A suspended, doubly clamped single wall carbon nanotube is characterized as
driven nano-electromechanical resonator at cryogenic temperatures.
Electronically, the carbon nanotube displays small bandgap behaviour with
Coulomb blockade oscillations in electron conduction and transparent contacts
in hole conduction. We observe the driven mechanical resonance in dc-transport,
including multiple higher harmonic responses. The data shows a distinct
negative frequency tuning at finite applied gate voltage, enabling us to
electrostatically decrease the resonance frequency to 75% of its maximum value.
This is consistently explained via electrostatic softening of the mechanical
mode.Comment: 4 pages, 4 figures; submitted for the IWEPNM 2013 conference
  proceeding

Huettel, A. K.

Kugler, S.

Schmid, D. R.

Stiller, P. L.

Strunk, C.

physica status solidi (b)

English

arXiv

A suspended, doubly clamped single wall carbon nanotube is characterized as driven nano‐electromechanical resonator at cryogenic temperatures. Electronically, the carbon nanotube displays small bandgap behavior with Coulomb blockade oscillations in electron conduction and transparent contacts in hole conduction. We observe the driven mechanical resonance in d.c.‐transport, including multiple higher harmonic responses. The data show a distinct negative frequency tuning at finite applied gate voltage, enabling us to electrostatically decrease the resonance frequency to 75% of its maximum value. This is consistently explained via electrostatic softening of the mechanical mode

Stiller, Peter L.

Kugler, Sabine

Schmid, Daniel R.

Strunk, Christoph

Hüttel, Andreas K.

University of Regensburg Publication Server

Negative frequency tuning of a carbon nanotube nano-electromechanical resonator under tension

A suspended, doubly clamped single wall carbon nanotube is characterized as driven nano-electromechanical resonator at cryogenic temperatures. Electronically, the carbon nanotube displays small bandgap behavior with Coulomb blockade oscillations in electron conduction and transparent contacts in hole conduction. We observe the driven mechanical resonance in d.c.-transport, including multiple higher harmonic responses. The data show a distinct negative frequency tuning at finite applied gate voltage, enabling us to electrostatically decrease the resonance frequency to 75% of its maximum value. This is consistently explained via electrostatic softening of the mechanical mode. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei

