube resonators. We capacitivelydrive the vibrations at ω drive near the resonant angular frequency ω by applying a constantvoltage V dcg and an oscillating voltage of amplitude V acg on the gate electrode. Central to thetechnique is that the oscillating voltage is amplitude modulated (AM) so that the resultingdisplacement near ω for not too large V acg is proportional to the driving amplitude, δz ( t ) = z vibra · cos( ω drive t − ϕ m ) · [1 − cos( ω AM t )] (2)where the amplitude modulation has a depth of 100 % and its angular frequency ω AM istypically 2 π × ω AM ); ϕ m is the phase difference between the displacement and the driving force. We applya constant voltage V dcsd to the source electrode and measure from the drain electrode thelow-frequency current I LF at ω AM with a lock-in amplifier (see supplementary section VIfor details). We show below that this technique allows us to measure the motion of δz eq associated with the symmetry breaking in nanotube resonators.We observe that I LF features a peak when ω drive is swept through a mechanical resonance(Fig. 2a); the mechanical resonance is also verified by directly measuring the vibrationalmotion using the frequency modulation (FM) mixing technique (Fig. 2b) [4]. The height I maxLF of the peak in I LF goes linearly to zero as V dcsd is decreased (Figs. 2c,d; black squares).These data show that the detected peak in I LF is related to the modulation of the nanotubeconductance δG = I LF /V dcsd at ω AM . It rules out an artefact related to the capacitive couplingbetween the gate and the source electrodes. (This coupling could result in a sizeable AMoscillating voltage at the source electrode, and could thus drive the resonator; but I LF wouldthen be independent of V dcsd .) 3e also observe a current peak when setting the reference (angular) frequency of thelock-in amplifier to 2 ω AM . The peaks measured at ω AM and 2 ω AM are similar in that theyappear at the same driving frequency and their heights depend linearly on V dcsd (Fig. 2d).However, the height of the peak measured at 2 ω AM is four times smaller. These observationssuggest that the measured peaks are related to a nonlinearity that scales as ( δz ( t )) andtherefore z vibra . (Indeed, if δz ( t ) ∝ − cos( ω AM t ), any physical quantity that is proportionalto ( δz ( t )) will be modulated at ω AM and 2 ω AM with a ratio of 4 between the amplitudesof the two components.)We estimate z vibra (cid:39) . . z vibra is inferred by comparing the signals on and away from resonance [16](supplementary section V).The peak in I LF is detected only for a fraction of the mechanical eigenmodes. In theresonator discussed thus far, the peak is observed for the second eigenmode but not forthe first one (by comparing I LF in Fig. 4a and the current in Fig. 4b obtained with theFM mixing technique). In all the five studied resonators, we find that about half of theeigenmodes feature a peak in I LF .We now discuss different possible origins of the peak in I LF . It could be related to thenonlinear capacitive coupling between the nanotube and the gate electrode, which leads toa ( δz ( t )) nonlinearity in the conductance of the nanotube. However, we estimate that thecurrent associated to this effect is I capaLF = 10 pA, which is 20 times smaller than the mea-sured value in Fig. 2c (supplementary section VII). Thus, we reject the ( δz ( t )) nonlinearityinduced by the capacitive coupling as the physical origin of the peak in I LF . Neither is the I LF peak attributed to the nonlinearity of the conductance in gate voltage [9], since it leadsto a current that is 3 orders of magnitude lower than that measured in Fig. 2c (supplemen-tary section VII). Another mechanism for the I LF peak could be the piezoresistance of thenanotube, whose dependence on the displacement is quadratic to a good approximation [17].The piezoresistance effect in nanotubes is by far strongest for positive gate voltages (whereelectrons tunnel from the p-doped regions of the nanotube near the metal electrodes intothe n-doped region of the suspended part of the nanotube [18, 19]). However, the observedheight of the peak in I LF can be as large for negative as for positive gate voltages (seeFig. 4a). We can thus rule out the piezoresistive effect.4e now consider that the peak in I LF is due to symmetry breaking of the vibrations.For the AM modulation ∝ [1 − cos( ω AM t )] the height of the peak in I LF depends on themaximal displacement of the equilibrium position δz eq as I maxLF = V dcsd V dcg ∂ V g G · ∂ z C g C g δz eq . (3)where δz eq is proportional to z vibra . Here, ∂ z C g is the derivative of the nanotube-gatecapacitance C g with respect to displacement; it is determined from Coulomb blockademeasurements at helium temperature (see supplementary section III). From the measured I maxLF in Fig. 2c, we get that δz eq = 0 .
