Computational study of geometry-dependent resistivity scaling in single-walled carbon nanotube films

We study the geometry-dependent resistivity scaling in single-walled carbon nanotube films as a function of nanotube and device parameters using Monte Carlo simulations. We first demonstrate that these simulations can model and fit recent experimental results on the scaling of nanotube film resistivity with device width. Furthermore, we systematically study the effect of four parameters; namely, tube-tube contact resistance to nanotube resistance ratio, nanotube density, nanotube length, and nanotube alignment on the film resistivity and its scaling with device width. Stronger width scaling is observed when the transport in the nanotube film is dominated by tube-tube contact resistance. Increasing the nanotube density decreases the film resistivity strongly and results in a higher critical exponent and a lower critical width. Increasing the nanotube length also reduces the film resistivity, but increases both the critical exponent and the critical width. In addition, the lowest resistivity occurs for a partially aligned rather than perfectly aligned nanotube film. Increasing the degree of alignment reduces both the critical exponent and the critical width. We systematically explain these observations, which are in agreement with previous experimental results, by simple physical and geometrical arguments. We also observe that, near the percolation threshold, the resistivity of the nanotube film exhibits an inverse power-law dependence on all of these parameters, which is a distinct signature of percolating conduction. However, the strength of resistivity scaling for each parameter is different, and it depends on how strongly a particular parameter changes the number of conduction paths in the film. Monte Carlo simulations, as presented in this paper, can help elucidate the effects of various parameters on percolating transport in films made up of one-dimensional conductors, which is an essential step towards understanding and characterizing the performance of these nanomaterials in electronic and optoelectronic devices.

Figure 1

(Color online) (a) A 2D random nanotube network generated using a Monte Carlo process for a device with device length $L=4\mathrm{\mu }\mathrm{m}$, device width $W=4\mathrm{\mu }\mathrm{m}$, $l_{CNT}=2\mathrm{\mu }\mathrm{m}$, and $n=4\mathrm{\mu }{\mathrm{m}}^{-2}$. (b) Atomic force microscope (AFM) image of a nanotube film etched into a series of lines with width and spacing of $500\mathrm{nm}$ by $e$-beam lithography and reactive ion etching.

Figure 2

(Color online) (a) Log-log plot of normalized nanotube film resistivity versus device width. The individual data points are the experimental results from our previous work (Ref. 23) for a nanotube film with device length $L=7$, 50, or $200\mathrm{\mu }\mathrm{m}$ (labeled by different symbols) and average thickness $t=15\mathrm{nm}$. The solid line represents our simulation best fit to the experimental data with $R_{ratio}=100$ and $n=2\mathrm{\mu }{\mathrm{m}}^{-2}$. The other simulation parameters are $L=7\mathrm{\mu }\mathrm{m}$, $l_{CNT}=2\mathrm{\mu }\mathrm{m}$, and $t=15\mathrm{nm}$. (b) Log-log plot of normalized resistivity versus device length $L$ for three different nanotube densities $n$, as labeled by different symbols in the plot, calculated using our simulation code for a device with $W=2\mathrm{\mu }\mathrm{m}$.

Figure 3

(Color online) Log-log plot of normalized resistivity versus device width for $R_{ratio}$ ranging from ${10}^{-2}$ to ${10}^4$, labeled by different symbols. The values of the critical exponent $\alpha$ extracted from the slopes of the $\rho$ vs $W$ curves near the percolation threshold (i.e., at small $W$) are 0.5, 0.8, 1.55, and 1.95 for $R_{ratio}={10}^{-2}$, 1, ${10}^2$, and ${10}^4$, respectively. The inset shows the critical exponent $\alpha$ as a function of the resistance ratio. The symbols are the simulation results, and the solid line represents a sigmoidal fit showing a smooth transition between $\alpha \approx 0.5$ and $\alpha \approx 2$.

Figure 4

(Color online) (a) Log-log plot of normalized resistivity versus device width for four nanotube densities $n$ ranging from 1 to $3\mathrm{\mu }{\mathrm{m}}^{-2}$, labeled by different symbols. The values of the critical exponent $\alpha$ extracted from the slope of the $\rho$ vs $W$ curves near the percolation threshold are 1.1, 1.35, 1.55, and 2.2 for $n=1$, 1.5, 2, and $3\mathrm{\mu }{\mathrm{m}}^{-2}$, respectively. (b) Percolation probability $P$ versus device width for $n$ ranging from 1 to $3\mathrm{\mu }{\mathrm{m}}^{-2}$. The inset shows the log-log plot of normalized resistivity versus nanotube density plotted for a device with $L=4\mathrm{\mu }\mathrm{m}$ and $W=4\mathrm{\mu }\mathrm{m}$.

Figure 5

(Color online) (a) Log-log plot of normalized resistivity versus device width for three nanotube lengths $l_{CNT}$ ranging from 1.5 to $4\mathrm{\mu }\mathrm{m}$, labeled by different symbols. Simulation parameters are the same as in Fig. 2a except for $L=4\mathrm{\mu }\mathrm{m}$. The values of the critical exponent $\alpha$ extracted from the slope of the $\rho$ vs $W$ curves near the percolation threshold are 1.0, 1.55, and 2.05 for $l_{CNT}=1.5$, 2, and $4\mathrm{\mu }\mathrm{m}$, respectively. (b) Log-log plot of normalized resistivity versus nanotube length for three different $n\text{−}{l}_{CNT}$ relationships. Simulation parameters are the same as in Fig. 2a, except $L=3\mathrm{\mu }\mathrm{m}$.

Figure 6

(Color online) Log-log plot of normalized resistivity versus device width at three nanotube alignment angles ${\theta }_a$ ranging from $18°$ (partially aligned) to $90°$ (completely random). The values of the critical exponent $\alpha$ extracted from the slope of the $\rho$ vs $W$ curves near the percolation threshold are 0.7, 1.35, and 1.55 for ${\theta }_a=18°$, $36°$, and $90°$, respectively. The inset shows the log-log plot of normalized resistivity versus nanotube alignment angle ${\theta }_a$ for a device with $W=2\mathrm{\mu }\mathrm{m}$.