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

Ashkan Behnam and Ant Ural*Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32611, USA

�Received 13 November 2006; revised manuscript received 30 January 2007; published 29 March 2007�

We study the geometry-dependent resistivity scaling in single-walled carbon nanotube films as a function ofnanotube and device parameters using Monte Carlo simulations. We first demonstrate that these simulationscan 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 tonanotube resistance ratio, nanotube density, nanotube length, and nanotube alignment on the film resistivity andits scaling with device width. Stronger width scaling is observed when the transport in the nanotube film isdominated by tube-tube contact resistance. Increasing the nanotube density decreases the film resistivitystrongly and results in a higher critical exponent and a lower critical width. Increasing the nanotube length alsoreduces the film resistivity, but increases both the critical exponent and the critical width. In addition, thelowest resistivity occurs for a partially aligned rather than perfectly aligned nanotube film. Increasing thedegree of alignment reduces both the critical exponent and the critical width. We systematically explain theseobservations, which are in agreement with previous experimental results, by simple physical and geometricalarguments. We also observe that, near the percolation threshold, the resistivity of the nanotube film exhibits aninverse power-law dependence on all of these parameters, which is a distinct signature of percolating conduc-tion. However, the strength of resistivity scaling for each parameter is different, and it depends on how stronglya particular parameter changes the number of conduction paths in the film. Monte Carlo simulations, aspresented in this paper, can help elucidate the effects of various parameters on percolating transport in filmsmade up of one-dimensional conductors, which is an essential step towards understanding and characterizingthe performance of these nanomaterials in electronic and optoelectronic devices.

DOI: 10.1103/PhysRevB.75.125432 PACS number�s�: 72.60.�g, 85.35.Kt, 73.63.Fg

I. INTRODUCTION

Single-walled carbon nanotube �CNT� networks and filmshave recently attracted significant research attention due totheir interesting properties. They are transparent, conductive,and flexible and have uniform physical and electronic prop-erties since individual variations in nanotube diameter andchirality1,2 are ensemble averaged.3–6 As a result, the repro-ducibility and reliability problems found in devices based onindividual nanotubes are solved, and CNT-film-based devicescan be mass produced in a cost-effective manner.3,6 Severalpromising applications of CNT films have recently beendemonstrated, such as thin film transistors,7,8 flexiblemicroelectronics,9–11 chemical sensors,12–15 and optoelec-tronic devices.16–20

One of the important characteristics of CNT films, whichneeds to be taken into account for any device application, isthat their resistivity depends strongly on nanotube and deviceparameters due to percolating conduction. For example, ithas been recently shown experimentally that resistivity ofnanotube networks depends strongly on device length, nano-tube density,4 and nanotube length in the film,21 and the re-sistivity of nanotube films depends strongly on the filmthickness.6 It has also been shown experimentally that theresistivity of carbon nanotube/polymer composites dependsstrongly on nanotube alignment in the composite.22 Further-more, we have recently reported experimental results dem-onstrating that the resistivity of nanotube films exhibits aninverse power-law dependence on device width and filmthickness near the percolation threshold.23

In order to investigate the physical and geometrical ori-gins of these experimental findings more systematically,

simulation and modeling techniques have to be employed.Although there have been several experimental reports onCNT films and networks, very little work has been done ontheir modeling and simulation. In a previous numerical simu-lation work, resistivity scaling with device length has beenstudied for different nanotube densities.24 In another work,the effect of nanotube alignment on the percolation probabil-ity of nanotube/polymer composites has been calculated bytwo-dimensional �2D� Monte Carlo simulations.22 However,a systematic study of the effects of various nanotube anddevice parameters on the CNT film resistivity and its scalingwith device width has not been reported previously.

