Automated circuit fabrication and direct characterization of carbon nanotube vibrations

Since their discovery, carbon nanotubes have fascinated many researchers due to their unprecedented properties. However, a major drawback in utilizing carbon nanotubes for practical applications is the difficulty in positioning or growing them at specific locations. Here we present a simple, rapid, non-invasive and scalable technique that enables optical imaging of carbon nanotubes. The carbon nanotube scaffold serves as a seed for nucleation and growth of small size, optically visible nanocrystals. After imaging the molecules can be removed completely, leaving the surface intact, and thus the carbon nanotube electrical and mechanical properties are preserved. The successful and robust optical imaging allowed us to develop a dedicated image processing algorithm through which we are able to demonstrate a fully automated circuit design resulting in field effect transistors and inverters. Moreover, we demonstrate that this imaging method allows not only to locate carbon nanotubes but also, as in the case of suspended ones, to study their dynamic mechanical motion.


Supplementary Note 1 Device fabrication and molecules deposition
Carbon nanotubes (CNTs) were grown using chemical vapor deposition (CVD) at 900 • C with 0.5/0.5 SLM flow of H 2 /CH 2 . The catalyst particles were deposited from ferritin solution onto predefined catalyst pads. Electrical contacts were deposited either before or after CNTs growth. For p-type devices Cr/Au 5/120 nm or Cr/Pt 5/40 nm were deposited 1-4 .
For n-type devices four different processes were used. The first was based on 50 nm of Al deposition capped by 50 nm of Au 5 . The second approach included Ca/Al 30/120 nm deposition 6 , and the third consisted of Sc 50 nm metal deposition 7 . The last approach, which was found to be the best, included Cr/Pt 5/40 nm electrical contacts and atomic layer deposition (ALD) of HfO 2 on top of the CNT 8 . The deposition temperature and pressure were 270 • C and 620 mTorr, and the HfO 2 thickness was ≈ 30 nm.
Powder CNTs were purchased from SWeNT and CoMoCAT T M and were dissolved in chloroform. After tip sonication for 20 min in pulse mode, they were dispersed on silicon wafer.
p-nitrobenzoic acid (pNBA) powder was purchased from Fluka. Two main methods were used for deposition of pNBA molecules on the CNTs. all the tubes that appeared in the SEM images were also seen in the optical images. However, we found that thick CNTs were decorated faster than thin CNTs. This observation hints for more subtle preferences between different tubes, such as tube chirality, or type (semiconducting vs metallic) and the possibility for tube sorting by these pNBA NCs. This issue deserves further research and will be addressed in the near future.
SEM images of suspended CNTs decorated with pNBA NCs are challenging since in high vacuum the pNBA molecules desorb more quickly than at ambient conditions. However, few attempts were successful and Supplementary Fig. 2e Fig. 3a) and on the right an enlarged image of the red marked area ( Supplementary Fig. 3b). Prior to the optical image we took an AFM scan of the NC marked in yellow circle in Supplementary Fig. 3b (Supplementary Fig. 3c), and present the cross section along the blue line of Supplementary Fig. 3c (Supplementary Fig. 3d). This result, which was found for other cases as well, shows that we can optically image a NC with lateral dimension of less than 100 nm and height of less than 25 nm along the CNT.
The lateral resolution between two adjacent CNTs was found to be ≈ 250 nm. Supple-

Supplementary Note 3 Marking with other molecules
Beside pNBA molecules, additional candidates were tested. Those which showed partial or full success are discussed in this section. Several features were found to be essential for the marking procedure. The first, and probably the most important is the existence of at least one benzene ring in the deposited molecules. The second, is its capability to diffuse on the cold substrate at ambient conditions, and the last, is the fact that they crystalize to solid phase at room temperature. Below we present our results in more detail. Supplementary The second molecule is 4-Chlorophenol which melts at relatively low temperature (40 • C) and evaporates very quickly. When we deposited these molecules, again from the liquid phase, onto SiO 2 substrates with CNTs, they formed spherical shapes all over the surface without any preferential adsorption toward the CNTs (Supplementary Figs. 10a, b). Nevertheless, when deposited over suspended CNTs, these molecules formed spherical NCs along the CNTs in a pearl like chain ( Supplementary Figs. 10b, c). After few minutes, these NCs disappeared both from the substrates and the suspended CNTs.
The third molecule, 2,4-dichlorophenoxyacetic acid, again has benzene ring and carboxylic acid group in its structure. These molecules are deposited from the liquid phase and adsorbed preferentially to both on-surface and suspended CNTs, as depicted in Supplementary   Figs. 11a-c. Nevertheless, the decorated NCs are less continuous along the suspended CNTs with comparison to pNBA NCs, and they did not mark all the on-surface CNTs. This result is very interesting, since it may suggest some sorting mechanism for these molecules, but further research is required to unveil it quantitatively.
The last molecule is 3,4,9,10-perylene tetracarboxylic dianhydride (PTCDA) which has several benzene rings. These molecules were deposited directly from the solid phase, as pNBA molecules. However, since they are very big (high molecular mass with respect to pNBA), when they were adsorbed onto the substrate, they could not diffuse much and formed NCs all over the substrate, without any preference ( Supplementary Fig. 12a). However, when deposited over suspended CNTs, PTCDA molecules formed NCs along the CNTs and made them optically visible (Supplementary Figs. 12b, c).

