Random stick network analysis of electronic transport in carbon nanotube thin films

Electronic transport in metallic carbon nanotube (CNT) thin films is investigated theoretically using a random stick network analysis combined with circuit theory. To discuss the dependence of the sheet conductance on the CNT alignment in the films, an angle φmax ( 0 ° ≤ φ max ≤ 90 ° ) is introduced as a new parameter to characterize the CNT alignment: φmax = 0° and φmax = 90° correspond to perfectly aligned and perfectly random network structures, respectively. The results indicate that the conductance exhibits a maximum around φmax = 55°. In addition, the conductance shows a local minimum around φmax = 30° because CNTs tend to intersect with incommensurate junctions that have a high contact resistance.

T ransparent conducting films have a wide-range of applications, such as flat displays, solar panels, and touch sensors. 1,2) Indium tin oxide (ITO) is the most widespread material for transparent conducting films, 3) however, ITO has some problems with respect to flexibility and mechanical strength, which are essential to the realization of future flexible devices. [4][5][6] As a potential candidate for flexible transparent conducting films, carbon nanotube (CNT) thin films have been attracting much attention, [7][8][9][10][11][12] because a CNT has a Young's modulus of ∼0.32-1.47 TPa, 13) a tensile strength of ∼100-600 GPa, 14) and a fracture strain of ∼10%-15%. 15) However, the electrical conductivity of CNT thin films is lower than that of ITO at present; therefore, it is desirable to enhance the electrical conductivity while maintaining flexibility and transparency. [16][17][18] Numerous experiments have been conducted by several groups to enhance the electrical conductivity of CNT thin films. [19][20][21] They have found that the electrical conductivity is strongly dependent on the alignment of CNTs in the film. Inspired by these experimental findings, theoretical studies to provide guidelines for the enhancement of electrical conductivity in CNT thin films have been attempted. 22) Reference 22 theoretically investigated the electrical conductivity of CNT thin films using a random stick network (RSN) model combined with circuit theory. The electrical conductivity was found to depend on the alignment of CNTs and a maximum electrical conductivity was shown for a certain alignment. 23,24) Although Behnam et al. assumed that the resistance at CNT-CNT interfaces (i.e., contact resistance) is independent of the contact angle, we propose that the contact angle dependence of the contact resistance has a strong influence on the conductivity of CNT thin films. In this work, we thus investigate the dependence of the electrical conductivity on the CNT alignment in thin films with different CNT densities using the RSN model combined with circuit theory. [25][26][27][28] In the present work, a CNT thin film is modeled as a RSN, where CNTs are assumed to be straight lines, as shown in Fig. 1(a). The square film is sandwiched between Electrode 1 and Electrode 2, where both the length L and the width W of the film are equal. The (10,10) CNTs, of which the length l CNT is 0.5 μm, are randomly distributed in the film and a periodic boundary condition is imposed on the width direction (y-axis direction). The CNT areal density σ, is defined as the number of CNTs per square micrometer (μm 2 ). It should be noted that the electrical conductivity is independent for a sufficiently large film. In the present simulation, the size of the film is taken to be W = L = 5 μm, which was determined to be sufficiently large to achieve this.
The CNT alignment is characterized by the alignment angle j max , because the angle between the axial direction of the CNT and the x-axis j, is assumed to vary randomly between −j max and +j max . The range of j max is from 0°to 90°, as shown in Fig. 1(b). For instance, when j max is 20°, most of the CNTs are aligned towards the x-axis direction, as shown in Fig. 1(c). On the other hand, when j max is 90°, the CNTs are completely randomly distributed, as shown in Fig. 1(d). The positions of CNTs and j are set randomly using the Mersenne Twister algorithm. 29) CNT networks are mapped on the RSN model to calculate the electrical conductivity of CNT thin films. Figure 2(a) represents a part of a RSN model consisting of three connected CNTs (CNT a,b,c ) and Fig. 2(b) gives an equivalent circuit corresponding to the RSN model. The equivalent circuit is described as a combination of CNT resistance R CNT and contact resistance R cont as shown in Fig. 2(b). R CNT is generally dependent on the CNT length l CNT and R cont is dependent on the contact angle Θ between two CNTs. 30) As for R CNT , the electrical resistance of a (10,10) CNT at room temperature (300 K) was calculated in our previous theoretical study as the length dependent actual resistance R CNT . 31) In our previous article, we showed that R CNT is proportional to l CNT up to 0.5 μm because R CNT = 9.6 l CNT + R q , where R q is the quantum resistance. For R cont , Ref. 30 investigated the contact resistance for two (10,10) CNTs as a function of the contact angle Θ, based on the tight-binding model. They found that R cont with commensurate stacking of the lattice of two CNTs is lower than that for incommensurate stacking. In the case of (10,10)-(10,10) CNT contact, whereas R cont exhibits local minimum values around j = 0, 60, 120, and 180°(commensurate stacking), it exhibits local maximum values around j = 30, 90, and 150°(incommensurate stacking).
In our simulation, we adopt data from Ref. 31 for R CNT . For example, to estimate R CNT of CNT b , as shown in Fig. 2(b), the distance d 12 between contact points 1 and 2 is evaluated, and R CNT is calculated based on R CNT = 9.6 d 12 (R q is ignored in this case). As for R cont , data in Fig. 2(c) are used. For instance, R cont at contact point 1, which is dependent on Θ ab between CNT a and CNT b , is estimated as a function of Θ, as shown in Fig. 2(c). Other R CNT and R cont are also estimated in a similar way.