18 nm and β = 4 . · m − s − using the relation β = ω δz eq /z vibra (see supplementary section VIII).This value for the symmetry breaking strength can be compared to the one estimated fromthe measurement of ω as a function of the oscillating driving force. Figures 4c and d showthat the peak in I LF shifts to lower frequency upon increasing the driving force. Disregardingthe cubic restoring force (which in nanotubes leads to the shift in the opposite direction, seeRef. [20]), we obtain from the shift in ω that β = 4 . · m − s − (supplementary sectionVIII). This value agrees with the one estimated from I maxLF , demonstrating that the peak in I LF is due to symmetry breaking of the vibrations.The strength of symmetry breaking can be made large in nanotubes, since it scales as β (cid:39) Eρ z s (cid:0) πL (cid:1) and the length L can be as short as 100 nm [7, 8]. This expression is derivedfor the fundamental mode of a rod (Eq. S6 in the supplementary information of Ref. [20]),and z s is the characteristic static displacement induced by the bending. Assuming that z s ranges from 1 to 10 nm, and using L = 1 . µ m and the graphite density ρ = 2300 kgm − and Young modulus E = 1 TP, we obtain β = 3 − · m − s − , which is consistentwith the value obtained from our measurements. The quadratic nonlinear force associatedto symmetry breaking is 3 orders of magnitude larger than the quadratic electrostatic force, − ∂ z C g ( V dcg ) / m · δz ( t ). The observed decrease of ω with the increasing resonant drivingin Fig. 4c and d indicates that the cubic nonlinear (Duffing) force has no substantial effecton the dynamics of the resonator. This points out that the actual static deformation z s is large compared to the vibration amplitude (because the dynamical cubic restoring forcescales as F (cid:39) F · δz ( t ) /z s ), thus supporting our above assumption that z s = 1 −
10 nm.The observation of a peak in I LF for only about half of the mechanical eigenmodesindicates that β varies from one eigenmode to the next. This is something expected from5he interplay between the shapes of the vibrational eigenmodes and the static deformationalong the nanotube if the static displacement is primarily in one plane. Our data suggestthat the static displacement is essentially perpendicular to the gate electrode. In such ageometry, the lowest-frequency eigenmode detected in Fig. 4b corresponds to the lowest-energy mode vibrating (essentially) parallel to the surface of the gate electrode, as shown inRef. [20]. A static deformation of the nanotube towards the gate electrode does not breakthe vibration symmetry of this mode, because the elastic tension inside the nanotube isequal for + δz and − δz . As a result, the amplitude of I LF should be weak, in agreementwith the measurements. The second eigenmode in Fig. 4b is assigned to the lowest-energymode vibrating in a direction (essentially) perpendicular to the gate electrode [20]. In thepresence of a static deformation towards the gate electrode, this mode experiences symmetrybreaking of vibrations. A peak shows up in I LF , as observed in Fig. 4a.Having shown that symmetry breaking leads to motion at (nearly) zero-frequency, wedemonstrate other connections between symmetry breaking and the mechanics of nanotuberesonators. A hallmark of nanotube resonators is that the resonance frequency can be widelytuned with V dcg . Symmetry breaking is expected to control this tunability in ω by an amount∆ ω = β∂ z C g mω ( V dcg ) (4)( V dcg is here offset so that V dcg = 0 when ω is minimum). We estimate that m (cid:39) . µ m) and using thetypical radius (1 . ω ( V dcg ) nearthe minimum of ω , we get that β = 3( ± · m − s − , which is close to the value estimatedabove. This result underscores that symmetry breaking is connected to the response of theresonance frequency to V dcg . We emphasize that Eq. (4) is only valid for not too large V dcg , asit is the leading-order term of the expansion of ω in V dcg . Here we find that Eq. (4) appliesin the whole range of V dcg that we studied, and that β and the bending of the nanotubeboth remain essentially constant within this range. This suggests that the bending is aconsequence of the mechanical tension built in during the fabrication process. In the future,it will be interesting to measure β as a function of V dcg for other nanotube resonators.In the presence of thermal vibrations, symmetry breaking leads to spectral broaden-ing [21]. Because the amplitude of thermal vibrations fluctuates in time, the nonlinearity-induced shift in ω (Fig. 4d) also fluctuates and, therefore, broadens the mechanical reso-6ance. The broadening in ω reads δω = 5 β k B T / mω (5)when the cubic restoring force is negligible compared to the quadratic one (supplementalsection X). Using β = 4 . · m − s − , we get δω = 2 π × . · Hz at room temperature.This corresponds to an apparent quality factor of 67, which is comparable to the value of (cid:39)
50 measured with the FM technique. We emphasize that this broadening is analogousto dephasing of two-level systems and qubits, which sets the characteristic time T . Themeasured broadening is not related to dissipation, so that the energy relaxation time could bemuch longer than 1 /δω (in fact, it is in this case that Eq. (5) gives the spectral broadening).For eigenmodes with a small β , the broadening can be due to the cubic restoring force [21].Mechanical resonances might be further broadened by the coupling between eigenmodes [21],as shown by recent simulations of nanotube resonators [22].We assumed in our analysis of I LF that the response of the amplitude of the vibrationalmotion is linear with the driving force. When the response becomes nonlinear at largedriving forces due to the restoring force nonlinearity, the ratio of I LF at ω AM and 2 ω AM isexpected to deviate from 4. Calculations show that the width of the peak in I LF remainsnearly constant upon varying the driving force, in contrast to the measurements in Fig. 4c.A general theory that incorporates nonlinearities in both the restoring force and damping[10, 23–26] as well as thermal vibrations is beyond the scope of this Letter. We note thatour new technique to measure the motion of the equilibrium position allows to study theresponse of the resonator over a broad parameter range in driving force.In conclusion, we demonstrate that symmetry breaking leads to a motion at nearly zero-frequency in response to resonant excitation of the vibrations. Our results indicate thatsymmetry breaking of vibrational modes also leads to such important dynamical propertiesas the apparent low quality factor of nanotube resonators at 300 K, and the shift of thevibration frequency in response to both (i) the static gate voltage and (ii) the amplitude ofthe oscillating driving force. A future strategy to improve the apparent Q at 300 K is to tune β with the gate voltage in order to compensate the spectral broadening due to symmetrybreaking with that due to the Duffing nonlinearity. Symmetry breaking is important forother vibrational systems of current interest, such as graphene resonators [10, 27–31] andlevitating particles [32–34]. Our new technique may help to reveal this effect in such systems.7ymmetry breaking also leads to mode mixing and to parametric resonance in response toadditive driving. This holds promise for a number of applications, such as controlled modemixing [35–37] and phase noise cancelation [38–40]. [1] Sazonova, V., Yaish, Y., ¨Ust¨unel, H., Roundy, D., Arias, T. A. & McEuen, P. L. Tunablecarbon nanotube electromechanical oscillator. Nature , 284-287 (2004).[2] Lassagne, B., Tarakanov, Y., Kinaret, J., Garcia-Sanchez, D. & Bachtold, A. Coupling me-chanics to charge transport in carbon nanotube mechanical resonators.
