Introduction
Owing to the promise of carbon nanotubes for application in nanoelectromechanical systems (NEMS)12, 13, the effects of mechanical deformations on their electronic properties have attracted great interest14. Linear electromechanical responses have been observed for axial15, 16, radial17 and flexural18 strain. Torsional electromechanical effects have been predicted by several groups8, 19, 20, but not yet observed. Conductance oscillations have been observed with magnetic fields6 and gating21, 22, but not with strain. To measure the conductance of carbon nanotubes under varying torsional strain, we built torsional nanotube-based NEMS as shown in Fig. 1, where a suspended multiwalled carbon nanotube is mechanically and electrically connected to a pair of electrodes and a small pedal in the middle. The nanotube is twisted by pressing against the pedal with an atomic force microscope (AFM) tip, and the electrodes allow us to simultaneously measure the two-terminal conductance across the nanotube. The devices were fabricated in an electron-beam lithography lift-off process, followed by wet etching and critical-point drying. Similar devices have been built with multiwalled5, 9, 23 and single-wall10 carbon nanotubes, and used as torsional pendulums9, 10 and nanorotors23, 24, but no torsion-dependent electrical measurements were reported.
Figure 1: Nanotube-based torsional NEMS for the study of torsional electromechanics of carbon nanotubes.
a, Schematic description of the device and measurement set-up. b, SEM image of a device (see Supplementary Information for a movie showing charging-induced reversible torsion). c, SEM image showing several devices (some devices are connected to electrodes and used for electromechanical measurements, and others are isolated and used only for mechanical characterization). d, Schematic description of the torque–torsion measurement and its relevant parameters.
Full size image (38 KB)Before the torsional electromechanical measurements, we characterized independently the torsional mechanical properties and the electronic properties of several nanotube devices. The former were studied by measuring force versus pedal-deflection curves at different points along the long axis of the pedal (Fig. 2a). The results agree with Archimedes' law of the lever (Fig. 2b), indicating that the pedal compliance is mostly due to nanotube torsion, rather than to bending or stretching. The corresponding torsional spring constants (see Supplementary Information for full data table) corresponded to the shear moduli that are expected for hollow cylinders, rather than for solid rods, indicating that torsion involves predominantly the outermost wall, in accordance with previous reports5 (inner walls were reported to be involved after a much larger number of actuations than were performed here). The elasticity of the torsion is evident both from the reversible and linear response to the force applied by the AFM tip, and from the reversible pedal deflection due to charging under a scanning electron microscope (see Supplementary Information, movie). The differential conductance as a function of bias in relaxed devices exhibits typical hyperbolic shapes (Fig. 2c), similar to those observed in pristine arc-grown multiwalled carbon nanotubes7, where conduction is ballistic across the outermost wall, increasing with bias as more channels become accessible. The low-bias differential conductance is smaller than the quantum conductance owing to the large contact resistance. From this preliminary characterization we conclude that both the torque and the current are predominantly carried by the outermost wall. Hence, our multiwalled nanotubes effectively act as large-diameter single-wall nanotubes, where the inner walls provide an inert mechanical support against bending and collapse.
Figure 2: Independent characterization of torsional mechanics and transport properties.
a, AFM image of a device showing the line of points selected for force–distance measurements. b, Archimedes' law-of-the-lever behaviour, where linear sample compliance Kp = kczc/z is given as a function of lever distance from nanotube x, where kc is the cantilever spring constant, zc is the cantilever deflection, and z is the pedal deflection, and fitted to 
/x2, where is the torsional spring constant. If bending is also considered, then Kp = (x2/ + kb–1)–1, where kb is the bending spring constant. However, kb–1 is significantly smaller than the other term, which means that most of the paddle deflection is due to torsion. c, Differential conductance as a function of bias, G(V), for several different devices. Devices A (blue), B (cyan) and C (yellow) are further analysed in Fig. 3.
The torsional electromechanical response of the carbon nanotubes was studied by repeatedly pressing and retracting an AFM tip on one point of the pedal (about halfway between the nanotube and the pedal edge), while monitoring the low-bias differential conductance using a lock-in amplifier. The applied a.c. bias amplitude (10 mV) was smaller than the nanotube bandgap (
30 meV), and the a.c. frequency (1 kHz) was significantly higher than the acquisition rate (loop rate, 0.2 Hz; 512 measurements per loop). The AFM probe was driven at resonance (70 kHz), with a sensing amplitude (60–80 nm) to detect its landing and detachment from the pedal by following the abrupt damping or reappearance of this amplitude, respectively. The torsion angle 
and the torque |T| exerted on the nanotube were monitored as = tan–1[(zp – zc)/x] and |T| = kczcx, where zp, zc, kc and x are the z-piezo extension, cantilever deflection, cantilever spring constant and lever arm length, respectively (Fig. 1d).
