Interrelation of elasticity and thermal bath in nanotube cantilevers
S. Tepsic, G. Gruber, C. B. Moller, C. Magen, P. Belardinelli, E. R. Hernadez, F. Alijani, P. Verlot, A. Bachtold
IInterrelation of elasticity and thermal bath in nanotube cantilevers
S. Tepsic, G. Gruber, C. B. Møller, C. Magén,
2, 3
P. Belardinelli, E. R. Hernández, F. Alijani, P. Verlot, and A. Bachtold ICFO - Institut De Ciencies Fotoniques, The Barcelona Institute ofScience and Technology, 08860 Castelldefels (Barcelona), Spain Instituto de Nanociencia y Materiales de Aragón (INMA),CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain Laboratorio de Microscopías Avanzadas (LMA),Universidad de Zaragoza, 50018 Zaragoza, Spain DICEA, Polytechnic University of Marche, 60131 Ancona, Italy Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), 28049 Madrid, Spain Department of Precision and Microsystems Engineering,3ME, Mekelweg 2, (2628 CD) Delft, The Netherlands School of Physics and Astronomy - The University of Nottingham,University Park, Nottingham NG7 2RD, United Kingdom
We report the first study on the thermal behaviour of the stiffness of individual carbon nanotubes,which is achieved by measuring the resonance frequency of their fundamental mechanical bendingmodes. We observe a reduction of the Young’s modulus over a large temperature range with a slope − (173 ± ppm/K in its relative shift. These findings are reproduced by two different theoreticalmodels based on the thermal dynamics of the lattice. These results reveal how the measuredfundamental bending modes depend on the phonons in the nanotube via the Young’s modulus.An alternative description based on the coupling between the measured mechanical modes and thephonon thermal bath in the Akhiezer limit is discussed. In engineering, thermoelasticity is central indetermining the elastic limits of structures rangingfrom large scale spacecrafts [1] and nuclear plants [2]down to nano-structured systems. A rich underlyingphenomenology emerges for small structures, includingdissipation [3, 4], fluctuations [5, 6], and torquegeneration [7, 8], which are key to the developmentof state-of-the-art nano- and micro-electromechanicaltechnologies [9, 10]. Thermoelasticity has also beenused with success in condensed matter physics, wherethermal measurements of the stiffness unveil the phasetransition of charge-density waves and superconductivityin transition metal dichalcogenides and high- T c superconductors [11–13]. From a fundamental point ofview, the thermal behaviour of the stiffness – quantifiedby the Young’s modulus – emerges from the non-trivialinterplay of the binding energy and the lattice dynamics.However, the effect of the thermal lattice dynamics onthe stiffness has remained elusive in individual nanoscalesystems due to experimental challenges related tomanipulating and measuring such small objects.In this work, we use the exquisite sensing capabilitiesof mechanical resonators based on nanoscale systems[14–29] to resolve the small effect associated with thethermal behaviour of their stiffness. Using the resonancefrequency measured by optomechanical spectroscopy,we estimate the Young’s modulus of micrometer-longnanotube cantilevers from room temperature down to afew Kelvins. These results agree with the temperaturedependence of the resonance frequency predicted bymolecular dynamics simulations, which take into accountthe lattice dynamics of the nanotube. Our measurements are also consistent with the Young’s modulus directlycomputed from a quasi-harmonic approximation of thefree energy of the phonon modes. This work not onlyshows how the stiffness of an individual nanotube isrelated to its phonons, but it also highlights the role ofthe phonon thermal bath in nanotube cantilevers, whichis a topic of importance in the field of nanomechanicalresonators [14–29].We use the single clamped resonator layout, where oneend of the nanotube is attached to a silicon chip andthe other end is free. This layout avoids prestress inthe nanotube built-in during fabrication, in contrast towhat may happen with the double clamped layout. As aresult, the restoring force is given solely by the bendingrigidity. This enables us to probe the Young’s modulus Y by measuring the resonance frequency, ω ∝ √ Y [32].Such a resonance-based methodology is also employedin thermoelasticity studies on larger scale systems [11–13, 33].We engineer a platinum particle at the free end ofthe nanotube, so that the resonator can be measuredby scattering optomechanical spectroscopy (Fig. 1a) [31].We grow the particle by focused electron beam-induceddeposition [30]. Figure 1b shows a scanning electronmicroscopy image of device A. Transmission electronmicroscopy (TEM) indicates that nanotubes can bemade from one to a few walls, with a median valueof two walls (Supplementary Material, Sec. I). Thevibrations are detected by measuring the backscatteredintensity from a
632 nm laser beam focused onto theparticle. Figure 1c shows the optomechanical spectrum ofdevice A. The resonance frequencies of the fundamental a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n x φ ( x ) L x m Pt / m tube x - S VV [ V / H z ] x - (c) Frequency [kHz] (a)
FIG. 1. (a) Schematic of the experimental setup. Thesample is placed at the waist of a strongly focused beam ofa He-Ne laser. The scattered light (not shown) is collectedin reflection by means of an optical circulator and furthersent on an avalanche photodetector. The two double arrowsrepresent the polarization of the fundamental mode doublet.(b) Device A imaged by scanning electron microscopy afterthe deposition of a platinum nanoparticle; the scale bar is µ m [30]. (c) Power spectra of the optical reflection fromdevice A showing the resonance of the thermal motion ofthe fundamental mode doublet at
300 K . The two spectraare recorded using different positions of the nanotube in thelaser waist to enhance the signal [31]. (d) Calculated profile ϕ ( x ) of the fundamental mode shape along the nanotubeaxis estimated for two different platinum particle massesnormalized by the nanotube mass. modes polarized in perpendicular directions are about . and . . The platinum particle does notaffect the restoring force nor the eigenmode shape of thetwo fundamental modes (Fig. 1d), in contrast to whathappens for higher frequency modes (SupplementaryMaterial, Sec. II). In this work we use low laser power sothat the resonance frequency is not affected by absorptionheating and optical backaction [31].We quantify Y = (1 . ± .