In this paper, we study the geometry-dependent resistivityscaling in CNT films using Monte Carlo simulations by ran-domly generating nanotubes on stacked 2D rectangularplanes. We first demonstrate that these simulations can modeland fit the recent experimental results on the scaling of nano-tube film resistivity with device width. Then, we systemati-cally study the effect of four parameters: namely, tube-tubecontact resistance to nanotube resistance ratio, nanotube den-sity, nanotube length, and nanotube alignment on the CNTfilm resistivity and its scaling with device width. We thenexplain these simulation results by simple physical and geo-metrical arguments. A better physical understanding ofgeometry-dependent percolating transport in single-walledcarbon nanotube films is essential for characterizing andevaluating their performance in potential electronic and op-toelectronic device applications.

II. COMPUTATIONAL METHOD

Simulation of the electrical properties of the nanotubefilm was performed by randomly generating the nanotubes

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using a Monte Carlo process. In particular, each nanotube inthe film is modeled as a “stick” with fixed length lCNT. Theposition of one end of the nanotube and its direction on a 2Dplane are generated randomly. This process is repeated untilthe desired value for the nanotube density n in the 2D layeris achieved. Additional 2D layers are generated using thesame approach to form a 3D nanotube film. An example of a2D nanotube layer produced by this method is shown in Fig.1�a�. For comparison, Fig. 1�b� shows an experimentalatomic force microscope �AFM� image of a nanotube filmthat we have etched into a series of lines with width andspacing of 500 nm by e-beam lithography and reactive ionetching, as described previously.25

After the 2D nanotube layer is generated, the locations ofthe junctions between pairs of nanotubes �which we call in-ternal nodes� and between the nanotubes and source anddrain electrodes �which we call boundary nodes� are deter-mined by the simulation code. It has been experimentallyobserved that the nanotubes in the film have random in planeorientations but are mostly ordered to lie in stacked planes.23

As a result, in the 3D film created by stacking several 2Dnanotube layers, it is assumed that only nanotubes in nearestneighbor 2D layers form junctions, and the locations of thesejunctions are also determined by the simulation code. Thevalue of the contact resistance at each tube-tube junctiondepends on whether the junction is metallic/semiconducting�MS�, semiconducting/semiconducting �SS�, or metallic/metallic �MM�.26 In particular, it has been shown that MSjunctions have a significantly larger contact resistance thanMM or SS junctions due to the fact that they form Schottkybarriers.26 In our simulations, each tube-tube junction ismodeled by an “effective” contact resistance RJCT regardless

of the type of junction, following the simplified approach ofRef. 24.

After the locations of the junctions are determined, coor-dinates of the neighbors of each internal node, which aredefined as other nodes that are in direct electrical contactwith that node, are located. By this definition, a node must beconnected by nanotube segments to all of its neighbors. Theresistance between two neighboring nodes can easily be de-termined by calculating the length of the nanotube segmentconnecting them. The resistance of a single-walled carbonnanotube segment, RCNT, as a function of its length l is cal-culated using the expression

RCNT = R0� + l

�, �1�

where R0 is the theoretical contact resistance at the ballisticlimit ��6.5 k�� and � is the mean free path, assumed to be1 �m in our simulations.27,28

Writing Kirchoff’s current law �KCL� at each internalnode for a mesh with n internal nodes, we get a set of nequations with n unknowns, where the n unknowns are thevoltages Vn at each node. The voltages applied to the sourceand drain electrodes set up the necessary boundary condi-tions. Once the voltage at each node is solved, the total cur-rent in the film is calculated by a summation over the cur-rents flowing into the drain boundary nodes. Finally, theresistance, and, as a result, the resistivity � of the nanotubefilm in the linear regime, is calculated by dividing the volt-age drop between the source and drain electrodes by the totalcurrent in the film.

For each data point presented in this paper, 200 or moreindependent nanotube film configurations were randomlygenerated and their results were averaged in order to removestatistical variations in the data calculated from different re-alizations of the nanotube film. In addition, the percolationprobability P, defined as the probability that the nanotubefilm is conducting �i.e., the probability of finding at least oneconducting path between the source and drain electrodes� isalso calculated to complement the resistivity data.

