The dependence of the optoelectrical properties of silver nanowire networks on nanowire length and diameter

In order to understand the optoelectrical properties of nanowire networks, it is necessary to understand the factors which control the transmittance and sheet resistance. In general, both parameters are controlled by the network thickness, t. For example, the transmittance can be expressed using [45]

$$\begin{eqnarray} &&T={\left(1+\frac{{Z}_{0}}{2}{\sigma }_{\mathrm{Op}}t\right)}^{-2}. \end{eqnarray} \tag{ 1 }$$

Here, Z₀ is the impedance of free space (377 Ω) and σ_Op is the optical conductivity, which like T is a function of wavelength, λ [9]. (We note that expansion of equation (1) shows it to be identical to the Lambert–Beer law, T = e^−αt, to first order. This shows σ_Op to be related to the absorption coefficient, α, by σ_Op ≈ α/Z₀.)

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

$$\begin{eqnarray} &&{R}_{\mathrm{s}}=({\sigma }_{\mathrm{DC},\mathrm{B}}t)^{-1} \end{eqnarray} \tag{ 2 }$$

(σ_DC,B is the bulk DC conductivity of the film i.e. the conductivity of a film which is thick enough such that the DC conductivity is thickness independent) to eliminate t, one obtains a relationship between T and R_s [16]:

$$\begin{eqnarray} &&T={\left(1+\frac{{Z}_{0}}{2{R}_{\mathrm{s}}}\frac{{\sigma }_{\mathrm{Op}}}{{\sigma }_{\mathrm{DC},\mathrm{B}}}\right)}^{-2}. \end{eqnarray} \tag{ 3 }$$

Here σ_DC,B/σ_Op can be considered a figure of merit, with high values giving the required properties (high T coupled with low R_s). However, while equation (3) tends to provide an excellent fit to (T,R_s) data for relatively thick networks of various nanoscale conductors, the agreement breaks down for thinner networks [23, 10, 46, 47]. As thin films are required to give high T, this can be a significant problem. Recently, we showed that equation (3) does not usually apply in the technologically relevant region around T = 90% [30]. This is because, for thin networks, the DC conductivity is no longer thickness independent but follows a percolation-like thickness dependence [48]: σ_DC ∝ (t−t_c)ⁿ, where t is the estimated thickness of the network, t_c is the critical thickness, i.e. the percolation threshold, and n is the percolation exponent. Approximating t − t_c ≈ t, this leads to a new relationship between T and R_s which applies to thin networks [30]:

$$\begin{eqnarray} &&T={\left[1+\frac{1}{\Pi }{\left(\frac{{Z}_{0}}{{R}_{\mathrm{s}}}\right)}^{1/(n+1)}\right]}^{-2} \end{eqnarray} \tag{ 4 }$$

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

$$\begin{eqnarray} &&\Pi =2{\left[\frac{{\sigma }_{\mathrm{DC},\mathrm{B}}/{\sigma }_{\mathrm{Op}}}{({Z}_{0}{t}_{\mathrm{min}}{\sigma }_{\mathrm{Op}})^{n}}\right]}^{1/(n+1)}. \end{eqnarray} \tag{ 5 }$$

Here, t_min is the thickness below which the DC conductivity becomes thickness dependent (i.e. equation (4) applies for thicknesses below t_min). Analysis of these equations shows that large values of Π, but low values of n, are required to achieve low R_s coupled with high T [30].

Inspection of equations (3)–(5) shows that the optical and electrical properties of nanowire networks are completely described by the parameters σ_DC,B/σ_Op, t_minσ_Op and n. Thus, in order to understand the dependence of network properties on wire dimensions, it will be necessary to measure the dependence of these properties (and the composite property, Π) on nanowire length and diameter.