28) TPa at roomtemperature from six devices by combining thermalmotion variance measurement and TEM imaging; theadvantage of this method is that it does not rely on ! " ’ ! ( ) *+ , - " $ % *+ : ; - *******4/01/23452/ <5=>30/=43?***********@A//1 *****0B>/ /CDE434EB= ****F/34E=G *********% **4F/203?****DBB?E=G *********$ ** **4F/203?****DBB?E=G *********% ** **4F/203?****DBB?E=G *********% ** ***>2EH/=****F/34E=G *********% ** ***>2EH/= FIG. 2. Resonance frequency and relative change of theYoung’s modulus of device A as a function of cryostattemperature. The legend indicates the direction of thetemperature sweep (cooling or heating), which fundamentalmode is measured, and whether the detected vibrations arethermal or driven with a piezo-actuator. the cantilever mass (Supplementary Material, Sec. III).The estimated Young’s modulus is similar to previouslypredicted and measured values [34–39]. This indicatesthat the contamination adsorbed on the nanotube surfacehas little contribution to the stiffness of the nanotube.The contamination, which is localized along someportions of nanotubes as observed by TEM, presumablyconsists of hydrocarbons adsorbed during their exposureto air and the particle growth. The typical stiffnessreported for such amorphous material is comparativelylow Y ≈ (50 – [40].Figure 2 shows the variation of the resonance frequencyof device A when sweeping the temperature T . Thevariation is remarkably similar for both fundamentalmodes, independent of the temperature sweep directionand of whether the motion is thermal or driven with apiezo-actuator. This variation of the resonance frequency ω = (cid:112) k/m is associated to the change of the springconstant k , which is linearly proportional to Y in thesingle clamped layout. We extract the relative shiftof the Young’s modulus from the relation ∆ Y ( T ) Y ( T min ) =2 ∆ ω ( T ) ω ( T min ) , where T min is the lowest temperature atwhich we record the vibrations. Figure 3a showsthe measurements of nine different devices. They allfeature the same trend with a reduction of the Young’smodulus when increasing temperature. The dependenceis essentially linear above about
100 K ; the slope averagedover devices is ∆ Y ( T ) /Y · /T = − (173 ± ppm/K.These measurements are related to neither the mass !!" ! " ! -. " $ -. ! " ! -. " $ -. -7- -@-A- -B-C- -D-=- -6-E-*F&)*G&8*: FIG. 3. Comparison of the relative change of the resonance frequency and the Young’s modulus between experiment (a)and theory (b) for different nanotubes. The theoretical results are obtained for different nanotube chiralities with eithermolecular dynamics (MD) simulations or quasi-harmonic approximation (QHA) calculations. The MD simulations and theQHA calculations quantify ∆ ω /ω and ∆ Y /Y , respectively. adsorbed on the nanotube nor the thermal expansionof the nanotube nor the combination of the Duffingnonlinearity and the thermal motion, as shown in Sec. IVof Supplementary Material.These measurements can be captured by moleculardynamics (MD) simulations of the nanotube cantileverdynamics. The temperature dependence of the resonancefrequencies of the lowest energy bending modes obtainedfrom the MD simulations behave in the same way as thosewe measure (Figs. 3a,b). The associated slope estimatedfor different nanotube chiralities leads to ∆ Y ( T ) /Y · /T = − (79 ± ppm/K, which is rather similar tothe measured value. This suggests that the thermalbehaviour of the Young’s modulus in our measurementsis related to the lattice dynamics of nanotubes.We employ a second method to directly computethe Young’s modulus from the energy dispersion of thenanotube phonon modes. For this, we evaluate the freeenergy F ( T, (cid:15) ) of the phonon modes at T and strain (cid:15) with the quasi-harmonic approximation, yielding Y ( T ) = 1 V ( T ) (cid:18) ∂ F ( T, (cid:15) ) ∂(cid:15) (cid:19) (cid:15) =0 , where V ( T ) is the equilibrium volume at thistemperature. The resulting Y ( T ) dependence is alsoconsistent with the measurements (Figs. 3a,b). The slopefor different chiralities is ∆ Y ( T ) /Y · /T = − (104 ± ppm/K. The variation of the slope is larger thanthat obtained with molecular dynamics; this differencemay be due to the infinite nanotube length and the purelylinear vibrational dynamics considered in the quasi-harmonic approximation method, while the lengths in themolecular dynamics simulations are much shorter, thatis, less than
40 nm . Overall, the experimental findingsare fairly consistent with both models considering thetypical differences between the values of Y of nanotubesobtained with different experimental and theoreticalmethods [34–39]. Both theoretical models are described in the Supplementary Material (Secs. V and VI).These results show how the measured fundamentalmechanical modes are linked to phonons via the Young’smodulus. An alternative way to describe this link isto consider the coupling of the measured mechanicalmodes with the thermal bath made of the phonons ofthe nanotube. It is likely that the phonon thermal bathin our experiments operate in the Akhiezer limit [41].Over the temperature range that we measure, the phononmodes in nanotubes with energy (cid:126) ω k similar to k B T have decay rates /τ k larger than ω , since τ k ≈ nswas measured for breathing modes at T = 5 K [42]and we estimate τ k to be typically in the − nsrange for the longitudinal and twist modes [43] (Sec. VIIof Supplementary Material). (The estimation of τ k forhigh-energy bending modes is complicated and beyondthe scope of this work.) This sets the Akhiezer limit ω τ k (cid:28) at least for the breathing, longitudinal, andtwist modes [44]. It involves three-phonon processes,where one vibration quantum of the measured mode isabsorbed together with the absorption and the emissionof high-energy phonons with frequencies ω k and ω k (cid:48) ,respectively. The sizeable decay rates of the high-energyphonons lead to uncertainty in their energy. This liftsto some extent the restriction associated with the energyconversation of the three-phonon process, ω = ω k − ω k (cid:48) ,which holds in the Landau-Rumer limit when ω τ k (cid:29) .For this reason, the resonance frequency reduction andthe relaxation in the Akhiezer limit are expected to belarger than that in the Landau-Rumer limit over thestudied temperature range. The thermoelastic limit [4]does not apply for nanotubes, since the model relieson phonons that locally reach thermal equilibrium atdifferent temperatures on the two sides of the beam cross-section, which is not realistic for such narrow resonators.It is expected that the phonon thermal bathsignificantly contributes to the measured dissipationvia the Akhiezer relaxation, since a thermal bath !!"