This invention was made with government support under DE-SC0001085 awarded by Department of Energy. The government has certain rights in the invention.
BACKGROUND OF THE INVENTION
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A. Field of the Invention
The present disclosure relates, generally, to transparent conducting materials (“TCMs”) and, more particularly, to hybrid TCMs including a polycrystalline film that is “percolation doped” with conductive nanostructures.
B. Description of the Related Art
Transparent conducting electrodes (TCEs) require high transparency and low sheet resistance for applications in photovoltaics, photodetectors, flat panel displays, touch screen devices and imagers. Indium tin oxide (ITO), or other transparent conductive oxides, have typically been used, and provide a baseline sheet resistance (RS) vs. transparency (T) relationship. However, ITO is relatively expensive (due to limited abundance of indium), brittle, unstable, inflexible. It increases in brittleness with aging and is chemically unstable under acid/base conditions. ITO transparency drops rapidly for wavelengths above 1000 nm, so it has poor transmittance in the near infrared. Furthermore, metallic-ion diffusion from ITO into thin barrier layers may result in parasitic leakage. These and other problems make ITO-based technologies non-ideal for applications such as thin film photovoltaics (“PVs”), flexible electronics, touch-screen displays, light emitting diodes, and the like.
A suitable replacement for ITO is desired therefore. However, since resistivity and transmittance are often fundamentally constrained by the intrinsic properties of a material, developing TCMs with both low sheet resistance (e.g., RS<10Ω/□) and high transmittance (e.g., T>90%) has been a persistent challenge. Various alternative TCMs to ITO have been explored, including, by way of example, networks of carbon nanotubes (“CNTs”) and networks of metal nanowires (“NWs”). In networks of silver nanowires (AgNWs nanonet) and single-wall carbon nanotubes (SWCNTs nanonet), for NW or CNT densities corresponding to 85-95% transparency (T), conduction is typically dominated by percolation through junctions with relative large tube-tube/NW-NW contact resistance (RNW-NW), resulting in a rapid increase in baseline sheet resistance (RS) (k Ω/□−G Ω/□, depending on the NW/NT) as T increases. Networks of only metallic nanowires exhibit sheet resistance of the order of kilo-ohm/□ and more. Approaches involving welding of the nanowires, thermal annealing under pressure, or electroplating decrease RS by improving RNW-NW, but it has been challenging to reduce overall RS below ≈30Ω/□, especially for broadband T at 90%. Moreover, micrometer-sized holes within the network add series resistance to devices that rely on vertical current transport such as LEDs and solar cells. Composite transparent conducting electrodes (TCEs) employing silver NWs with another conducting polymer such as PEDOT:PSS and a combination of TiO2 nanoparticles with PEDOT:PSS have recently been demonstrated with sheet resistances of 12Ω/□ at average T of 86% over wavelengths 350-800 nm and 15Ω/□ at T550 nm of 83% respectively. The conducting polymer and TiO2 nanoparticle primarily reduce the tube-tube contact resistance.
Other alternative TCMs to ITO have been explored, including chemical vapor deposited (“CVD”) polycrystalline graphene (“poly-graphene” or “PG”) films, including single layer graphene (SLG) and few-layer graphene (FLG). “Single-crystal” graphene, such as that obtained by exfoliation from highly ordered pyrolytic graphite (HOPG) crystals, exhibits several interesting physical phenomena, including an RS lower than ITO, at a given optical transparency. Single-layer graphene (SLG) or few-layer graphene provide sufficiently high transparency (≈97% per layer) to be a potential replacement for ITO. However, the exfoliated approach yields samples that are too small for practical applications, and large-area synthesis approaches, including chemical vapor deposition (CVD), typically involving growth on copper foil and subsequent transfer to an arbitrary substrate, produces grain sizes typically ranging from a few micrometers to a few tens of micrometers, depending on the specific growth conditions. The resulting films have relatively high sheet resistance due to small grain sizes and high-resistance grain boundaries (HGBs).
While these potential ITO replacements each resolve several practical issues associated with ITO, their respective RS−T curves are not significantly different from that of ITO (as shown in FIG. 9). To achieve technologically relevant sheet resistance values (e.g., RS<20Ω/□), the density of a network of CNTs or NWs must significantly exceed the percolation threshold. These high densities of CNTs or NWs, however, reduce the transmittance of such TCMs considerably. Moreover, even with low RS, vertical current collection in PV cells is compromised by current crowding at the small-area interface between a network of CNTs or NWs and the bulk emitter layer. Meanwhile, experimental data suggests that there is a fundamental limitation to the sheet resistance and transmittance of pure poly-graphene films, making it difficult for poly-graphene to compete successfully with ITO.
