document: networks with randomly distributed single walled carbon nanotubes ( swcnts ) are emerging as novel materials for various applications , particularly as electronic materials . 
   @xcite this creates a need for the accurate determination of their fundamental electrical properties . 
   swcnt networks can be viewed as a two - dimensional network of conducting one - dimensional rods.@xcite these rods are either metallic or semiconducting with large dimension ratio(length / diameter @xmath2 ) , which leads to the following interesting behavior.@xcite since the nanotube - nanotube junction resistance is much larger than the nanotube resistance itself,@xcite such a network can be seen as a system with randomly distributed barriers for electrical transport . for a sub - monolayer network , 
   the junction resistance will dominate the overall resistance and the network resistance shows percolation behavior with nonlinear thickness dependence.@xcite for swcnt networks with thickness in the tens of nanometers , e.g. with tens of layers of swcnts , the conductance through the metallic tubes will dominate the conduction , and leads to metallic behavior of the overall resistance.@xcite    the frequency - dependent and electric - field - dependent conductivity have been investigated for individual swcnt@xcite and some types of swcnt networks.@xcite burke and co - workers measured the microwave conductivity of individual swcnts and investigated their operation as a transistor at 2.6 ghz.@xcite for swcnt networks with 30 nm thickness , our previous work finds the ac conductivity is frequency - independent up to an onset frequency @xmath3 of about 10 ghz , beyond which it increases with an approximate power law behavior.@xcite for thicker films with thickness in the range of tens of micrometers , p. peit _ 
   _ found that conductivities at dc and 10 ghz are almost the same.@xcite m. fuhrer _ 
   _ observed the nonlinearity of the electric field - dependent conductivity of relatively thick nanotube networks , and they claimed that the charge carriers can be localized by disorder in the swcnts with an approximate length scale @xmath4 of 1.65@xmath5m.@xcite v. skakalova _ 
   _ studied the current - voltage(_i - v _ ) characteristics of individual nanotubes and nanotube networks of varying thickness , and discussed the modulation of _ i - v _ characteristics by a gate - source voltage.@xcite however there is a lack of frequency and electric field dependent conductivity investigation for systematic variation of network densities , and also there is no correlation study between the onset frequency @xmath0 and the length scale determined from nonlinear transport . 
   the frequency- and electric - field - dependent conductivity investigation of different network densities is not only helpful to build a comprehensive understanding of the electrical transport properties of swcnt networks , but is also important for the sake of applications of these networks.@xcite for example , when using swcnt networks to construct high speed transistors , or as transparent shielding materials , knowledge of their frequency dependent electric transport properties are required . 
   also use of swcnt networks as inter - connects in circuits requires understanding of their electric field - dependent properties .    in this paper 
   , we investigated the frequency- and electric - field - dependent conductivity for swcnt networks with systematically varying densities . 
   we find that the onset frequency @xmath0 extracted from the frequency dependence measurement increases with the film thickness , while the length scale extracted from the electric field dependent measurement decreases at the same time . 
   using the measured @xmath0 and the extracted length scale @xmath1 for different films , we developed an empirical relation between the two transport measurement methods . 
   swcnt networks on polyethylene terephthalate ( pet ) substrate are prepared by the spraying method.@xcite arc - discharged nanotube powder from carbon solution inc . 
   was dispersed in water with sodium dodecyl sulphate surfactants . 
   the tubes were purified with nitric acid and left with a variety of functional groups.@xcite after the nanotubes were sprayed on the substrate , the samples were rinsed in water thoroughly to wash away the surfactant . 
   the network density , defined as the number of nanotubes per unit area , was controlled by the solution concentration(0.1 - 1 mg / ml ) and the spraying time(10 - 60 seconds ) . 
   the nanotubes formed bundles with size of 4 - 7 nm and length of 2 - 4 @xmath5m.@xcite the thicknesses of the films range from submonolayer to 200 nm corresponding to room temperature sheet conductance ranging from @xmath6 to @xmath7 s / square . 
   fig.[fig : figuresample ] shows sem images of films with two different coverage densities . 
   it is convenient to characterize the films by their sheet conductance rather than conductivity , because of the inhomogeneous nature of the conducting pathways and thus the difficulty in assigning a film thickness , particularly for submonolayer films . 
