Modeling carrier density dependent charge transport in semiconducting carbon nanotube networks

Charge transport in a network of only semiconducting single-walled carbon nanotubes is modeled as a random-resistor network of tube-tube junctions. Solving Kirchhoff's current law with a numerical solver and taking into account the one-dimensional density of states of the nanotubes enables the evaluation of carrier density dependent charge transport properties such as network mobility, local power dissipation, and current distribution. The model allows us to simulate and investigate mixed networks that contain semiconducting nanotubes with different diameters, and thus different band gaps and conduction band edge energies. The obtained results are in good agreement with available experimental data.

Figure 1

2D Network of 1D sticks as a geometric model for a random SWNT network (gray). The magnified inset (left) illustrates the junctions (red) between single nanotubes (blue) and their conversion into a random resistor network (right inset) with junctions acting as resistors and SWNTs as nodes.

Figure 2

Density of states (first van Hove singularity, first conduction subband) of a single (6,5) SWNT as calculated within the tight binding approximation including trigonal warping effects (circles). [47, 48] The fitted first van Hove singularity (gray line) is used for modeling the SWNT network. To include energetic disorder, the band edge of each SWNT is shifted by a random energy drawn from a Gaussian distribution.

Figure 3

Normalized mobility versus charge carrier density for a network of (6,5) SWNTs without energetic disorder (simulation box: $20\times 20\mu {\mathrm{m}}^2$, SWNT length: $1.5\pm 0.9\mu \mathrm{m}$, linear SWNT density: $7\mu {\mathrm{m}}^{-1})$. The dotted line is a guide to the eye.

Figure 4

Network mobilities for different disorder strengths (simulation box size: $20\times 20\mu {\mathrm{m}}^2$, SWNT length: $1.5\pm 0.9\mu \mathrm{m}$, linear SWNT density: $7\mu {\mathrm{m}}^{-1})$: (a) Normalized charge carrier mobility versus carrier density. (b) Peak mobility versus disorder strength. Dotted lines are guides to the eye.

Figure 5

Network mobilities for different simulation box sizes (energetic disorder: $0\mathrm{meV}$, SWNT length: $1.5\pm 0.9\mu \mathrm{m}$, linear SWNT density $7\mu {\mathrm{m}}^{-1}$ and $15\mu {\mathrm{m}}^{-1})$: (a) Normalized peak mobility for square boxes. (b) Normalized peak mobility for rectangular boxes with constant box area. Dotted lines are guides to the eye.

Figure 6

Network mobilities for different SWNT densities (SWNT length $1.5\pm 0.9\mu \mathrm{m}$, simulation box size $20\times 20\mu {\mathrm{m}}^2)$: (a) Normalized mobility versus charge carrier density (without energetic disorder). (b) Peak mobility, normalized to the mobility ${\mu }_0$ for the minimum SWNT density, versus SWNT density for 0, 30, and $45\mathrm{meV}$ disorder strength. Dotted lines are guides to the eye.

Figure 7

Normalized carrier mobilities for different SWNT length distributions (no energetic disorder, SWNT density $7\mu {\mathrm{m}}^{-1}$, simulation box size $20\times 20\mu {\mathrm{m}}^2)$. Mobility versus charge carrier density and peak mobility versus length standard deviation for fixed SWNT mean length of $1.3\mu \mathrm{m}$ (a),(d), mobility versus charge carrier density and peak mobility versus mean SWNT length for fixed SWNT length standard deviation of $0.5\mu \mathrm{m}$ (b),(e), and for fixed ratio between the mean length and length standard deviation of 2 (c),(f). Dotted lines are guides to the eye.

Figure 8

Histograms of the junction power dissipation $\left(P\right)$ distribution (with color-coded visualization as insets) for different network densities and energetic disorder strengths (see subfigure titles). The SWNT length distribution is $1.1\pm 0.5\mu \mathrm{m}$. The carrier densities are those at the peak mobilities $(n_a=1.0\times {10}^{11}{\mathrm{cm}}^{-2},n_b=1.3\times {10}^{11}{\mathrm{cm}}^{-2},n_c=1.8\times {10}^{11}{\mathrm{cm}}^{-2},n_d=2.5\times {10}^{11}{\mathrm{cm}}^{-2},n_e=3.1\times {10}^{11}{\mathrm{cm}}^{-2}$, and $n_f=4.5\times {10}^{11}{\mathrm{cm}}^{-2})$.

Figure 9

(a) Normalized charge carrier mobility versus charge carrier density. The dotted line is a guide to the eye. (b)–(f) Visualization of current paths through a network with $45\mathrm{meV}$ energetic disorder at different charge carrier densities $n$ as labeled in (a). The 10% of SWNTs that carry the most current in the network have been colored (other SWNTs are light gray) and given a thickness proportional to the logarithm of the current (in a.u.) they carry. Simulation box sizes are $20\times 20\mu {\mathrm{m}}^2$.

Figure 10

Experimental [52] and simulated normalized mobility versus charge carrier density for (6,5) SWNT networks. Simulated results are given for the cases without energetic disorder and an energy disorder of 45 meV. The SWNT density was estimated from the experiments to be $30\mu {\mathrm{m}}^{-1}$. Dotted lines are guides to the eye.

Figure 11

Experimental [20] (a) and simulated (b) peak mobilities (energetic disorder: $45\mathrm{meV}$, simulation box size $20\times 20\mu {\mathrm{m}}^2)$ for two different SWNT length distributions of a (6,5) SWNT network. Dotted lines are guides to the eye.

Figure 12

Comparison of experimental [22] (open symbols) to simulated (closed symbols with error bars) results for the share in the current of each nanotube species in a mixed network of five chiralities: (a) (7,5) and (8,6), (b) (7,6), and (c) (8,7), (9,7). Dotted lines are guides to the eye.

Figure 13

(a) Comparison of experimental [22] to simulated carrier density dependence of the mobility in a mixed network of five chiralities (density: $10\mu {\mathrm{m}}^{-1}$, length distribution: $1.0\pm 0.5\mu \mathrm{m})$. Dotted lines are guides to the eye. (b)–(f) Simulated current maps at the carrier densities indicated in (a), with nanotubes that carry more than the average current per tube highlighted [red: (9,7) SWNTs, black: (7,5) SWNTs]. Gray lines represent SWNTs that carry less than the average current. The thickness of each drawn nanotube is proportional to the relative current $I_i/I_{\mathrm{avg}}$ it carries.