! ! " & ’( ) * + ,!!%-!%!! -! !!-!! ./01/23452/’(6+738 ""! 9"!, -! $-!$$! ! " ’( ) * + !"!$!!"%!$,! ,!!%-!%!! -! !!-!! ./01/23452/’(6+7:8 FIG. 4. (a) Temperature dependence of the mechanicallinewidth for device A. The black line is the average ofdifferent temperature traces (red lines). (b) Temperaturedependence of the linewidth for all the measured nanotubes,from device A at the top to device I at the bottom; theassociated resonance frequencies are
54 kHz ,
96 kHz ,
194 kHz ,
77 kHz ,
57 kHz ,
108 kHz ,
44 kHz ,
58 kHz ,
48 kHz . The arrowsindicate peaks in dissipation. results in a resonance frequency reduction as well asdissipation, both of them being related through theKramers–Kronig relations [45]. Figures 4a,b show themeasured temperature dependence of the mechanicallinewidth of the different measured devices. Themeasurements feature one or two peaks of dissipationat some specific temperatures. These observed peakscould arise from the Akhiezer relaxation. The Akhiezerdissipation rate depends in a complicated way on thenumber of phonon modes with energy (cid:126) ω k (cid:46) k B T , theirpopulation, and their decay rate [44]. The temperaturedependence of the Akhiezer dissipation rate could featureone or more peaks in dissipation, especially since thephonon density of states varies up and down as a functionof energy [43, 46] and the temperature behaviour ofthe decay rate changes for different phonon modes. Inaddition, the dissipation peaks could emerge at differenttemperatures for different nanotube chiralities, since the phonon energy dispersion is chirality dependent. Themeasured peaks in dissipation cannot be described by themodel that is used in the literature [47, 48] to quantifydissipation due to defects. See Secs. VIII and IX ofSupplementary Material for further discussion on theAkhiezer dissipation and dissipation due to defects.In conclusion, we report the first experimental study ofthe temperature dependence of the Young’s modulus ofa nanoscale system. The measurements are consistentwith theoretical predictions based on the nanotubelattice dynamics. This indicates that the phononthermal bath plays an important role in the dynamicsof nanotube cantilevers, including thermal vibrationalnoise, dissipation, and resonance frequency reduction.Further theoretical work is needed to compute theAkhiezer relaxation in nanotubes beyond the modelsused so far, where a single decay rate is employed forall the high-frequency phonon modes [48–50]. Thismay be achieved with a microscopic theory [44] takinginto account the phonon energy dispersion [46] and theenergy decay of high-frequency phonons [43]. It willbe interesting to see whether such a model leads todissipation peaks at specific temperatures as observed inour work.We thank Mark Dykman and Andrew Fefferman forenlightening discussions. This work is supported byERC advanced (grant number 692876), ERC PoC (grantnumber 862149), Marie Skłodowska-Curie PROBIST(grant number 754510), the Cellex Foundation,the CERCA Programme, AGAUR (grant number2017SGR1664), Severo Ochoa (grant number SEV-2015-0522), MICINN (grant number RTI2018-097953-B-I00and PGC2018-096955-B-C44), the Fondo Europeo deDesarrollo Regional, the grant MAT2017-82970-C2-2-R of Spanish MINECO and the project E13_17Rfrom Aragon Regional Government (ConstruyendoEuropa desde Aragón). F.A. acknowledges supportfrom European Research Council (ERC) starting grantnumber 802093. [1] E. A. Thornton, Thermal structures for aerospaceapplications (American Institute of Aeronautics andAstronautics, 1996).[2] Z. Zudans, T. C. Yen, and W. H. Steigelmann,
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I. DEVICE FABRICATION AND STRUCTURAL CHARACTERIZATION
The carbon nanotubes were grown on silicon substrates via chemical vapor deposition. A Zeiss Auriga scanningelectron microscope (SEM) was used to select suitable nanotube cantilevers. The SEM is equipped with a gasinjection system, which was used to deposit platinum particles at the apex of the nanotubes for their optomechanicalfunctionalization [1, 2]. Figure 5(a) shows a pristine nanotube cantilever (device A). Figure 5(b) shows the samecantilever after the deposition of a Pt particle. The free end of the cantilever is blurred in the SEM images due tothe thermally driven motion. The displacement profile was measured by a SEM line trace across the nanotube atthe tip. Figure 5(c) shows the observed Gaussian distribution in the secondary electron current I SE , as expectedfor thermal vibrations [3]. The displacement variance σ = (87 . was obtained from a fit of the data. Thespring constant k = 5 . × − N / m was determined from the equipartition theorem k = k B T /σ where k B is theBoltzmann constant and T is the temperature [3]. The mass of the deposited Pt particle was controlled during itsgrowth by monitoring the mechanical resonance frequency of the lowest flexural mode of the nanotube; the thermalvibrations were measured by pointing the electron beam onto the apex of the nanotube in spot mode while recordingthe noise of I SE [2]. The initial effective mass of the nanotube was m ∗ = 243 ag and the mass of the depositedparticle visible in Fig. 5(b) was m Pt = (3 . ± .
