The dependence of the optoelectrical properties of silver nanowire networks on nanowire

length and diameter

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Nanotechnology 23 (2012) 185201 (9pp) doi:10.1088/0957-4484/23/18/185201

The dependence of the optoelectricalproperties of silver nanowire networks onnanowire length and diameter

Sophie Sorel1, Philip E Lyons1, Sukanta De1, Janet C Dickerson2 andJonathan N Coleman1

1 Centre for Research on Adaptive Nanostructures and Nanodevices and School of Physics,Trinity College Dublin, Dublin 2, Ireland2 Seashell Technology, 3252 Holiday Court Suite 115, La Jolla, CA 2037, USA

E-mail: colemaj@tcd.ie

Received 3 February 2012Published 13 April 2012Online at stacks.iop.org/Nano/23/185201

AbstractWe have characterized the optoelectrical properties of networks of silver nanowires as afunction of nanowire dimension by measuring transmittance (T) and sheet resistance (Rs) fora large number of networks of different thicknesses fabricated from wires of differentdiameters (D) and lengths (L). We have analysed these data using both bulk-like andpercolative models. We find the network DC conductivity to scale linearly with wire lengthwhile the optical conductivity is approximately invariant with nanowire length. The ratio ofDC to optical conductivity, often taken as a figure of merit for transparent conductors, scalesapproximately as L/D. Interestingly, the percolative exponent, n, scales empirically as D2,while the percolative figure of merit, 5, displays large values at low D. As high T and low Rsare associated with low n and high 5, these data are consistent with improved optoelectricalperformance for networks of low-D wires. We predict that networks of wires with D = 25 nmcould give sheet resistance as low as 25 �/� for T = 90%.

S Online supplementary data available from stacks.iop.org/Nano/23/185201/mmedia

(Some figures may appear in colour only in the online journal)

1. Introduction

Transparent conductors (TCs) are materials which can beformed into thin films with high transparency yet lowsheet resistance and are an important part of modernelectronics [1]. These materials are most commonlyencountered as transparent electrodes in displays. For suchapplications, industry requires films with transparency T >

90% in the visible region and sheet resistance Rs < 100 �/�[2]. However, some applications such as electrodes in solarcells would benefit from even higher performance materials,capable of achieving T > 90% and Rs < 10 �/� [3].Currently, the most common TC materials and the onlyones to approach these stringent requirements are the dopedmetal oxides, most commonly indium tin oxide (ITO) [4].However, due to depleted supplies of indium, the price

of ITO has increased dramatically in recent years, raisingdoubts over its future usefulness. ITO also suffers fromtechnical problems; its brittleness makes it incompatiblewith applications in flexible displays [5]. In addition, thecost of scaling deposition to large areas, coupled with therelatively high sheet resistances reported for low temperaturegrowth [6, 7], make ITO unsuitable for future large areaplastic electronics. Thus, many believe doped metal oxides areprobably not the preferred TC for future applications.

A large component of the search for an ITO replacementhas been the study of networks of nanoconductors [8] suchas carbon nanotubes [9–20], graphene [21–29] and metallicnanowires [23, 30–42]. The advantage of these networksis that they retain their conductivity under flexing [23,10, 11], an important consideration given that many futuredisplays are likely to reside on flexible plastic substrates.

10957-4484/12/185201+09$33.00 c© 2012 IOP Publishing Ltd Printed in the UK & the USA

Nanotechnology 23 (2012) 185201 S Sorel et al

In addition, these networks can be deposited cheaply fromsolution over large areas by spraying [13, 18, 36, 22].However, of these materials, only metallic nanowire networkshave come close to surpassing minimum industry standards,displaying sheet resistance for T = 90% as low as Rs =

49 �/� for spray cast networks [36] and Rs = 10 �/� forspin coated networks [39]. However, while these networksare extremely promising, there is still much room forimprovement. Recently, we showed that careful control of thespray deposition parameters leads to more uniform networkswith significantly improved optoelectrical properties [36].However, another route to improved performance would beto change the physical dimensions of the wires themselves.For example, it has been predicted that by reducing thewire diameter and increasing the wire length the networkconductivity can be significantly improved [43, 44]. Thisshould translate into reductions in Rs for networks thin enoughto have high T .

However, such improvements have not been clearlydemonstrated experimentally. In addition, it is not clearwhether increasing the nanowire conductivity by reducingnanowire diameter would translate into better optoelectricalperformance. This is largely because it is not known howvarying the nanowire dimensions will affect the opticalproperties of the network. For example, if the networkabsorbance increases with decreasing wire diameter, theadvantages of any conductivity increases would be largelynegated. Thus, it is critical to understand how both the opticaland electrical properties of nanowire networks depend on thedimensions of the nanowires used.

