The Electro-Optical Performance of Silver Nanowire Networks

Networks of metallic nanowires have the potential to meet the needs of next-generation device technologies that require flexible transparent conductors. At present, there does not exist a first principles model capable of predicting the electro-optical performance of a nanowire network. Here we combine an electrical model derived from fundamental material properties and electrical equations with an optical model based on Mie theory scattering of light by small particles. This approach enables the generation of analogues for any nanowire network and then accurately predicts, without the use of fitting factors, the optical transmittance and sheet resistance of the transparent electrode. Predictions are validated using experimental data from the literature of networks comprised of a wide range of aspect ratios (nanowire length/diameter). The separation of the contributions of the material resistance and the junction resistance allows the effectiveness of post-deposition processing methods to be evaluated and provides a benchmark for the minimum attainable sheet resistance. The predictive power of this model enables a material-by-design approach, whereby suitable systems can be prescribed for targeted technology applications.

www.nature.com/scientificreports www.nature.com/scientificreports/ Modification of the PVP surface layer can give resistive switching memory effects 24,25 , or enhance the thermal and chemical stability of the Ag NWNs 17,26,27 .
PVP is necessary during the synthesis process and stabilises the nanowires in solution 28 . Optimisation of the R jxn value in NWNs typically involves post-processing techniques such as thermal treatment 29 , mechanical pressing 30 , cold welding 31 , optically induced welding 32 , electrochemical Ostwald ripening 33 or electrical stressing 34 , each of which can dramatically lower the R s . However, in the absence of an electro-optical predictive tool, the effectiveness of these processing methods has been difficult to assess, compare and prescribe for specific applications.
We previously introduced a computational approach to describe the conduction properties of metallic NWNs using a multi-nodal representation (MNR) model which calculates the R s considering the contributions associated with NW junctions (R jxn ) and the NW segments (with inner resistances given by R in ) between them 23 . One of the computational implementations for this model is available in the Supporting Information. Incorporating the inner-wire resistance (which depends on the NW material and diameter) is important but often overlooked. It allows the skeletal resistance of the network (in the limit when R jxn → 0) to be determined, for which the resulting network is comprised of ballistic NW junctions, representing the ultimate conductivity of the NWN 23 . It is well known that increasing the NW length to diameter aspect ratio (AR) of the NWs results in lower R s values and a larger T 35,36 . Thus far, electro-optical models of Ag NWNs have used empirical expressions to describe how T depends on R s (which itself depends explicitly on material resistivity (ρ), R in , R jxn and AR); approaches that typically require fitting parameters 35,37,38 , or using other empirically sourced quantities to achieve agreement between experimental and simulated data 13,20,39,40 . The T-R s curve has a characteristic shape, which was highlighted by Mutiso et al. 38 who fit an empirical expression with good agreement to a percolative model derived from thin-films 40 . However, none of these semi-empirical approaches are predictive, nor can they accurately describe a wide range of NWNs.
In this work we use the Mie light scattering theory (MLST) of NWNs described by Khanarian et al. to predict the transmittance as a function of the diameter of the NWs and the surface fraction coverage 41 . MLST is an exact theory which has no fitting parameters and is only dependent on the wavelength of incident light, the NW diameter, and the optical constants of the NW material. We build upon the electrical MNR model by incorporating a first principles approach based on MLST of NWs to determine the electro-optical performance of the NWN. This fully predictive model establishes the limits of Ag NWN performance and faithfully captures the behaviour of experimental data from the literature over a wide range of NW ARs. Importantly, this tool can be used to engineer NW systems for different applications in a true materials-by-design approach, allowing an effective comparison of different NWN processing methods that will facilitate the adoption of NWN films in current and next-generation devices.