1) fg . All discussed samples were fabricated as described above. Themechanical properties of the samples that were optomechanically characterized at low temperature (devices A-I) aresummarized in Table I.We performed high-resolution transmission electron microscopy (HRTEM) to assess the microscopic structure ofnanotube cantilevers. The samples were fabricated on silicon aperture windows using the identical procedure asoutlined above. HRTEM imaging was conducted using a Thermo Fisher Titan Cube 60-300, equipped with an imageaberration corrector CETCOR from CEOS. The microscope was operated at
80 kV to minimize beam damage and (a)(b) − x [µm] I SE [ a . u .] DataFit= 87.4 nm σ (c) FIG. 5. Device A imaged by SEM (a) before and (b) after deposition of a Pt nanoparticle; the scale bars are µ m . (c)Secondary electron signal I SE across the apex of the nanotube; from a Gaussian fit the displacement variance σ = (87 . is determined.TABLE I. Mechanical properties of the nanotube cantilever devices discussed in the main text. These include the length l ,the standard deviation of the thermal displacement σ , the spring constant k , and the mass ratio m ∗ between the Pt particleand the nanotube as defined in Supplementary Section II.Device l ( µ m) σ (nm) k (N/m) m ∗ A 8.2 87.4 . × − . × − . × − . × − . × − . × − . × − . × − . × − achieve a spatial resolution below . Å . Figure 6 shows atomically resolved images obtained for different devicesnear the clamping point where the thermal displacement is negligible. The devices shown are a single wall device,a seven wall device, and a triple wall device. The latter device was also characterized optomechanically at lowtemperature before conducting the HRTEM experiments and is referred to as device C in the main text and in tableI. The amorphous material visible in Figs. 6 (a) and (c) presumably consists of hydrocarbons adsorbed during theirexposure to air and the particle growth [2].Using such HRTEM images, we determined the number of walls and the associated diameters for six different devicesranging from single wall to seven wall nanotubes. Table II shows the diameters obtained by HRTEM together withother parameters obtained by SEM. The calculation of the Young’s modulus in the table is outlined in Section III. (a) (b) (c)
FIG. 6. HRTEM images recorded near the clamping point of single wall device T1 (a), seven wall device T6 (b) and triple walldevice C (c). In order to enhance the signal-to-noise ratio multiple images were overlaid and averaged. All image dimensionsare
10 nm by
10 nm .TABLE II. Properties of different nanotube cantilevers functionalized with platinum particles. Cantilever length l and springconstant k were obtained by SEM imaging. Number of walls N and associated diameters d i were obtained by HRTEM. TheYoung’s modulus Y was calculated as described in section III.Device N l ( µ m) k (N/m) d i (nm) Y (TPa)T1 1 . ± . . ± . × − d = 3 . ± .
14 1 . ± . T2 2 . ± . . ± . × − d = 3 . ± . , d = 2 . ± .
16 0 . ± . T3 2 . ± . . ± . × − d = 3 . ± . , d = 2 . ± .
09 0 . ± . T4 2 . ± . . ± . × − d = 3 . ± . , d = 2 . ± .
22 1 . ± . C 3 . ± . . ± . × − d = 5 . ± . , d = 5 . ± . , d = 4 . ± .
11 1 . ± . T6 7 . ± . . ± . × − d = 6 . ± . , d = 5 . ± . , d = 4 . ± . , . ± . d = 4 . ± . , d = 3 . ± . , d = 2 . ± . , d = 2 . ± . II. EIGENMODES AND SPRING CONSTANT OF A CANTILEVER WITH ADDED MASS AT THEFREE ENDA. Model
The Euler-Bernoulli partial differential equation (PDE) that describes the motion y ( x, t ) of a vibrating beam is ∂ y∂t + Y IρA ∂ y∂x = 0 . (1)In Eq. 1, Y is the Young’s modulus, I is the second moment of the cross-sectional area A , and ρ is the density of thecarbon nanotube (CNT) with length l . Solution of Eq. 1 is y ( x, t ) = cos ω n t [ c cos α n x + c sin α n x + c cosh α n x + c sinh α n x ] , (2)with radial frequency ω n = cα n and c = (cid:114) Y IρA . In Eq. 2, α n is the wave number whereas c , ..c are constants thatwill be determined by satisfying boundary conditions. In the presence of a particle with mass m bead at the free end, theboundary conditions to satisfy become: y | x =0 = ∂y/∂x | x =0 = 0 , and ∂ y/∂x | x = l = 0 , − Y I ∂ y∂x | x = l = m bead c α n y x = l [4], in which the effect of the bead’s rotary inertia is neglected. Implementing these conditions in Eq. 2 leads to thefollowing characteristic equation cos Ω n cosh Ω n + 1 + m ∗ Ω n (sinh Ω n cos Ω n − sin Ω n cosh Ω n ) = 0 , (3)where eigenvalues Ω n = α n l are solutions of Eq.3 with m ∗ = m bead /m beam . The eigenmodes associated with theeigevalues can then be obtained as Φ n ( x ) = cos(Ω n x ) − cosh(Ω n x ) − cos(Ω n ) + cosh(Ω n )sin(Ω n ) + sinh(Ω n ) (sin(Ω n x ) − sinh(Ω n x )) . (4)Figure 7 shows the variation of the first three eigenfrequencies as a function of m ∗ . When the ratio between the massof the bead at the free end and the mass of the beam becomes large, the mode shapes approach those of a beamclamped at one end and hinged at the other. The mode shapes for an increasing m ∗ are shown in Figs. 8. The profileof the fundamental eigenmode