The aim of this work is to understand the effect ofnanowire dimensions on the optical and electrical propertiesof nanowire networks. To this end, we prepared a rangeof networks of silver nanowires with varying diameter andlength. Measurements of T and Rs were analysed using bothbulk and percolative models to give quantities such as thebulk DC conductivity, the optical conductivity (related to theabsorption coefficient), the percolative figure of merit andpercolation exponent (see below). This allowed us to quantifythe dependence of optical and electrical properties on bothnanowire length and diameter.

1.1. Theoretical background

In order to understand the optoelectrical properties ofnanowire networks, it is necessary to understand the factorswhich control the transmittance and sheet resistance. Ingeneral, both parameters are controlled by the networkthickness, t. For example, the transmittance can be expressedusing [45]

T =

(1+

Z0

2σOpt

)−2

. (1)

Here, Z0 is the impedance of free space (377�) and σOp is theoptical conductivity, which like T is a function of wavelength,λ [9]. (We note that expansion of equation (1) shows it to beidentical to the Lambert–Beer law, T = e−αt, to first order.

This shows σOp to be related to the absorption coefficient, α,by σOp ≈ α/Z0.)

By combining equation (1) with the definition of sheetresistance (for a bulk-like film),

Rs = (σDC,Bt)−1 (2)

(σDC,B is the bulk DC conductivity of the film i.e. theconductivity of a film which is thick enough such that theDC conductivity is thickness independent) to eliminate t, oneobtains a relationship between T and Rs [16]:

T =

(1+

Z0

2Rs

σOp

σDC,B

)−2

. (3)

Here σDC,B/σOp can be considered a figure of merit, withhigh values giving the required properties (high T coupledwith low Rs). However, while equation (3) tends to providean excellent fit to (T,Rs) data for relatively thick networks ofvarious nanoscale conductors, the agreement breaks down forthinner networks [23, 10, 46, 47]. As thin films are requiredto give high T , this can be a significant problem. Recently,we showed that equation (3) does not usually apply in thetechnologically relevant region around T = 90% [30]. This isbecause, for thin networks, the DC conductivity is no longerthickness independent but follows a percolation-like thicknessdependence [48]: σDC ∝ (t − tc)n, where t is the estimatedthickness of the network, tc is the critical thickness, i.e. thepercolation threshold, and n is the percolation exponent.Approximating t − tc ≈ t, this leads to a new relationshipbetween T and Rs which applies to thin networks [30]:

T =

[1+

15

(Z0

Rs

)1/(n+1)]−2

(4)

where we denote 5 as the percolative figure of merit (FoM),

5 = 2[σDC,B/σOp

(Z0tminσOp)n

]1/(n+1)

. (5)

Here, tmin is the thickness below which the DC conductivitybecomes thickness dependent (i.e. equation (4) applies forthicknesses below tmin). Analysis of these equations showsthat large values of 5, but low values of n, are required toachieve low Rs coupled with high T [30].

Inspection of equations (3)–(5) shows that the opticaland electrical properties of nanowire networks are completelydescribed by the parameters σDC,B/σOp, tminσOp and n. Thus,in order to understand the dependence of network propertieson wire dimensions, it will be necessary to measure thedependence of these properties (and the composite property,5) on nanowire length and diameter.

2. Experimental procedure

Silver nanowires were synthesized by Seashell Technologies(www.seashelltech.com/) and supplied as suspensions inisopropyl alcohol (C = 12.5 mg ml−1 as measured bythermogravimetric analysis). A small volume of the dispersionwas diluted to 0.001 mg ml−1 with Millipore water. In

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Nanotechnology 23 (2012) 185201 S Sorel et al

Table 1. Measured parameters for each wire type. Note that thevalues of σDC,B/σOp and 5 given in this table are as-measured andnot rescaled to represent the case when L = 5 µm.

D (nm) L (µm) σDC,B/σOp Tx (%) 5 n

61 4.0 140 37 42 0.5964 3.4 151 67 56 0.6788 4.4 153 61 20 1.53

127 8.7 205 83 30 2.44

general, this was subjected to 5–20 min low power sonicationin a sonic bath (Model Ney Ultrasonic).

Silver nanowire films were prepared by vacuum filtrationof the above dispersions using porous mixed cellulose esterfilter membranes (MF-Millipore membrane, mixed celluloseesters, hydrophilic, 0.22 µm, 47 mm). The deposited filmswere dried on a hotplate (50 ◦C) followed by a wet transferto a polyethylene terephthalate (PET) substrate using heatand pressure [20]. The cellulose filter membrane was thenremoved by treatment with acetone vapour and subsequentacetone liquid baths followed by a methanol bath [20]. Thefilm was 36 mm in diameter.

Scanning electron microscopy measurements were madeusing a Zeiss Ultra scanning electron microscope. Chargingwas avoided by transferring the AgNW film from the cellulosemembrane to a silicon substrate coated with a thin goldfilm. Optical transmission spectra were recorded using a CaryVarian 6000i, with a sheet of PET used as the reference. Sheetresistance measurements were made using the four probetechnique with silver electrodes of dimensions and spacingstypically of the order of millimetres in size and a Keithley2400 source meter.

