Determination and mapping of diameter and helicity for single-walled carbon nanotubes using nanobeam electron diffraction

The atomic structures of 124 single-walled carbon nanotubes, described by their diameter and helicity or equivalently by the two chiral indices $(u,v)$ that define the perimeter of each nanotube, have been determined unambiguously by nanobeam electron diffraction. A mapping of 70 possible nanotubes in the range of 1.20–1.65 nm in diameter with all possible helicities has been constructed experimentally for a carbon nanotube sample produced by arc discharge. Among the total 124 nanotubes characterized experimentally, 58 nanotubes of different structure have been identified. By examining the histogram of occurring helicities, we find that, while certain nanotubes were observed slightly more often than others, the overall feature showed a rather uniform distribution in occurrence. Basing on a nucleation-and-growth model, we suggest that the uniform distribution of helicity be originated from the weak dependence on helicity of the formation energy of carbon nanotubes, while the growth prefers slightly the structure with helicity 15°–30° for which the addition of carbon dimers would facilitate the growth of carbon nanotubes.

Figure 1

Calculated electron diffraction pattern of carbon nanotube of chiral indices (14,10). The two hexagons represent the primary {100}-type reflections from graphene which form principal layer lines labeled as $L_1,L_2$, and $L_3$ as indicated in the figure. The intensities are elongated in the direction perpendicular to the tubule axis and the center of each reflection also shifts outward compared with the corresponding graphene reflection.

Figure 2

Characterization of a chiral SWNT (14,10). (a) HRTEM image of the SWNT. The nanotube diameter, $1.60\pm 0.08\mathrm{nm}$, was measured directly from the two dark lines. (b) Corresponding NBED pattern. The helicity of this nanotube is measured $24.50°\pm 0.08°$ from the layer line spacings. (c) Digitized diffraction intensity profile of the equatorial layer line in (b). The distance $2R$ between the two peaks $\left(X_5\right)$ indicated by arrows is used to measure the nanotube diameter. The overlapped intensity profile is the calculated electron intensities expressed by the module square of the zero-order Bessel function $J_0\left(\pi dR\right)$.

Figure 3

(a) All 458 nanotube species with diameter from 0.8 to 3.0 nm. Solid circles and triangles represent metallic and seminconducting nanotubes, respectively. (b) Diameter distribution of the 124 individual SWNTs examined experimentally. A Gaussian fit of the diameter distribution was represented by the black curve, in which the peak is around 1.38 nm with a standard deviation of 0.1 nm. (c) Portion of the shaded region in (a), which has been characterized experimentally. 58 out of the 70 nanotube species were observed, among which 21 are metallic and 37 are semiconducting.

Figure 4

Helicity distribution of the 124 individual SWNTs characterized experimentally. (a) Histogram showing the number of nanotube species (gray column) and the number of the observed individual nanotubes (black column) with different helicities, plotted in intervals of 2°. (b) Histogram showing normalized number (normalized by dividing the observed number of nanotubes by the number of nanotube species at each helicity) of characterized individual nanotubes with different helicities. Nanotubes of helicities 15°–30° were slightly favored, attributed to the higher growth rate of these nanotubes.

Figure 5

Schematic illustrating the faster growth rate of nanotubes of helicities 15°–30°, where additions of newly arrived dimers (dark-gray colored) would facilitate higher growth rate. The dotted lines denote new bonds.