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Nano Lett. , 5054-5059 (2011). Acknowlegements
We acknowledge support from the European Union through the RODIN-FP7 project,the ERC-carbonNEMS project, and a Marie Curie grant (271938), the Spanish state(FIS2009-11284), the Catalan government (AGAUR, SGR), and the US Army ResearchOffice.
Author contributions
A.E. fabricated the devices and carried out the measurements. J.M. participated in themeasurements. M.I.D. provided support with the theory and wrote the theoretical part ofSupplementary Information. All the authors contributed to writing the manuscript. M.I.D.and A.B. conceived the experiment. A.B. supervised the work.Correspondence and requests for materials should be addressed to A.B.([email protected]) 11
IG. 1: | Effect of curvature in a nanotube resonator. a , Symmetry breaking of therestoring potential U ( z ). The equilibrium position depends on the energy of the resonator mode. b , Schematic of a curved resonator. The dashed line represents the static profile of the resonator,that is, when it does not vibrate. The plain lines show the profiles for + δz and − δz . c , Theresonator studied consists of a carbon nanotube suspended over a trench between source (S) anddrain (D) electrodes. A gate electrode (G) is defined at the bottom of the trench. The trench hasa width of 1 . µ m and a depth of ∼
350 nm. d , The vibrational motion is amplitude modulated at ω AM (upper schematic). As a result, the equilibrium position is modulated with the same period(lower schematic). IG. 2: | Characterization of the low-frequency current. a , I LF as a function of ω drive . V acg = 0 .
53 mV, V dcsd = 10 mV, and V dcg = − .
45 V. b , Mechanical vibrations detected with theFM technique [4]. V acsd = 1 . V dcg = − .
45 V. c , I LF versus ω drive with V acg = 1 . V dcsd = 10 mV, and V dcg = − . d , I maxLF as a function of V dcsd measured at ω AM (black squares)and 2 ω AM (open squares). V acg = 2 . V dcg = − . IG. 3: | Vibrational motion measured with the 2-source mixing technique. a, X quadrature and b, Y quadrature of the current measured with the lock-in amplifier. I Xvibra consistsof a current proportional to the real part of the vibrational amplitude in addition to a purely electri-cal background current. I Yvibra is proportional to the imaginary part of the vibrational displacement. V acg = 1 . V acsd = 0 . V dcg = − . IG. 4: | Response of the low frequency current to static and oscillating forces. a , I LF as a function of ω drive and V dcg . V acg = 0 .
53 mV, V dcsd = 10 mV. Color bar: 0 (white) to 280 pA (darkred). The background signal varies with V dcg ; this variation likely has a purely electrical origin.The number of measurement points is kept as low as possible so that resonances are captured withabout 3 points along the frequency axis. b , Current as a function of ω drive and V dcg measuredwith the FM technique. V acsd = 2 . c , Measuredlineshapes of I LF as a function of ω drive for different V acg . V acg = 4 .
2, 3 .
5, 2 .
5, 1 .
6, 1, and 0 . d , Resonance shift extracted from the data in c (black dots) and from FMmeasurements (open squares). upplementary Material for: Symmetry breaking in a mechanicaloscillator made from a carbon nanotube A. Eichler , , J. Moser , , M. I. Dykman , and A. Bachtold , ICFO - Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860Castelldefels, Barcelona, Spain, Institut Catal`a de Nanotecnologia, Campus de la UAB, E-08193 Bellaterra, Spain, and Department of Physics and Astronomy, Michigan State University, East Lansing,Michigan 48824, USA
I. GENERAL EXPRESSION FOR THE LOW-FREQUENCY CONDUCTANCE
We consider the conductance of a suspended nanotube (supplementary references [1, 2]) in thepresence of a gate voltage that has a large DC component and a small AC component at a frequencyclose to the frequency of the eigenvibrations of the nanotube.We assume that ( i ) the nanotube conductance is a function only of the total charge q of thenanotube, ( ii ) the charge distribution along the nanotube is independent of the gate voltage V g , and( iii ) the system is in the adiabatic limit, i.e. the vibration dynamics is much slower than the electrondynamics. Then q is related to V g by the capacitance C g . We consider the effect on conductance ofthe bending mode of the nanotube which is polarized in the direction z perpendicular to the gate.Based on the above assumptions we write the conductance as G ( q ( t )) (cid:39) G ( q ) + ∂ q G δq ( t ) + 12 ∂ q G [ δq ( t )] + . . . . (6)For the temperatures used in our experiments, where the Coulomb blockade effect is insignificant,the term ∝ δq is comparatively small. In what follows we will disregard it. Incorporating thisterm will not change the qualitative results (see section VII).The charge increment δq ( t ) is a function of the time-dependent (AC) increment of the gatevoltage δV g ( t ) and the (AC) vibrational displacement δz ( t ), which is the displacement of thenanotube at the antinode of the vibrational mode with the largest amplitude, for a given mode.For small | δV g | and | δz | δq ( t ) (cid:39) ∂ V g q δV g ( t ) + ∂ z q δz ( t ) + 12 ∂ V g q δV g ( t ) + ∂ z ∂ V g q δz ( t ) δV g ( t ) + 12 ∂ z q δz ( t ) . (7) he coefficients in this expression have a simple form in the case where the charge q is relatedto the gate voltage by the gate capacitance C g , which itself depends on the displacement of thenanotube. We then have ∂ V g q = C g , ∂ z q = V dcg ∂ z C g , ∂ z ∂ V g q = ∂ z C g , ∂ z q = V dcg ∂ z C g (8)whereas we can set ∂ V g q = 0, assuming the capacitor to be linear. In Eq. (8) V dcg