The model can be improved in several ways if a moreprecise simulation is required. First, our model ignores non-linear effects at high bias voltages28 by assuming a fixednanotube resistance independent of bias voltage. This depen-dence of RCNT on Vn can be included by solving the currentcontinuity equation self-consistently with the Poison equa-tion in a recursive manner. Second, metallic and semicon-ducting nanotubes could be separately tracked and differentcontact resistance values could be used for SS, MM, and MSjunctions instead of an “effective” contact resistance RJCT.Furthermore, a more accurate model can be developed forMS Schottky junctions by taking into account the variationof the depletion region in the semiconducting nanotube withthe voltage at the contact node.26 Finally, a Gaussian lengthdistribution could be used for the nanotubes instead of afixed “effective” length lCNT.29

Our simulation results presented below show that, despitetheir relative simplicity, the models we have used can capturethe essential physics of the experimentally observed resistiv-

FIG. 1. �Color online� �a� A 2D random nanotube network gen-erated using a Monte Carlo process for a device with device lengthL=4 �m, device width W=4 �m, lCNT=2 �m, and n=4 �m−2. �b�Atomic force microscope �AFM� image of a nanotube film etchedinto a series of lines with width and spacing of 500 nm by e-beamlithography and reactive ion etching.

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ity scaling in single-walled nanotube films, and they can pro-vide valuable physical insights into the effects of variousnanotube and device parameters on the geometry-dependentresistivity scaling in these films.

III. RESULTS AND DISCUSSION

Figure 2�a� shows the normalized resistivity of the CNTfilm as a function of device width W. The symbols are theexperimental data points from our previous work23 for ananotube film with device length L=7, 50, and 200 �m andaverage thickness t=15 nm. The solid line represents the the-oretical fit to the experimental data using our simulations. InFig. 2�a�, both the theoretical and experimental resistivityvalues have been normalized by dividing the absolute valueof resistivity at each data point by its value at large W, wherethe resistivity saturates at a constant minimum value �min.The same normalization procedure is used for all figures inthis paper. When several resistivity curves are shown in a

single figure, all curves are normalized by the �min of onlyone of them, preserving their relative resistivity magnitudes.

In our simulations, we have used a device length of L=7 �m, as explained in detail below, and nanotube length oflCNT=2 �m, since the average nanotube length in the nano-tube films used in our experimental work is estimated to bearound 2 �m based on AFM images.23 In addition, we haveestimated the average “thickness” �i.e., vertical height� of asingle nanotube layer in the nanotube film to be �3 nm,which is about twice the average diameter of the CNT’smaking up the film. This estimation takes into account theextra volume that interwoven nanotubes occupy. As a result,five stacked 2D nanotube layers were used to model the ex-perimental film thickness of t=15 nm. With those nanotubeand device values fixed, two fitting parameters were used tomatch the experimental data: �i� nanotube density n pernanotube layer and �ii� resistance ratio �Rratio�, which wedefine as the ratio of the tube-tube junction contact resistanceRJCT to the theoretical contact resistance at the ballistic limitR0:

Rratio =RJCT

R0. �2�

By this definition, Rratio�1 implies tube-tube junctionresistance-limited transport, whereas Rratio�1 implies nano-tube resistance-limited transport in the film.

As seen in Fig. 2�a�, below a critical width WC of around2 �m, the resistivity starts to increase strongly above itsminimum saturation value, and near the percolation thresh-old the experimental data points can be characterized by aninverse power law of the form