Results
The electrical performance of percolating NWNs depends significantly on the properties of the constituent NWs. Physical properties such as length and diameter determine the ultimate conductance potential of the network. The electrical performance of a Ag NWN was simulated using the MNR model for a given AR and wire density, setting ρ = 22.6 nΩm 23 , and R jxn = 11 Ω corresponding to the median value of the experimentally optimised distribution of junction resistances in Ag/PVP systems (see Fig. 2c) 42 . For the purposes of simulation, the NWs were considered as rigid rods. Singular values of NW length and diameter were used in all computations, although the MNR model is also able to account for more realistic aspects of networks, e.g. dispersion in physical parameters such as length, diameter, R jxn and the presence of "outlier" junctions. Here, our goal is to avoid unnecessary complexity and show the raw capabilities of the model in describing and predicting real world performance without any fitting parameters; the flexibility of the model enables the incorporation of dispersion and other disorder elements in a straightforward fashion. Figure 2(a) shows plots of the simulated Ag NWNs at a density of 0.1 NW/µm 2 for ARs of 200, 400, 600 and 800; which corresponds to T values of 99.4%, 98.8%, 98.2% and 97.6%, respectively. The diameter of the NWs was fixed at 30 nm and the simulation box was set at 50 × 50 µm in each case. In Fig. 2(b), the power law dependence of the R s on NW density is clear for all AR values plotted, consistent with percolative and closed form models of electrical transport within NWNs 22,38,43 . The importance of including the material resistance in the calculation of the R s is highlighted in Fig. 2(c), where R s is calculated as a function of the R jxn for a junction dominated approach (JDA, dashed black line) NWs have no internal resistance. The MNR model which accounts for the resistance of the wire segments between junctions is shown as a solid black line. When R jxn → 0, the MNR model provides the ultimate limit of the network R s , which for an AR of 200 with a density of 0.84 NW/µm 2 (T = 95%) is ~20 Ω/□. The distribution on the lower half of the panel represents the range of experimentally measured R jxn values for Ag NWs 42 . As the R jxn increases, the R s increases linearly, therefore NWNs must be subjected to processing steps after network formation to reduce the contact resistance between the NWs. These results highlight the importance of R jxn optimisation in high performance NWNs, moreover, it shows the significant contribution of the R in to the overall R s , and that the true upper bounds of electrical performance can only be determined when the inner wire resistance is considered.
To estimate the T of a NWN, we begin by considering the extinction coefficient at normal incidence, C ext , which represents the amount of light scattered and absorbed by a single NW from Mie theory 44 . The area fraction (AF) describes the projection of the NWs per unit area of the substrate which is defined as the density, N, per unit area multiplied by the length (L) and diameter (D) of the NWs.
When the thickness of the network is comparable to the diameter of the NWs, the transmittance, T, as derived by Khanarian et al. can be expressed as 41 , , the average R s was calculated as a function of the R jxn using the MNR model, and when the inner resistance is neglected from the calculations in the junction dominated approach (JDA). Below the main plot is a histogram of the experimentally measured R jxn distribution of Ag NWs from Bellew et al. 42 .

AF C ext
The flow diagram in Fig. 3 describes the implementation of the combined MNR and MLST models to simulate the electro-optical properties of NWNs. The only necessary inputs to the model are the NW diameter and length. The model then calculates the corresponding network density and determines the R s given the values of ρ and R jxn . Thus the model not only predicts the performance of a particular network but given experimental T-R s data for a network of known D and L values it can predict the average R jxn . The only way to alter the R s for such a network (at a specific value of T) is to vary the R jxn value used in the MNR model, which, as previously discussed, can be influenced by different processing techniques. A discussion and analysis of previously reported, empirically derived, electro-optical models is presented in the Supplementary Information as Figs S1 and S2. This data shows that the MNR MLST model describes the expected shape of not only experimentally obtained T-R s data, but can accurately describe the synthetic data generated from previously reported semi-empirical models. Figure 4(a-d) shows the MNR calculated plots of the T-R s for Ag NWNs with various AR values and for mean R jxn values of 11 Ω, 100 Ω, 1000 Ω and in the case of perfect junctions with a resistance of 0 Ω. Each point in Fig. 4 was calculated by following the process outlined in Fig. 3 where ρ is fixed, and R jxn is varied. Figure 4(a) highlights the importance of optimising R jxn in the technologically relevant T region. It is important to note that when AR > 200 ( Fig. 4(a,b)) at a T = 90%, the simulations predict that the R s will be <100 Ω/□, which is acceptable for touch screen applications, however, solar cells and OLED electrodes require much lower R s (~10 Ω/□) which can only be achieved by R jxn optimisation (R jxn < 100 Ω) of highly transparent networks (T > 95%). The truncation of the simulation in Fig. 4(c,d) is due to the prohibitive computational requirements to calculate sufficiently dense networks with T < 93% for AR = 600 and T < 95% for AR = 800 samples. The linear dependence of T with respect to the network density is plotted in Fig. S3, a dependence that is experimentally observed and has been theoretically derived by Ainsworth et al. 39 .