physica status solidi, 13 June 2013Negative frequency tuningof a carbon nanotube nano-electromechanical resonator undertensionP. L. Stiller*,1, S. Kugler1, D. R. Schmid1, C. Strunk1, A. K. Hu¨ttel11 Institute for Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, GermanyReceived XXXX, revised XXXX, accepted XXXXPublished online XXXXKey words: carbon nanotube, resonator, NEMS∗ Corresponding author: e-mail peter.stiller@ur.de, Phone: +49 941 943 1615A suspended, doubly clamped single wall carbon nan-otube is characterized as driven nano-electromechanicalresonator at cryogenic temperatures. Electronically, thecarbon nanotube displays small bandgap behaviour withCoulomb blockade oscillations in electron conductionand transparent contacts in hole conduction. We observethe driven mechanical resonance in dc-transport, includ-ing multiple higher harmonic responses.The data shows a distinct negative frequency tuning at fi-nite applied gate voltage, enabling us to electrostaticallydecrease the resonance frequency to 75% of its maxi-mum value. This is consistently explained via electro-static softening of the mechanical mode.Copyright line will be provided by the publisher1 Introduction As has been shown in many bothroom temperature [1,2] and low temperature measure-ments [3–5], suspended carbon nanotubes display excel-lent properties as mechanical resonator systems. Low-temperature measurements have displayed mechanicalquality factors up to Q ∼ 105 [3,6] at frequencies inthe megahertz to gigahertz range. Multiple higher harmon-ics have been observed with frequencies up to 39GHz[7]. One additional feature of particular interest of car-bon nanotube nano-electromechanical resonators is thehigh tunability of the resonance frequency: the particularcombination of high Young’s modulus [8,9], low diameterand low mass make it possible to tune a carbon nanotube“beam resonator” all the way from the case of a hangingchain to a rope with high tensile load [10,2] by applyingelectrostatic forces via gate voltages alone. As expectedfrom the bulk beam model, the resonance frequency typi-cally has a minimum around zero applied gate voltage, andincreases at finite voltage. We report here on a particularresonator where we have observed a strong manifestationof the opposite behaviour: the resonance frequency canbe tuned down to ∼ 75% of its maximum value. Similarbehaviour has been observed in literature for various ma-terial systems [11,12,6], in particular also for the case ofnanotubes with low tension or slack [13].2 Sample fabrication Our devices are fabricatedusing standard lithography techniques. Base material is ahighly positive doped silicon wafer with a thermally grown550 nm silicon dioxide layer on top. Figure 1(a) displaysan SEM image of the electrode geometry used in our mea-surements. We achieve suspended and contamination-freecarbon nanotubes by chemical vapor deposition (CVD)growth over predefined contacts separated by etchedtrenches [14,15,5]. As contact material, we use a 40 nmthick, co-sputtered layer of a rhenium / molybdenum al-loy [5,16]. This material also serves as etch mask for thetrench definition. Various geometries are used in differentchip structures; in the case discussed here, the trench be-tween the nanotube contacts was 500 nm wide and 220 nmCopyright line will be provided by the publisher2 P. L. Stiller et al.: Negative frequency tuning of a CNT resonator				Figure 1 (a) SEM image (before catalyst deposition and nanotube CVD growth) of the electrode geometry also used forthe sample described here. (b) Schematic of the low temperature electronic transport measurement setup; as cryostat, anOxford Instruments helium-3 evaporation system was used. In the measured sample the gate finger electrode (see also (a))could not be used because of a lithographic defect.deep. Within the trench, an additional finger gate was de-fined, see Figure 1(a); in the measurement it could not beused because of a lithographic defect. For characterizationand selection, devices are only tested electrically at roomtemperature. The low temperature transport experimentswith suitable samples are conducted at Tbase = 300mK ina helium-3 evaporation cooling system. Here, in additionto a typical Coulomb blockade measurement setup [17],a radio frequency signal for mechanical excitation can beapplied contact-free by an antenna nearby in the cryostat[3]. Figure 1(b) depicts a schematic of this measurementsetup: we apply a bias voltage and measure the resultingcurrent; a gate voltage is used for varying the electrochem-ical potential and thereby also the charge of the carbonnanotube.3 Basic electronic characterization Figure 2(a)displays a low-temperature (T ≃ 300mK) measurementof the current I(Vg) through our carbon nanotube sam-ple as a function of applied gate voltage Vg, for constantVsd = 2mV. The sample displays the typical behaviour ofa small bandgap carbon nanotube. On the hole conductionside (Vg . 2.2V) we only observe few oscillations ofthe current and a subsequent fast transition into an opentransport regime [19]. Note, however, that the overall re-sistance remains high (R ≃ 180 kΩ at Vg = 0), indicatingeither a high series resistance in the leads or a multi-dotstructure. On the electron conduction side (Vg & 2.6V)we observe sharp Coulomb oscillations, however no clearshell structure can be discerned. By additionally varyingthe bias voltage Vsd we obtain the typical “Coulomb dia-mond” stability diagram, see Fig. 2(b). Multiple, stronglygate-dependent inelastic cotunnelling lines without clearpattern can be seen [20,21], hinting at a potential struc-ture more complex than a single minimum, e.g., involvingtrap states near the leads or potential irregularities alongthe nanotube. Also in nonlinear transport no regular shellstructure can be observed. Of note in the measurement ofFig. 