   = 2.5 in = 3.8 in    the corbino reflectometry setup , which is ideal for broadband conductivity measurements of resistive materials,@xcite was used to investigate the conductivity of the swcnt networks . 
   the measurement procedures are similar to our previous work.@xcite in order to measure the conductivity in as broad a frequency range as possible , two test instruments are used , the agilent e8364b network analyzer ( covering 10 mhz to 50 ghz ) and the agilent 4396b network analyzer ( covering 100 khz to 1.8 ghz ) , giving five and half decades of frequency coverage . the electric field dependent conductivity was measured with a standard 4-probe method in a well - shielded high precision dc transport probe.@xcite 
   the data of conductance vs. frequency are shown in fig.[fig : figurecond](a ) . 
   the films decrease in thickness from sample 1 ( 200 nm ) to sample 5 ( sub - monolayer ) . 
   for all the films , the real parts of the conductance keep their dc value up to a characteristic frequency and start to increase at higher frequency . 
   this kind of behavior has been widely observed for disordered systems.@xcite similar behavior has been found for carbon nanotube polymer composites close to the percolation threshold , in which the extended pair approximation model was applied to describe the observed phenomena.@xcite the carbon nanotube networks in fig . 
   [ fig : figurecond](a ) have densities well above the percolation threshold . 
   however , since the junction resistances between different tubes are much larger than the resistance of the tubes themselves,@xcite swcnt networks above the percolation threshold can still be seen as systems with randomly distributed barriers for electrical transport . in this case 
   the extended pair approximation model can be used to describe the real part of the ac conductance : @xmath8 where @xmath9 is the dc conductance , @xmath10 , @xmath11 is a constant and @xmath0 is the onset frequency.@xcite    = 3.2 in = 4.8 in    the obtained frequency dependent conductance in fig.[fig : figurecond](a ) fit well to the extended pair approximation model . fig . 
   [ fig : figurecond](b ) shows the relation between the fit film onset frequency and the dc sheet conductance . 
   the onset frequency changes from @xmath12 to @xmath13 rad / sec as the sheet conductance increases from @xmath6 to @xmath7 s / square . 
   the onset frequency of the ac conductance increases as the dc conductance increases . 
   the solid line fig.[fig : figurecond].(b ) has slope one , implying a linear relation between the onset frequency and their dc sheet conductance , or @xmath14 , over about 4 decades . 
   we also measured the frequency - dependent conductance of ultra - thin sub - monolayer nanotube networks , which are fabricated via the filtration method.@xcite we choose chloroform as the solvent instead of water to avoid the washing steps which can easily destroy the sub - monolayer film structure . 
   the network density is controlled by the concentration and volume of the solvent used.@xcite the conductance of these films are measured by a 4284a precision lcr meter , covering the frequency range from 20 hz to 1 mhz . for these films just above the percolation threshold , their frequency dependent conductance also fit well to the extended pair approximation model . in fig . 
   [ fig : swcntfig4 ] , we plot the onset frequency versus their dc sheet conductance . for comparison , we also present results of kilbride _ 
   et al._,@xcite for polymer - nanotube composites as well as the data from fig.[fig : figurecond](b ) for our films prepared by the spray method . 
   the three solid lines in fig . 
   [ fig : swcntfig4 ] all have slope one ( @xmath15 ) . here 
   an interesting result is that although the intercepts are very different , the slopes of the three different samples are essentially the same . 
   the difference in intercept is due to the different types of nanotubes . 
   different carbon nanotube sources have different bundle sizes and lengths , which causes the conductance difference . 
   the polymer - cnt composite has much smaller dc conductance than the other two , due to the separation of swcnts by polymer that leads to charge transfer barriers . despite these differences in detail 
   , there is a universal relationship between onset frequency and dc conductance . 
   = 3.4 in = 2.8 in    the linear proportionality @xmath16 is one of the characteristic features of ac conductance of disordered solids.@xcite for disordered systems , the universality of ac conductance has been studied by many researchers since its discovery in the 1950 s , and is usually referred to as taylor - isard scaling . 
   @xcite this inspires us to investigate the scaling behavior of the conductance and frequency for swcnt networks . 
   similar to kilbride _ 
   s work@xcite , taking the dc conductance @xmath9 and the onset frequency @xmath0 as scaling parameters , we plotted the ac conductance data for all samples in fig . 
   [ fig : swcntscaling ] . 
   the data show a scaling behavior with all data sets falling on the same master curve described by @xmath17 and @xmath18 . 
   = 3.4 in = 2.8 in    in order to explain the observed onset frequency change with the swcnt volume fraction for polymer - nanotube composites close to percolation threshold , kilbride _ 
   defined a characteristic length scale , the `` correlation length '' @xmath19.@xcite for our samples , which are well above the percolation threshold , we can similarly define the correlation length @xmath19 as the distance between connections in the sample , i.e. , the distance between junctions of the multiply - connected swcnt network . 
   when one measures the conductance at a given frequency , there is a typical probed length scale . at low frequencies ( corresponding to long time periods ) 
   , the carriers travel long distances in one - half ac cycle and the experiment investigates longer length scales . at high frequencies , the carriers travel short distances in one half ac - cycle and 
   the probed length scale is shorter . in the absence of an applied dc electric field , and assuming a random walk , the probed length scale @xmath20.@xcite    for low frequency , @xmath21 , the probed length scale spans multiple junctions of the swcnt network . 
   the junction resistances between different tubes are much larger than the resistances of the tubes themselves . 