� � W−�, �3�

where � is the critical exponent for width scaling. As we willshow in detail later, changing the resistance ratio affects onlythe critical exponent, whereas changing the nanotube densitychanges both the critical exponent and the critical width. Asa result, to fit the experimental data of Fig. 2�a�, first, thenanotube density was varied to match the critical width WC,and then the resistance ratio was varied to match the criticalexponent �. The best fit shown by the solid line in Fig. 2�a�was obtained with n=2 �m−2 per nanotube layer and Rratio=100. The extracted density value is consistent with the den-sity of nanotubes estimated from AFM images of the nano-tube films used in the experimental work.23 The error bar onthe nanotube density n extracted using the fitting procedureis ±15%. The extracted value for the resistance ratio can alsobe compared to estimates from previous work. In particular,it has been shown by several experiments that the tube-tubejunction resistance is larger than the resistance of the nano-tubes themselves by a factor of around 30–70 for SS andMM junctions and by a factor of at least an order of magni-tude higher than that for MS junctions.26 Since MS junctionsare more resistive, they do not contribute to conduction asmuch as SS and MM junctions do. As a result, Rratio definedin Eq. �2� is expected to be higher than but close to the range30–70 observed for SS and MM junctions. The value ofRratio=100 we have extracted using the fitting procedure is

FIG. 2. �Color online� �a� Log-log plot of normalized nanotubefilm resistivity versus device width. The individual data points arethe experimental results from our previous work �Ref. 23� for ananotube film with device length L=7, 50, or 200 �m �labeled bydifferent symbols� and average thickness t=15 nm. The solid linerepresents our simulation best fit to the experimental data withRratio=100 and n=2 �m−2. The other simulation parameters are L=7 �m, lCNT=2 �m, and t=15 nm. �b� Log-log plot of normalizedresistivity versus device length L for three different nanotube den-sities n, as labeled by different symbols in the plot, calculated usingour simulation code for a device with W=2 �m.

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consistent with these observations. The experimentally ob-served value of � could still be matched by the simulationeven when Rratio was changed by ±50%, putting a bound onthe sensitivity of the � fit to the value of Rratio.

It is clear from Fig. 2�a� that the simulation results agreewell with the experimental data. However, the decrease inresistivity for widths above 2 �m observed experimentally�although not as strong as below W�2 �m� is not capturedby the simulations. This discrepancy can be explained by thepresence of some nanotubes much longer than 2 �m in theexperimental nanotube films, whereas in the simulations allnanotubes were assumed to have a fixed length lCNT=2 �m.As we will show explicitly later, when lCNT increases, WCmoves to higher widths. As a result, the decrease in resistiv-ity observed experimentally is consistent with the existenceof some much longer nanotubes in the film. Furthermore,experiments on thicker films have shown that, even in thepresence of some longer nanotubes, the resistivity eventuallysaturates for device widths above 20 �m.23

It can also be seen from the experimental data in Fig. 2�a�that for longer devices, the inverse power-law behaviorseems to hold up to a wider device width. This can be ex-plained by the fact that for longer devices, there is a higherprobability that some conduction paths are eliminated sooneras the device width shrinks, resulting in a larger criticalwidth WC. However, the number of experimental data pointsis too few for a conclusive fit for each separate length, andthe simulation time becomes prohibitively long for devicelengths above approximately 10 �m. Due to these limita-tions, we have chosen to use the smallest length device �L=7 �m� for our simulations and fit all the experimental datapoints with a single inverse power-law curve. Although thisis an approximation, it can still capture the essential physicsof the experimentally observed resistivity scaling in single-walled nanotube films.

Figure 2�b� illustrates normalized resistivity versus devicelength L for three different nanotube densities calculated us-ing our simulation code for a device with W=2 �m. Allother simulation parameters are the same as those in Fig.2�a�. Unless otherwise mentioned, the same device and nano-tube parameters will be used in the remainder of the paper. Itis evident from Fig. 2�b� that the resistivity starts to decreasefrom its maximum constant value when the device length Lbecomes smaller, in agreement with previous experimentaland theoretical results.4,24 At large L, each conduction pathconsists of many nanotubes in series, and total length andnumber of junctions of each conduction path vary linearlywith the device length. As a result, the total film resistancealso varies linearly with L and the film behaves like a homo-geneous material with a constant resistivity. On the otherhand, when L becomes comparable to the nanotube length,the statistical distribution of nanotubes in the film can resultin short conduction paths consisting of only a few nanotubesconnecting source to drain, decreasing the total resistance ofthe film. Figure 2�b� shows that the effect of L on resistivityis more pronounced when the nanotube density is lower. Thisis because if the nanotube density is low, the number ofconduction paths between source and drain decreases drasti-cally as the device length increases, which increases the re-sistivity more strongly. This effect has also been observed in