3. Results and discussion

3.1. Scanning electron microscopy of wires and films

In order to understand the effect of nanowire dimensions onthe optical and electrical properties of nanowire networkswe acquired a number of batches of silver nanowiresfrom Seashell Technologies. These had mean diameters, D,between 61 and 127 nm and mean lengths between 3.4and 8.7 µm (see SI for histograms (available at stacks.iop.org/Nano/23/185201/mmedia), figures 1(A) and (B) forimages and table 1 for data). These wires were provided asdispersions in isopropanol and could easily be formed intonetworks by vacuum filtration on to porous membranes [23].These networks could then be transferred onto PET orglass substrates by standard methods [20]. Dependingon the combination of dispersion volume and nanowireconcentration, the deposited nanowire mass can be controlled,resulting in the control of network thickness. Examples ofboth sparse and thick networks are shown in figures 1(C)and (D) respectively. Once these networks have beendeposited, it is straightforward to characterize their electricaland optical properties.

Figure 1. SEM images of (A) low diameter silver nanowires(D = 61 nm), (B) large diameter silver nanowires (D = 127 nm),(C) a sparse network and (D) a thick network.

3.2. Dependence of electrical properties on nanowire length

In order to understand the dependence of electrical and opticalproperties on both D and L, it will be necessary to treatthese parameters separately. To do this, we take nanowireswith a particular mean diameter (D = 84 nm) and use sonicenergy to cut them up to give a number of sets of wires,each with the same mean diameter but with different meanlengths. Specifically, a dispersion of nanowires with initiallength of 6 µm was sonicated in a sonic bath for a period of60 min. At various times aliquots were removed. The wireswere deposited on a substrate and the length measured bySEM. Shown in figure 2(A) is the measured mean nanowirelength as a function of sonication time, τ . It is clear from thesedata that the length follows a power law: L ∝ τ−0.34. This

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Nanotechnology 23 (2012) 185201 S Sorel et al

Figure 2. (A) Mean wire length as a function of sonication time.(B) Bulk DC conductivity of thick films (t ∼ 2–3 µm) as a functionof mean nanowire length. The dashed line represents the best powerlaw fit. Inset: linear scale. The solid line described the best linear fit:σDC,B = 1.16× 1012L. (C) Optical conductivity, measured for thinfilms (t ∼ 200–300 m) as a function of nanowire length.

exponent is reasonably close to the value of −0.5 expectedassuming that sonication induced scission is responsible forthe cutting process [49, 50]. In addition, for each sonicationtime both a thick film (2–3 µm) and a thin film (200–300 nm)was prepared by vacuum filtration. The thicknesses of thesefilms were estimated by breaking the films and imaging thecross section by SEM. We note that this method, like allmethods of thickness measurement for nanostructured thinfilms, is somewhat unreliable. However, in this experimentthe variation in conductivity over the range of nanowirelengths was large enough that the thickness-related errorin conductivity was acceptable. For the thick films, thesheet resistance was measured and σDC,B calculated usingequation (2). As all films were many times thicker than thenanowire diameter, they are expected to display bulk-likeelectrical properties [30]. Figure 2(B) shows that σDC,Bincreases monotonically as a function of L. We attribute thescatter to the difficulties in measuring the film thickness.Hecht has argued that the conductivity should scale withnanowire length as a power law: σDC,B ∝ Lα with exponentin the range 0 ≤ α ≤ 2.48 [51]. Indeed by studying networksof carbon nanotubes, Hecht found α = 1.46. It is clear fromfigure 2(B) that a power law is appropriate for our data but

Figure 3. Transmittance spectra of networks of nanowires withfour different diameters.

with exponent α = 0.9 ± 0.2. In fact this result is consistentwith a linear relationship as shown in the inset of figure 2(B).For the rest of this paper we will assume a linear relationship:σDC,B ∝ L. For the thin films, we measured the opticaltransmittance (550 nm) and used the estimated thickness tocalculate σOp using equation (1). These data are shown infigure 2(C) and strongly suggests the optical conductivity tobe independent of nanowire length.

3.3. Measurements of T and Rs for networks of wires withdifferent dimensions

Once the effect of L is understood, it is necessary to investigatethe effect of D on the optoelectrical properties of nanowirenetworks. In order to ascertain the diameter dependence,we prepared four sets of networks, corresponding to thefour wire types (61 nm ≤ D ≤ 127 nm, table 1). Each setconsisted of networks with a range of different thicknesses.Although it is difficult to accurately measure the networkthickness, we estimate the mean thicknesses to vary from∼10 nm to ∼8 µm. (We note that the thinnest networks havemean thickness lower than the mean wire diameters. Thismean thickness is defined as the thickness of a continuousfilm with density equal to that of the network. Becausethe networks are very porous, these mean thicknesses canbe much lower than the wire diameter.) In each case wemeasured the sheet resistance and transmittance (550 nm).The transmittance spectra are shown in figure 3 and are typicalof AgNW networks. The transmittance at 550 nm is plotted asa function of Rs in figure 4. In all cases we see an increase intransmittance coupled with an increase in sheet resistance asthe network thickness decreases. These data sets can now beanalysed using a number of models.

