Mapping Time-Dependent Conductivity of Metallic Nanowire Networks by Electrical Resistance

Metallic nanowire (NW) networks have attracted great attention as promising transparent conductive materials thanks to the low sheet resistance, high transparency, low cost production and compatibility with flexible substrates. Despite many efforts have been devoted to investigating the conduction mechanism, a quantitative characterization of local electrical properties of nanowire networks at the macroscale still represents a challenge. In this work, we report on the investigation of local electrical properties and their evolution over time of Ag NW networks by means of electrical resistance tomography (ERT). Spatial correlation of local conductivity properties and optical transparency revealed that the non-scanning and rapid ERT technique allows to probe local electrical inhomogeneities in the NW network, differently from conventional measurement techniques such as van der Pauw and four-point probe. In addition, ERT mapping over time was employed for in situ monitoring the evolution of Ag NW networks conductivity, elucidating the dependence of the degradation of local electrical properties under ambient exposure on the initial conductivity. Our results shed light on the importance of the characterization of local electrical properties of NW networks where uniformity and stability represent the main challenges to overcome for their use as transparent conductive materials.


INTRODUCTION
Metallic nanowire (NW) networks combining high electrical conductivity, high transparency, high flexibility and stretchability are considered emerging candidates as transparent conductive materials (TCMs), being particularly promising for cost-effective replacement of indium tin oxide (ITO) thanks to the compatibility with solution-based processing, large area deposition techniques and high throughput production. [1][2][3][4][5][6][7][8][9][10] Due to these characteristics, metallic NW networks realized with a bottom-up approach have been exploited for a wide range of applications including transparent electrodes (TEs) for flexible electronics [11][12][13][14][15] , photovoltaics 16,17 , film heaters 18,19 , electronic textiles 20 as well as for the realization of electromagnetic and optoelectronic devices for sensors and information communications technology (ICT) [21][22][23][24][25] . For all these applications, the characterization of electronic conduction properties of metallic NW networks over different length scales represents a key aspect not only for unveiling the conduction mechanism but also to optimize their performances depending on the target application. Despite many efforts have been devoted to the investigation of the conduction properties of metallic NW network at the micro/nanoscale, characterization of network conductivity at the macroscale still represents a challenge. At the nanoscale, devices based on single NWs and single NW junctions have been realized to investigate electrical properties of single building blocks of the network. [26][27][28][29] Instead, NW network conduction properties at the microscale have been investigated by means of conductive atomic force microscopy (C-AFM) and voltage contrast imaging in scanning electron microscopy (SEM). 23,30 Even though these techniques can be employed to identify NWs that actively contribute to the network conductivity allowing direct investigation of current paths and making possible to correlate electrical properties with the network morphology, these methods allow to investigate conduction properties of NWs over a limited area of tens of microns. On the other hand, electrical properties of NW networks at the macroscale are usually assessed by means of two point probe (2PP) 27,31 or four point probe (4PP) measurements 12,32,33 . These techniques allow to evaluate the overall conductivity of the NW network without providing details concerning the spatial distribution of electrical properties across the network. Understanding local electrical properties is even more important than evaluating the overall conductivity for a wide range of applications, since network inhomogeneities are related to propagation of cracks, point of failure and local overheating phenomena that limit device performances even causing device breakdown. 1,34,35 For example, Kim et al. 35 by infrared imaging reported that inhomogeneous networks with self-aggregated NW islands lead hot spots in the sample due to localized heating effects. The Joule heating problem of NW networks was analyzed by Khaligh et al. 36 that reported that the presence of local hot spots is responsible for an acceleration of NW degradation leading to electrode failure. Also, Sannicolo et al. 34 reported that local inhomogeneities can lead to a nonuniform distribution of the electric field when the NW network is biased, influencing failure dynamics during propagation of cracks. In case of NW networks integrated in solar cells, these inhomogeneities can lower device efficiency by increasing the series resistance. It is worth noticing also that, since the NW network conductivity is related to network density, local inhomogeneities of electrical properties can be correlated also to local variations of optical transparency. 37 With the aim of characterizing spatial distribution of electrical properties at the macroscale, conduction in NW networks was investigated by means of infrared thermography. 38 Despite being a promising tool for visually identify conductive channels and local hot spots, this technique fails in providing information on local NW network conductivity properties. As an alternative, Sannicolo et al. 34 reported one-probe electrical mapping (1P-mapping) by scanning a tip probe across a biased NW network to locally measure the electrical potential, allowing spatial mapping of the voltage equipotential lines in the device and providing information about sample homogeneity. However, i) the time for acquiring an electrical map with one-probe technique scales with the sample area, thus limiting its scalability and ii) this technique fails in quantitative estimation of local conductivity across the sample. In addition, both infrared thermography and 1P-mapping necessarily require the realization of proper NW network devices involving the realization of metal electrodes. In this framework, a scalable, fast and quantitative technique to probe local conductivity properties of metallic NW networks over large area is currently missing.
In this work, we report on mapping of metallic Ag NW network conductivity over large scale (~cm 2 ) by means of electrical resistance tomography (ERT). Differently from conventional techniques adopted for electrical characterization of NW networks, this non-scanning technique allows a quantitative and traceable to the international system of units (SI) characterization of local conduction properties of metallic networks and its evolution over time. In case of nearly uniform Ag NW network samples, ERT results are in good accordance with electrical conductivity obtained by conventional 4PP and van der Pauw (vdP) measurements. More interestingly, ERT allows to probe electrical inhomogeneities in NW networks as discussed by spatially correlating local conductivity and optical transparency. Moreover, ERT was employed to track in situ the spatial distribution of the network conductivity over time, showing that the degradation of local conductivity properties under ambient exposure depends on the initial conductivity distribution of the network. In particular, more pronounced variations of conductivity are correlated to sample areas characterized by lower initial conductivity. Besides showing that ERT represents a versatile technique to obtain conductivity maps, these results shed new light on the importance of characterizing local conduction properties of metallic NW networks and their evolution over time.
In this framework, ERT mapping results particularly promising for the development of next-generation transparent conductive materials and electronic devices based on metallic nanowire networks where uniformity and stability represent the most important technical challenges to overcome.