previous theoretical and experimental work.4,24

In addition to matching the experimental data, we havealso used numerical simulations to systematically study theeffects of four parameters: namely, resistance ratio, nanotubedensity, nanotube length, and nanotube alignment on theCNT film resistivity and its scaling with device width.

A. Effect of resistance ratio

Figure 3 shows the normalized resistivity versus devicewidth for Rratio ranging from 10−2 to 104. It is evident fromFig. 3 that increasing Rratio increases the critical exponent �,but does not have a significant effect on the critical widthWC, since WC is only determined by nanotube density andgeometrical parameters. In particular, � increases from 0.52to 1.95 when Rratio changes from 10−2 to 104. When Rratio isvery low, nanotubes form low-resistance contacts with eachother and the film resistivity depends mainly on the totallength of the conduction paths. On the other hand, whenRratio is very high, the film resistivity mainly depends on thenumber of contacts in the conduction paths. When the devicewidth is reduced, the number of tube-tube junctions in theremaining conduction paths increases significantly due to therandom angular distribution of the remaining nanotube seg-ments, whereas the lengths of the remaining conductionpaths do not increase that strongly. As a result, increasingRratio results in a larger value of the critical exponent �.

To illustrate this point further, the inset of Fig. 3 depicts �as a function of Rratio. In the nanotube resistance-limitedtransport regime, � is constant at a value of about 0.5. Itstarts to increase as the resistance ratio increases above 1 andfinally saturates at a value of around 2 in the junctionresistance-limited transport regime. Since � also depends onseveral other parameters, such as nanotube density, length,and alignment, the lowest and highest values obtained fromthe inset of Fig. 3 do not give the absolute limits on �.

FIG. 3. �Color online� Log-log plot of normalized resistivityversus device width for Rratio ranging from 10−2 to 104, labeled bydifferent symbols. The values of the critical exponent � extractedfrom the slopes of the � vs W curves near the percolation threshold�i.e., at small W� are 0.5, 0.8, 1.55, and 1.95 for Rratio=10−2, 1, 102,and 104, respectively. The inset shows the critical exponent � as afunction of the resistance ratio. The symbols are the simulationresults, and the solid line represents a sigmoidal fit showing asmooth transition between ��0.5 and ��2.

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However, the inset does show that, other parameters beingconstant, the critical exponent exhibits a minimum and amaximum at the two limiting cases of the resistance ratio.

B. Effect of nanotube density

Figure 4�a� shows the normalized resistivity versus devicewidth for different nanotube densities ranging from 1 to3 �m−2. It is evident from this figure that as the densityincreases, three events take place: First, the resistivity of thefilm decreases. Adding more nanotubes to the film increasesthe number of conducting paths and decreases the averagepath length between source and drain and, as a result, de-creases the resistivity. Second, the critical width WC shifts tolower widths. At higher densities, there are more conductionpaths which have narrower width distributions. As a result,for dense films, width scaling effects become visible only atsmaller widths, which results in a smaller value of WC. Fi-nally, the critical exponent � increases. This is due to theincreased number of paths per width �relative to the total

number of paths� removed from the film because of thehigher nanotube density. In other words, for a film withhigher nanotube density, resistivity starts to increase at asmaller WC, but the rate of power law increase is muchfaster.

Figure 4�b� depicts these trends from a different point ofview by plotting the percolation probability P, defined ear-lier, versus width for different densities ranging from 1 to3 �m−2. For all densities, the probability of having a conduc-tion path goes from 1 at large widths to 0 at very smallwidths. However, as the width decreases, the percolationprobability transition profile from 1 to 0 is quite different fordifferent densities. At higher densities, the transition starts atsmaller widths, which indicates a smaller WC, but the transi-tion slope is steeper, which indicates a higher value of �.