is the DCgate voltage, which is assumed to be large compared to δV g ( t ).We will consider an AC modulation δV g ( t ) (with amplitude V acg ) at frequency ω drive close tothe eigenfrequency of the nanotube ω . If this modulation is not too weak, the major contributionto the AC displacement δz is the one induced by this resonant modulation whereas the thermaldisplacement can be disregarded.The quantity of immediate interest to us is the quasi-static change of the conductance in responseto δV g . As explained in the main text, to detect this change we consider a periodic signal withslowly modulated amplitude, δV g ( t ) = V acg ( t ) cos ω drive t, V acg ( t ) = V (1 − cos( ω AM t )) , ω AM (cid:28) ω drive . (9)There are several contributions to the low-frequency response of the conductance to the modulation(9). To study them we first estimate the response of the resonator to the modulation assumingthat the resonator dynamics is linear, δz = δz lin . The linearized equation of motion in the simplestcase of viscous friction reads δ ¨ z lin + 2Γ δ ˙ z lin + ω δz lin = F d ( t ) cos( ω drive t ) m , F d ( t ) = ∂ z C g V dcg V acg ( t ) (10)where m is the mass of the nanotube, 2Γ = ω /Q is the decay rate of the oscillator with qualityfactor Q , and F d ( t ) is the AC driving force amplitude. In what follows we assume that the frequencyof the amplitude modulation, ω AM , is small compared to the decay rate Γ, so that the inducedvibrations adiabatically follow V acg ( t ). Then δz lin ( t ) = A lin ( ω drive , t ) cos( ω drive t − φ ) , A lin ( ω drive , t ) = F d ( t ) /m (cid:113)(cid:0) ω − ω drive (cid:1) + 4Γ ω drive ,φ = arctan (cid:18) ω drive ω − ω drive (cid:19) . (11)From Eqs.(9), and (11) it follows that all terms that have a quadratic dependence on δV g and δz in Eq. (7) have a slowly varying part, which oscillates with period 2 π/ω AM . If the distancebetween the gate electrode and the nanotube is h (it is of the order of the depth of the trench,
350 nm), then ∂ z C g ∼ C g /h . Then in the linear approximation the amplitude on resonance is A res = F d / m Γ ω ∼ C g V acg V dcg /m Γ ω h . The slowly varying parts of the last two terms in Eq. (7)are ∼ C g V acg A lin /h, C g V dcg ( A lin ) /h . (12)A simple estimate shows that, for the device parameters, the second term is much larger than thefirst for resonant driving, which means that the term ∝ δV g δz in Eq. (7) should be disregarded inthe analysis of the low-frequency conductance. II. NONLINEAR RESPONSE OF THE VIBRATIONAL MODE
In the linear approximation [Eq. (11)], the term ∝ δz ( t ) in the expression for the charge andthus the conduction modulation [Eq. (7)] are oscillating at high frequencies ω drive , ω drive ± ω AM .However, the vibrations of the nanotube are nonlinear, and this leads to the onset of slowly varyingterms in the displacement δz ( t ). To find these terms we write the part of the capacitive energyand the internal energy of the mode that is nonlinear in δV g and δz , H nlc = − ∂ z C g δV g δz − ∂ z C g V dcg δV g δz + 13 mβδz + 14 mγδz + . . . . (13)The first two terms in this expression describe the nonlinear capacitive energy, whereas the lasttwo terms refer to the nonlinear part of the vibrational energy. We emphasize that the termwhich is cubic in δz is present only because the mode lacks inversion symmetry: this term is theindication of symmetry breaking (it corresponds to a force that is quadratic in δz ). Such symmetrybreaking may result from the gate voltage which bends the nanotube. Therefore we expect that β depends on V dcg . On the other hand, the term ∝ γ is the familiar Duffing nonlinearity, which hasbeen known to play an important role in the vibrational dynamics of nanotubes (supplementaryreferences [1, 2]).We emphasize again that δz refers to the maximal displacement for the considered mode in the z -direction, i.e., toward the gate electrode. More generally, for bending modes, one should thinkof the displacement δ r as a function of length l along the nanotube ( δ r locally transverse to d l ).Then, for example, the term that leads to ( m/ βδz in Eq. (13) would be written as a tripleintegral over the length H sym − brk = 13 m ˜ β (cid:90) dl dl dl f ijk ( l , l , l ) δr i δr j δr k . unction f here is nonzero only for a nanotube with broken symmetry, i.e., where the energychanges if one replaces δ r → − δ r . The term ∝ δz in Eq. (13) is obtained if one substitutes δ r ( l )with the solution of the harmonic problem, for the considered mode. In general, in nanotubes withbroken symmetry, the coupling between different modes leads to an energy that is cubic in thedisplacements of the modes.A simple calculation shows that, to leading order, the first 3 terms in Eq. (13) give the slowlyvarying terms in δz ( t ) of the form δz slow ( t ) ≈ ∂ z C g V acg ( t ) mω + ∂ z C g V dcg V acg ( t ) A lin ( ω drive , t )2 mω cos φ − β A lin ( ω drive , t ) ω (14)Here, the first term is very much smaller than the second term for typical device parameters; theratio of these terms is of the same order of magnitude as the ratio of the terms in Eq. (12). We note,however, that on exact resonance, ω drive = ω , and we have cos φ = 0. Therefore either | δz slow | displays an extremely narrow and extremely deep dip as a function of ω drive , which is expected for β →
0, or the dominating term in Eq. (14) is the last term, which comes from the broken inversionsymmetry.It is necessary also to look at the ratio of the contributions to the conductance modulation ofthe second term in δz slow in Eq. (14) and the term ∂ z qδz in Eq. (7). One can easily see thatthis ratio is ∼ Γ /ω = (2 Q ) − (cid:28)