Supplementary Note 4 Marking single and few layers graphene
This marking procedure is applicable also for graphene samples, as may be expected. The method is effective for exfoliated graphene, CVD graphene, nanoribbons, and also for making graphene optically visible on different substrates which do not support constructive interference as exists for example for graphene on 285 nm SiO 2 9 .
Supplementary Fig. 13 clearly shows CVD graphene patterned in the shape of electrode on 285 nm SiO 2 . In the bright field images single layer graphene is visible thanks to the oxide thickness. In the dark field images the graphene patterns are visible only due to the decoration of the NCs. Supplementary Fig. 14

Supplementary Note 5 Sublimation rate
In this section we derive the temporal dependence of the sublimation process of pNBA molecules which desorb from the CNT surface. This analysis follows the study of Sambles et al. 10 . The sublimation rate depends on the partial pressure of the pNBA molecules in the gas phase in the vicinity of the nano-crystal (NC). Assuming spherical geometry of the NC, with radius r, one can employ Kelvin equation (Supplementary Eq. 1) for evaluating the partial pressure, P r , in the gas phase adjacent to the NC surface. In this relation P ∞ is the partial pressure above flat surface, γ is the surface energy of the NC, ρ is its mass density, M r is its molecular weight, T the temperature, and R is the gas constant.
Defining n v as the number of molecules leaving a unit area of the NC per second, and V a as the volume of a single molecule, results with relation between the temporal change of the NC radius and the outgoing particle flux: dr/dt = n v V a . Kinetic theorem tells us that the number of vapour molecules colliding the NC surface per unit area per second is n c = nc/4 where n is the number of molecules per unit volume in the vapour, andc = √ 8RT /πM r is their average speed.
It is assumed that the rates of condensation and evaporation are independent, thus, at dynamic equilibrium, n v = αn c , where α is the fraction of colliding particles which become integral part of the solid phase. Combining these relations, and assuming that the vapour behaves as a mono-atomic perfect gas, i.e., n = P/k B T , yields When the radius of curvature is r → ∞ the last equation reduces to Dividing the last two equations, results with a differential equation for r(t) which describes the sublimation process of the pNBA NCs: where the ratio between the two partial pressures was replaced by the Kelvin equation The surface energy of a NC is lower than its bulk value, and depends on the NC size 11 .
This arises from the reduction of the cohesive energy due to the increase in surface atoms compared with bulk atoms as the size decreases. A common expression which takes this effect into account reads the following: where γ 0 is the bulk value, and h is the size of the unit cell of the sublimed material. Since r decreases with time, Supplementary Eqs. 4 and 5 reduce to which has the following solution where r 0 is the initial radius of the NC,

Supplementary Note 6 Electrical measurements
One of the most rewarding observation of this study is the simplicity of making CNTs devices utilizing this method. The marking procedure can be performed either for a complete circuit, in order to identify the locations of the existing tubes, or, for the design of a new circuit according to the optical images. We have used this method and fabricated more than 100 CNT based devices, with more than 95% success rate. The high yield is attributed to the knowledge of the exact location of the complete CNT network and the possibility to place our electrodes on optimal location, where the tubes are straight, far from other tubes, and do not split to additional tubes. Supplementary Figs. 18a, b depict two histograms of the total resistance and per micron length of the CNT devices that were fabricated and measured during this study assisted by the described method. The left histogram includes CNT devices with variety of metallic electrodes, and lengths (most of them are above 5 µm in length). As evident from the graph, more than 50% of the devices have total resistance which is less than 200 kΩ, and if we normalize the resistance per micron length we obtain Supplementary Fig. 18b. Here, we notice that more than 50% of the devices have resistance per micron which is less than 50 kΩ. These results are extremely good and are well within the high-end group of existing devices.