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

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In order to understand the dependence of electrical and optical properties on both D and L, it will be necessary to treat these parameters separately. To do this, we take nanowires with a particular mean diameter (D = 84 nm) and use sonic energy to cut them up to give a number of sets of wires, each with the same mean diameter but with different mean lengths. Specifically, a dispersion of nanowires with initial length of 6 µm was sonicated in a sonic bath for a period of 60 min. At various times aliquots were removed. The wires were deposited on a substrate and the length measured by SEM. Shown in figure 2(A) is the measured mean nanowire length as a function of sonication time, τ. It is clear from these data that the length follows a power law: L ∝ τ^−0.34. This exponent is reasonably close to the value of  − 0.5 expected assuming that sonication induced scission is responsible for the cutting process [49, 50]. In addition, for each sonication time both a thick film (2–3 µm) and a thin film (200–300 nm) was prepared by vacuum filtration. The thicknesses of these films were estimated by breaking the films and imaging the cross section by SEM. We note that this method, like all methods of thickness measurement for nanostructured thin films, is somewhat unreliable. However, in this experiment the variation in conductivity over the range of nanowire lengths was large enough that the thickness-related error in conductivity was acceptable. For the thick films, the sheet resistance was measured and σ_DC,B calculated using equation (2). As all films were many times thicker than the nanowire diameter, they are expected to display bulk-like electrical properties [30]. Figure 2(B) shows that σ_DC,B increases monotonically as a function of L. We attribute the scatter to the difficulties in measuring the film thickness. Hecht has argued that the conductivity should scale with nanowire length as a power law: σ_DC,B ∝ L^α with exponent in the range 0 ≤ α ≤ 2.48 [51]. Indeed by studying networks of carbon nanotubes, Hecht found α = 1.46. It is clear from figure 2(B) that a power law is appropriate for our data but with exponent α = 0.9 ± 0.2. In fact this result is consistent with a linear relationship as shown in the inset of figure 2(B). For the rest of this paper we will assume a linear relationship: σ_DC,B ∝ L. For the thin films, we measured the optical transmittance (550 nm) and used the estimated thickness to calculate σ_Op using equation (1). These data are shown in figure 2(C) and strongly suggests the optical conductivity to be independent of nanowire length.

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Once the effect of L is understood, it is necessary to investigate the effect of D on the optoelectrical properties of nanowire networks. In order to ascertain the diameter dependence, we prepared four sets of networks, corresponding to the four wire types (61 nm ≤ D ≤ 127 nm, table 1). Each set consisted of networks with a range of different thicknesses. Although it is difficult to accurately measure the network thickness, we estimate the mean thicknesses to vary from ∼10 nm to ∼8 µm. (We note that the thinnest networks have mean thickness lower than the mean wire diameters. This mean thickness is defined as the thickness of a continuous film with density equal to that of the network. Because the networks are very porous, these mean thicknesses can be much lower than the wire diameter.) In each case we measured the sheet resistance and transmittance (550 nm). The transmittance spectra are shown in figure 3 and are typical of AgNW networks. The transmittance at 550 nm is plotted as a function of R_s in figure 4. In all cases we see an increase in transmittance coupled with an increase in sheet resistance as the network thickness decreases. These data sets can now be analysed using a number of models.

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As described in section 1, it is thought that nanostructured networks only display thickness independent (i.e. bulk-like) DC conductivities for thickness above some critical value, t_min. In contrast, thinner films are controlled by percolation and have DC conductivity which varies with thickness. As a result, we can fit the thicker (i.e. low T and low R_s) films with bulk-like theory as described by equation (3) (see dashed lines in figure 4) and the thinner (i.e. high T and high R_s) films with percolation theory as described by equation (4) (see solid lines). In all cases good fits are obtained. From these fits, we can get σ_DC,B/σ_Op, Π and n (table 1). Indirectly, we can also find t_minσ_Op. These parameters will be described in subsequent sections.

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From the fits of equation (3) to the data in figure 4, we can obtain σ_DC,B/σ_Op for each nanowire diameter, finding values between 140 and 205 as reported in table 1. These values are in line with those found recently by analysing [47] literature data for networks of metallic nanowires using equation (4), giving values of σ_DC,B/σ_Op between 83 and 453 [31, 34]. However, our values cannot be compared directly with each other, as they are for nanowires with different lengths. To make them comparable we use the knowledge that σ_DC,B ∝ L to rescale the data to represent wires with the same length i.e. L = 5 µm (we note that σ_Op is invariant with L). The rescaled σ_DC,B/σ_Op data are shown as a function of D in figure 5. These data can be fit well to a power law σ_DC,B/σ_Op ∝ D^−β with β = 0.8 ± 0.2 (see figure caption for equation of fit curve). This rather low value of β implies that the optoelectrical properties of nanowire networks vary rather slowly with D. We had predicted [30] this exponent to be considerably larger (i.e. β = 3) based on theoretical estimates that σ_DC,B ∝ D⁻³ [52]. That this is not the case suggests that either the predicted D dependence of σ_DC,B is wrong or that σ_Op also scales with D. Combining this measurement of β with the length dependence discussed previously allows us to approximately write σ_DC,B/σ_Op ∝ L/D.