1. Therefore the leading-order contribution to the scaled slowlyvarying conductance is δG ≈ ∂ q GV dcg δz ( t ) (cid:18) ∂ z C g − ∂ z C g βω (cid:19) . (15)Here, bar means averaging over the period of fast oscillations 2 π/ω drive .Experimentally, the easiest way to separate the two contributions to δG in Eq. (15) is byestimating ∂ z C g , ∂ z C g , and β from independent measurements. In the following sections, wedemonstrate how this estimate is done for our device. The estimate indicates that the first termin the bracket in Eq. (15) is too small to account for our measurements. We also estimate β byanalyzing the shift of ω as a funtion of z vibra (the amplitude of δz ( t )). We find that this latterestimate is in good agreement with our measurement. Therefore, the major effect is coming fromthe symmetry breaking of the vibrations. In the main text and in the following, we refer to theslow motion δz slow in terms of a (quasi-static) shift of the equilibrium position, δz eq . II. ELECTRICAL CONDUCTANCE AND CAPACITANCE OF THE NAN-OTUBE
In Figure 5a, we show the electrical conductance G of the nanotube device presented in themain text as a function of the constant gate voltage V dcg at a temperature of 65 K. We find thistrace to be reproducible over a timescale of weeks (a current annealing procedure is performedevery day to counter the effects of contamination with residual gas particles). The conductance ofa nanotube depends on its charge carrier density, which is controlled by V dcg . The voltage couplesto the nanotube through the capacitance C g , which we can easily determine: in the Coulombblockade regime, the separation between two conductance peaks is given by ∆ V g = eC g , where e isthe electron charge. From the measurement in Fig. 5b, we get C g = 12 aF, which is in agreementwith an estimation based on the device geometry: C g = 2 π(cid:15) L ln(2 d/r ) . (16)Here, (cid:15) is the vacuum permittivity, L = 1 . µ m is the nanotube length, and d = 350 nm is theequilibrium distance between the nanotube and the gate electrode. Since we cannot measure thediameter of the nanotube due to the large surface roughness of the electrodes in the studied device,we use a typical value for the radius ( r = 1 . ∂ z C g and ∂ z C g by differentiating Eq. (16) and get ∂ z C g = 5 . ∂ z C g = 21 µ F/m .From the measurements of the resonance frequency as a function of V dcg in Fig. 4b of the maintext, we obtain a voltage offset of 0 .
45 V, which corresponds to the work function difference betweenthe nanotube and the gate electrode. This offset in V dcg is included in all the estimates. However,the values of V dcg that we indicate in the main text and the supplementary information are alwaysthe voltages that are applied to the gate electrode. IV. MEASUREMENTS OF VIBRATIONAL MOTION
We discuss first the frequency mixing (FM) technique [4]. A driving voltage V acsd is applied tothe source electrode. Modulating the frequency (with a modulation rate of 671 Hz and a frequencydeviation of 100 kHz) results in a current ( I F M ) at 671 Hz. The gate electrode is biased with V dcg to tune the resonance frequencies. This technique has a low current background and is typically IG. 5:
Nanotube conductance as a function of static gate voltage. a,
Nanotube con-ductance at 65 K as a function of the constant gate voltage V dcg applied to the gate electrode. b, Nanotube conductance at 650 mK. The device is in the Coulomb blockade regime. The spacingbetween consecutive conductance peaks is ∆ V g = eC g , where e is the electron charge.more sensitive than the 2-source technique, so we preferentially use it to detect the eigenmodes ofa nanotube resonator. The main drawback of the FM technique is that the measured signal is notproportional to z vibra , but to the derivative with respect to the frequency of the real part of thedisplacement.In the 2-source technique [1], we apply a driving voltage V acg to the gate in addition to a DCvoltage V dcg . The motion of the nanotube is detected by applying a second, smaller voltage V acsd to the source. The two oscillating voltages are slightly detuned, and the mixing current I mix ismeasured at the detuning frequency (typically δω/ π = 10 kHz). When the displacement is writtenas z ( t ) = Re [˜ z ( ω )] cos( ωt ) + Im [˜ z ( ω )] sin( ωt ), the mixing current I mix measured with the 2-sourcetechnique has the form [16] I mix = 12 V acsd ∂ V g G (cid:18) V acg cos( δωt − ϕ E ) + V dcg ∂ z C g C g R e [˜ z ( ω )] cos( δωt − ϕ E )+ V dcg ∂ z C g C g I m [˜ z ( ω )] sin( δωt − ϕ E ) (cid:19) (17)where G is the conductance of the nanotube, and ϕ E is the phase difference between the voltagesapplied to source and gate. For a properly tuned phase of the lock-in amplifier, the out-of-phasecomponent of the lock-in amplifier output, Y , corresponds to the imaginary part of the resonantdisplacement [third term in Eq. (17)], whereas the in-phase component, X , corresponds to the realpart of the resonant displacement [second term in Eq. (17)] added to a background (first term inEq. 17) that weakly depends on frequency near resonance with a given mode (we note that it can ave contributions from other modes). For the modulation frequency ω close to resonance, we getfor the Y -component of the mixing current I mix , which we denote as I Yvibra , I Yvibra = 12 · V acsd V dcg · ∂ V g G · ∂ z C g C g · z vibra , (18)where z vibra is the amplitude of resonant forced vibrations. For the considered small V acg , z vibra isproportional to the amplitude of V acg . I Yvibra can be conveniently read out from the measurement.