Figure 3 shows the results obtained from three different devices (see Supplementary Information for additional graphs and full data tables). The arrows in the torque–torsion graphs (Fig. 3, second row) indicate the angle at which the tip starts to press the pedal and the nanotube starts to be twisted (negative angles correspond to measurements where the tip has not yet touched the paddle or has already retracted, so the actual torsion angle of the nanotube is zero, unless the pedal is pulled back by the tip because of adhesion in the retraction). When twisting the nanotube up to small torsion angles, < 30° (device A), the torque increases linearly with a slope that remains constant in consecutive press–retract loops, indicating that each suspended nanotube segment is elastically twisted, with a torsional spring constant equal to half of this slope. The conductance shows a reversible decrease and increase on both pressing and retracting the tip, suggesting the beginning of an oscillation (see Supplementary Information for additional data). When twisting the nanotube to large torsion angles, = 75° (devices B and C), the torque shows an initial linear increase, followed by a narrow region of softening, and then an abrupt and irreversible drop. This behaviour indicates a transition from an elastic regime to a plastic regime, ending with the eventual torsional failure of the nanotube outermost wall. In the elastic torsion regime, the conductance displays a series of oscillations as a function of torsion angle, with periodicities of 14–20°. The oscillations begin when the nanotube starts to be twisted, and eventually end with an abrupt and irreversible drop, coinciding with torsional failure. AFM images before and after the experiment show the irreversible deflection of the pedal, confirming the torsional failure. The drastic conductance drop upon torsional failure supports the previous assumption that the current is predominantly carried by the outermost wall. It is important to note that in all the devices the conductance varies only at positive angles, when the nanotube is actually twisted, but remains constant at negative angles, when the nanotube is not twisted.
Figure 3: Torsional electromechanical measurements for three representative devices A, B and C.
The upper three rows display the simultaneous measurements of sensing amplitude (Asens), torque (|T|) and low-bias differential conductance (G), respectively, as a function of the torsion angle (). The bottom row displays the relative change in resistance (
R/R0) as a function of torsion angle (). In the first column, repeated experiments are indicated by blue (1st), green (2nd), and red (3rd). In second and 3rd columns, red is the ingoing and blue the outgoing curve. Curve colours follow the red-green-blue sequence by order of acquisition. In the top row, upwards and downwards midline arrowheads indicate pressing and retracting curves. The insets in the top row for device B show AFM images before (right) and after (left) torsional failure. In the second row, tilted straight lines in the torque–torsion curves show the slopes of the elastic regime, and in the second and third rows, black arrows show the angles at which the actual torsion of the nanotube starts (negative values correspond to measurements before the tip touches the pedal). The inset scheme in the second row for device B represents the positions of tip and pedal. In the bottom row, black curves represent theoretical fits to the relative resistance change (R/R0 versus ). Device A is assumed metallic based on the overall decrease in conductance upon torsion, whereas B and C are assumed semiconducting based on the initial decrease and increase in conductance. Dipped peaks in some of the theoretical curves arise from a phase lag of 120° between the periodic metal–semiconductor transitions expected for the two nanotube segments on either side of the pedal as they are simultaneously twisted in opposite directions.
The conductance oscillations as a function of torsion can be theoretically explained, as graphically represented in Fig. 4, by the geometric and the electronic structure of carbon nanotubes2. The latter derives from the electronic structure of the graphene sheet, with a quantization of the two-dimensional wavevector into a series of one-dimensional sub-bands by the periodic boundary condition of the cylindrical geometry, Chk = 2
j, where Ch is the circumferential vector, k is the wavevector and j = 0, 
1, 2, 3, ... is a quantum number associated with each sub-band. The first Brillouin zone (BZ) of graphene is a perfect hexagon in whose corners resides the Fermi level. Depending on the diameter and chirality of the nanotube, which determine Ch, the sub-bands may, or may not, include the BZ corners, in which case the nanotube is metallic or semiconducting, respectively3. Upon mechanical deformation, the BZ becomes distorted8, and its corners can move away from or closer to the nearest allowed sub-bands. This opens or closes a bandgap, leading to changes in the conductance. Following the Yang–Han model8, the graphene BZ corners are shifted by torsional strain from the initial kF0 to kF = kF0 + 
kF. The circumferential component of this shift (perpendicular to the sub-bands) is kFc = 
sin(3
0)/aC–C, where 0 is the initial nanotube chirality, aC–C is the C–C bond length and is the torsional strain. The torsional strain is directly related to the torsion angle by = r/ℓ, where r and ℓ are the nanotube radius and length, respectively. Previous models considered small-diameter single-wall carbon nanotubes, where kFc is smaller than the sub-band spacing 
k = 1/r, and hence the bandgap changes linearly with torsion8. Here we show that this linear regime should break down for large diameters, where the nanotube becomes metallic at each torsion angle for which the corner of the distorted BZ—that is, kF()—crosses any sub-band allowed by the periodic boundary condition ChkF() = 2j. Therefore we can predict that the nanotube will undergo periodic metal–semiconductor transitions, with an angular period = aC–Cℓ/r2 sin 30. The conductance oscillation periods observed in our devices agree with this prediction for chiralities ranging between 0 = 14° and 30°.