As described in section 1, it is thought that nanostructurednetworks only display thickness independent (i.e. bulk-like)DC conductivities for thickness above some critical value,tmin. In contrast, thinner films are controlled by percolationand have DC conductivity which varies with thickness. Asa result, we can fit the thicker (i.e. low T and low Rs) filmswith bulk-like theory as described by equation (3) (see dashedlines in figure 4) and the thinner (i.e. high T and high Rs)

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Nanotechnology 23 (2012) 185201 S Sorel et al

Figure 4. Transmittance-sheet resistance curves for the fournanowire types used in this study. In each case, the data can bedivided into two regimes, the bulk-like regime and the percolativeregime. These regimes have been fitted using equation (3) (dashedline) and equation (4) (solid line) respectively. Illustrated infigure 3(A) is the meaning of the quantity Tx discussed in the text.

films with percolation theory as described by equation (4) (seesolid lines). In all cases good fits are obtained. From thesefits, we can get σDC,B/σOp, 5 and n (table 1). Indirectly, wecan also find tminσOp. These parameters will be described insubsequent sections.

3.4. Dependence of bulk parameters on nanowire diameter

From the fits of equation (3) to the data in figure 4, we canobtain σDC,B/σOp for each nanowire diameter, finding valuesbetween 140 and 205 as reported in table 1. These values arein line with those found recently by analysing [47] literaturedata for networks of metallic nanowires using equation (4),giving values of σDC,B/σOp between 83 and 453 [31, 34].However, our values cannot be compared directly with eachother, as they are for nanowires with different lengths. Tomake them comparable we use the knowledge that σDC,B ∝ Lto rescale the data to represent wires with the same length i.e.

Figure 5. The conductivity ratio, σDC,B/σOp, extracted from thebulk fits in figure 2 plotted as a function of mean nanowire diameter.Because the mean wire lengths are different for all samples, the datahas been rescaled to represent the ratios expected for sample ofwires with the same mean length of 5 µm. This rescaling was basedon the linear relationship between σDC,B and L observed infigure 2(B). The dashed line is a power law fit to the data;σDC,B/σOp = 4.27× 10−4D−0.79 (D in metres).

L = 5 µm (we note that σOp is invariant with L). The rescaledσDC,B/σOp data are shown as a function of D in figure 5.These data can be fit well to a power law σDC,B/σOp ∝

D−β with β = 0.8 ± 0.2 (see figure caption for equationof fit curve). This rather low value of β implies that theoptoelectrical properties of nanowire networks vary ratherslowly with D. We had predicted [30] this exponent to beconsiderably larger (i.e. β = 3) based on theoretical estimatesthat σDC,B ∝ D−3 [52]. That this is not the case suggests thateither the predicted D dependence of σDC,B is wrong or thatσOp also scales with D. Combining this measurement of βwith the length dependence discussed previously allows us toapproximately write σDC,B/σOp ∝ L/D.

Ideally, it would be of interest to measure the diameterdependence of σDC,B and σOp independently. The mostobvious way to do this is to measure the film thickness anduse the T and Rs data coupled with equations (1) and (2) tofind σDC,B and σOp for each wire type. However, as indicatedabove, it is difficult to accurately measure the thickness ofsuch networks. In particular the small range of wire diametersavailable means that variation of σDC,B and σOp with diameteris not likely to be enough to overcome the thickness-relatederrors. In fact during this work we found that the errorsassociated with our film thickness measurements were solarge as to make any comparative analysis of σDC,B andσOp completely unreliable. Thus, we found it impossible toaccurately measure the diameter dependence of σDC,B andσOp.

However, it is worth noting that, as described above, wedo not need to know σDC,B and σOp individually. In fact, onceσDC,B/σOp is known, the system can be fully described solong as tminσOp and n are known. As we will see, n can befound from fitting the data with equation (4). This meanswe need to find a way to estimate find tminσOp. We cando this by noting that where both bulk-like and percolativebehaviour are observed, it is possible to obtain information

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Nanotechnology 23 (2012) 185201 S Sorel et al

Figure 6. Proxy for optical conductivity as a function of meannanowire diameter. The dashed line is a power law fit to the data;σOptmin = 3.98× 10−16D−1.77 (SI units).

from the measured transmittance (Tx) where the transitionfrom bulk-like to percolative occurs (see figure 4(A) for anexplanation of this parameter). Noting that the film thicknesswhen this transition occurs is tmin, we can rewrite equation (1)in this special case as

σOptmin =2Z0(T−1/2

x − 1). (6)

As Tx can be read directly from figure 4 (see table 1) it isstraightforward to obtain σOptmin for each wire type. Thesedata are shown in figure 6 as a function of wire diameter. Itis clear from these data that σOptmin scales with D as a powerlaw with exponent 1.8 ± 0.6 (see figure caption for equationof fit curve).