V. ESTIMATION OF VIBRATION AMPLITUDE
Equation (17) allows estimating the vibration amplitude of the resonator for resonant drivingby comparing the out-of phase current on resonance, I Yvibra , to the background far from resonance, I Xoff − res [1, 16]. Using Eq. (16), we get that z vibra (cid:39) d · ln (cid:18) dr (cid:19) I Yvibra I Xoff − res V acg V dcg (19)The measurement in Fig. 3 yields a value of z vibra = 2 . V acg = 1 . VI. DETECTION OF THE MOTION OF THE EQUILIBRIUM POSITION
In the following, we explain in more detail the technique we develop to detect the motion of theequilibrium position of the nanotube resonator due to symmetry breaking. We drive the resonatorwith an amplitude modulated (AM) driving force, which causes an AM vibrational motion (Eq. 2of the main text). The amplitude change of the vibration is quasi-adiabatic from the point of viewof the resonator (we checked that the result is independent of the modulation period 2 π/ω AM upto 0.1 ms.) In the presence of AM modulation the quadratic nonlinear force F ∝ δz ( t ) leads tothe oscillation of the nanotube equilibrium position, δz eq , as illustrated in Fig. 1d of the main text.The conductance G of the nanotube depends both on the voltage that is applied to the gateelectrode ( V g ) and on the capacitance between the gate electrode and the nanotube ( C g ). We canrewrite Eqs. 6-8 for the change in conductance as δG ( t ) = ∂ V g G · δV g ( t ) + ∂ C g G · δC g ( t ) . (20)In the analysis of the low-frequency conductance the first term on the right hand side can beneglected because the voltage that we apply to the gate electrode has no term at the frequency of nterest ω AM / π (we have verified this using a signal analyzer). Taking into account the analysisof Sec. I, we then write δG ( t ) = ∂ V g G · V dcg ∂ z C g C g δz ( t ) . (21)The slow oscillation of the conductance is caused by the motion of the equilibrium position dueto symmetry breaking. As shown in Fig. 1d of the main text, for comparatively weak resonantmodulation, where the AM vibrational motion is of the form of δz ( t ) = z vibra · cos( ω drive t ) · [1 − cos( ω AM t )], the equilibrium position δz eq ∝ [ δz ( t ) ] oscillates with period 2 π/ω AM . The vibrationsare nonsinusoidal, δz eq ( t ) = δz eq (cid:20) − cos ω AM t + 34 + 14 cos 2 ω AM t (cid:21) . (22)The amplitude of the low-frequency current oscillation measured at ω AM is then simply I LF = V dcsd · δG = V dcsd V dcg (cid:12)(cid:12)(cid:12)(cid:12) ∂ V g G · ∂ z C g C g (cid:12)(cid:12)(cid:12)(cid:12) δz eq . (23)This current corresponds to the second term in the bracket of Eq. (15). By comparing Eq. (23)to Eq. (18), we can express the low-frequency current at resonance as I maxLF = 2 I Yvibra · δz eq z vibra · V dcsd V acsd (24)provided that I LF and I Yvibra are measured with the same driving force. (The driving force comesfrom the gate voltage oscillation at frequency ω ; however, in the 2-source technique used to measure I Yvibra the voltage amplitude is not modulated; we note again that we study the regime where V acg is comparatively small.) VII. ESTIMATION OF THE LOW-FREQUENCY CURRENTS DUE TO THE CA-PACITIVE NONLINEARITY AND THE CONDUCTANCE NONLINEARITY
In this section, we first estimate the current that is expected due to the nonlinearity of thecapacitance with respect to displacement, ∂ z C g [see Eqs. (7), (8), and (15)]. From Eqs. (7) and(8), the current at ω AM that we expect due to the capacitive nonlinearity in the presence of a DCbias voltage ( V dcsd ) has the form I capaLF = V dcsd · δG = 12 V dcsd V dcg · ∂ V g G · ∂ z C g C g z vibra . (25) omparing Eq. (25) to Eq. (18), we can express this current in terms of the 2-source mixing currentmeasured on resonance in Fig. 3 of the main text. We get I capaLF = I Yvibra · V dcsd V acsd · ∂ z C g ∂ z C g z vibra . (26)where the currents are taken at the resonance frequency and for the same amplitude V acg . Thisrelation is useful, because it depends on a small number of parameters. Using I Yvibra = 37 pA fromFig. 3b, V dcsd = 10 mV, V acsd = 0 . ∂ z C g = 5 . ∂ z C g = 21 µ F/m , and z vibra = 2 . I capaLF = 9 . ± I LF = 219 pA that we measured in Fig. 2c.Next, we estimate the current that is expected due to the nonlinearity of the conductance withrespect to the gate voltage, ∂ V g G , which is described by the last term in Eq. (6). From Eqs. (7)and (8), the current at ω AM is I condLF = 12 V dcsd ( V dcg ) · ∂ V g G · ( ∂ z C g /C g ) z vibra . (27)Using V dcsd = 10 mV, V dcg = − . ∂ z C g = 5 . C g = 12 aF, and z vibra = 2 . ∂ V g G =54 µ S/V , we get that I condLF = 0 .