Figure 4: Theoretical model for the torsional electromechanical quantum oscillations in carbon nanotubes.
a, First Brillouin zone of graphene (green hexagon), with the sub-bands (black parallel lines) allowed in the nanotube. b, Magnification near a corner (circled in a), showing its simultaneous shifts on the two oppositely twisted nanotube segments, kF(+) (red) and kF(–) (blue), and their sub-band crossing points, where the condition for metallicity ChkF() = 2j(j = 0, 1, 2, 3, ...) is fulfilled. c,d, Simultaneous bandgap shifts due to torsion, 
Eg(+) (red) and Eg(–) (blue), for initially semiconducting (c) and metallic (d) nanotubes of radius 13 nm.
The changes in conductance with torsion can be related to the induced change in the bandgap, which affects the barrier height for thermally activated transport. We modelled the torsional piezoresistance of the device (that is, the relative change in resistance with torsion) by assuming serial contributions16 from the two suspended nanotube segments on either side of the pedal, which are simultaneously twisted through the same angle but in opposing directions (Fig. 4, red and blue). Solving the Landauer–Büttiker formula25 (see Supplementary Information for details), gives
where E() is the torsion-induced change in the activation energy, A is an attenuation factor due to contact resistance, scattering, doping and thermal smearing, Rc is the contact resistance, h and e are Planck's constant and the electron charge, respectively, |t|2 is the transmission probability, and E0 is the initial activation energy. This last energy depends on the initial bandgap and possible effects of doping and Schottky barriers26 (see Supplementary Information). In any case, the torsion-induced change in activation energy is half the bandgap change, E() = Eg()/2 (Fig. 4c,d).
We verify the consistency of our measurements with this model using a semi-quantitative analysis, where the experimental amplitudes of Fig. 3 are fitted to equation (1) (black curves in bottom row of Fig. 3). The fitting parameter A is consistent with reasonable energy barriers, high contact resistances and low transmission probabilities, which may be due to the morphology of the contacts (see Supplementary Information, table). The electromechanical effects can be observed at room temperature because the maximum bandgap change with torsion is larger than, or comparable to, the thermal energy kBT, and hence significantly affects the Fermi–Dirac statistics, even if some doping may exist and increase the attenuation factor (see Supplementary Information). In fact, both p- and n-doped multiwalled carbon nanotubes have been observed to coexist in arc-discharge samples27, and two-channel conductance has been reported when pristine samples were used7, which means that doping can also be very small. This could not be confirmed in our devices because gating is not technically feasible owing to the large distance from the silicon wafer, the low dielectric constant of air, and the electrostatic deflection of the pedal. The statistical ensemble of data available is limited by the difficulty of finding devices that are both elastically resilient and electrically well-connected, in addition to the effect of tip-pedal capillary adhesion forces in air, which results in the overtwisting or breaking apart of many devices during scanning or experiments. However, oscillations were observed in all the devices that were elastically resilient, but never in irreversibly twisted ones. The acquisition time is long (5 s per loop), so the frequencies of the oscillations (3.7–8.2 Hz) are very low compared with any source of periodic noise. In the elastic regime, there is no correlation between the electromechanical oscillations and the torque–torsion response, which is linear. The possible electromechanical effect of tension (see Supplementary Information) was also estimated to be linear, and significantly smaller than that of torsion. Theoretical studies have also predicted that changes in the registry between the walls can affect the probability of scattering when more than one wall participates in conduction28. Although this effect cannot be completely ruled out, the fact that in our case most of the current is carried by the outer wall makes it relatively unlikely.
Another interesting aspect of the results concerns the dynamics of torsional failure of the nanotubes, which was accurately monitored by following the conductance drop beyond a critical torsion angle. Considering the measured torsion angles of failure f and the torsional spring constants , we determined the torsional strength of carbon nanotubes as 
= f M, where f = rf/ℓ is the torsional strain of failure and M = 2ℓ/[r4 –(r – r)4] is the single-wall shear modulus, r being the interwall distance. The obtained values 10 GPa (see Supplementary Information) are one order of magnitude larger than the shear strength of carbon fibres29. The maximum torque that a carbon nanotube can withstand has been predicted to be |T|max = fr2 per wall, where f = 6 N m–1 (ref. 30). Our observed values (Fig. 3; see Supplementary Information, table) agree with this prediction for the failure of only the outermost wall.
In summary, we have studied for the first time the torsional electromechanical response of carbon nanotubes, and found an oscillatory behaviour that is consistent with the metal–semiconductor periodic transitions predicted by theory. Each oscillation involves a different sub-band defined by the wavevector quantization around the nanotube. This is a conceptually interesting system where a quantum number is forced to leap by pressing on a pedal. In this respect, carbon nanotubes could act as torsional transducers in NEMS, where the oscillatory behaviour would resemble the response of an interferometric sensor. From these measurements, we also determined for the first time the torsional strength at failure of carbon nanotubes, which is an important parameter of these unique building-blocks for nanotechnology.