3.5. Dependence of percolation parameters on nanowirediameter

The low thickness (high T and high Rs) portions of thecurves in figure 4 were fitted using percolation theory asdescribed by equation (4). From these fits, one obtains thepercolation exponent, n, and the percolative figure of merit,5, for each wire type. These data are plotted as a functionof wire diameter in figure 7. Figure 7(A) shows that thepercolation exponent increases with increasing diameter from∼0.6 to ∼3.5. We note that lower values of n lead to lowerRs, coupled with higher T . Again this suggests that lowD wires are desirable. Empirically this appears to be veryclose to a quadratic dependence (see figure caption for fitequation). It is well known that the percolation exponentcan deviate from its universal value of 2 in the presenceof a distribution of inter-wire junction resistances with themagnitude of the deviation scaling with the details of thedistribution [53–56]. It is likely that a disordered network willhave a broad distribution of junction resistances and so largern. Indeed, such a relationship has recently been observed forspray cast AgNW networks where n was observed to scalelinearly with the network non-uniformity [36]. In light of this,our data suggests that the network non-uniformity tends toincrease with increasing wire diameter. This clearly suggests

Figure 7. Percolation fit constants as a function of mean wirediameter, D. (A) The percolation exponent, n, and (B) thepercolative figure of merit, 5. In (A) the data has been fitted to anempirical power law; n = 9.12× 1013D1.96 (D in metres) In (B) thedata has been rescaled to represent the values expected for samplesof wires with the same mean length of 5 µm (this was done bymultiplying the original values of 5 by (5 µm/L)1/(n+1)). Thedashed line in (B) represents a continuous function calculated usingequation (5) incorporating the empirical fit curves shown infigures 5, 6 and 7(A).

that for networks in the percolative regime, low diametersare advantageous because low values of n lead to betteroptoelectrical properties but also because they lead to moreuniform networks.

The percolation fits also give values for 5 as shownin table 1. However, as with other data these values mustbe corrected for the varying length. We rescaled the data torepresent the values expected for samples of wires with thesame mean length of 5 µm (this was done by multiplying theoriginal values of 5 by (5 µm/L)1/(n+1)). The rescaled dataare shown in figure 7(B) as a function of D. Here 5 increasessharply as D decreases, reaching ∼70 for D ∼ 60 nm. Thathigher values of 5 lead to lower Rs, coupled with higherT , again suggests that low D wires give better optoelectricalproperties. We note that values of 5 approaching 70 areextremely high. Recently, we analysed literature data fornetworks of metallic nanowires using equation (4). Thehighest value of 5 we found was 48 for networks of coppernanowires [47].

As described above, 5 is a composite parameter definedby equation (5). The more fundamental parameters controlling5 are σDC,B/σOp and tminσOp and n. However, for each ofthese parameters we have generated empirical expressionsby fitting the data in figures 5, 6(B) and 8(A) (seecaptions for equations). These expressions can be substitutedinto equation (5) to give a semi-empirical expression for 5 as

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Nanotechnology 23 (2012) 185201 S Sorel et al

Figure 8. The estimated sheet resistance associated with a film oftransmittance of 90%, RT=90%

s , plotted as a function of mean wirediameter. The data points represent RT=90%

s calculated two ways:using values of 5 and n found for each wire type using equation (7)and using values of σDC,B/σOp, σOptmin and n using equation (8)(in each case the values used were those rescaled to representsamples of 5 µm long nanowires). The solid line represents acontinuous function calculated using equation (8) incorporating theempirical fit curves shown in figures 5, 6 and 7(A).

a function of D. This is shown in figure 7(B) as the dashedline and clearly shows the rapid increase in5 for D < 80 nm.

3.6. Resistance for 90% transparency

The data described above show that lower values of D givebetter values of n and 5, with respect to optoelectricalapplications. However, having two parameters describingthe performance is not ideal. A single FoM which coulddirectly be linked to performance would be much moredesirable. We note that for industrial applications thetransmittance generally needs to be above 90% while thesheet resistance must be as low as possible. With this inmind, a useful parameter is the sheet resistance of a film withT = 90%, RT=90%

s . As networks with T = 90% generallyreside in the percolative regime [47], this can be found byrearranging equation (4):

RT=90%s = Z0(0.0545)−(n+1) (7)

(here the factor of 0.054 comes from setting T = 90% inthe factor (T−1/2

− 1)). Alternatively, we can write RT=90%s

in terms of more fundamental quantities by combiningequations (4) and (5) and reorganizing the resultant expressionto give

RT=90%s ≈

Z0

0.11σDC,B/σOp

(Z0tminσOp

0.11

)n

(8)