For a general 3D percolation problem, it has been shownthat, near the percolation threshold, the resistivity exhibits aninverse power law dependence on density given by

� � �n − nC�−, �4�

where nC is the critical density at the percolation thresholdand is the critical exponent for density.30 We have alsostudied this dependence as shown in the inset of Fig. 4�b�,where normalized resistivity versus density is plotted for adevice with L=4 �m and W=4 �m. It is evident from thisinset that the resistivity as a function of density obeys Eq. �4�near the percolation threshold with a critical exponent =2.5 extracted from the simulation data. When more nano-tubes are added to the film, the number of conducting pathsincreases and the average path length between source anddrain decreases, both of which reduce the film resistivity.Furthermore, the rate of change of resistivity decreases athigh density values, since adding more nanotubes to an al-ready dense film is less likely to introduce a significant num-ber of new conduction paths or reduce the length and thenumber of junctions in existing paths.

C. Effect of nanotube length

It is clear from Figs. 2�a� and 2�b� that the devicegeometry-dependent resistivity scaling behavior in nanotubefilms is observed when the device length or width becomescomparable to the length of individual nanotubes making upthe film. Figure 5�a� shows normalized resistivity versus de-vice width for three different nanotube lengths ranging fromlCNT=1.5 to 4 �m, for a device with L=4 �m. It is evidentfrom this figure that as lCNT increases, the critical width WCmoves to higher widths and the critical exponent increases.For tube-tube junction resistance-limited transport �i.e., highRratio�, when lCNT is longer, the conduction paths have awider width distribution. Therefore, as W is decreased, theconduction paths start to get disconnected at higher values ofWC. Furthermore, due to the increased number of paths perwidth removed from the film because of the higher nanotubelength, the critical exponent also increases for films withlonger lCNT once the width decreases below WC.

We have also studied resistivity scaling as a function ofnanotube length for three different length-density relation-ships, as shown in Fig. 5�b�. In the first case, where the

FIG. 4. �Color online� �a� Log-log plot of normalized resistivityversus device width for four nanotube densities n ranging from 1 to3 �m−2, labeled by different symbols. The values of the criticalexponent � extracted from the slope of the � vs W curves near thepercolation threshold are 1.1, 1.35, 1.55, and 2.2 for n=1, 1.5, 2,and 3 �m−2, respectively. �b� Percolation probability P versus de-vice width for n ranging from 1 to 3 �m−2. The inset shows thelog-log plot of normalized resistivity versus nanotube density plot-ted for a device with L=4 �m and W=4 �m.

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density is kept constant �at a value of 2 �m−2�, the resistivityincreases sharply as lCNT decreases. Nanotubes with shorterlengths have a lower chance to make junctions and form acontinuous path between source and drain, which results in ahigher resistivity. Below a critical nanotube length, thisstrong resistivity scaling with lCNT can be fit by an inversepower law of the form �� lCNT

− , where =5.6 is the criticalexponent for nanotube length scaling extracted from thesimulation data. In the second case, the resistivity versusnanotube length is plotted when the density–nanotube-lengthproduct is kept constant. Physically, this corresponds to thecase when the net weight of the nanotubes vacuum filtered toform the film is kept constant while the nanotubes are cutinto smaller lengths. Figure 5�b� shows that the resistivitystill increases with decreasing nanotube length in this case,but with a critical exponent of 2.5, which is less than theprevious case. Although the increase in nanotube density inthis case decreases the resistivity as explained previously, itis not enough to compensate for the increase in resistivitydue to the decrease in lCNT. In the third case, the product of

density and the square of the nanotube length �i.e., nlCNT2 � is

kept constant. In this case, the resistivity remains almost con-stant, which indicates that the resistivity increase due to re-duced lCNT is balanced by the resistivity decrease due tohigher n. As a result, the resistivity dependence on lCNT andn near the percolation threshold can be fit by an inversepower law of the form �� �nlCNT