   @xcite hence the conductance is small and equal to the dc conductance . as the frequency increases , @xmath22 becomes smaller than @xmath19 , @xmath23 , and the junction resistances have less contribution to the total resistance . 
   the intrinsic properties of the swcnt then dominate the measurement . 
   therefore the observed conductance increases as the frequency increases . 
   the transition frequency is expected to be that where carriers scan an average distance of order of the correlation length . 
   @xcite    the swcnt film can be treated as a random walk network , in which the probed length scale at a given frequency goes as @xmath24 . then with the onset frequency @xmath0 of samples we are able to estimate their correlation length @xmath25 . as the density of the swcnt networks increases , there are more junctions and the average distance between connections , the correlation length @xmath19 , becomes smaller and thus the onset frequency @xmath0 increases . 
   this is consistent with the behavior shown by the data in fig . 
   [ fig : figurecond ] . 
   from the measured frequency dependent conductivity , we observed that the correlation length @xmath19 varies as the thickness of the film increases . 
   due to the large resistance of junctions between different tubes , the carriers in the swcnt networks are easier to move on small length scales than on larger length scales . 
   this phenomenon inspires us to investigate whether the transport properties are nonlinear as a result . according to the works of m. s. fuhrer _ 
   _ , the localization behavior of a swcnt network can be studied by measuring the electric field dependent nonlinearity of the conductivity . 
   @xcite generally , increasing both the temperature and the electric field reduces the effects of localization . 
   hence a system typically displays a characteristic electric field at which nonlinear conductivity begins to appear . through measurement of the characteristic field 
   @xmath26 one can determine a length scale of the system : @xmath27 where @xmath1 corresponds to the size of the regions with good conductivity , and @xmath28 is the energy gained by a carrier in one conducting region on average.@xcite . 
   for our swcnt films @xmath29 depends on the average distance between connections and thus should vary with sample thickness . for relatively thick 
   films(@xmath30 ) with low sheet resistance , fuhrer _ et al . 
   _ found a temperature - independent length scale @xmath31 m in a swcnt film.@xcite    here we investigated the electric field - dependent conductance for a different class of films . 
   the studied samples were cut into rectangular pieces with length @xmath32 mm and width @xmath33 mm . 
   gold contact leads , which completely cover the width of the samples , were deposited on the surfaces of the samples to form a standard four probe pattern . 
   the measured swcnt networks have dimensions of @xmath34mm@xmath353 mm , between the two voltage leads . because swcnt films have a large in - plane thermal conductivity , @xcite and the probe used to perform the measurement 
   was specially designed to reduce heating effects@xcite we believe that the heating effect can be ignored in our measurement range . 
   this was confirmed by observing the voltage response of pulsed current input of several samples in the time domain with an oscilloscope . 
   = 3.2 in = 4.8 in    the dc resistance vs. electric field of a submonolayer film(sample 5 ) at different temperatures is shown in fig.[fig : swcntfig5](a ) . 
   we can see clearly that there is nonlinear behavior even at room temperature . as sketched in fig.[fig : swcntfig5](a ) , we extracted @xmath26 for each temperature , which is the characteristic electric field at which nonlinear conductance begins to appear.@xcite fig.[fig : swcntfig5](b ) shows the extracted @xmath26 vs temperature for this sample . as temperature increases , 
   @xmath26 also increases . 
   roughly , @xmath26 and temperature have a linear relationship and satisfy the equation @xmath36 , with @xmath37 . 
   note that this is much larger than the @xmath4 obtained in ref.@xcite . 
   this suggests that a different mechanism for hindering transport is active here , namely poor contacts between bundles of swcnts .    to investigate the dependence of @xmath29 on the film thickness , we measured the electric field - dependent conductance for samples with various thicknesses . 
   we found that the nonlinear behavior strongly depends on the thickness of the film . at low temperature , 
   all the samples show nonlinear behavior of resistance on electric field , and the critical field @xmath26 is larger for thicker samples and smaller for thin samples . at room temperature , 
   all the other samples are purely ohmic(linear ) in electric fields up to 100 v / cm , except for the thinnest sample , sample 5 , which showed electric field - dependent conductance at all temperatures . through the critical field @xmath26 
   , we extracted the @xmath29 for different density films , shown in fig.[fig : swcntfig6 ] as red - dots . 
   clearly @xmath29 depends on the density of the swcnt networks . for the thicker film with larger swcnt density , which has larger sheet conductance , 
   the @xmath29 is smaller . 
   the lower density films with larger average distance between the junctions has larger @xmath29 . 