Supplementary Note 7 Vibrational analysis
The The differential equations, describing the in-plane vibration of a curved planar beam are given by 14 (8) where u = u(p, t), and w = w(p, t). Here (u,w) are the tangential and normal components of the beam displacement, p is the coordinate along the beam, and t represents time.
For the out of plane vibration (orthogonal to the x-y plane) the following equations hold 15 : where ν = ν(p, t), and β = β(p, t). Here (ν, β) are the displacement in the z direction, and the axial rotation 15 , respectively. G is the shear modulus, J is the moment of rotation, and In order to solve these equations, first, we eliminate the time from the two sets of equations (using separation method) and obtain two boundary-value problems. Afterwards, we solve the boundary-value problems using finite element method (FEM) with Galerkin procedure 14 . The cubic (Hermit) FEM approximation was used for w(p) and ν(p), and linear FEM approximation was used for u(p) and β(p) 16 . The experimental vibrational modes were automatically measured by matlab code which captured the dark field image of the vibrating tube and analyzed its shape for each frequency at the measured range. Gaussian fit was employed for fitting the intensity profile and the vibrating amplitude was extracted from the full width half max ( Supplementary Fig. 19). Fig. 5e and Supplementary Fig. 20c depict two examples of the experimental data and the obtained fits according to the FEM.
The slack of each tube was extracted from the optical image, and the two fitting parameters were only r 1 and r 2 . The random distribution of the outer shell is essential for obtaining excellent agreement between the finite element approach and the experimental data.
For the electrostatic actuation, both dc and ac electrical forces are applied between the vibrating tubes and the metallic probe, which serves as an external gate. The constant voltage between the tube and the gate creates tension along the tube, which affects its resonance frequencies. Supplementary Fig. 24a presents the frequency increase of the fundamental resonance mode as the external dc gate voltage increases. As before, since the tube contains nonhomogeneous distribution of NCs, a simple phenomenological model which interpolates between the bending behavior at low external fields and the elastic behavior at high biases will not be adequate 17 . However, our FEM which takes into account prestressed beam con-figuration, is appropriate for these cases, as well. Again, the original slack is found from the optical image, and with only two fitting parameters, r 1 and r 2 , we could nicely fit the experimental results, for the whole bias range (Supplementary Fig. 24a). The electric force which is exerted on the tube depends on the capacitance coupling between the gate and the CNT. Since the external probe is shorter than the tube length the standard expression for the capacitance between the tube and the metallic plane, C g (z) = 2πϵ 0 L/ log(2z/r 0 ), where z is the vertical distance between the tube and the gate, is not adequate. A common approach for the current configuration is based on approximating the metallic probe to cut cone, and adding together the contributions of the face and side wall capacitances as plotted in Supplementary Fig.26. Exact electrostatic solution using COMSOL simulation agrees very well with our approximate model. The total capacitance and the total electric force are given by The residual tensions which are tuned by the applied dc voltage, as well as the vibrational modes are calculated by the FEM. The resulted resonances of the first mode as a function of the gate voltages are plotted in Supplementary Fig. 24a, with r 1 and r 2 as the two fitting parameters. As evident, the agreement between the experimental data and the theoretical prediction is excellent, but the data do not follow the expected behavior of homogenous CNT 13 . Supplementary Fig. 24b   CNTs with different random distribution, r 2 , but the same r 1 and slack. As evident, for increasing slack (s) and amplitude of the random shell (r 2 ) the calculated vibrational modes differ significantly from the prediction for the homogenous tube 13 .
For large ac excitations the frequency response reflects non linear behavior and hysteretic shape. Examples are depicted in Fig. 6 and Supplementary Fig. 22 where u is the in-plane motion, and T 0 is the prestress tension, or residual tension arising from the dc gate voltage. For convenience, we choose the following boundary conditions: where ξ 2 = EI/T 0 , and λ 2 = ρ v Aω 2 /T 0 . The solution for Supplementary Eq. 13 is given where, Using the following relation, ξ 2 k 2 2 = 1 + ξ 2 k 2 1 , and the secular equation we receive an implicit equation for k 1 . For high dc gate voltages, ξ 2 = EI/T 0 ≪ 1, thus we can expand Supplementary Eq. 17 in power of ξ, and find the leading terms of k 1 , k 2 , λ = k 1 k 2 ξ, and finally the resonance frequency, f 1 = ω 1 /2π, of the fundamental mode, i.e., ) . (20) and the potential energy is the following where T 0 is the residual tension. The equation of motion within the lump model is written as follows: where F dc and F ac are the external dc and ac applied electric forces. When the beam displacements are not negligible with respect to the distance between the tube and the external probe (z), one should expand the external dc force in powers of u. Such expansion together with rearrangement of Supplementary Eq. 23 yield the following expression: where m = LρS/2 and In the calculations below, we have used the full expression for the different F ′ i s in Supplementary Eq. 30 up to the fourth powers in ξ. However, for the simplicity of typing we note only the leading terms of these expressions for the approximated case where L g → L and L m → 0, i.e., where r eff is the effective shell radius which takes into account the total mass average of the suspended beam. The static contribution of the external force will modify the equilibrium position of the beam, hence, induces residual tension along the tube. The first and third terms (F 1 and F 3 ) will modify the linear resonance frequencies and the non-linear behavior, respectively. The overall dynamic equation can be written as follows: The solution for Supplementary Eq. 31 is found using the rotating frame approximation 18 .
Briefly, one assumes external excitation of the form exp(iω p t) and tries the following ansatz u t = (a − a * )/(Γ − Γ * ), where a = A exp(iω p t), and Γ = −γ + i √ω 0 2 − γ 2 . In this notation we included the shift of the resonance frequency inω 0 , i.e.,ω 0 2 =ω 2 0 (1 − β). Plugging this approximation into the equation of motion and keeping only terms which change slowly with respect to ω p results with the following implicit equation for A where κ = 3α/8/(ω 0 2 − γ 2 ) 3/2 , and β 3 = γ 3 ω p /8/(ω 0 2 − γ 2 ) 3/2 . The solution for |A| is found from the following non linear equation and the final temporal dependence of the tube reads the following The linear and non-linear behaviors of the tube displacement were fitted to the maximal dis- The transition from hardening to softening, as depicted in Fig. 6, can be calculated within the Duffing lumped model. At low pressure according to our fitting (Fig. 6d), the nonlinear term α is found to be α = −0.56 ± 0.07 10 20 s −2 m −2 . According to our model both the frequency and the nonlinear spring constant are affected by the electrostatic force (Supplementary Eq. 30). Both terms depend on several parameters such as slack, probe-tube distance, and tube length, all of which can be extracted from the optical image. The only two fitting parameters are r 1 and r 2 , which are found from the dependance of the resonance mode on the external dc voltages at low excitation amplitude using our FEM (similar data as depicted in Supplementary Fig. 24). All together, we obtained from the Duffing lumped