In a physical NWN sample, it is impossible to probe individual R jxn values via experimental means. A strength of the MNR model is that it provides insights into the average contribution of the R jxn to the measured R s . By combining MNR and MLST models, we can begin to benchmark experimental data developing a materials-by-design approach to NWN-based TCs. Setting a theoretical benchmark for NWN systems allows a better comparison of synthesis methods, deposition procedures and specific post-processing techniques. For example, thermal annealing can hugely improve the R s of as-deposited networks, but the anneal temperature and time must be chosen carefully and will depend on NW diameter and the thermal properties of the substrate. Figure 5  www.nature.com/scientificreports www.nature.com/scientificreports/ the standard deviation of the R s for 10 simulated networks, and the vertical spread in T due to the uncertainty in the NW diameter value when calculating C ext . After the annealing step, the T-R s curve is better described by R jxn = 11 Ω, suggesting the annealing treatment has produced highly optimised TC films with extremely low R s for that particular AR. However, even at this optimised R jxn value, the NWNs fail the R s requirement for photovoltaic applications and barely reaches the requirements for screen/lighting technologies (see Fig. 1).
Another example of NWN post processing benchmarking is shown in Fig. 5(b) using the data from Liu et al. 31 . In their study, moisture-induced capillary-forces were shown to cause a self-limiting cold welding of the NW junctions, hence reducing the R s . The hollow square data points show the T-R s data of the as-prepared samples. The effect of the moisture treatment significantly reduces the R s of the Ag NWNs and causes the network to adopt the "expected shape" curve which is an important indicator of the performance of the network as predicted by the MNR MLST models (cf. Fig. 4). We can apply the MNR MLST models to determine the R jxn value needed to describe the network (comprised of NWs with AR = 222). The resulting blue shaded curve for the moisture treated samples has the expected T-R s shape, and is well described by R jxn = 750 Ω. The failure of the as-prepared film to exhibit the same shape as the moisture treated samples suggest that the network connectivity is poorly established or that there is a significant spread in R jxn values. The power of MNR MLST is that it can determine the R jxn values necessary to describe the measured R s values in the as-prepared films -the spread is between 20 kΩ and 40 kΩ allowing a rapid evaluation of processing techniques used to form the network, which hitherto was not possible.
While it is obvious that moisture treatment has made a significant improvement to the measured R s , it has not produced the most optimised NWN yet. MNR MLST can predict the effect of additional optimisation. By decreasing the R jxn to a value of 11 Ω (red shaded curve, Fig. 5(b)), the simulations suggest further room for improvement. In-situ resistance measurements during thermal annealing for NWNs of a similar diameter suggest that an annealing temperature of 200 °C is required to realise the most conducting NWN films 46   www.nature.com/scientificreports www.nature.com/scientificreports/ flexibility of the NWs and the diameter dependent persistence length which is known to play an important role in the connectivity of the networks 47,48 . At this point, for the sake of simplicity, our simulations only account for rigid rods and fixed persistence lengths. Nonetheless, the current model is flexible and can be extended to incorporate these elements.