is basically unchanged when increasing m ∗ , in contrast to what happens for the othereigenmodes.The equivalent spring constant associated with the free-end CNT deflection for the n-th eigenmode, k n , can becalculated as follows [5]: k n = Y Il (cid:82) (Φ (cid:48)(cid:48) n ( x )) dx Φ n (1) . (5)By letting I = π (cid:0) d g + dg (cid:1) / in Eq. 5, with g and d the thickness and the diameter of the CNT, respectively, theexplicit form of k n becomes k n = π Ω n Y (cid:0) d g + dg (cid:1) l − Ω n cos(2Ω n ) + Ω n cosh(2Ω n ) + 4Ω n sin Ω n sinh Ω n − n sinh Ω n + sin(2Ω n ) cosh Ω n (sin Ω n cosh Ω n − cos Ω n sinh Ω n ) + 2 cosh Ω n (cid:0) sin Ω n − cos Ω n sinh Ω n (cid:1) (sin Ω n cosh Ω n − cos Ω n sinh Ω n ) . (6)The effect of the added particle at the free end of the CNT on the standard deviation equation can be now obtainedusing the equipartition theorem: σ n = k B Tk n . (7)The expression in the special case of m ∗ = 0 reduces to Eq.27 of [6]: σ n = 32 kl TπY ( d g + dg ) Ω n . (8) FIG. 7. Influence of m ∗ on the first three eigenfrequencies of a beam with added mass ( m bead ) at the free end. FIG. 8. Influence of m ∗ on the first three eigenmodes: (a) Φ ( x ) ; (b) Φ ( x ) ; (c) Φ ( x ) . Functions normalized such that (cid:82) Φ n ( x ) = 1 . The resultant standard deviation σ of the cantilever can be obtained by summing up all the independent contributionsof eigenmodes. Considering the first 10 flexural modes the expression becomes: σ = N =10 (cid:88) n =1 σ n = 0 . k B l T ( d g + dg ) Y . (9)The numerical coefficient in Eq. 9 is function of the number of modes considered in the summation N and dependson the influence of the added mass. This is illustrated in Figure 9 and reported in Tab. III. The standard deviationof the cantilever is primarily given by that of the fundamental eigenmode independently of the particle mass at thefree end. FIG. 9. Variation of the coefficient of Eq. 9 as a function of m ∗ and while considering a different number of modes N in thesummation.TABLE III. Numerical values for the coefficient of the standard deviation in Eq. 9 when taking into account the fundamentaleigenmode only ( N = 1 , middle column) and the first ten eigenmodes ( N = 10 , right column). m ∗ σ Y (cid:0) d g + dg (cid:1) k B l T N (cid:88) n =1 σ n Y (cid:0) d g + dg (cid:1) k B l T B. Experiment
The model of the previous subsection indicates that the platinum particle does not affect the restoring force northe eigenmode shape of the two fundamental eigenmodes, which are polarized in perpendicular directions, while theshapes of the higher frequency eigenmodes are strongly modified by the platinum particle. For the higher frequencyeigenmodes, the displacement amplitude at the free end is suppressed to zero when the particle has a larger mass thanthe nanotube. For this reason, our detection method based on the reflection at the free end can only measure thetwo fundamental modes. This is what we observe in Fig. 10 for device A. The resonances of the fundamental modedoublet are clearly visible, whereas the resonance frequencies of the second bending mode doublet are expected to beabout
900 kHz but cannot be detected. S VV [ - ∙ V / H z ] Frequency [Hz]
FIG. 10. Power spectrum of the optical reflection from device A undergoing thermal motion at
300 K . The two near-degeneratepeaks are associated with the fundamental modes polarized in perpendicular directions. The spectrum is shown over a narrowerfrequency range in Fig. 1c of the main text. The nonlinearity in the detection results in higher harmonics of these modes, whichare marked in gray.
III. ESTIMATION OF Y We determine the Young’s modulus at T = 300 K for various nanotube cantilevers. We use the geometricalparameters determined by SEM and HRTEM as described in Sec. I (see table II). The spring stiffness k of a nanotubecantilever composed of N concentric shells is the sum of the spring constant k i of each shell, k = N (cid:88) i =1 k i , (10)where we assume that the interaction between the concentric shells has negligible contribution to the spring stiffness.We determine Y of a nanotube cantilever with N shells from its measured spring stiffness, its length, and the diameter d i of each shell, using Y = 0 . kl (cid:80) Ni =1 ( d i g + g d i ) , (11)where we assume that all the shells have the same Young’s modulus and the wall thickness is g = 0 .
34 nm . Thisexpression can be obtained from Sec II.Figure 11 shows the resulting Y for the six measured devices plotted as a function of the cantilever length. Theerror bars represent the standard error ∆ Y for each measurement, which is determined by expanding Eq. 11 andcalculating the propagation of the measurement uncertainties in l , k and d (see table II). The solid line is the meanYoung’s modulus ¯ Y = (cid:80) Y /N = 1 .
06 TPa whereas the dashed lines indicate the confidence intervals ∆ ¯ Y . The latteris estimated by summing the standard error of the Y value of the different cantilevers and the mean of their standarderror ∆ Y divided by √ N , which yields ∆ ¯ Y = ± .