(here the factor of 0.11 comes from setting T = 90%in the factor 2(T−1/2

− 1)). This expression displays anexponential relationship between RT=90%

s and n, underliningthe importance of n and so network uniformity. We haveused both these expressions to calculate RT=90%

s using thedata described above for each wire type (figure 8). Althoughthere is some scatter and considerable uncertainty, these data

clearly show that lower diameter wires give better valuesof RT=90%

s . To make the diameter dependence clearer, wehave used the empirical expressions for σDC,B/σOp andtminσOp and n described above to generate a semi-empiricalexpression for RT=90%

s as a function of D. This is shownas the solid line in figure 8. These data clearly show thatvery low values of RT=90%

s can be expected if wires withvery low D can be produced. The curve also decreasesat high D due to the reduction in σOp with increasing D.However, networks of wires with such high D would exhibitsignificant haze [31] and be extremely non-uniform andso be unsuited to practical applications. We note that theanalysis described above does not include the fact that theconductivity of metallic nanowires begins to fall as D isreduced below 50 nm [57]. Measurements have shown thenanowire conductivity to fall by a factor of 2 as D is reducedfrom 50 to 25 nm [57, 58]. Incorporating this effect meansthat the actual value of RT=90%

s may be up to a factor of 2higher than that predicted in figure 8 depending on the balanceof nanowire and junction resistances. While figure 8 predictsa value of RT=90%

s = 12 �/� for D = 25 nm, this meansthe true value at this diameter may be as high as RT=90%

s =

25 �/�. However, we note that we have not optimizedthe inter-nanowire junction resistance in this work. It caneasily be shown that RT=90%

s scales linearly with the junctionresistance [59]. Thus, improvements in processing whichresult in a reduction in junction resistance will reduce thesevalues even further. We suggest that the exceptionally lowvalue of RT=90%

s = 10�/� reported recently [39] stems froma combination of low D and low junction resistance. Thusmoving to low diameter nanowires would give considerableperformance enhancement. In addition, networks of AgNWsare usually strongly affected by haze due to light scatteringby the wires. This effect should be significantly reducedby moving to lower D nanowires. In addition, the networkwould be considerably more uniform leading to more reliableperformance, especially for small pixel applications.

4. Conclusion

We have measured the transmittance and sheet resistance fora large number of networks of silver nanowires with differentlengths and diameters. We find that, while the bulk-like DCconductivity of the networks scales linearly with wire length,the optical conductivity is invariant with wire length. Theratio of DC to optical conductivity, often used as a figure ofmerit, scaled with wire diameter and length approximatelyas σDC,B/σOp ∝ L/D. We found that, in all cases, networkswithin the technologically relevant regime of T > 90% weredescribed by percolation theory. In this case, the dependenceof T on Rs is controlled by two parameters, the percolationexponent, n, and the percolative figure of merit, 5. We foundn to decrease with decreasing D while 5 increased withdecreasing D. In both cases, this behaviour points to betteroptoelectrical performance for networks of lower diameterwires. We could check this by calculating the expected sheetresistance for a network with T = 90%. We found thisquantity to fall rapidly with decreasing D. We estimate that,

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Nanotechnology 23 (2012) 185201 S Sorel et al

by preparing networks of wires with D = 25 nm, values aslow as RT=90%

s = 10 �/� could be obtained.

Acknowledgments

We acknowledge the Science Foundation Ireland fundedcollaboration (SFI grant 03/CE3/M406s1) between TrinityCollege Dublin and Hewlett Packard, which has allowed thiswork to take place. JNC is also supported by an SFI PI award.

References

[1] Gordon R G 2000 Criteria for choosing transparent conductorsMRS Bull. 25 52

[2] Zhou Y H, Zhang F L, Tvingstedt K, Barrau S, Li F H,Tian W J and Inganas O 2008 Investigation on polymeranode design for flexible polymer solar cells Appl. Phys.Lett. 92 23

[3] Rowell M W and McGehee M D 2011 Transparent electroderequirements for thin film solar cell modules EnergyEnviron. Sci. 4 131–4

[4] Granqvist C G and Hultaker A 2002 Transparent andconducting ITO films: new developments and applicationsThin Solid Films 411 1–5

[5] Leterrier Y, Medico L, Demarco F, Manson J A E, Betz U,Escola M F, Olsson M K and Atamny F 2004 Mechanicalintegrity of transparent conductive oxide films for flexiblepolymer-based displays Thin Solid Films 460 156–66

[6] Kim H, Horwitz J S, Kushto G P, Kafafi Z H andChrisey D B 2001 Indium tin oxide thin films grown onflexible plastic substrates by pulsed-laser deposition fororganic light-emitting diodes Appl. Phys. Lett. 79 284–6

[7] Zheng J P and Kwok H S 1993 Low-resistivity indium tinoxide-films by pulsed-laser deposition Appl. Phys. Lett.63 1–3

[8] Hecht D S, Hu L B and Irvin G 2011 Emerging transparentelectrodes based on thin films of carbon nanotubes,graphene, and metallic nanostructures Adv. Mater.23 1482–513

[9] Dan B, Irvin G C and Pasquali M 2009 Continuous andscalable fabrication of transparent conducting carbonnanotube films ACS Nano 3 835–43