2(b), however, are the rounded regions marked witharrows – here, electromechanical feedback leads to self-driving of the mechanical motion in absence of an externalrf driving signal [4,5,18].4 Driven mechanical resonator measurementsA mechanical resonance detection measurement is shownin Figure 3(a). The bias voltage Vsd = 2mV and the gatevoltage Vg = 3.234V are kept constant, the frequency ofan applied rf-signal (compare Fig. 1(b)) is varied acrossa large range at constant nominal generator power P =2 dBm. Since in this setup the cabling used for the radio-frequency signal is not particularly optimized, the actualpower transmitted to the sample varies over the observedfrequency range, leading to the large-scale oscillatory be-haviour in Fig. 3(a). Mechanical resonances of the carbonnanotube show up as sharp spikes in the recorded cur-rent (for details of the detection mechanism, see e.g. [3]).We observe several resonance frequencies ranging from72.8 MHz to 552.1 MHz. Fig. 3(b) displays an exemplarydetail zoom measurement of the resonance in Fig. 3(a)around f = 358.5MHz. The width of the measured peakcorresponds to a quality factor of Q = f/∆f = 11800;maximum quality factors observed on this device wereQmax ≃ 72000 at T ≃ 300mK.Figure 3(c) shows selected extracted peak positionsplotted as a function of an assigned mode number of themechanical carbon nanotube resonator. The peaks form asequence of approximately integer multiple frequencies;the dependence of the resonance frequency on the modenumber can be fitted linearly, leading to a good agreementfor ffit,n = n (71.4±0.4)MHz. This indicates the presenceof a string under tension, since for the case of a hangingchain, where the bending rigidity dominates mechanicalbehaviour, higher vibration modes do not occur at integermultiples of the fundamental frequency [10,2,13]. The ob-servation is consistent with recent measurements on shortnanotube segments [7].Copyright line will be provided by the publisherpss header will be provided by the publisher 3 		           Figure 2 (a) dc current as function of applied gate voltage Vg for constant Vsd = 2mV, at T ≃ 300mK. The device displaystransparent behaviour in the hole regime (Vg . 2.2V), a small bandgap, and irregular Coulomb oscillations in electronconduction (Vg & 2.6V). (b) Differential conductance dI/dVsd (numerically obtained from a dc-current measurement) asfunction of gate voltage Vg and bias voltage Vsd, in the few-electron regime (logarithmical color scale). The yellow arrowsmark regions of nano-electromechanical feedback [4,5,18].     					    	      	 !"	" Figure 3 (a) Time-averaged dc current through the carbon nanotube as function of the frequency of the rf driving signal,for constant Vsd = 2mV, Vg = 2.234V, and nominal rf generator power P = 2 dBm. Black arrows mark the resonancefeatures used in the plot of (c); additionally observed resonances are marked with their frequency. (b) Detail from (a)(higher resolution measurement): exemplary resonance trace around f = 358.5MHz. The peak width corresponds to aquality factor Q = 11800. (c) Selected resonance frequencies from measurements as in (a) as function of assigned modenumber. A sequence of harmonics can be observed; the linear fit results in ffit,n = n (71.4± 0.4)MHz.In addition, in the plot of Fig. 3(a) further resonancesappear. These are each marked in the plot with the reso-nance frequency. While a mechanical origin of these fea-tures is likely because of the sharp frequency response, adetailed identification cannot be made from our data.5 Gate voltage dependence of mechanical reso-nance Varying both the driving frequency f and the gatevoltage Vg enables to trace the resonant features in a 2Dplot. This is done in Fig. 4 for a wide gate voltage rangefrom -12 V to 10 V. We have chosen the first harmonic fre-quency for this evaluation since it provides the best signalto noise ratio. As already stated, the expected behaviourwould be an increase of the mechanical resonance fre-quency for increasing gate voltages: electrostatic force onthe nanotube leads to a deflection towards the back gateand thereby to increased mechanical tension [1,2]. Fig. 4,however, displays a strong negative curvature of the reso-Copyright line will be provided by the publisher4 P. L. Stiller et al.: Negative frequency tuning of a CNT resonator 	    Figure 4 Background: numerical derivative dI/df of the measured time-averaged dc current as function of external rfdriving frequency f and gate voltage Vg, for nominal driving power P = 0 dBm and bias voltage Vsd = 2mV. The colorscale has been adapted per partial plot to obtain optimal contrast. The manually extracted resonance peak positions for thefirst harmonic frequency f2 as function of gate voltage Vg, which form the raw data for curve fitting, are overlaid as yellowcrosses. The black dotted line corresponds to a parabolic fit, see equation 2.nance frequency; at Vg = −12V far in the hole conduc-tion regime the resonance frequency is reduced to approx-imately 75% of its maximum value. For electron conduc-tion, the same effect emerges symmetrically.A decrease of resonance frequency has been observedpreviously in literature in measurements on a suspendedmetallized SiC beam and on a doubly clamped InAsnanowire resonator [11,12] and also in carbon nanotubemechanical resonators [6,13]. In particular, in a dual-gatesetup, negative tuning was observed for a weakly tensionednanotube in [13]. It is explained via so-called electrostaticsoftening of the vibration [6,11–13]. For this effect, a vi-bration of the carbon nanotube towards and away from thegate is required: within one oscillation period the deriva-tive C ′g(h) = dCg/dh