Scaling Law in Carbon Nanotube Electromechanical Devices

We report a method for probing electromechanical properties of multiwalled carbon nanotubes (CNTs). This method is based on atomic force microscopy measurements on a doubly clamped suspended CNT electrostatically deflected by a gate electrode. We measure the maximum deflection as a function of the applied gate voltage. Data from different CNTs scale into an universal curve within the experimental accuracy, in agreement with a continuum model prediction. This method and the general validity of the scaling law constitute a very useful tool for designing actuators and in general conducting nanowire-based nanoelectromechanical systems.

Figure 1

(a) Schematic picture of a nanoelectromechanical doubly clamped suspended CNT three-terminal device. The distance between the CNT and the back-gate electrode is labeled as $H$. $L$ denotes the suspension length and $D$ the external diameter of the CNT. The deviation from the straight line is denoted by $y(x)$. (b) SEM image of a typical device used in AFM experiments (scale bar: 500 nm). The CNT is connected with two metallic electrodes, used both as conducting electrodes and anchor pads. The heavily $n$ doped Si substrate (${10}^{19}{\mathrm{cm}}^{-3}$) acts as a back gate.

Figure 2

Feedback voltage applied to the piezoelectric element $V_{\mathrm{piezo}}$ as a function of the gate voltage $V_G$ at two different positions on device #1 ($L=600\mathrm{nm}$, $D=10\mathrm{nm}$): 1 at the middle of the CNT; 2 on the sacrificial oxide of the silicon substrate (see inset). In the first case, increasing $|V_G|$ produces the deformation of the CNT, the cantilever tip has more room to oscillate, and the oscillation amplitude increases. To maintain constant the oscillation amplitude of the tip (tapping mode), a positive $V_{\mathrm{piezo}}$ is applied by the feedback loop. Conversely, in the second case, increasing $|V_G|$ increases the attractive electrostatic force between the tip and the substrate, the oscillation amplitude decreases and a negative $V_{\mathrm{piezo}}$ is applied by the feedback loop. The actual CNT deflection is obtained by subtracting from the curve measured at position 1 the one measured at position 2 previously divided by the relative permittivity of the silicon dioxide (dotted line).

Figure 3

(a) Maximum deflection $u_{\mathrm{MAX}}$ as a function of $V_G$ for devices #1 ($L=600\mathrm{nm}$, $D=10\mathrm{nm}$), #2 ($L=770\mathrm{nm}$, $D=13.5\mathrm{nm}$), and #3 ($L=480\mathrm{nm}$, $D=10\mathrm{nm}$). Dashed line: guide to the eye that has $V_G^2$ dependence. (b) $u_{\mathrm{MAX}}$ as a function of $V_G$ rescaled using Eq. (4) (see text) for four different devices: #1, #2, #3, and #4 ($L=730\mathrm{nm}$, $D=15\mathrm{nm}$). Inset: calculated Young’s modulus obtained from the scaling constant $K$ [Eq. (4)] for the four devices. Error bars are estimated from the uncertainty on $L$ and $D$.

Figure 4

(a) Calculated maximum deflection $u_{\mathrm{MAX}}$ as a function of $V_G$ for different $L$, $D$ parameters: $a$ ($L=1200\mathrm{nm}$, $D=13\mathrm{nm}$), $b$ ($L=1000$ $\mathrm{nm}$, $D=13\mathrm{nm}$), $c$ ($L=800\mathrm{nm}$, $D=10\mathrm{nm}$), $d$ ($L=800\mathrm{nm}$, $D=15\mathrm{nm}$), $e$ ($L=800\mathrm{nm}$, $D=20\mathrm{nm}$), $f$ ($L=500\mathrm{nm}$, $D=13\mathrm{nm}$), $g$ ($L=800\mathrm{nm}$, $D=8\mathrm{nm}$). (b) Same results rescaled according to Eq. (4) (see text). Dashed line: analytical result obtained for induced stress $T=0$.