It is therefore desired to produce an alternative to ITO that simultaneously exhibits high transparency and a technologically relevant sheet resistance value (e.g., RS<20Ω/□).
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OF THE INVENTION
According to one aspect of the invention, a hybrid transparent conducting material (TCM) comprises a granular polycrystalline film and a layer, on the granular polycrystalline film, comprising a plurality of randomly dispersed conductive nanostructures. The conductive nanostructures are in contact with, or adjacent, the polycrystalline film. The granular polycrystalline film preferably is an atomic monolayer, and the granular polycrystalline film preferably is a polycrystalline graphene film. Most preferably, the polycrystalline film is an atomic monolayer of polycrystalline graphene.
The conductive nanostructures preferably are metallic nanowires, more preferably silver nanowires. In one embodiment, each of the conductive nanostructures has a length greater than 1 μm and a cross-sectional dimension of less than 1 μm. Preferably, average length of the conductive nanostructures is greater than average grain diameter in the granular polycrystalline film, and average distance between the conductive nanostructures is greater than average length of the conductive nanostructures.
In one aspect of the invention, density of the plurality of conductive nanostructures randomly dispersed in the granular polycrystalline film is below a percolation threshold. Preferably, density of the plurality of conductive nanostructures randomly dispersed on the polycrystalline film is at most sixty percent of the percolation threshold. In some embodiments, the conductive nanostructures do not form a percolating network for charge carriers in the polycrystalline film.
In the hybrid TCM of the invention, preferably the granular polycrystalline film and the nanowire layer separately each have a sheet resistance of 20 ohms per square or greater. The hybrid TCM has a sheet resistance below twenty ohms per square. The hybrid TCM of the invention preferably has a transmittance above ninety percent for solar radiation. Preferably the transparent electrode has a sheet resistance below twenty ohms per square and a transmittance above ninety percent for solar radiation.
In the hybrid TCM, the number of conductive nanostructures preferably is less than one half the number of grains in the granular polycrystalline film, more preferably the number of conductive nanostructures is less than one fourth the number of grains in the granular polycrystalline film.
In some embodiments, the transparent electrode may comprise a plurality of stacked layers, where each of the plurality of stacked layers comprises polycrystalline graphene that is percolation doped with metallic nanowires. The transparent electrode may have a sheet resistance below twenty ohms per square and a transmittance above ninety percent for solar radiation.
A photovoltaic cell is provided according to the invention which comprises a transparent electrode comprising polycrystalline graphene that is percolation doped with metallic nanowires, wherein the metallic nanowires do not form a percolation network for charge carriers across the transparent electrode. Because the nanowires by themselves have poor electrical contact they do not provide good percolation transport apart from the combination with the graphene. The metallic nanowires are at sufficiently low density that they do not form a percolation network for charge carriers across the transparent electrode. The photovoltaic cell can comprise a transparent electrode comprising a plurality of stacked layers, with each of the plurality of stacked layers comprising polycrystalline graphene that is percolation doped with metallic nanowires.
A liquid crystal display according to the invention comprises a transparent electrode comprising polycrystalline graphene that is percolation doped with metallic nanowires.
BRIEF DESCRIPTION OF THE DRAWINGS
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The foregoing advantages and features of the invention will become apparent upon reference to the following detailed description and the accompanying drawings, of which:
FIG. 1A illustrates one embodiment of a poly-graphene microstructure having uniform square grains (“square”).
FIG. 1B illustrates one embodiment of a poly-graphene microstructure having uniform hexagonal grains (“hex1”).
FIG. 1C illustrates one embodiment of a poly-graphene microstructure having perturbed hexagonal grains with Gaussian size distribution (“‘hex2”).
FIG. 1D illustrates one embodiment of a poly-graphene microstructure having random grains with normal size distribution (“rand1”).
FIG. 1E illustrates one embodiment of a poly-graphene microstructure having random grains with log-normal size distribution (“rand2”).
FIG. 1F illustrates a simplified one-node model for the uniform square grains of FIG. 1A.
FIG. 2A illustrates a grain size distribution for the poly-graphene microstructures of FIG. 1C.
FIG. 2B illustrates a grain size distribution for the poly-graphene microstructure of FIG. 1D
FIG. 2C illustrates a grain size distribution for the poly-graphene microstructure 1E.
FIG. 3 is a graph of normalized sheet conductance for the poly-graphene microstructures of FIGS. 1A-E as a function of sample length, for three different percentages of high-resistance grain boundaries (“GBs”).
FIG. 4 is a graph of normalized conductivity for the poly-graphene microstructures of FIG. 1A-E as a function of the percentage of high-resistance GBs for a relatively long sample.
FIG. 5 illustrates the percolation of charge carriers through a poly-graphene film including high-resistance and low-resistance GBs.