   = 2.6 in    although the localization behavior may be affected by the properties of individual swcnts @xcite , the above result shows that the observed behavior is more likely a consequence of poor electric conduction through inter - bundle junctions . as the thickness and density increase , there are more and more connections , which decreases the length scale @xmath29 , as shown in fig.[fig : swcntfig6 ] .    when the film thickness increases to a large value , the density of the swcnts and connections of the film will finally saturate and stop increasing . for those very thick swcnt films ( also called mats ) , their correlation length @xmath19 , and also the length scale @xmath29 , as well as the onset frequency , 
   will be independent of the thickness , but depend on the properties of individual bundles , which include both the disorder of the individual nanotube and also the geometry of the bundles . 
   hence the difference of swcnt sources and purities can change the onset frequency . 
   we believe this is the reason that our previous work found that the ac conductivity follows the extended pair approximation model with an onset frequency @xmath0 about 10 ghz , while peit _ 
   _ found that swcnt networks have the same conductivities at dc and 10 ghz for swcnt networks with thickness in the range of tens of micrometers.@xcite 
   the frequency dependent conductivity shows that the onset frequency @xmath0 increases with the sample thickness and gives an estimation of the correlation length @xmath19 of the sample , @xmath38 . 
   @xmath19 and @xmath39 decrease as the sample thickness increases . 
   the dc nonlinear conductivity measurement shows that the length scale @xmath29 of the samples also decreases with increasing sample thickness . 
   qualitatively @xmath39 and @xmath29 have the same dependence on the sample thickness . based on the above consideration 
   , it is reasonable to expect some correlation between the onset frequency @xmath0 and the length scale @xmath29 , and we proposed the following relation , @xmath40 where @xmath41 is a fitting parameter , which can be obtain from our experiment data , @xmath42 . 
   fig.[fig : swcntfig6 ] shows the comparison of the nonlinear transport length scale @xmath29 measured directly through electric - field dependent dc conductance measurements(red - dot ) and the right - hand side of eq . 
   [ eq : emperequation ] from the frequency dependent measurement(black - star ) . 
   they are consistent with each other . in this way we obtain an empirical relation between the frequency - dependent and the electric field - dependent conductivity . 
   the parameter @xmath41 should depend on the diffusion properties of carriers in the swcnt films.@xcite writing @xmath43 , where d is a diffusion constant and @xmath44 is the corresponding time scale , we take @xmath45 . assuming the einstein relation @xmath46 , we can estimate the mobility of carriers on the swcnt network , which is on the order of @xmath47 . 
   this is much smaller than that of the individual single walled carbon nanotube , which can be on the order of @xmath48 at room temperature.@xcite in the swcnt networks , due to the randomly distributed barriers for electrical transport , the mobility is expected to be much smaller than the swcnt itself . 
   the carbon nanotube film - based transistors have reported mobilities on the order of @xmath49,@xcite which is consistent with our estimation . to get a deeper physical understanding of these networks , more investigation and theoretical work are needed . 
   for example , measuring the temperature dependence of the frequency - dependent conductivity for swcnt networks with various densities would be one approach.@xcite 
   to conclude , we systematically studied the frequency and electric field dependent conductivity of swcnt films of various thicknesses . 
   we found the poor interbundle junctions affect the transport behavior of the carriers in the swnct films , which causes strong thickness dependence of characteristic length scales . 
   a thinner film has larger correlation length @xmath19 and length scale @xmath29 , so it has a smaller onset frequency @xmath0 , whereas a thicker film with smaller correlation length @xmath19 and length scale @xmath29 , has a larger onset frequency @xmath0 . 
   an approximate empirical formula relating the onset frequency @xmath0 and nonlinear transport length scale establishes contact between the frequency and the electric field dependent conductivity , which helps us to understand the electric transport properties of the swcnt films . 
   + the authors would like to thank m. s. fuhrer , c. j. lobb , weiqiang yu and bing liang for their help and discussions on this work . 
   this work has been supported by center for nanophysics and advanced materials of the university of maryland and the national science foundation grant nos . 
   ( dmr-0404029 ) , ( dmr-0322844 ) , and ( dmr-0302596 ) .
summary: we present measurements of the frequency and electric field dependent conductivity of single walled carbon nanotube(swcnt ) networks of various densities . 
   the ac conductivity as a function of frequency is consistent with the extended pair approximation model and increases with frequency above an onset frequency @xmath0 which varies over seven decades with a range of film thickness from sub - monolayer to 200 nm . 
   the nonlinear electric field - dependent dc conductivity shows strong dependence on film thickness as well . 
   measurement of the electric field dependence of the resistance r(e ) allows for the determination of a length scale @xmath1 possibly characterizing the distance between tube contacts , which is found to systematically decrease with increasing film thickness . 
   the onset frequency @xmath0 of ac conductivity and the length scale @xmath1 of swcnt networks are found to be correlated , and a physically reasonable empirical formula relating them has been proposed . 
   such studies will help the understanding of transport properties and benefit the applications of this material system .