[10] De S et al 2009 Transparent, flexible, and highly conductivethin films based on polymer–nanotube composites ACSNano 3 714–20

[11] Doherty E M, De S, Lyons P E, Shmeliov A, Nirmalraj P N,Scardaci V, Joimel J, Blau W J, Boland J J andColeman J N 2009 The spatial uniformity andelectromechanical stability of transparent, conductive filmsof single walled nanotubes Carbon 47 2466–73

[12] Fanchini G, Miller S, Parekh L B and Chhowalla M 2008Optical anisotropy in single-walled carbon nanotube thinfilms: implications for transparent and conductingelectrodes in organic photovoltaics Nano Lett. 8 2176–9

[13] Geng H Z, Kim K K, So K P, Lee Y S, Chang Y andLee Y H 2007 Effect of acid treatment on carbonnanotube-based flexible transparent conducting films J. Am.Chem. Soc. 129 7758

[14] Green A A and Hersam M C 2008 Colored semitransparentconductive coatings consisting of monodisperse metallicsingle-walled carbon nanotubes Nano Lett. 8 1417–22

[15] Hecht D S, Heintz A M, Lee R, Hu L B, Moore B,Cucksey C and Risser S 2011 High conductivity transparentcarbon nanotube films deposited from superacidNanotechnology 22 075201

[16] Hu L, Hecht D S and Gruner G 2004 Percolation intransparent and conducting carbon nanotube networks NanoLett. 4 2513–7

[17] Manivannan S, Ryu J H, Jang J and Park K C 2010 Fabricationand effect of post treatment on flexible single-walled carbonnanotube films J. Mater. Sci.-Mater. Electron. 21 595–602

[18] Scardaci V, Coull R and Coleman J N 2010 Very thintransparent, conductive carbon nanotube films on flexiblesubstrates Appl. Phys. Lett. 97 023114

[19] Shin D W, Lee J H, Kim Y H, Yu S M, Park S Y andYoo J B 2009 A role of HNO3 on transparent conductingfilm with single-walled carbon nanotubes Nanotechnology20 475703

[20] Wu Z C et al 2004 Transparent, conductive carbon nanotubefilms Science 305 1273–6

[21] Becerril H A, Mao J, Liu Z, Stoltenberg R M, Bao Z andChen Y 2008 Evaluation of solution-processed reducedgraphene oxide films as transparent conductors ACS Nano2 463–70

[22] Blake P et al 2008 Graphene-based liquid crystal device NanoLett. 8 1704–8

[23] De S, King P J, Lotya M, O’Neill A, Doherty E M,Hernandez Y, Duesberg G S and Coleman J N 2009Flexible, transparent, conducting films of randomly stackedgraphene from surfactant-stabilized, oxide-free graphenedispersions Small 6 458

[24] Eda G, Fanchini G and Chhowalla M 2008 Large-areaultrathin films of reduced graphene oxide as a transparentand flexible electronic material Nature Nano 3 270–4

[25] Green A A and Hersam M C 2009 Solution phase productionof graphene with controlled thickness via densitydifferentiation Nano Lett. 9 4031–6

[26] Mattevi C, Eda G, Agnoli S, Miller S, Mkhoyan K A, Celik O,Mostrogiovanni D, Granozzi G, Garfunkel E andChhowalla M 2009 Evolution of electrical, chemical, andstructural properties of transparent and conductingchemically derived graphene thin films Adv. Funct. Mater.19 2577–83

[27] Wang X, Zhi L J and Mullen K 2008 Transparent, conductivegraphene electrodes for dye-sensitized solar cells NanoLett. 8 323–7

[28] Wu J B, Becerril H A, Bao Z N, Liu Z F, Chen Y S andPeumans P 2008 Organic solar cells withsolution-processed graphene transparent electrodes Appl.Phys. Lett. 92 263302

[29] Zhu Y W, Cai W W, Piner R D, Velamakanni A andRuoff R S 2009 Transparent self-assembled films ofreduced graphene oxide platelets Appl. Phys. Lett.95 103104

[30] De S, King P J, Lyons P E, Khan U and Coleman J N 2010Size effects and the problem with percolation innanostructured transparent conductors ACS Nano4 7064–72

[31] Hu L B, Kim H S, Lee J Y, Peumans P and Cui Y 2010Scalable coating and properties of transparent, flexible,silver nanowire electrodes ACS Nano 4 2955–63

[32] Lee J Y, Connor S T, Cui Y and Peumans P 2008Solution-processed metal nanowire mesh transparentelectrodes Nano Lett. 8 689–92

[33] Madaria A R, Kumar A, Ishikawa F N and Zhou C W 2010Uniform, highly conductive, and patterned transparent filmsof a percolating silver nanowire network on rigid andflexible substrates using a dry transfer technique Nano Res.3 564–73