of the capacitance Cg betweengate and resonator as function of the distance h betweenthem is modulated, and thereby also the electrostatic forceFel ∝ C′g(h) between gate and resonator. Linearizing atsmall deflection ∆h from the equilibrium position, theelectrostatic force modulation can be written using thesecond derivative C ′′g(h) = d2Cg/dh2 as∆Fel ∝ ∆h C′′g(h) (1)and be described via an effective spring constant contribu-tion.Assuming that the built-in tension in the carbon nan-otube device is dominant at low gate voltage Vg, we ap-proximate that the mechanical tension and thereby thepurely mechanical spring constant does not change in theobserved range of Vg. Following [6], we can then approxi-matef(Vg) = fmax − β(Vg − Vg,0)2, (2)where we define with m the mass and L the length of thenanotubefmax =12√T0mL. (3)The curvature of the parabola f(Vg) is connected to C′′g,viaβ = fmaxC ′′gL4pi2T0. (4)The black dotted line in Fig. 4 corresponds to aparabolic fit using Eq. 2 with the parameters fmax =146.9MHz, Vg,0 = 1.4V, and β = 0.192MHz/V2.Assuming a mass of the carbon nanotube m = 0.17 ×10−21 kg and a nanotube length equal the trench widthL = 500 nm, these values lead to T0 = 7.3 pN andC ′′g= 7.5× 10−7 F/m2.As can be seen in Fig. 4, the parabolic fit does notaccurately represent the functional dependence of f(Vg).Copyright line will be provided by the publisherpss header will be provided by the publisher 5Several possible mechanisms can contribute here. Eq. 2 as-sumes that the charge on the nanotube is proportional tothe applied gate voltage, which in particular does not holdwithin and close to the electronic band gap. In addition themechanical tension varies, leading to additional contribu-tions to f .The case of finite tension described here differsmarkedly from the observation of electrostatic soften-ing in [13]. There, an “end gate” to the side of the de-vice was required to obtain motion in the direction of thegate electrode. Here, the occurrence of strong electrostaticsoftening indicates the vibration of the tensioned stringtowards the back gate. However, because of the complexstructure geometry including a non-functional additionalgate electrode (cf. Figure 1(a)), further evaluation provesto be difficult.As a consistency check, we calculate the distance be-tween carbon nanotube and gate required to obtain theabove value of C ′′g= 7.5 × 10−7 F/m2, using the sim-ple model of a thin beam with radius r above an infiniteconductive plane. Assuming a nanotube radius r = 2 nm,and L and C ′′gfrom above, we obtain a required distance invacuum between nanotube and gate of h = 1µm. In spiteof the many approxmations used, this result is indeed ofthe correct order of magnitude, compared to a trench depthbelow the carbon nanotube of 220 nm and an addition-ally remaining silicon oxide layer of 380 nm. The obtainedvalue for h is an upper limit since we neglect a deflection-induced increase in mechanical tension and thereby springconstant, which will have to be overcome by the electro-static softening as well.6 Conclusion We measure electronic and mechani-cal properties of a high-quality factor carbon nanotube vi-brational resonator at cryogenic temperatures. Both quan-tum dot behaviour and multiple driven mechanical res-onances are observed. A sequence of higher harmonicsoccurs approximately at multiple integers of a base fre-quency, indicating that the nanotube resonator is under ten-sion. The gate voltage dependence of the resonance fre-quency of the first harmonic f2(Vg) displays a distinct neg-ative curvature; the frequency can be tuned from a max-imum f2(Vg = 1.4V) = 145.9MHz down to f2(Vg =−11.7V) = 110.2MHz. We successfully model this byelectrostatic softening. Future experiments may address vi-bration direction and mode shape by providing more com-plex gate and thereby electric field geometries.Acknowledgements The authors would like to thank theDeutsche Forschungsgemeinschaft (Emmy Noether grant Hu1808/1-1, SFB 631 TP A11, GRK 1570) and the Studienstiftungdes deutschen Volkes for financial support.References[1] V. Sazonova, Y. Yaish, H. U¨stu¨nel, D. Roundy, T. A. Arias,and P. L. McEuen, Nature 431(7006), 284 (2004).[2] B. Witkamp, M. Poot, and H. van der Zant, Nano Lett.6(12), 2904 (2006).[3] A. K. Hu¨ttel, G. A. Steele, B. Witkamp, M. Poot, L. P.Kouwenhoven, and H. S. J. van der Zant, Nano Letters 9(7),2547–2552 (2009).[4] G. A. Steele, A. K. Hu¨ttel, B. Witkamp, M. Poot, H. B.Meerwaldt, L. P. Kouwenhoven, and H. S. J. van der Zant,Science 325(5944), 1103–1107 (2009).[5] D. R. Schmid, P. L. Stiller, C. Strunk, and A. K. Hu¨ttel,New Journal of Physics 14(8), 083024 (2012).[6] A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. 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P. Kouwenhoven, C. M. Marcus, P. L. McEuen,S. Tarucha, R. M. Westervelt, and N. S. Wingreen,Electron transport in quantum dots, (Kluwer, 1997).[18] O. Usmani, Y. M. Blanter, and Y. V. Nazarov, Phys. Rev. B75(19), 195312 (2007).[19] W. Liang, M. Bockrath, D. Bozovic, J. H. Hafner, M. Tin-kham, and H. Park, Nature 411, 665 (2001).[20] S. De Franceschi, S. Sasaki, J. M. Elzerman, W. G. van derWiel, S. Tarucha, and L. P. Kouwenhoven, Phys. Rev. Lett.86, 878–881 (2001).[21] K. Goß, S. Smerat, M. Leijnse, M. R. Wegewijs, C. M.Schneider, and C. Meyer, Phys. Rev. B 83, 201403 (2011).Copyright line will be provided by the publisher

P. L. Stiller

S. Kugler

D. R. Schmid

C. Strunk

A. K. Hüttel

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Negative frequency tuning of a carbon nanotube nano-electromechanical
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Negative frequency tuning of a carbon nanotube nano-electromechanical resonator

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