28 TPa .1 μ m] Y [ T P a ] T1 (1 wall)T2 (2 walls)T3 (2 walls)T4 (2 walls)C (3 walls)T6 (7 walls)
FIG. 11. Y determined from SEM and HRTEM for six different nanotube cantilevers. The error bars represent the standarderror for each measurement. The solid black line marks the mean value ¯ Y = 1 .
06 TPa of all the measurements and the dashedblack lines indicate the corresponding confidence intervals ∆ ¯ Y = ± .
28 TPa . IV. DISCUSSION ON THE ORIGIN OF THE TEMPERATURE DEPENDENCE OF THE RESONANCEFREQUENCY
In the main text, we discuss the measured temperature dependence of the resonance frequency in terms ofthe variation of Y . Here, we consider other possible mechanisms but we show that they cannot account for ourmeasurements.The measured T dependence of ω could originate from the variation of the mass m adsorbed on the nanotube.However, mass adsorption, which occurs when lowering T , would lead to a reduction of ω [2, 7–9], which is just theopposite of what is measured. Moreover, we do not observe any hysteresis in ω when cooling the device from
300 K to cryogenic temperatures and then heating it back to
300 K , which shows that temperature-induced mass adsorptionand desorption plays a negligible role [9]. Thus, the measured variation of ω ( T ) is not accounted for by adsorbedmass changes.The measurements could be related to the change in the length of the nanotube when the thermal environmentis varied, since the spring constant depends on the nanotube length as k ∝ l − . However, the measured resonancefrequency reduction at room temperature ∆ ω ( T = 300 K ) /ω (cid:39) − . × − for device A is much larger than thepredicted reduction ∆ ω ( T = 300 K ) /ω = − . × − based on the longitudinal expansion of the nanotube indifferent thermal conditions obtained from our molecular dynamics simulations for a (8,8) CNT. The predicted relative (8,8) FIG. 12. Predicted relative change in stiffness with respect to temperature induced by the nanotube elongation. MD simulationsfor a (8,8) CNT fully clamped at one end and free on the other. k ( T ) /k K associated with the thermal expansion of the nanotube as a function of temperature isshown in Fig. 12. The results are obtained by calculating the elongation of the nanutube for different thermalisationtemperatures. Here we assume that the stiffness ratio k ( T ) /k K is proportional to the cube of the function ( l K /l ( T )) in which l K and l ( T ) are the length of the CNT at K and at temperature T , respectively. The decreasing behaviourreported in Fig. 12 suggests that the nanotube stretches with the increase in temperature. Overall, this shows thatthe thermal expansion is not the cause of the measured ω ( T ) reduction.Another possible origin could be the nanotube resonance frequency change that arises from the combination of theDuffing nonlinearity and the thermal motion. Figure 13a shows the variation of the resonance frequency as a functionof driven vibrational amplitude, which allows us to quantify the Duffing constant using ∆ ω = 38 γ eff ω (cid:10) x vibra (cid:11) , (12)The driven amplitude is calibrated following the procedure described in Ref. [1]. The combination of the Duffingnonlinearity and the thermal vibrations leads to a linear temperature dependence of the resonance frequency, ∆ ω = 38 γ eff ω (cid:10) x th (cid:11) , (13)Figure 13b shows that the slope of the expected dependence is positive, in contrast to what we measure. Moreover,the frequency shift ∆ ω ( T = 300 K ) /ω = 3 . × − is much smaller in magnitude than the measured value ∆ ω ( T = 300 K ) /ω (cid:39) − . × − . This shows that the Duffing nonlinearity together with the thermal vibrationscannot describe our experimental findings. !"! ! " % &’ ( ) * +&,-! . *:4; -<.-
We report molecular dynamics simulations of the Brownian motion of carbon nanotubes over a finite temperaturerange. Simulations are carried out in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)software [10] for single layer CNTs of different chirality. In Figure 14 we showcase the geometry of one such CNT.To account for atom-atom interactions, we use the Tersoff potential [11] with optimized parameters for latticedynamics and phonon thermal transport [12]. We note that this potential is commonly used for simulating atomicinteractions and predicting mechanical properties of carbon-based nanomaterials [13, 14].To track the Brownian motion of the nanotube, the system is initially relaxed to ensure equilibrium at the minimumpotential state. The minimization of the total potential energy is performed via the Polak-Ribiere conjugate gradientalgorithm [15]. The starting point for the minimization procedure is the initial configuration of the atoms, and thepotential energy of the system is considered to be in a local minimum when its energy is less than × − eV orwhen the forces are less than × − eV/Å. After the relaxation, at one end, the translational degrees of freedom areconstrained for all atoms for a length of 5 nm (see Figure 14). This constraint is applied to obtain a CNT cantilever.3
20 nm0.5 nm 1.084 nm
FIG. 14. (8,8) CNT with a total of 2624 atoms. Atoms at one end are clamped for a length of 0.5 nm. The (8,8) CNT has aradius of 0.542 nm and a total length of 20 nm.
FIG. 15. Temperature fluctuation during the thermalization phase for a (8,8) CNT. For the thermostat temperature of 300 Kwe obtain a mean of 300.031 K and standard deviation of 5.69 K.
