NANO LETTERS
Predicting the Results of Chemical Vapor Deposition Growth of Suspended Carbon Nanotubes
2009 Vol. 9, No. 5 1806-1811
Matthew S. Marcus,†,‡,§ Jason M. Simmons,*,†,§,| Sarah E. Baker,‡ Robert J. Hamers,‡ and Mark A. Eriksson*,† Department of Physics, Department of Chemistry, UniVersity of Wisconsin-Madison, Madison, Wisconsin Received December 10, 2008; Revised Manuscript Received February 16, 2009
ABSTRACT The successful growth of suspended carbon nanotubes is normally based on purely empirical results. Here we demonstrate the ability to predict the successful suspension of nanotubes across a range of trench widths by combining experimental growth data with a theoretical description of nanotube mechanics at the growth temperature. We show that rare thermal oscillations much larger than the rms amplitude combined with the large nanotube-substrate adhesion energy together are responsible for unsuccessful nanotube suspensions. We derive an upper limit on the number of deleterious nanotube-substrate interactions that can be tolerated before successful growth becomes impossible, and we are able to accurately explain literature reports of suspended nanotube growth. The methodology developed here should enable improved growth yields of suspended nanotubes, and it provides a framework in which to analyze the role of nanotube-substrate interactions during nanotube growth by chemical vapor deposition.
The use of chemical vapor deposition (CVD) to grow suspended carbon nanotubes1-4 is an efficient process that has become especially attractive with recent realizations of nanotube electromechanical systems.5-7 The optical and electronic properties of nanotubes are strongly dependent on the local interactions of the nanotubes with the surrounding environment.8-10 Of particular importance is the question of whether the nanotubes are suspended in space between contacts or are in direct contact with the underlying substrate. Previous studies have shown that suspending the nanotubes reduces conductance hysteresis and 1/f noise by separating the nanotube from surface charge traps11,12 and, though saturation currents in suspended nanotube transistors are suppressed compared to transistors on flat substrates due to the lack of heat dissipation to the substrate,13 the absence of substrate quenching enhances photoluminescence and Raman yields.14,15 In addition, suspended nanotubes are ideal for probing quantum effects such as Aharonov-Bohm-type quantum interference in coherent nanoelectromechanical systems (NEMS).16,17 Yet, there remains little understanding of the factors that control whether the nanotubes are freely * To whom correspondence should be addressed. E-mail: (J.M.S.)
[email protected]; (M.A.E.)
[email protected]. † Department of Physics. ‡ Department of Chemistry. § These authors contributed equally to this work. | Present address: NIST Center for Neutron Research. 10.1021/nl803726b CCC: $40.75 Published on Web 04/01/2009
2009 American Chemical Society
suspended or whether they are in contact with the underlying substrate. Here we demonstrate that thermal vibrations during the CVD growth process are a significant factor causing nanotubes to become stuck to the lower substrate. We have fabricated framework suspensions with a large range of suspension gaps. For a suspension height of 80 nm, we find that all of the observed nanotubes that are connected to more than one ridge-top are suspended when the gap has a width of 500 nm or less. In contrast, nanotubes collapsed to the substrate for all suspension gaps greater than 2.5 µm. For intermediate suspension gaps, we find that the probability for collapse increases from 0 to 1 as the width increases from 500 nm to 2.5 µm. We demonstrate that thermal oscillations provide ample opportunity for doubly connected single wall nanotubes to interact with the substrate and become stuck to the surface. Although the rms amplitude of thermal oscillations of single wall carbon nanotubes at the 900 °C growth temperature does not exceed 20 nm for a 1 µm long nanotube,18 we show that it is actually the relatively rare thermally driven oscillation fluctuations much larger than the rms amplitude that are most important in suspended nanotube growth. Adhesion energies for nanotubes to silicon surfaces are quite strong, making it energetically favorable, should the nanotube come into contact with the lower substrate, to bind nanotubes to the bottom of the trench. Because nanotube oscillation frequencies are large, on the order of 10-100
Figure 1. (a) Scanning electron micrograph and schematic cross section showing the growth of suspended nanotubes. The bright vertical lines are raised ridge-tops and the dark areas are the trench-bottoms. Nanotubes that appear bright in the SEM image are suspended across the trenches and dark nanotubes have collapsed to the bottom. For the individual nanotube that bridges all four trenches, the dark arrow indicates trenches where the nanotube remains suspended and the light arrow shows where the nanotube has become stuck to the substrate. (b) Tilted view SEM image (∼45°) showing a suspended (dark arrow) and a trapped (light arrow) nanotube across the same trench.