[34] Madaria A R, Kumar A and Zhou C W 2011 Large scale,highly conductive and patterned transparent films of silvernanowires on arbitrary substrates and their application intouch screens Nanotechnology 22 245201

8

Nanotechnology 23 (2012) 185201 S Sorel et al

[35] Rathmell A R, Bergin S M, Hua Y L, Li Z Y andWiley B J 2010 The growth mechanism of coppernanowires and their properties in flexible, transparentconducting films Adv. Mater. 22 3558–63

[36] Scardaci V, Coull R, Lyons P E, Rickard D andColeman J N 2011 Spray deposition of highly transparent,low-resistance networks of silver nanowires over large areasSmall 7 2621–8

[37] Wu H, Hu L B, Rowell M W, Kong D S, Cha J J,McDonough J R, Zhu J, Yang Y A, McGehee M D andCui Y 2010 Electrospun metal nanofiber webs ashigh-performance transparent electrode Nano Lett.10 4242–8

[38] Liu C-H and Yu X 2011 Silver nanowire-based transparent,flexible, and conductive thin film Nanoscale Res. Lett.6 6–75

[39] Leem D-S, Edwards A, Faist M, Nelson J, Bradley D D C andde Mello J C 2011 Efficient organic solar cells withsolution-processed silver nanowire electrodes Adv. Mater.23 4371

[40] Lyons P E, De S, Elias J, Schamel M, Philippe L, Bellew A T,Boland J J and Coleman J N 2011 High-performancetransparent conductors from networks of gold nanowiresJ. Phys. Chem. Lett. 2 3058–62

[41] Rathmell A R and Wiley B J 2011 The synthesis and coatingof long, thin copper nanowires to make flexible, transparentconducting films on plastic substrates Adv. Mater. 23 4798

[42] Yang L Q, Zhang T, Zhou H X, Price S C, Wiley B J andYou W 2011 Solution-processed flexible polymer solar cellswith silver nanowire electrodes ACS Appl. Mater. Inter.3 4075–84

[43] Hecht D, Hu L B and Gruner G 2006 Conductivity scalingwith bundle length and diameter in single walled carbonnanotube networks Appl. Phys. Lett. 89 133112

[44] Lyons P E, De S, Blighe F, Nicolosi V, Pereira L F C,Ferreira M S and Coleman J N 2008 The relationshipbetween network morphology and conductivity in nanotubefilms J. Appl. Phys. 104 044302

[45] Dressel M and Gruner G 2002 Electrodynamics of Solids:Optical Properties of Electrons in Matter (Cambridge:Cambridge University Press)

[46] King P J, Khan U, Lotya M, De S and Coleman J N 2010Improvement of transparent conducting nanotube films by

addition of small quantities of graphene ACS Nano4 4238–46

[47] De S and Coleman J N 2011 The effects of percolation innanostructured transparent conductors MRS Bull.36 774–81

[48] Stauffer D S and Aharony A 1994 Introduction to PercolationTheory (London: Taylor and Francis)

[49] Hennrich F, Krupke R, Arnold K, Rojas Stutz J A,Lebedkin S, Koch T, Schimmel T and Kappes M M 2007The mechanism of cavitation-induced scission ofsingle-walled carbon nanotubes J. Phys. Chem. B111 1932–7

[50] Khan U, O’Neill A, Lotya M, De S and Coleman J N 2010High-concentration solvent exfoliation of graphene Small6 864–71

[51] Hecht D, Hu L B and Gruner G 2006 Conductivity scalingwith bundle length and diameter in single walled carbonnanotube networks Appl. Phys. Lett. 89 133112

[52] Lyons P E, De S, Blighe F, Nicolosi V, Pereira L F C,Ferreira M S and Coleman J N 2008 The relationshipbetween network morphology and conductivity in nanotubefilms J. Appl. Phys. 104 044302

[53] Balberg I 1987 Tunnelling and nonuniversal conductivity incomposite-materials Phys. Rev. Lett. 59 1305–8

[54] Balberg I 1998 Limits on the continuum-percolation transportexponents Phys. Rev. B 57 13351–4

[55] Grimaldi C and Balberg I 2006 Tunneling and nonuniversalityin continuum percolation systems Phys. Rev. Lett.96 066602

[56] Johner N, Grimaldi C, Balberg I and Ryser P 2008 Transportexponent in a three-dimensional continuumtunneling-percolation model Phys. Rev. B 77 174204

[57] Bid A, Bora A and Raychaudhuri A K 2006 Temperaturedependence of the resistance of metallic nanowires ofdiameter >= 15 nm: applicability of Bloch–Gruneisentheorem Phys. Rev. B 74 035426

[58] Wiley B J, Wang Z H, Wei J, Yin Y D, Cobden D H andXia Y N 2006 Synthesis and electrical characterization ofsilver nanobeams Nano Lett. 6 2273–8

[59] Nirmalraj P N, Lyons P E, De S, Coleman J N andBoland J J 2009 Electrical connectivity in single-walledcarbon nanotube networks Nano Lett. 9 3890–5

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