The sheet conductance of the films is calculated as follows. When the voltage drop V is applied between Electrode 1 and Electrode 2, the total current I flowing through the film is calculated using the circuit simulation software LTspice IV ver. 4.231 (Linear Technology, USA). 32) The total resistance of the CNT network, R, is calculated using Ohm's law, R = V/I, and the conductance is then given by (1/R)(W/L). Repeating these procedures for various configurations of CNTs, we can obtain the averaged sheet conductance 1 s for the samples. In this simulation, all data are obtained by taking the ensemble average for N = 100 samples with respect to σ and j max . Figure 3(a) shows the dependence of G s on j max for various σ. For a fixed σ, G s has a maximum near j max = 55°, while G s shows a dip near j max = 30°, as indicated by the red arrows. In the following, we describe the dependence of G s on j max for three distinct regimes: j max ≈ 0°, j max ≈ 90°, and j max ≈ 30°.
Here we consider the G s behavior near j max = 0°. As shown in Fig. 3(b), G s is closely zero in the range of  j j  < 0 max c , where j c is the percolation threshold. In this region, CNTs in the film are almost perfectly aligned along the x-axis and most of the CNTs do not intersect each other. As a result, there are few conduction paths that connect Electrode 1 and Electrode 2, so that On the other hand, when j max exceeds j c , G s increases steeply because the conduction paths begin to form. For a fixed j max , j c decreases with increasing σ as detailed in our supplementary materials, available online at stacks.iop.org/ APEX/12/055006/mmedia. This is because conduction paths can be formed with lower j c as σ increases.
Here, we consider the dependence of G s on j max based on percolation theory for the RSN model. According to the theory, the behavior of G s near j c can be described by: where t j is the critical exponent and a j is a prefactor that is used as a fitting parameter for the numerical data in Fig. 3(b). The dashed curves for the best fit for t j and a j are shown in Fig. 3(b), where we obtain t j = 1.33, which is in agreement with the universal value predicted by percolation theory. 33)   Therefore, the behavior of G s near j c can be explained by percolation theory.
Here we focus on the G s behavior near j max = 90°. Figure 3(a) shows that G s decreases with increasing j max in the range of To show the decrease in G s quantitatively, we introduce a new quantity that represents the CNT alignment: where S(j, j max ) is the probability distribution for j and is an orthogonal projection of the CNT length l CNT along the x-axis for a certain CNT. j ( ) X CNT max represents the average CNT length along the xaxis for a fixed j max , which is obtained by averaging X CNT (j) weighted by S(j, j max ). j is distributed uniformly in the range of   j j j -; max max therefore, j ( ) X CNT max can be calculated as: j ( ) X CNT max represents the CNT alignment well. For instance, when X CNT is a maximum (j max = 0°), all CNTs are perfectly aligned along the x-axis. On the other hand, when X CNT is a minimum (j max = 90°), the CNTs are randomly orientated.
We now consider the effect of CNT alignment on G s in terms of j ( ) X CNT max . When X CNT is small, more CNTs forming a connection between Electrode 1 and Electrode 2 are required than aligned CNTs. In this case, CNTs form the conduction paths that contain more contact points than aligned CNTs, and G s decreases with X CNT . Figure 3(a) shows that G s can be well fitted by near j max = 90°(a x and β x are fitting parameters) as shown by the solid curves. Thus, G s decreases monotonically with increasing j max due to the effect of the decrease in X CNT .
We next consider the physical origin of the dips indicated by the red arrows in Fig. 3(a) near j max = 30°. The dips are due to the peaks in R cont around Θ = 30°and 150°in the case of (10,10)-(10,10) CNT contact, which was not previously taken into account. 22) To clarify this, we compare the dependence of G s on j max for our model and the constant R cont model previously reported. 22) In the constant model, R cont is independent of the contact angle and takes a fixed value. We also calculate the dependence of G s on j max with the constant R cont model for the same range of σ and show the result in our supplementary materials as shown in Fig. S3. The similarity between two models is that G s has the maximum by the adjustment of CNT alignment. On the other hand, no dips in G s are seen in the constant R cont model (see Fig. S3 in our supplementary materials). Near j max = 30°, most of CNTs tend to intersect with incommensurate  junctions to eventually form a high-resistance network, which causes the dips at j max ≈ 30°. To clarify this quantitatively, we introduce the average value of the contact resistance R cont for a network with j max : where F (Θ, j max ) is the histogram of Θ with respect to j max . Figure 4(a) shows F (Θ, j max ) for σ = 200 μm −2 . The peak of the histogram seems to correspond to j max : when j max = 15°, 25°, and 35°, the peak of the histogram is located at Q ≈ 15°, 25°, and 35°, respectively. Figure 4(b) shows j ( ) R cont max for σ = 200 μm −2 . R cont has a maximum near j max = 30°, which corresponds to the dip in G s in Fig. 3(a). Therefore, the dips are due to the R cont dependence on Θ.
We theoretically investigated the effects of the CNT-CNT contact resistance on the sheet conductance of CNT thin films using the RSN model combined with circuit theory. The averaged sheet conductance has a maximum when j max ≈ 55°. This is consistent with the results obtained in a previous study 22) and our supplementary materials. However, considering the CNT-CNT contact angle dependence of the contact resistance, we revealed that dips appear at a specific alignment angle. This is because CNTs tend to intersect with incommensurate junction and form a high-resistance network. We therefore suggest that for the design of highconductance CNT thin films, it is essential that CNTs do not intersect with the incommensurate junctions.