Once the equilibrium position is obtained, Newton’s equations are integrated using the velocity-Verlet algorithm, witha time-step dt = 0 . fs to determine the variation of the position and velocity of the atoms.To account for the thermal effects, the system is then equilibrated in a constant volume and temperature ensemble(NVT). The temperature is first brought to a certain value and then kept constant by applying the Nose-Hooveralgorithm that thermostats the translational velocity of atoms [16]. The algorithm for the thermalisation is appliedfor 10 ns to ensure that a stable temperature is obtained (see Fig.15).Once thermal equilibrium is reached, the vibration response is studied in an energy conserving ensemble (NVE). Inthis context, the thermal fluctuations of the CNT are monitored for 50ns discarding an initial transient response of10 ns, and the coordinates of all atoms are saved every 2.5 ps.To obtain the resonance frequencies of the CNT, we compute the FFT of the extracted time signals from moleculardynamics. An example of one such FFT averaged over all atoms is shown in Figure 16(a) for a (8,8) CNT at 50 K.The thermal influence on the mechanics of the CNT is obtained by tracking the natural frequencies as a function ofthe thermostat temperature. The relative change of the square of the frequencies for the first three flexural modes forthe (8,8) CNT cantilever is shown in Figure 16(b); this quantity is equal to the relative change of the spring constant ∆ k ( T ) /k . The first three flexural modes highlight the same reduction with respect to the variation of the thermalbath. The staircase behaviour of the first flexural mode is due to the insufficient resolution in frequency (i.e. 20 MHz).Our results are compared to experimental measurements in Fig. 3 of the main text.We remark that the spectral analysis performed to extract the thermal behaviour of the system, does not allowfor an immediate classification of the natural modes of the system and their associated eigenfrequencies. However,it is possible to unravel spatial information of the nanotube from the time response data via the proper orthogonaldecomposition (POD) method. The details of this technique can be found in [17] and are briefly described in Sec. V A.Using POD, we can identify the eigenmodes corresponding to the resonance peaks of Figure 16(a). The mode shapesfor the first three flexural modes obtained via POD for a (8,8) CNT at 50 K are reported in Figure 17. The procedureoutlined above has been repeated in the temperature range T ∈ [5 , K, for three different chiralities namely (5,10),(8,8), and (10,10), and the results are shown in Figure 3(b).4 -2 -2 -3 -2 -0.03-0.025-0.02-0.015-0.01-0.0050 First flexural modeSecond flexural modeThird flexural mode
FIG. 16. a) The averaged frequency spectrum of all atoms for the (8,8) CNT at T = 50 K. . b) Relative change of the squareof frequency with temperature for the first three flexural modes of the CNT from 5K to 330 K with a temperature incrementof 5 K. The staircase behaviour of the first flexural mode is due to the insufficient resolution in frequency
FIG. 17. Mode shapes obtained via proper orthogonal decomposition at T = 50 K. (a) First flexural mode. (b) Second flexuralmode. (c) Third flexural mode. Modes are amplified ten times for visualization. Colormap for the norm of x and y displacement.
A. Proper orthogonal decomposition
The MD simulations provide the time response in a vector u comprising the position of M − atoms. The time historyconsists of N snapshots of the motion as [ u ( t ) , u ( t ) , . . . , u ( t N )] . We remove the time average (mean values) of theresponses by obtaining the time-varying part, x ( t i ) = u ( t i ) − mean ( u ) . To extract the proper orthogonal modes ofvibrations, a discrete matrix X is first built such that each row corresponds to a time response of one atom and eachcolumn corresponds to a snapshot of the CNT at a specific time as: X = (cid:2) x ( t ) x ( t ) · · · x ( t N ) (cid:3) = x ( t ) · · · x ( t N ) ... . . . ... x M ( t ) · · · x M ( t N ) , (14)where x i ( t j ) is the response of the i − th atom at time t j . Once matrix X is constructed, the orthogonal modes areobtained by using the singular-value decomposition (SVD) of the discrete matrix. The SVD operator decomposes X as: X = U Σ V ∗ , (15)where U is an M × M real or complex unitary matrix, Σ is a M × N rectangular diagonal matrix with non-negativereal diagonals σ i that are the singular values of X , and V is an N × N real or complex unitary matrix, with V ∗ being its conjugate transpose. The columns of U and V are the so-called left-singular and right-singular vectors of X ,respectively. Among these matrices, U corresponds to proper orthogonal modes of vibration that can linearly obtainall the snapshots of the motion with minimum error. Using this matrix we can identify the modes corresponding tothe peaks seen in Figure 16 and report them in Figure 17.5 VI. QUASI-HARMONIC APPROXIMATION
The elastic constants of a solid are defined as appropriate derivatives of the free energy with respect to strain tensorcomponents [18]. In particular, the Young’s modulus of a nanotube along the axial direction can be computed from: Y ( T ) = 1 V ( T ) (cid:18) ∂ F ( T, (cid:15) ) ∂(cid:15) (cid:19) (cid:15) =0 , (16)where F ( T, (cid:15) ) is the free energy at temperature T and strain (cid:15) , and V ( T ) is the equilibrium volume at thattemperature. It is frequent to assume that the temperature dependence of elastic constants is small and close to theirzero-temperature value, which amounts to substituting the internal energy in place of the free energy in Eq. (16).However, in this work we are particularly interested in the temperature-dependence of the Young’s modulus. To thisend we resort to a quasi-harmonic approximation of the free energy: F ( T, (cid:15) ) ≈ F ( (cid:15) ) + F vib ( T, (cid:15) ) = F ( (cid:15) ) + k B T (cid:88) n k ln (cid:20) (cid:18) (cid:126) ω n k ( (cid:15) )2 k B T (cid:19)(cid:21) , (17)where F ( (cid:15) ) is the free energy at zero temperature (i.e. the potential energy) at (cid:15) strain, ω n k ( (cid:15) ) is the frequency ofvibrational mode n at reciprocal lattice vector k calculated at strain (cid:15) . The nanotube phonon frequencies have beencalculated using a tight-binding model [19] employing the PHON package [20] (Fig. 18). From the free energy wecan obtain the temperature-dependent Young’s modulus via Eq. (16). Our results are compared with experimentalmeasurements in Fig. 3 of the main text. FIG. 18. (a) panel shows the phonon band structure and vibrational density of states for the (10,10) nanotube (see text). (b)panel shows the same information for the (26,0) nanotube.
