Electrical conductivity of nanorod-based transparent electrodes: Comparison of mean-field approaches

We mimic nanorod-based transparent electrodes as random resistor networks (RRNs) produced by the homogeneous, isotropic, and random deposition of conductive zero-width sticks onto an insulating substrate. We suppose that the number density (the number of objects per unit area of the surface) of these sticks exceeds the percolation threshold, i.e., the system under consideration is a conductor. We computed the electrical conductivity of random resistor networks versus the number density of conductive fillers for the wire-resistance-dominated case, for the junction-resistance-dominated case, and for an intermediate case. We also offer a consistent continuous variant of the mean-field approach. The results of the RRN computations were compared with this mean-field approach. Our computations suggest that, for a qualitative description of the behavior of the electrical conductivity in relation to the number density of conductive wires, the mean-field approximation can be successfully applied when the number density of the fillers $n>2n_c$, where $n_c$ is the percolation threshold. However, note the mean-field approach slightly overestimates the electrical conductivity. We demonstrate that this overestimate is caused by the junction potential distribution.

Figure 1

The conductive linear wire is placed in an external electrical field with a coordinate-dependent potential $V\left(x\right)$. The wire is characterized by its resistance per unit length, $R$, and its leakage conductivity per unit length, $G$.

Figure 2

Comparison of the dependencies of the electrical conductivity, $\sigma$, on the number density, $n$, for $L=32$, $R_{\text{s}}=0$, $R_{\text{j}}=1$ arbitrary units. Solid symbols correspond to our simulation results. The solid line corresponds to the least squares fit $\sigma /{\sigma }_{\text{j}}=0.019+0.21(n-n_{\text{c}})+0.026{(n-n_{\text{c}})}^2$. The dashed curve corresponds to the MFA (12).

Figure 3

Comparison of the dependencies of the electrical conductivity, $\sigma$, on the number density, $n$, for $L=32$, $R_{\text{s}}=1$ arbitrary units, $R_{\text{j}}=1$ arbitrary units. Solid symbols correspond to our simulation results. The solid line corresponds to the least-squares fit $\sigma /{\sigma }_{\text{j}}=-4.15+0.417(n-n_{\text{c}})$. The dashed curve corresponds to formula (12).

Figure 4

Comparison of the dependencies of the electrical conductivity, $\sigma$, on the number density, $n$, for $L=32$, $R_{\text{s}}=1$ arbitrary units, $R_{\text{j}}=0$. Solid symbols correspond to our simulation results. The solid line corresponds to the linear least squares fit $\sigma /{\sigma }_{\text{s}}=-0.26+0.49(n-n_{\text{c}})$. The dashed curve corresponds to formula (12).

Figure 5

Finite-size effect for a junction-resistance-dominated case.

Figure 6

Relative deviation between $\frac{2}{\langle {\lambda }_k\rangle l}tanh\left(\frac{\langle {\lambda }_k\rangle l}{2}\right)$ and $\langle \frac{2}{{\lambda }_{k}l}tanh\left(\frac{{\lambda }_{k}l}{2}\right)\rangle$ for different ratios between the stick resistance and junction resistance.

Figure 7

The normalized potential, $U/\mathrm{\Delta }U$, of each junction of the network is plotted against its normalized position, $x/L$, in one particular sample at $n-n_{\text{c}}=10$ for the junction-resistance-dominated case. The potential difference, $\mathrm{\Delta }U$, is applied along the $x$ axis. A line is presented for comparison. Inset: Distribution of the junction potentials relative to the line $U/\mathrm{\Delta }U=x/L$.

Figure 8

Distribution of the electrical current in a sample (solid fill) in comparison with the prediction of the mean-field approach (hatching) for the case when $R_{\text{s}}=R_{\text{j}}=1$ (arbitrary units); $n-n_{\text{c}}=10$.

Figure 9

Dependencies of the electrical current, $i$, on the position in the conductive wire, where 0 corresponds to the wire center. The current is averaged over all the wires in one sample. The dashed curves correspond to the evaluation obtained within the mean-field approach while the solid curves correspond to the simulation. The wire resistance and the junction resistance are equal, $R_{\text{s}}=R_{\text{j}}=1$ (arbitrary units); $n=15.6$ (two bottom curves) and $n=100$ (two upper curves).