NW network fabrication and characterization
Metallic NW networks were fabricated by drop casting Ag NWs with diameter of about 115 nm and length of 20-50 μm in isopropyl alcohol suspension (from Sigma-Aldrich) on a square crystal quartz (z-cut) substrate of 10×10 mm 2 . Chemical and structural properties of Ag NWs employed in this work were analysed in our previous work. 22 Different metallic NW network densitites were obtained by diluting NWs in different volumes of isopropyl alcohol suspension. By controlling the concentration of Ag NWs in the suspension and by fixing the volume deposited on the 10×10 mm 2 substrate to 13.8 μl it was possible to estimate the areal mass density (AMD). The volume of suspension deposited on the substrate was optimized to obtain homogeneous distribution of NWs all over the sample. In order to obtain an intentionally nonuniform distribution of NWs over the 10×10 mm 2 sample, drop casting was performed by keeping the quartz substrate tilted. The normalized NW density was estimated by using the relation = eV and using the C 1s peak position (284.8 eV) as calibration.

ERT measurement setup and map reconstruction
Electrical measurements were performed with a setup that consists of a Keysight 34980A multifunction unit equipped with a switch matrix module (Keysight 34933 read matrix module), a Keithley 2602B source-meter and an Agilent 34461A digital multimeter. Samples were contacted with a fixture provided of spring-mounted needle probes described in a previous work. 40 The needle probes have a round tip with a contact section of about 40 μm in diameter. The probe diameter was observed to provide reliable electrical contacts in NW networks with AMD considered in our work (60 mg/m 2 -181 mg/m 2 ). The distance between nearby contact needles along an edge is 2 mm while the distance between nearby contacts at the corner is 2.12 mm. The sample is preliminarily loaded in an engraved plastic support to keep it in the correct position. The support is loaded in the fixture that is provided of an actuator lever, which allows to land the probe array on the sample. The probes make physical contact to the NW network (with a mechanical force of 0.15 N) at its boundaries, at 500 μm from its edges (a schematization of the contact geometry is reported in Supporting Information S1). After that the sample is connected to the electrical measurement setup, the switch matrix defined the stimulation-measurement pattern consistently with the adjacent protocol as discussed in Supporting Information S2. The measurement accuracy was estimated by taking into account both standard deviation of repeated measurements and instrument accuracy from one-year specification given by the manufacturer.
The time evolution of Ag NW network conductivity was performed in air (relative humidity in the range of 47-56 %) at room temperature of 22 (1) °C without moving the needle probes.
The ERT image reconstruction process involves the minimization of a functional in σ: applied to ERT 42 . The value of λ was selected by means of the so-called L-curve method. 43 The value of λ chosen by the method is dependent on all the individual transresistance measurements performed, in turn dependent on the whole conductivity map of the specific sample. We identified a fast routine for the determination of λ that minimizes the additional computation effort required by the L-curve method. 44 It is worth noticing that the L-curve method allows to find the optimal balance between reconstruction error and amount of regularization (determined by ) with an objective criterion and using only the information already included in eq. (1). No additional information other than the one included in the measurements and in the finite element model is needed to apply the L-curve method. The problem (1) was discretized to a finite element method (FEM) with elements reasonably dense not to affect the maps resolution. The image reconstruction algorithm that run on the FEM was based on one of the available Gauss-Newton absolute iterative solvers of the EIDORS 3.7.1 library. The problem (1) was discretized to a FEM with elements reasonably dense not to affect the map resolution. EIDORS' routines were also used to graphically render the solution (see Supporting Information S3 for details on the image reconstruction   process).