MHz,18-21 high amplitude thermal oscillations occur frequently over the course of a typical CVD growth (∼5-10 min) and cause a much larger number of pinned nanotubes than might otherwise be expected from a simple comparison of the rms amplitude to the suspension height. Analysis of the sticking behavior yields an empirical criterion for the successful growth of suspended carbon nanotubes as well as yielding information about the nanotube-substrate interaction. The model developed here accurately explains independent literature reports of suspended nanotube growth, indicating that the model is sufficiently quantitative to offer predictive capability. Further, though the model is explicitly developed for the CVD growth of carbon nanotubes, it is sufficiently generic to be extended to studies of the growth and stability of other low-dimensional nanostructures including nanowires and atomically thin nanosheets such as graphene. Experiments. Arrays of trenches were fabricated on thermally oxidized silicon wafers using electron beam lithography and reactive ion etching. The etch depth was 80 nm and the suspension widths vary from 140 nm to 2.5 µm. After the trenches were prepared, the substrates were dipped in a solution of Fe(NO3)3, a common nanotube catalyst,22,23 and single wall nanotubes were grown in a CVD reactor at 900 °C using methane feedstock. The nanotubes that were grown are predominately connected to the top of the ridges on one or both ends. Nanotubes that were attached to just one ridge typically have grown from the ridge top to the lower substrate, after which the growth has terminated. Some of the singly connected nanotubes demonstrated growth along the lower substrate, however the high adhesion energy to the silicon substrate (discussed below) effectively prevents the nanotube from leaping off the substrate to become connected to a second ridge. Thus, nanotubes that are connected to two ridge tops must have begun as suspended structures. In order to analyze the role of oscillations, the work presented here focuses only on those nanotubes that are connected to at least two ridgetops. Figure 1 shows typical examples of nanotube growth on a trench array. The nanotube indicated by arrows in Figure 1a extends across many suspension gaps and is adjacent to several nanotubes that are suspended between just two ridges. In this example, the indicated nanotube is suspended in the Nano Lett., Vol. 9, No. 5, 2009
Figure 2. The probability that a nanotube, which contacts two or more ridge-tops, is stuck to the substrate between ridges, plotted as a function of the suspension length. Error bars are determined from binomial statistics, ∆P ) P(L)[1 - P(L)]/N, where N is the number of nanotubes in each bin. Error bars for the end points are ill defined in binomial statistics and thus are not shown. The lines are fits to the data based on the model described in the text. The solid red line yields the best least-squares fit to the data and the dashed lines represent the (1σ confidence levels.
three regions marked with the dark arrows, and it has collapsed to the lower substrate level in the region marked with the light arrow. Fully suspended nanotubes can be distinguished from those attached to the lower substrate by the use of tilted SEM imaging and by examining the relative intensity in the SEM images, because suspended nanotubes appear brighter during SEM imaging (Figure 1).24 An initially surprising result of studies of many such samples is that all doubly connected nanotubes in our samples are suspended when the trench spacing is small ( h) ) (2)
where L is the suspension length, F2D is the surface mass density of the graphene sheet, β ) 4.73 is a numerically calculated constant based on the boundary conditions, and I(dt) is a model dependent cross-sectional moment of area that depends on the nanotube diameter.18,19,26,27 Modeling the nanotube as a thin hollow cylinder, the moment of area is I(dt) )
π G d (d 2 + Gw2) 8 w t t
(3)
where Gw is the cylinder wall thickness. Though there is a wide range of experimental and theoretical values reported for the Young’s modulus of a carbon nanotube, from 20 GPa to 5 TPa, much of the discrepancy of the reported moduli is dependent on the choice of effective nanotube wall thickness, a value which has varied between 0.66 and 3.4 Å.18-20,28,29 Since there is no consensus on the correct values for the modulus and thickness, we use instead the in-plane bending stiffness, also called the surface Young’s modulus, which is defined as Ys)YGw. This product has been shown theoretically to have a consistent value of Ys ∼ 0.36 TPa.nm (∼360 J/m2)30,31 and can explain most of the experimentally determined moduli. Using this convention, the product YI that appears in both the oscillation frequency and amplitude can be written as π π π YI ) Y Gwdt(dt2 + Gw2) ) Ysdt(dt2 + Gw2) ≈ Ysdt3 8 8 8
(4)
On the basis of the range of wall thicknesses used in the literature, the final approximation introduces an error (∼Gw2/dt2) between 0.2 and 5% for a nanotube with a 1.6 1808
(5)