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Ideally, it would be of interest to measure the diameter dependence of σ_DC,B and σ_Op independently. The most obvious way to do this is to measure the film thickness and use the T and R_s data coupled with equations (1) and (2) to find σ_DC,B and σ_Op for each wire type. However, as indicated above, it is difficult to accurately measure the thickness of such networks. In particular the small range of wire diameters available means that variation of σ_DC,B and σ_Op with diameter is not likely to be enough to overcome the thickness-related errors. In fact during this work we found that the errors associated with our film thickness measurements were so large as to make any comparative analysis of σ_DC,B and σ_Op completely unreliable. Thus, we found it impossible to accurately measure the diameter dependence of σ_DC,B and σ_Op.

However, it is worth noting that, as described above, we do not need to know σ_DC,B and σ_Op individually. In fact, once σ_DC,B/σ_Op is known, the system can be fully described so long as t_minσ_Op and n are known. As we will see, n can be found from fitting the data with equation (4). This means we need to find a way to estimate find t_minσ_Op. We can do this by noting that where both bulk-like and percolative behaviour are observed, it is possible to obtain information from the measured transmittance (T_x) where the transition from bulk-like to percolative occurs (see figure 4(A) for an explanation of this parameter). Noting that the film thickness when this transition occurs is t_min, we can rewrite equation (1) in this special case as

$$\begin{eqnarray} &&{\sigma }_{\mathrm{Op}}{t}_{\min }=\frac{2}{{Z}_{0}}({T}_{x}^{-1/2}-1). \end{eqnarray} \tag{ 6 }$$

As T_x can be read directly from figure 4 (see table 1) it is straightforward to obtain σ_Opt_min for each wire type. These data are shown in figure 6 as a function of wire diameter. It is clear from these data that σ_Opt_min scales with D as a power law with exponent 1.8 ± 0.6 (see figure caption for equation of fit curve).

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The low thickness (high T and high R_s) portions of the curves in figure 4 were fitted using percolation theory as described by equation (4). From these fits, one obtains the percolation exponent, n, and the percolative figure of merit, Π, for each wire type. These data are plotted as a function of wire diameter in figure 7. Figure 7(A) shows that the percolation exponent increases with increasing diameter from ∼0.6 to ∼3.5. We note that lower values of n lead to lower R_s, coupled with higher T. Again this suggests that low D wires are desirable. Empirically this appears to be very close to a quadratic dependence (see figure caption for fit equation). It is well known that the percolation exponent can deviate from its universal value of 2 in the presence of a distribution of inter-wire junction resistances with the magnitude of the deviation scaling with the details of the distribution [53–56]. It is likely that a disordered network will have a broad distribution of junction resistances and so larger n. Indeed, such a relationship has recently been observed for spray cast AgNW networks where n was observed to scale linearly with the network non-uniformity [36]. In light of this, our data suggests that the network non-uniformity tends to increase with increasing wire diameter. This clearly suggests that for networks in the percolative regime, low diameters are advantageous because low values of n lead to better optoelectrical properties but also because they lead to more uniform networks.

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The percolation fits also give values for Π as shown in table 1. However, as with other data these values must be corrected for the varying length. We rescaled the data to represent the values expected for samples of wires with the same mean length of 5 µm (this was done by multiplying the original values of Π by (5 µm/L)^1/(n+1)). The rescaled data are shown in figure 7(B) as a function of D. Here Π increases sharply as D decreases, reaching ∼70 for D ∼ 60 nm. That higher values of Π lead to lower R_s, coupled with higher T, again suggests that low D wires give better optoelectrical properties. We note that values of Π approaching 70 are extremely high. Recently, we analysed literature data for networks of metallic nanowires using equation (4). The highest value of Π we found was 48 for networks of copper nanowires [47].