/ �−, where /�2. In gen-eral, the value of the ratio / depends on other device andnanotube parameters, such as L, W, and Rratio. However, forlarge L, large W, and a high Rratio, such as the third curveplotted in Fig. 5�b�, / equals approximately 2. In fact, ifwe divide the value of =5.6 extracted from the first curvein Fig. 5�b� by the value of =2.5 extracted from the inset inFig. 4�b�, we get �2.2, which is very close to 2. Further-more, this value of / physically makes sense because in-creasing the nanotube length increases the number of nano-tubes in a 2D area element proportional to the square of thelength, whereas increasing the density increases this numberonly linearly. It has been previously shown that the nanotubedensity at the percolation threshold is proportional to lCNT

2 inan isotropic 2D network.4

D. Effect of nanotube alignment

Finally, we study the effect of alignment �i.e., in-planeangular orientation� of nanotubes in the film on the resistivityand its scaling with device width. We define �a as the “nano-tube alignment angle,” which is a measure of the degree ofnanotube alignment in the film. Nanotubes in our simulationare generated at random angles � with respect to the horizon-tal axis in the range −�a����a, where 0��a�90°. When�a=90°, nanotubes are completely randomly distributed,whereas when �a=0°, they are perfectly aligned. Figure 6shows normalized resistivity versus width for three differentnanotube alignment angles: namely, �a=18°, 36°, and 90°. It

FIG. 5. �Color online� �a� Log-log plot of normalized resistivityversus device width for three nanotube lengths lCNT ranging from1.5 to 4 �m, labeled by different symbols. Simulation parametersare the same as in Fig. 2�a� except for L=4 �m. The values of thecritical exponent � extracted from the slope of the � vs W curvesnear the percolation threshold are 1.0, 1.55, and 2.05 for lCNT=1.5,2, and 4 �m, respectively. �b� Log-log plot of normalized resistivityversus nanotube length for three different n-lCNT relationships.Simulation parameters are the same as in Fig. 2�a�, except L=3 �m.

FIG. 6. �Color online� Log-log plot of normalized resistivityversus device width at three nanotube alignment angles �a rangingfrom 18° �partially aligned� to 90° �completely random�. The valuesof the critical exponent � extracted from the slope of the � vs Wcurves near the percolation threshold are 0.7, 1.35, and 1.55 for�a=18°, 36°, and 90°, respectively. The inset shows the log-log plotof normalized resistivity versus nanotube alignment angle �a for adevice with W=2 �m.

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is evident from Fig. 6 that the normalized resistivity WC and� all change as the nanotubes become more aligned �i.e., �abecomes smaller�. The value of �min initially decreases as �agoes from 90° to 36° because aligned nanotubes help to formconduction paths with fewer junctions and shorter lengthsbetween the source and drain electrodes. Surprisingly, how-ever, the resistivity starts to increase when the degree ofalignment in the film is increased even further �i.e., when�a=18°�. In that case, each nanotube forms too few junctionswith its neighbors, because nanotubes mostly lie parallel toeach other. Therefore, many existing conduction paths areeliminated and resistivity is increased.

Furthermore, it can be seen from Fig. 6 that as the nano-tube alignment angle �a decreases, both WC and � decrease.This is because as the nanotubes become more aligned, thewidth distributions of conduction paths in the film becomenarrower. As a result, width scaling becomes visible only atsmaller widths, which decreases WC, and relatively fewerconduction paths per width are removed, which decreases �.In short, although �min first decreases then increases as �adecreases, WC and � decrease monotonically with decreasing�a.