The probability that the nanotube oscillates with an amplitude larger than the suspension height h (i.e., that the center of the nanotube interacts with the bottom of the trench) is
and ω 1 β2 f) ) 2π 2π L2
2
∫
∞
h
( )
h 1 P(y)dy ) erfc 2 √2δ
(6)
Finally, the rate at which the nanotube interacts with the substrate, known as the attempt frequency, is fattempt ) P(y > h) f
(7)
A key physical feature is that the attempt frequency is a strong function of nanotube diameter and suspension length, and it spans an enormous range for the various nanotubes formed during CVD growth. For example, a 1.6 nm diameter nanotube that is suspended across a 1 µm wide, 80 nm deep trench has an attempt frequency of approximately 0.001 s-1, or about 1 attempt during a 10 min growth. In contrast, the same nanotube suspended across a 1.5 µm trench has an attempt frequency over one million times larger, ∼3000 s-1. This large range of attempt frequencies is a critical component of the model and leads to the predictions discussed below. Analysis of Oscillations for Ensembles of Nanotubes. To relate the observed sticking probability in Figure 2 to the number of attempts, it is important to remember that CVD produces a range of nanotube diameters. Each nanotube that is grown during the CVD process will have a slightly different oscillation frequency and amplitude probability distribution. In addition, a nanotube suspended across several trenches could potentially have different behavior at each trench due to variations in the nanotube chirality along its length.32 To account for these variations, we make use of the distribution of nanotube diameters that result from our process conditions. We have measured this distribution using atomic force microscopy and have plotted this as a histogram for over 100 nanotubes in the upper inset to Figure 3. The distribution is well fit by a Gaussian (dotted line in figure) Nano Lett., Vol. 9, No. 5, 2009
N(dt) )
1 -(dt e √πσ
- dj)2/σ2
(8)
with a mean diameter dj of 1.6 nm with width σ of 0.9 nm. Since the oscillation amplitude is larger for small diameter nanotubes (δ ∼ dt-3/2), small diameter nanotubes will interact with the substrate more often and are thus more likely to become stuck at the bottom of the trench. For an ensemble of nanotubes, the probability that a nanotube will become stuck at the trench bottom P(L) is dependent on the number of small diameter nanotubes produced in the growth. Assuming the same modulus for all nanotubes, we can then associate the measured sticking probability P(L) with the proportion of nanotubes that have a diameter less than some limiting value dl dl
P(L) )
∫ N(d )dd t
t
)
0
[ ( ) ( )]
dl - dj dj 1 erf + erf 2 σ σ
(9)
which can be inverted to yield (Figure 3)
[
dl ) σ inverf 2P(L) - erf
( )] dj σ
+ dj
(10)
For a given suspension length, this model predicts that nanotubes with diameters smaller than dl will be trapped by the substrate and those with larger diameters will remain suspended. For example, in Figure 2, a sticking probability of 0.5 corresponds to a limiting diameter dl equal to the mean diameter dj. By analyzing the behavior of a nanotube with the limiting diameter for each suspension length, we can extract an upper bound on the number of attempts required for the nanotube to become stuck to the substrate during growth. Setting dt ) dl in eqs 1, 2, and 4, we can use eqs 6 and 7 to calculate the upper bound on the attempt frequency and the total number of attempts necessary for the nanotube to become trapped in our CVD process. The result of this calculation is presented in the lower inset to Figure 3. Intriguingly, the maximum number of nanotube-substrate interactions that can occur before the nanotube becomes stuck to the substrate is between 104 and 107 times during the 10 min CVD growth, which is a surprising number of interactions given the disparity between the rms amplitude and the trench height. Further, this upper limit is approximately the same for all suspension lengths. This is consistent with the description of the nanotube as a thermal oscillator in that there is a fixed amount of energy (∼kbT), independent of the suspension length, that must be dissipated for the nanotube to become stuck to the substrate. Though the calculated maximum number of interactions spans 3 orders of magnitude (104-107), this range is relatively narrow when compared to the range of interactions that are possible given our diameter distribution and trench widths. The minimum number of interactions, calculated for a typical 1.6 nm diameter nanotube across a 0.5 µm trench, is less than one per growth, yielding a lower bound on the interaction rate Nano Lett., Vol. 9, No. 5, 2009
Figure 3. Calculated limiting diameter, that is the largest nanotube diameter that becomes stuck to the lower substrate for a given suspension length. Error bars are propagated from the data in Figure 2. (upper inset) AFM measured diameter distribution. (lower inset) Maximum number and rate of nanotube-substrate interactions that result in the nanotube with diameter dl becoming stuck to the substrate between trenches. Because this number is an upper bound, only positive error bars are shown. Lower error bars descend to the axis.