19 pA.
VIII. ESTIMATION OF THE SYMMETRY BREAKING STRENGTH β In the presence of the AM vibrational motion of the form of δz ( t ) = z vibra · cos( ω drive t ) · [1 − cos( ω AM t )] and in the limit of small δz eq (such that the restoring force can be approximated bythe spring force mω ), δz eq is related to the parameter β as ω δz eq = | β | z vibra . (28)Using Eq. (28) together with Eq. (24), we arrive at the relation I maxLF = 2 I Yvibra · V dcsd V acsd · | β | ω · z vibra . (29)Here, again the currents are taken at the resonance frequency and for the same amplitude V acg .This relation is useful, because it depends on a small number of parameters that, in addition,are well characterized. With I Yvibra = 37 pA from Fig. 3b, V dcsd = 10 mV, V acsd = 0 . ω =2 π ·
51 MHz, z vibra = 2 . I maxLF = I LF = 219 pA measured in Fig. 2c of the main text, weget δz eq = 0 .
18 nm and β = 4 . · m − s − .An alternative estimation of β is possible using the fact that the symmetry breaking forceleads to a shift of the resonance frequency with driving force. We have measured the shift of the esonance frequency as a function of the vibrational amplitude with the FM technique as well aswith our new detection technique. The values measured with the two methods agree well and areplotted in Fig. 4d of the main text. We use the relation (supplementary reference [3])∆ ω = 38 γ eff ω z vibra , (30)where ∆ ω is the shift of the resonance frequency and γ eff = γ − ω − β (31)[ γ is the coefficient of the Duffing nonlinearity, see Eq. (13).] We estimate γ eff = − . · m − s − from Eq. (30) by inserting z vibra = 2 . ω = 2 π ·
51 MHz, and ∆ ω (cid:39) − π ·
150 kHz from Fig. 4dat the driving voltage V acg = 1 . β = 4 . · m − s − from Eq. (31) by assumingthat γ is negligible. A nonzero value of γ > β . IX. THE DEPENDENCE OF THE VIBRATION FREQUENCY ON DC GATEVOLTAGE
Here, we show that the quadratic nonlinearity of the restoring force also leads to a shift of thenanoresonator frequency ∆ ω due to a dc gate voltage. We start with equation of motion m∂ t z + 2 m Γ ∂ t z + mω z = − mβz − mγz + 12 ∂ z C g ( V dcg ) + 12 ∂ z C g ( V dcg ) z (32)where we assume that the displacement z is small. Taking z as the sum of a static contributionand an oscillating contribution, we get in first order in ( V dcg ) ∆ ω = 12 mω [ ∂ z C g βω − ∂ z C g ]( V dcg ) . (33)The first term in the bracket, which depends on the symmetry breaking strength β , leads to theincrease of ω with V dcg ( β is usually positive). This is the behavior observed for a large majorityof nanotube resonators. By contrast, the second term leads to the decrease of ω with V dcg . Thisis observed occasionally and is attributed to nanotubes with a large built-in tension.Interestingly, the bracket of Eq. (33) is the same as that of Eq. (15) where the two termscorrespond to the low-frequency currents induced by symmetry breaking and by the nonlinearcapacitive coupling, respectively. In the resonator discussed in the main text, ω increases with V dcg . This further supports our finding that the peak in I LF is attributed to symmetry breaking. IG. 6:
Mechanical resonance at room temperature.
Mechanical vibration of the samenanotube device as in the main text measured at 250 K with the FM technique. The resonancewidth ω / πQ can be conveniently read out from the separation between the two minima that areflanking the main peak [4]. We obtain Q ∼ V dcg = − . V acg = 5 mV.We obtain from Fig. 4b in the main text that the prefactor a in the relation ∆ ω = a ( V dcg ) ranges from 4 · to 8 · ; we offset V dcg so that V dcg = 0 when ω is minimum. From the lengthof the nanotube, we estimate that the effective mass is (cid:39) β (cid:39) ± · m − s − . This value is comparable to the values estimatedabove with different methods. X. FLUCTUATION-INDUCED SPECTRAL BROADENING
Thermally-induced spectral broadening can be understood from Eq. (32) if one incorporates intothe right-hand side of this equation a random force f T ( t ) that describes thermal noise. This noisecomes from the same coupling to a thermal reservoir that leads to the friction force ∝ Γ ∂ t z . Fromthe fluctuation-dissipation relation the noise is δ -correlated, with (cid:104) f T ( t ) f T ( t (cid:48) ) (cid:105) = 4 m Γ k B T δ ( t − t (cid:48) ).To gain a qualitative insight into the broadening we assume that the resonator vibrates as z vibra cos( ωt + φ ) with frequency ω close to ω . We now look at the overall displacement as z ( t ) = z vibra cos( ωt + φ ) + δz ( t ) and linearize Eq. (32) with respect to δz ( t ). The left-hand-side will havethe same form as for z ( t ), i.e., it will describe a resonator with coordinate δz ( t ) and eigenfrequency ω . In the right-hand-side, however, there will be a term − mγ [ z vibra cos( ωt + φ )] δz ( t ). Whenaveraged over the period 2 π/ω , this term leads to the shift of the vibration frequency for δz ( t ) ofthe form ω → ω + 3 γz vibra / ω . This is the well-known frequency shift of a nonlinear oscillator ith vibration amplitude; a systematic treatment (supplementary reference [3]) shows that, if z vibra is the amplitude of eigenvibrations, to the lowest order in z vibra the frequency shift is 3 γz vibra / ω ,cf. Eq. (30).We now note that the vibrations z vibra (cos ωt + φ ) are in fact eigenvibrations induced by noise.They have random phase φ and also random amplitude. The distribution of this amplitude is ofthe Boltzmann form, ∝ exp( − mω z vibra / k B T ) for weak resonator nonlinearity. The spread of thevibration amplitudes leads to the effective spread of the vibration eigenfrequencies, with typicalwidth δω = 3 γ eff k B T / mω ; (34)we have replaced here γ with γ eff to allow for the renormalization of γ by the quadratic-nonlinearityterm, see Eq. (31).Figure 6 shows the resonance line-shape measured at 250 K. From the mechanical bandwidth,measured between the two minima that are flanking the main peak, the apparent quality factor is (cid:39)