The inset in Fig. 6 illustrates the effect of alignment onfilm resistivity. The normalized resistivity versus nanotubealignment angle �a is plotted in this figure for W=2 �m. It isevident from this inset that the resistivity slowly decreases as�a is reduced and reaches a minimum value at �a�45°. As�a is reduced even further, the resistivity starts to increase.The reason for this behavior is the same as that discussed for�min in Fig. 6. In other words, the resistivity minimum occursfor a partially aligned, rather than perfectly aligned, nano-tube film. The location of this resistivity minimum can de-pend on other device and nanotube parameters, and there-fore, should be calculated for each device conditionseparately.

The inset of Fig. 6 also shows that, at small alignmentangles near the percolation threshold, the resistivity exhibitsan inverse power-law dependence on �a given by

� � �a− , �5�

where =2.9 is the critical exponent for nanotube alignmentextracted from the slope of the log-log plot. This strong scal-ing with �a is due to the fact that as the nanotubes alignparallel to each other, many conduction paths are eliminated,which increases the resistivity significantly. These results arein agreement with recent experimental work on the effect ofnanotube alignment on percolation conductivity in carbonnanotube/polymer composites.22

Our simulation results illustrate clearly that decreasing thedevice width W, decreasing the density n, decreasing thenanotube length lCNT, and aligning the nanotubes �i.e., de-creasing the nanotube alignment angle �a� all cause the nano-tube film to approach the percolation threshold. Furthermore,near the percolation threshold, the resistivity of the nanotubefilm exhibits an inverse power-law dependence on all ofthese parameters. In other words, regardless of how we ap-proach the percolation threshold, we observe an inversepower-law behavior, which is a distinct signature of perco-lating conduction.

However, the strength of resistivity scaling for each pa-rameter, represented by the corresponding critical exponent,is different. This strength depends on how strongly a particu-lar parameter changes the conduction paths in the film. Fur-thermore, these parameters are not completely independent.For example, as we have demonstrated explicitly, thestrength of resistivity scaling with device width depends onnanotube density, length, and alignment.

IV. CONCLUSIONS

In summary, we have used a Monte Carlo simulation plat-form to model percolating conduction in single-walled car-bon nanotube films. We have demonstrated that this simplemodel can fit the experimental results on resistivity scalingas a function of device width. In addition, we have demon-strated that geometry-dependent resistivity scaling in single-walled carbon nanotube films depends strongly on nanotubeand device parameters. In particular, we have studied theeffect of four parameters: namely, resistance ratio, nanotubedensity, length, and alignment on resistivity and its scalingwith device width. Stronger width scaling is observed whenthe transport in the nanotube film is dominated by the tube-tube contact resistance. Increasing the nanotube density de-creases the film resistivity strongly and results in a highercritical exponent � for width scaling and lower critical widthWC. Increasing the nanotube length also reduces the filmresistivity, but increases both � and WC. In addition, the low-est resistivity occurs for a partially aligned, rather than per-fectly aligned, nanotube film. Increasing the degree of align-ment reduces both � and WC. We have systematicallyexplained these observations, which are in agreement withprevious experimental work, by simple physical and geo-metrical arguments. We have also observed that, near thepercolation threshold, the resistivity of the nanotube film ex-hibits an inverse power-law dependence on all of these pa-rameters, which is a distinct signature of percolating conduc-tion. However, the strength of resistivity scaling for eachparameter, represented by the corresponding critical expo-nent, is different.

Our simulation results presented here are not limited tocarbon nanotubes, but are applicable to a broader range ofproblems involving percolating transport in networks, com-posites, or films made up of one-dimensional conductors,such as nanowires and nanorods. Monte Carlo simulations,as presented in this paper, can help elucidate the effects ofvarious parameters on percolating transport mechanisms infilms made up of one-dimensional conductors. This would bean essential step towards understanding and characterizingthe performance of these nanomaterials in electronic and op-toelectronic devices.

ACKNOWLEDGMENTS

The authors would like to thank Jing Guo for fruitful dis-cussions. This work was funded by a Southeastern Center forElectrical Engineering Education �SCEEE� junior faculty de-velopment award.

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*Author to whom correspondence should be addressed. Electronicmail: antural@ufl.edu

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