Network optimisation can be graphically described by the combined MNR and MLST model. In Fig. 5(c,d), the R s is calculated as a function of the R jxn for the data presented in Fig. 5(a,b) at T = 95%. This linear relationship has been previously reported by our group when first implementing the MNR model and now serves as a roadmap for predicting the ultimate performance of NWN materials 23 . The linear decrease in the R s assumes a decrease in the resistance of all junctions in the network, however in reality, some paths within the network may not be conducting initially and may require one of the processing steps previously discussed. As these additional paths become conducting, the simple linear relationship shown here may be curved or stepped 49,50 . The two datasets in Fig. 5(a,b) (two further examples for AR 306 and 760 are presented in Fig. S18) initially had R jxn which were predicted to be higher than the optimised value. The ability of the model to separately consider junction and inner-wire resistances allows for the ultimate performance of the NWN to be determined, which occurs when R jxn → 0 Ω. This allows an estimate of how close a NWN film is to having perfect interwire contacts, and enables a rigorous and quantitative analysis of post-processing techniques. While the tunability of the R s in metallic NWNs www.nature.com/scientificreports www.nature.com/scientificreports/ via a combination of AR, NW density, material choice and R jxn makes these materials attractive to numerous applications, the presence of a R jxn between wires will always limit performance. Perfect lossless junctions may not be achievable in solution deposited NWNs but are a feature of seamless junction networks such as crackled template networks 51,52 . conclusions In this work, we combined two computational methods to deliver the first fully predictive model of both the electrical and optical performances of metal nanowire networks. The multi-nodal representation (MNR) model which calculates the sheet resistance (R s ) of the nanowire networks (NWNs) considers both the resistance contribution of the nanowire segments and the nanowire (NW) junctions. Using experimentally measured resistivity and junction resistance values for the case of Ag NWNs, we show how the R s depends on the nanowire length/ diameter aspect ratio (AR). The inner-wire resistance is also important as it determines the lowest attainable R s , and the inclusion of the skeletal resistance allows the magnitude of the junction resistance between overlapping wires, R jxn , to reveal the level of optimisation of the NWN through post-processing steps. The Mie light scattering theory (MLST) model describes the optical transmittance of NWNs according to its fundamental physical properties and network density. This simple but robust model achieves excellent agreement with experimental data over a wide range of NW ARs. The results of this work show that simulation of NWNs is an important tool in benchmarking the efficacies of post-processing methods, and offers a strategic approach to exploring the potential applications of NWN materials and guiding the synthesis of systems for specific needs. NWNs are well suited as replacements for ITO (tin-doped indium oxide) in a wide variety of current and emerging flexible devices. The development of a predictive model for these materials is an important step towards a materials-by-design approach for transparent conductor applications.

Methods
MNR simulations were implemented in the Python language, the code for which is available in the Supplementary Information. The networks were generated according to the input parameters of wire diameter, wire length, wire density (number of wires per unit area) and simulation box length, which defines the squared area of the box where wires are randomly placed. The simulation box was always set larger than two times the wire length. Ag NW material resistivity is ρ = 22.6 nΩm, and the R jxn was varied according to the experiment. The MNR voltage grid scheme, which is described in detail in reference 23 of the manuscript maps the spatial coordinates of the interwire connection points, and assigns either an interwire junction resistance R jxn , or an inner-wire segment with a resistance calculated by R in = ρl/A, where l is the length of the segment and A is the cross-sectional area of the wire. The corresponding resistance matrix is solved using Kirchhoff 's circuit law to obtain the R s of the sample. The number of representative NWN samples of the ensemble was set to 10. The program output the average R s , and the standard deviation of the R s .
The C ext is calculated using the MatScat 53 (Mie theory for infinite cylinders) implementation by Schäfer et al. 54 and depends only on the NW diameter and the optical constants for the metal. The refractive index (n) and extinction coefficient (k) used for Ag is, n = 0.13936, k = 3.5604 at λ = 546 nm 55 . From Equations 2 and 3, the NWN density corresponds to a T value. T-R s data of NWNs across a wide variety of aspect ratios was gathered from 17 publications, where the NW lengths and diameters were reported. The T values reported by these publications were converted into NWN densities which were calculated by MNR through the process outlined in Fig. 3. Where a spread in diameter values was reported, the upper and lower bounds described a variation in C ext and hence T which is displayed as the shaded areas on the T-R s graphs.

Data Availability
All data generated or analysed during this study are included in this published article (and its Supplementary  Information Files), the datasets are also available from the corresponding author on reasonable request.