VII. ESTIMATION OF PHONON DECAY RATES
We estimate the decay rates for different phonon modes using the expressions derived by de Martino et al.[21]. Forthe longitudinal phonon modes the decay rate due to phonon-phonon interactions is given by τ − L = (cid:126) πρr (cid:40) (cid:112) k ph r / coth (cid:18) (cid:126) v L k ph k B T (cid:19) + √ exp (cid:18) − (cid:126) v L √ rk B T (cid:19) sinh (cid:18) (cid:126) v L k ph (cid:19)(cid:41) (18)where (cid:126) is the reduced Planck constant, ρ = 3 . × − kg / m , r is the nanotube radius, k ph is the phonon wavenumber, and v L = 1 . × m / s is the longnitudinal speed of sound. For the twist phonon modes the decay rate is τ − T = (cid:126) ρ (cid:18) v T v L (cid:19) / / ( k ph r ) / πr (cid:40) coth (cid:18) (cid:126) v T k ph k B T (cid:19) + 2 / (cid:18) v T v L k ph r (cid:19) / exp (cid:18) − (cid:126) v T √ v L rk B T (cid:19) sinh (cid:18) (cid:126) v T k ph k B T (cid:19)(cid:41) (19)6where v T = 1 . × m / s . We calculate the decay rates for different phonon energies E ph . The wave number is k = E ph / (cid:126) v L for longitudinal and k = E ph / (cid:126) v T for twist phonons and r = 1 nm . Figs. 19 (a)-(b) show the respectivetemperature dependencies of τ L and τ T .
50 100 150 200 250 300050100150 T [K] τ L [ n s ] E ph = 4 meV E ph = 8 meV E ph = 15 meV E ph = 25 meV 50 100 150 200 250 3000100200300400500600700 T [K] τ T [ n s ] E ph = 4 meV E ph = 8 meV E ph = 15 meV E ph = 25 meV FIG. 19. Temperature dependence of the phonon decay times of the longitudinal (left) and twist phonon modes (right) fordifferent phonon energies E ph . VIII. AKHIEZER DISSIPATION
The general expression of the Akhiezer dissipation rate Γ Akh of a mechanical eigenmode can be found in Eq. 70of Ref. [22]. We do not reproduce it here, since the description of the different terms would be rather long. Thisexpression is derived for nonlinear dissipation, but the linear Akhiezer dissipation rate can be obtained from thisEq. 70 when using the proper V α for the coupling between the mechanical resonator and the high-energy phononmodes.The temperature enters in Γ Akh through ( i ) the number of phonon modes with energy (cid:126) ω k (cid:46) k B T , ( ii ) the thermalnumber ¯ n k of quanta for each phonon mode, and ( iii ) the decay rate /τ k of each phonon mode. When increasing thetemperature, the contributions ( i ) and ( ii ) enhance Γ Akh , while the contribution ( iii ) lowers it. Moreover, when thedecay rate of a phonon mode is much shorter than the period of the mechanical eigenmode, ω τ k (cid:29) , this mode doesnot contribute to Γ Akh . Since the density of states of phonons varies up and down as a function of energy (Fig. 18),it might well happen that Γ Akh depends in a non-monotonic way on temperature, resulting in peaks in dissipation.
IX. DISSIPATION DUE TO DEFECTS
We show in this section that the measured temperature dependence of the dissipation cannot be described by themodel that is used in the literature [23, 24] to quantify dissipation due to defects. Peaks in dissipation when sweepingtemperature is often attributed to microscopic defects. These defects are modelled by double-well potentials withbarrier height V and asymmetry ∆ between the two wells. At the high temperature of our experiments, the passagefrom one well to the other well is thermally activated with a characteristic time τ d = τ d exp ( V /k B T ) , (20)with /τ d the attempt rate to overcome the barrier. Assuming that all defects have similar V , ∆ , and τ d , a peakin dissipation occurs when the characteristic rate /τ d of the defects matches the mechanical resonance frequency, /τ d = ω . Our measurements in Fig. 4b of the main text show dissipation peaks at different temperatures. Using thevalues of these temperatures together with /τ d = ω , we construct a plot of τ d as a function of T (Fig. 20a). Despitethe relatively large spread in the values of τ d in Fig. 20a, the data cannot be described by an exponential behaviour,suggesting that exp V /k B T ∼ in the measured temperature range in order to force a reasonable description of7the data by Eq. 20. Such an analysis would lead to an unrealistically long τ d ∼ × − s, considering that τ d istypically in the − s – − s range [23–25]. If we were considering two or three different types of defects, each ofthem with well defined characteristics V , ∆ , and τ d , we would also obtain τ d in the µ s range. Therefore, the modelbased on defects with narrow characteristics distribution cannot account for our measurements. Another possibilitywith the double-well potential model is to assume a broad distribution of the defect characteristics V and ∆ [23–25].A peak in dissipation can be obtained in a specific parameter space region. The peak always features a negativecurvature between T = 0 K and the peak temperature (Fig. 20b), which is just the opposite of what is observed in ourexperiments (Fig. 4 of the main text). Overall, our measurements cannot be explained by the double-well potentialmodel with neither a narrow nor a broad defect characteristics distribution. −7 −6 −5 τ d [ s ] T [K] 0 50 100 150 200 250 30000.20.40.60.81 T [K] Γ [ a . u .] FIG. 20. (a) Characteristic time τ d to overcome the barrier height as a function of temperature, obtained from all the measureddevices as explained in the text. (b) Mechanical dissipation due to a distribution of defects as a function of temperature,calculated using Eq. 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