Supplementary Note 8 Image processing
Dark field optical microscopy images of CNTs decorated with pNBA NCs form bright lines on a dark background. Using image processing techniques to detect nanotubes in those images is challenging due to the large amount of granular noise and the fact that nanotubes lines are usually faint and noncontinuous. For this aim, we use a two-stage approach. First, a Canny edge detector 20 is used to extract edges that form a rough estimation of the nanotubes layout. The Canny edge detector was selected due to its ability to detect a wide range of edges in the images. Then, we apply a set of post-processing techniques to the binary image of edges in order to remove noise and detect continuous curves. A morphological closing operation 21 is applied in order to fill gaps at the edges. The structuring element used for morphological closing is relatively small so that irrelevant elements situated close to a nanotube will not be merged with the nanotube curve. Then, connected component analysis 22 is used to remove these irrelevant elements. Different properties, such as the area and bounding ellipse, are calculated for each connected component. Nanotubes (and parts of nanotubes) can be distinguished due to their long and narrow bounding ellipse, and due to their large area compared with irrelevant isolated elements. After imposing threshold for these properties, the remaining components consist mainly of disconnected nanotube parts and large irrelevant elements that are not close to the nanotubes. This allows for another iteration of morphological closing and threshold screening, this time with a larger structuring element that connects the nanotube parts, and higher thresholds that remove the remaining irrelevant elements. In the resulting binary image, pixels belonging to a nanotube are white and all other pixels are black. With this mask it is possible to mark the nanotubes in the original image and identify the starting and ending points of each nanotube, as shown in Fig. 7a.
After applying the image processing analysis to the optical images, a complete map of the CNTs location is found. Then, a layout of the circuit design is automatically obtained by imposing several design rules, such as electrodes separation, pads size, and single tube in the junction. An example of this process is presented in Supplementary Fig. 27, where inverter based on p and n type CNTFETs was planned. Supplementary Fig. 27a presents the CNT layout and the inverter design, all automatically obtained by our homemade computer code. The process includes three steps of e-beam lithography. In the first step strips for oxygen plasma were defined in order to remove the unnecessary tubes. In the second step electrodes for the two CNTs were fabricated. Next, we deposit 100 nm of SiO 2 on top of one CNT which suppose to remain p type at the end of the process. Than, a thin layer of HfO 2 was deposited all over the sample, and was etched from the device pads and from the area with the deposited SiO 2 . Last, a quick wet etching in buffer oxide etch (BOE) removed the protected SiO 2 layer and rendered the device back again to be p-type FET.
Supplementary Fig. 27b depicts the resulted inverter device that was design in Supplementary Fig. 27b. Alternative approach was based on two different metals for the p and n type