Percolation scaling of $1/f$ noise in single-walled carbon nanotube films

We use Monte Carlo simulations and noise modeling to study the scaling of $1/f$ noise in single-walled carbon nanotube films as a function of device parameters and film resistivity. Despite its relative simplicity, this computational approach provides a general framework for the characterization of $1/f$ noise in nanotube films and explains previous experimental observations. We consider noise sources due to both tube-tube junctions and nanotubes themselves. By comparing the simulation results with the experimental data, we find that the noise generated by tube-tube junctions dominates the total nanotube film $1/f$ noise. Furthermore, we systematically study the effect of device length and film thickness on the $1/f$ noise scaling in nanotube films in order to demonstrate that the simulation results are in good agreement with the available experimental data. Our results further show that the $1/f$ noise amplitude depends strongly on device dimensions, nanotube degree of alignment, and the film resistivity, following a power-law relationship with resistivity near the percolation threshold after properly removing the effect of device dimensions. We also find that the critical exponents associated with the noise-resistivity and noise-device dimension relationships are not universal invariants, but rather depend on the specific parameter that causes the change in the resistivity and $1/f$ noise, and the values of the other device parameters. Since $1/f$ noise is a more sensitive measure of percolation than resistivity, these simulations not only provide important fundamental physical insights into the complex interdependencies associated with percolation transport in nanotube networks and films, but also help us understand and improve the performance of these nanomaterials in potential device applications, such as nanoscale sensors, where noise is an important figure of merit.

Figure 1

(Color online) (a) Atomic force microscope (AFM) image of a nanotube film with an approximate thickness of $t=15\text{nm}$ where nanotubes are randomly distributed. (b) A 2D nanotube network generated using Monte Carlo simulations for a device with length $L=4\mu \text{m}$, width $W=4\mu \text{m}$, nanotube length $l_{\text{CNT}}=2\mu \text{m}$, and nanotube density per layer $n=10\mu {\text{m}}^{-2}$. Semiconducting and metallic nanotubes are shown in light and dark color (cyan and blue color online), respectively. The inset illustrates the alignment angle ${\theta }_a$, which defines the angle range within which nanotubes can be generated to form partially aligned CNT films in simulations.

Figure 2

(a) Log-log plot of the noise amplitude normalized by resistance $\left(A/R\right)$ versus device length for a single-layer nanotube network. Experimental data from 2D nanotube networks of Snow et al. (Ref. 17) is shown by the filled circles. Our simulation data points for single-layer devices with $W=2\mu \text{m}$, $l_{\text{CNT}}=2\mu \text{m}$, $n=5\mu {\text{m}}^{-2}$, and $L$ ranging from 2 to $20\mu \text{m}$ are shown by the open circles. The dashed line fit to the experimental data reported by Snow et al. (Ref. 17) has a critical exponent of $\sim -1.3$, which is same as the exponent obtained from the fit to the simulation data for $8<L<20\mu \text{m}$. The inset shows log-log resistivity versus device length for the same device as the main panel. Resistivity rolloff is obvious at small device lengths. (b) Log-log plot of noise amplitude normalized by resistance $\left(A/R\right)$ versus device length for multilayer nanotube films. Filled circles represent our own experimental measurements (Ref. 37) of CNT film devices with $\sim 15\text{nm}$ thickness, $50\mu \text{m}$ and $1000\mu \text{m}$ device length, and device widths ranging from 2 to $50\mu \text{m}$. Open circles and squares are our simulation data points for devices with a film thickness of $t=16\text{nm}$ (eight layers) and $t=6\text{nm}$ (three layers), respectively, where the other simulation parameters are $W=2\mu \text{m}$, $l_{\text{CNT}}=2\mu \text{m}$, $n=1.25\mu {\text{m}}^{-2}$, and $L$ ranging from 2 to $14\mu \text{m}$. The extracted critical exponents from the dashed line fits to these two simulation data sets above $L>6\mu \text{m}$ are $-1.9$ and $-0.8$, respectively. The inset shows the distribution of $A$ in 500 simulated devices, all with $L=2\mu \text{m}$, $t=16\text{nm}$, and the other parameters same as above. The data can be fit by a log-normal distribution as shown. (c) Log-log plot of computed $A\times L$ versus resistivity for the device shown in part (b) with $t=16\text{nm}$. The change in resistivity is a result of the change in device length. The extracted critical exponent of the dashed line fit is 0.4. The inset shows log-log plot of $A\times L$ versus resistivity for the same device as in the main panel, but without any noise sources at the tube-tube junctions. The extracted critical exponent is $-2.9$ in this case.

Figure 3

(a) Log-log plot of computed $A$ versus $W$ for a device with $L=5\mu \text{m}$, $t=16\text{nm}$, and other parameters same as in Fig. 2b. There are two separate scaling regimes. The extracted exponent of the dashed line fit for large widths (where resistivity is constant) is $-1.1$. The extracted exponent of the dashed line fit for small widths is $-5.6$. The inset shows log-log plot of resistivity versus device width for the same device as in the main panel. (b) Log-log plot of computed $A\times W$ versus resistivity. The change in resistivity is a result of change in device width. The extracted critical exponent in this case is 1.7. The inset shows log-log plot of $A$ versus resistivity for the same device.

Figure 4

(a) Log-log plot of computed $A\times t$ versus resistivity. The change in resistivity is a result of change in device thickness. The extracted critical exponent in this case is 1.8. The left inset shows log-log plot of resistivity versus device thickness for the same device. The right inset shows log-log plot of $A\times t$ versus resistivity for the same device, but without any noise sources at the tube-tube junctions. The extracted critical exponent is $-0.8$ in this case. (b) Log-log plot of computed $A$ versus resistivity. The change in resistivity is a result of change in the nanotube alignment angle ${\theta }_a$. It is evident that CNT films with the same resistivity values can have two different noise amplitudes, depending on their alignment angles. A power-law fit to the data points (dashed line) with noise amplitudes $A$ higher than $7\times {10}^{-7}$ yields a critical exponent of 1.3. The inset shows the log-log plot of resistivity versus alignment angle for the same device, in which the resistivity minimum at about $45°$ is evident.