vdP, 2PP and 4PP measurement setup
The same setup and dataset of ERT measurements was exploited for applying the vdP method, by taking into account of four terminal configuration which allowed the use of a simple formulation for this technique without numerical corrections. 40

Mapping NW network conductivity at the macroscale
Metallic NW networks were realized by drop-casting Ag-NWs in suspension on a 10×10 mm 2 quartz substrate, as reported in Figure 1a  passive electrical network in absence of an external magnetic field. The impedance matrix is then used as input for the ERT reconstruction of the conductivity map. 40,46,47 At the macroscale, conduction properties are expected to be isotropic due to the random orientation of NWs (according to ref. 48 ) and the network can be assumed as a continuous conductive medium. Indeed, despite the microscopic structure, it was shown that the approximation as a continuous conductive medium holds over a scale of a few NW length when the normalized density D > 2Dc, where Dc=5.63 is the normalized density at the percolation threshold. 39 In our case, this approximation is reasonable since in all the considered samples the normalized density was estimated to be D > 25.
Also, the approximation as a continuous medium is in agreement with resistance grid network models, where the effective resistance measured in between 3 or 4 lattice spacings apart well matches with that of a continuous uniform sheet. 49 In the continuous approximation, the electrostatic problem can be described by the Laplace equation: ill-posed, ill-conditioned and nonlinear problem that was solved by means of optimization techniques with numerical methods (details in the Experimental section). 40,46,47 In brief, the sample geometry and electrode positions were considered in a finite element model and conductivity maps were retrieved using a Gauss-Newton method to minimize eq. (1). The finite element solution is interpolated and then discretized on a grid of 100×100 pixels. Figure 1g reports the ERT conductivity map of a metallic NW network obtained by taking the impedance matrix of Figure 1e as input for the numerical solver. As it can be observed, ERT allows for a spatial-resolved characterization of the local electrical properties of NW networks at the macroscale over large area. Note that the acquisition of the whole set of measurements necessary for ERT reconstruction was performed in less than 300 s with a measurement accuracy of the transresistance better than the 0.5 % (an uncertainty budget is available in Sec 2.2.3 of ref. 50 ). It is worth noticing that the measurement speed can be even higher, which can be of interest depending on the final application, though at the cost of a lower accuracy. Being a non-scanning technique, an important aspect is that the measurement speed does not depend on the area under investigations. Concerning the spatial resolution of the ERT conductivity maps, it is necessary to take into account that it depends mainly on the amount of available boundary measurements and, thus, is limited by the number of contacts: features smaller that the distance in between electrodes are smeared out. 40 Thus, the spatial resolution of mapping can be further increased by increasing the number of contact electrodes. In this context, it is important to remark the effective scalability of the ERT technique that allows to map conductivity of metallic NW networks over large area by properly engineering the number of electrodes and their distance, taking into account that the inter-electrode distance have to be adjusted depending on the target spatial resolution.

Probing electrical inhomogeneities
In order to demonstrate the applicability of ERT to probe electrical inhomogeneities in NW networks, an intentionally non-uniform NW network was realized by drop-casting Ag-NWs in suspension on a tilted substrate in order to obtain a gradient of the NW concentration across the sample with accumulation of NWs at one corner of the sample. As reported in Figure 4a, ERT revealed that the gradient of NW density resulted in a gradient of conductivity across the sample, with high conductivity in the upper-left angle where NWs were accumulated. In order to correlate the conductivity map to the morphological properties of the network, transmittance spectra of the Ag NW network were recorded in selected regions along the observed gradient of conductivity.
Transmittance spectra acquired on the spots marked in Figure 4a are reported in Figure 4b.  Transmittance spectra of the NW network acquired on sample areas evidenced in panel a.

Time evolution of NW network conductivity
An interesting aspect of ERT as a non-scanning technique is that it allows to investigate the  In order to investigate degradation of electrical properties at the nanoscale, morphological and structural changes in Ag NWs were monitored by transmission electron microscopy (TEM). After being exposed to air at ambient conditions for 7 days, TEM revealed the presence of silver sulfide (Ag2S) nanoclusters on the NW surface (Figure 6 a-c) that were not initially observed after that the NW were extracted from the solution (not shown). In the electron diffraction pattern in Figure   6d reflections from Ag2S (-112) can be identified along with the main reflections from Ag (respectively JCPDS card no. 89-3840 and 89-3722). The presence of Ag2S after air exposure was corroborated also by chemical analysis by means of X-ray photoelectron spectroscopy (XPS) as reported in Supporting Information S12. As discussed by Elenchiguerra et al. 52 , the formation of Ag2S is driven by the presence of reduced-sulphur gases that, even if their presence in atmosphere is very low, are sufficient to initiate the corrosion process. Due to its insulating nature, the formation of Ag2S on the NW surface is responsible for an increase of the resistance at cross-point NW junctions. Reportedly, the formation of this compound is responsible for the creation of bumps on surface and even breaks of NWs. 52