of 1.6 × 10-3 s-1. At the opposite extreme, the smallest diameter nanotube measured by AFM (0.7 nm) suspended across the longest trench (2.67 µm) has a calculated number (rate) of interactions of ∼5 × 108 (∼9 × 105 s-1), or about an order of magnitude larger than top of the calculated range of interactions resulting in a nanotube becoming trapped. Given this 8-order of magnitude range of possible interaction numbers, our extracted range for the critical number of interactions is relatively narrow in breadth and in fact leads to reasonable predictions, as we show below. To yield a more rigorous upper bound on the number of deleterious nanotube-substrate interactions, we can simulate all of our observations using only the AFM-measured diameter distribution as input. First, the number of attempts is calculated for a range of nanotube diameters and suspension lengths, using the experimental CVD conditions and trench height of 80 nm. The result of this calculation is shown in Figure 4a. As expected, for a given suspension length, larger diameter nanotubes have fewer substrate interactions during the CVD process. Constant Nattempt contours within Figure 4a are then inverted numerically to extract the theoretical limiting diameter as a function of suspension length. The probability that a nanotube with diameter dl will become stuck to the substrate is then calculated, generating curves of P(L) at fixed Nattempt. These curves are then compared to the experimental results in Figure 2. Using leastsquares fitting of a range of Nattempt values, the best fit is obtained for Nattempt ) Nmax ) 9 × 104. This contour is shown as the solid line in Figure 2 and in Figure 4a. The dashdotted lines in the each figure, taken at Nattempt ) 104 and Nattempt ) 106, represent the (1σ confidence interval from the least-squares fitting. Nanotubes with diameters and suspension lengths beneath the solid line should remain suspended throughout the 10 min CVD process, while those above the line will become trapped to the lower substrate. 1809
We note that Nmax is an upper limit since the nanotube must first grow across the trench and could become stuck before the end of the growth time. Though the time required by the nanotube to bridge the trench is not precisely known, nucleation times of 10 s and growth rates on the order of 100 µm/min have been reported, making the role of suspension time a small correction.33 To demonstrate the usefulness of this result, it is important to compare the analysis developed here with the results of independent growth experiments that have been reported in the literature. Although most reports focus only on successfully suspended nanotubes, Son et al.34 specifically provide data about nanotubes that become stuck to the substrate for long trench widths. Using the 300 nm suspension height from Son et al., the corresponding number of nanotube-substrate interactions as a function of nanotube diameter and suspension length is plotted in Figure 4b, assuming a 10 min growth time. Overlaid on the figure are the results from Son et al.;34-36 for 1.5 and 3 µm suspension lengths, a ∼1.4 and a ∼2 nm nanotube remain suspended (white circles) while the same nanotubes become trapped on the lower substrate for a 6 µm length (black circles). In addition, the dashed box shows a range of suspended nanotube diameters and suspension lengths that were reported by Mann et al.37 for the same 300 nm suspension height. The excellent agreement of the reported results with our calculation demonstrates the predictive power of the analysis presented here. Implicit in the analysis presented in this paper is the assumption that nanotubes can indeed become stuck if they interact with the substrate. We close by justifying this assumption by calculating typical adhesion and strain energies. For the nanotube to remain stuck after interacting with the surface, the increased strain energy must be offset by the adhesion energy gained by interacting with the substrate. The elastic strain energy stored in the nanotube is Eel ) (1/2)c1u2, where c1 is the effective spring constant (c1 ) 192(YI/L3)) and the position u is taken to be the suspension height of 80 nm.19 For a 1.6 nm diameter nanotube with a suspension length of 1.3 µm, the elastic energy is
Ed )
96[(π/8)Ysdt3] 3
L
h2 ≈ 1.8 eV
(11)