50. We change the driving force by a factor up to 4 and we do not observe a variation of thebandwidth. Using β = 4 . · m − s − and Eq. (31), we get an apparent quality factor of 67 fromEq. (34).The spread of the eigenfrequencies (34) leads to a broadening of the resonator spectrum. Weemphasize that this broadening is not related to the vibration decay, it is a result of the interplayof the resonator nonlinearity and fluctuations. Moreover, since the distribution of the squaredvibration amplitude, and thus of the vibration eigenfrequency, is exponential, the spectrum isasymmetric. The overall spectrum in the presence of nonlinearity and fluctuations, on the onehand, and decay, on the other hand, is determined by the ratio of δω and the decay-inducedbroadening Γ. It can be obtained in an explicit form for an arbitrary δω/ Γ [21]. The frequencyspread of the type (34) can come also from the nonlinear coupling of the considered mode to othermodes of the resonator [21]. In the context of carbon nanotubes, this latter mechanism has recentlyattracted significant attention [22].Whereas the internal nonlinearity of the resonator leads to the change of the shape of thespectrum with increasing temperature, this is not the case for the nonlinearity associated to thequadratic dependence of the capacitance on the resonator displacement (second term in in Eq. (13)). IG. 7:
Measurements for a second nanotube resonator.
This device has the same geo-metrical layout as the one discussed in the main text. All measurements are performed at 65 K. a , Current as a function of ω drive and V dcg measured with the FM technique. V acg = 1 . b , I LF versus ω drive with V acg = 0 .
53 mV, V dcsd = 10 mV, and V dcg = 2 . c , X quadrature and d , Y quadrature of the current measured with the 2-source mix-ing technique. I Xvibra consists of a current proportional to the real part of the vibrational amplitudein addition to a purely electrical background current. I Yvibra is proportional to the imaginary partof the vibrational displacement. V acg = 1 . V acsd = 0 . V dcg = 2 . b . XI. ADDITIONAL DEVICE
In this section, we present data from a second nanotube device. We verify with atomic forcemicroscopy that the trench width and depth are the same as for the device presented in the maintext (1 . µ m and 350 nm, respectively). The roughness of the metal electrodes do not allow ameasurement of the nanotube diameter. We therefore use the same estimates for the capacitanceand the mass as for the first device ( C g = 12 aF, ∂ z C g = 5 . ∂ z C g = 21 µ F/m , and (cid:39) V dcg = 2 . I LF and find a peak at the resonance frequency (Fig. 7b). In order to determine thevibration amplitude z vibra , we also measure I Xvibra and I Yvibra with the 2-source mixing technique(Fig. 7c and d). We obtain z vibra = 8 . . z vibra directlyfor the lower driving voltage used to measure I LF . In order to compare the two sets of data, weassume z vibra ∝ V acg , thus obtaining the scaled value z vibra = 2 . V acg = 0 .
53 mV.We perform the same analysis as for the main device to identify the origin of I LF . We estimatethe currents due to the nonlinearities in the capacitance and in the electrical conductance, finding I capaLF = 4 . I condLF = 0 .
21 pA. Both values are far below the measured I LF = 374 pA. Inaddition to C g , ∂ z C g , and ∂ z C g mentioned above, we use here V dcsd = 10 mV, V acsd = 0 . ∂ V g G = 19 µ m/V . For the analysis, we use an offset for V dcg such that V dcg = 0 when ω is lowest.As for the vibration amplitude, we use a scaled value I Yvibra = 30 pA to account for the differencein the driving voltages.We now consider symmetry breaking of the vibrations as the origin of the peak in I LF . UsingEq. (29), we calculate β = 1 . · m − s − . This value can be compared to the one we obtainfrom the dependence of ω on V dcg . From Eq. (33), we get β = 5 . · m − s − . For this device,we have not studied the dependence of ω on V acg .The fluctuation-induced spectral broadening δω , expected from the two values of β , leads to aquality factor at room temperature that lies between 7 .
1] Meerwaldt, H. B., Steel, G. A. & van der Zant, H. S. J. in “
Fluctuating Nonlinear Oscillators ”,ed. by M. Dykman (OUP, Oxford 2012), p. 312.[2] Moser, J., Eichler, A., Lassagne, B., Chaste, J., Tarakanov, Y., Kinaret, J., Wlson-Rae, I. &Bachtold, A. ibid ., p.341.[3] Landau, L. D. & Lifshitz, E. M.
Mechanics (Elsevier, Amsterdam 2004).(Elsevier, Amsterdam 2004).