As described above, Π is a composite parameter defined by equation (5). The more fundamental parameters controlling Π are σ_DC,B/σ_Op and t_minσ_Op and n. However, for each of these parameters we have generated empirical expressions by fitting the data in figures 5, 6(B) and 8(A) (see captions for equations). These expressions can be substituted into equation (5) to give a semi-empirical expression for Π as a function of D. This is shown in figure 7(B) as the dashed line and clearly shows the rapid increase in Π for D < 80 nm.

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The data described above show that lower values of D give better values of n and Π, with respect to optoelectrical applications. However, having two parameters describing the performance is not ideal. A single FoM which could directly be linked to performance would be much more desirable. We note that for industrial applications the transmittance generally needs to be above 90% while the sheet resistance must be as low as possible. With this in mind, a useful parameter is the sheet resistance of a film with T = 90%, ${R}_{\mathrm{s}}^{T=90\% }$. As networks with T = 90% generally reside in the percolative regime [47], this can be found by rearranging equation (4):

$$\begin{eqnarray} &&{R}_{\mathrm{s}}^{T=90\% }={Z}_{0}(0.054 \Pi )^{-(n+1)} \end{eqnarray} \tag{ 7 }$$

(here the factor of 0.054 comes from setting T = 90% in the factor (T^−1/2 − 1)). Alternatively, we can write ${R}_{\mathrm{s}}^{T=90\% }$ in terms of more fundamental quantities by combining equations (4) and (5) and reorganizing the resultant expression to give

$$\begin{eqnarray} &&{R}_{\mathrm{s}}^{T=90\% }\approx \frac{{Z}_{0}}{0.11{\sigma }_{\mathrm{DC},\mathrm{B}}/{\sigma }_{\mathrm{Op}}}{\left(\frac{{Z}_{0}{t}_{\min }{\sigma }_{\mathrm{Op}}}{0.11}\right)}^{n} \end{eqnarray} \tag{ 8 }$$

(here the factor of 0.11 comes from setting T = 90% in the factor 2(T^−1/2 − 1)). This expression displays an exponential relationship between ${R}_{\mathrm{s}}^{T=90\% }$ and n, underlining the importance of n and so network uniformity. We have used both these expressions to calculate ${R}_{\mathrm{s}}^{T=90\% }$ using the data described above for each wire type (figure 8). Although there is some scatter and considerable uncertainty, these data clearly show that lower diameter wires give better values of ${R}_{\mathrm{s}}^{T=90\% }$. To make the diameter dependence clearer, we have used the empirical expressions for σ_DC,B/σ_Op and t_minσ_Op and n described above to generate a semi-empirical expression for ${R}_{\mathrm{s}}^{T=90\% }$ as a function of D. This is shown as the solid line in figure 8. These data clearly show that very low values of ${R}_{\mathrm{s}}^{T=90\% }$ can be expected if wires with very low D can be produced. The curve also decreases at high D due to the reduction in σ_Op with increasing D. However, networks of wires with such high D would exhibit significant haze [31] and be extremely non-uniform and so be unsuited to practical applications. We note that the analysis described above does not include the fact that the conductivity of metallic nanowires begins to fall as D is reduced below 50 nm [57]. Measurements have shown the nanowire conductivity to fall by a factor of 2 as D is reduced from 50 to 25 nm [57, 58]. Incorporating this effect means that the actual value of ${R}_{\mathrm{s}}^{T=90\% }$ may be up to a factor of 2 higher than that predicted in figure 8 depending on the balance of nanowire and junction resistances. While figure 8 predicts a value of ${R}_{\mathrm{s}}^{T=90\% }=12~\Omega /\square$ for D = 25 nm, this means the true value at this diameter may be as high as ${R}_{\mathrm{s}}^{T=90\% }=25~\Omega /\square$. However, we note that we have not optimized the inter-nanowire junction resistance in this work. It can easily be shown that ${R}_{\mathrm{s}}^{T=90\% }$ scales linearly with the junction resistance [59]. Thus, improvements in processing which result in a reduction in junction resistance will reduce these values even further. We suggest that the exceptionally low value of ${R}_{\mathrm{s}}^{T=90\% }=10~\Omega /\square$ reported recently [39] stems from a combination of low D and low junction resistance. Thus moving to low diameter nanowires would give considerable performance enhancement. In addition, networks of AgNWs are usually strongly affected by haze due to light scattering by the wires. This effect should be significantly reduced by moving to lower D nanowires. In addition, the network would be considerably more uniform leading to more reliable performance, especially for small pixel applications.