Previous AFM studies of the adhesion of nanotubes to a substrate indicate that the adhesion energy is approximately Eb ≈ (-0.53 + 0.86dt) eV/nm, where the diameter is measured in nanometers.38 For the same 1.6 nm diameter nanotube Eb ≈ 0.9 eV/nm, and as little as 2 nm of the nanotube length needs to adhere to the substrate to balance the strain. Finally, it is important to note that the calculated number of attempts necessary for nanotubes to become stuck shown in Figure 4 is specific to a certain range of CVD conditions; the exact number of attempts necessary is likely to depend on the details of the CVD process and is something that should be calibrated. For example, the critical number of nanotube-substrate interactions can depend on nucleation and growth times, temperature, gas flow dynamics and damping, the structural quality of the nanotubes that are grown, as well as the effects of surface dissipation.39-41 Nanotube sticking could be partially mitigated through the use of other substrates or processing conditions. Though the mechanisms of nanotube-surface energy dissipation are poorly understood, the framework developed here can in fact be applied to other process conditions in order to shed light on the specific nanotube-substrate interactions. On a practical level, once applied to a specific CVD process, one can predict the fate of nanotubes with a specified diameter and can tune the suspension length and height as well as the growth time so as to minimize the number of interactions with the substrate in order to successfully grow suspended nanotubes. In addition to its applicability to nanotube growth, the model we present is general enough to be applicable to other low dimensional nanomaterials such as nanowires and graphene sheets. Further, the model could be used to predict the useful lifetime of devices involving suspended nanotubes or nanowires, particularly in cases where the active element is at elevated temperatures. Summary. We have demonstrated that large-amplitude thermal oscillations are a critical factor controlling the contact
Figure 4. Predicted number of nanotube-substrate interactions (color scale) as a function of the nanotube diameter and suspension length for trench heights of (a) 80 nm and (b) 300 nm. The solid line marks the contour where Nattempt ) Nmax, determined by least-squares fitting to the experimental data, while the dashed lines enclose the (1σ confidence region. Also included in (b) are results from the literature. The shaded box includes nanotubes that were successfully suspended from Mann et al. (ref 37). The white (black) circles are nanotubes that were suspended (stuck) based on Son et al. (ref 34). 1810
Nano Lett., Vol. 9, No. 5, 2009
between nanotubes and the underlying substrate. Although the thermal rms amplitudes are relatively small, high oscillation frequencies permit a significant number of large amplitude deviations away from this average. Using the framework developed here and the measured diameter distribution of nanotubes for these process conditions, we calculate that a nanotube requires approximately 9 × 104 substrate interactions to become trapped at the bottom of a trench. The model developed can be applied to other CVD growth conditions and offers predictive power for future developments in nanotube NEMS. Applying this framework to other processes should yield important information about the intrinsic mechanical properties of nanotubes as well as the detailed nanotube-substrate interactions and energy dissipation mechanisms. Acknowledgment. The authors would like to acknowledge Paul Kienzle for many useful discussions and funding from the NSF CAREER program under Grant DMR-0094063, the NSF MRSEC program under Grant DMR-0520527, and the NSF NSEC program under Grant DMR-0425880. References (1) Cassell, A. M.; Franklin, N. R.; Tombler, T. W.; Chan, E. M.; Han, J.; Dai, H. J. Am. Chem. Soc. 1999, 121, 7975. (2) Franklin, N. R.; Dai, H. AdV. Mater. 2000, 12, 890. (3) Homma, Y.; Kobayashi, Y.; Ogino, T.; Yamashita, T. Appl. Phys. Lett. 2002, 81, 2261. (4) Yang, B.; Marcus, M. S.; Keppel, D. G.; Zhang, P. P.; Li, Z. W.; Larson, B. J.; Savage, D. E.; Simmons, J. M.; Castellini, O. M.; Eriksson, M. A.; Lagally, M. G. Appl. Phys. Lett. 2005, 86, 263107. (5) Fennimore, A. M.; Yuzvinsky, T. D.; Han, W. Q.; Fuhrer, M. S.; Cumings, J.; Zettl, A. Nature (London) 2003, 424, 408. (6) Sazonova, V.; Yaish, Y.; Ustunel, H.; Roundy, D.; Arias, T. A.; McEuen, P. L. Nature (London) 2004, 431, 284. (7) Garcia-Sanchez, D.; San Paulo, A.; Esplandiu, M. J.; Perez-Murano, F.; Forro, L.; Aguasca, A.; Bachtold, A. Phys. ReV. Lett. 2007, 99, 085501. (8) LeRoy, B. J.; Lemay, S. G.; Kong, J.; Dekker, C. Appl. Phys. Lett. 2004, 84, 4280. (9) Marcus, M. S.; Simmons, J. M.; Castellini, O. M.; Hamers, R. J.; Eriksson, M. A. J. Appl. Phys. 2006, 100, 084306. (10) Simmons, J. M.; In, I.; Campbell, V. E.; Mark, T. J.; Leonard, F.; Gopalan, P.; Eriksson, M. A. Phys. ReV. Lett. 2007, 98, 086802. (11) Kim, W.; Javey, A.; Vermesh, O.; Wang, Q.; Li, Y.; Dai, H. Nano Lett. 2003, 3, 193. (12) Lin, Y.-M.; Tsang, J. C.; Freitag, M.; Avouris, P. Nanotechnology 2007, 18, 295202. (13) Pop, E.; Mann, D.; Cao, J.; Wang, Q.; Goodson, K.; Dai, H. Phys. ReV. Lett. 2005, 95, 155505.
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