FIG. 6 illustrates the percolation of charge carriers through a poly-graphene film including high-resistance and low-resistance GBs. where the poly-graphene film has been “percolation doped” with metallic nanowires.
FIG. 7 is a graph of normalized conductivity for pure poly-graphene and for two hybrid TCMs as a function of the percentage of high-resistance GBs, for different NW densities.
FIG. 8A illustrates one embodiment of a poly-graphene sample having perturbed hexagonal grains.
FIG. 8B illustrates a poly-graphene sample percolation doped with metallic NWs.
FIG. 8C illustrates a potential profile of the poly-graphene sample of FIG. 8A.
FIG. 8D illustrates a potential profile of the poly-graphene sample of FIG. 8B.
FIG. 8E is a graph of simulated transmittance as a function of incident light wavelength for a regularized network of NWs for two different NW densities.
FIG. 9 is a graph of transmittance as a function of sheet resistance for illustrative embodiments of the presently disclosed hybrid TCMs as well as various conventional TCMs.
FIG. 10 is a flow chart of the manufacturing method for TCM according to the invention.
FIG. 11a shows SLG transfer process of CVD graphene grown on copper and the fabrication process flow of hybrid films (Hybrid 1).
FIG. 11b shows SLG transfer process of CVD graphene grown on copper and the fabrication process flow of hybrid films (Hybrid 2).
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OF SPECIFIC EMBODIMENTS
While the concepts of the present disclosure are susceptible to various modifications and alternative forms, specific exemplary embodiments thereof have been shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that there is no intent to limit the concepts of the present disclosure to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives consistent with the present disclosure and appended claims.
References in the specification to “one embodiment,” “an embodiment,” “an illustrative embodiment,” etc., indicate that the embodiment described may include a particular feature, structure, or characteristic, but every embodiment may not necessarily include the particular feature, structure, or characteristic. Moreover, such phrases do not necessarily refer to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is submitted that it is within the knowledge of one skilled in the art to effect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described. Publications and patent documents mentioned in the present specification are incorporated by reference herein in their entirety. In particular, the article “Co-Percolating Graphene-Wrapped Silver Nanowire Network for High Performance, Highly Stable, Transparent Conducting Electrodes” (Chen et al., Advanced Functional Materials, 2013), its Supporting Information, and articles referenced therein all are incorporated in their entirety by reference.
In the drawings, specific arrangements or orderings of schematic elements may be shown for ease of description. However, it should be understood by those skilled in the art that the specific ordering or arrangement of the schematic elements in the drawings is not meant to imply that a particular order or sequence of processing, or separation of processes, is required. Further, the inclusion of a schematic element in a drawing is not meant to imply that such element is required in all embodiments or that the features represented by such element may not be included in or combined with other elements in some embodiments.
The present invention uses a polycrystalline film, preferably a single layer graphene (SLG) film. SLG sheets are produced by chemical vapour deposition, and are a single atom thick layer of graphene which has a structure with grains and grain boundaries (GB). The granular nature of the graphene means that it forms a semi-continuous layer of material. It is weakly conductive.
There is an increasing effort to fundamentally understand the structure of graphene GBs and the impact of GBs on mechanical strength as well as electronic transport. The present inventors have classified GBs broadly into two categories, namely high resistance grain-boundaries (HGBs) and low resistance grain-boundaries (LGBs). Both HGBs and LGBs contribute to the resistance in a SLG film, but it is the HGBs which severely limit the (percolating) electronic transport, as indicated by best reported RS of =≈250-700Ω/□ (and 2-6 kΩ/□ for each layer in typical SLG/FLG). In typical CVD graphene grown on copper with average grain size of ≈1 μm, the present inventors have recently estimated the percentage of HGBs by using percolation theory through HGBs to interpret the resistance of circular transfer length measurement (CTLM) devices.
A number of groups have proposed methods or reported initial experimental results for improving the transport properties by GB engineering or doping. CVD grown under high methane flow rate provides very uniform coverage over the copper foil and better inter-domain stitching, allowing control of RS. While the inter-grain four-terminal RS changes between ≈3-9 kΩ as the carrier density is modulated, the intragrain resistance scales up by a constant factor of 1.4 for the same modulation in carrier density, indicating scattering by GBs even for best quality poly-SLG. The most impressive result to date was for large scale CVD graphene by roll-to-roll production, in which the as-grown SLG RS (≈250Ω/□) was reduced to ≈125Ω/□ by wet-chemical doping, and four layers were stacked to provide 30Ω/□ at 90% transparency and at 550 nm (T550 nm≈90%). However, the chemical doping decreases RS only by a factor of 2 because while conductivity of individual grain is enhanced, the percolating resistance is still dominated by the HGBs (FIG. 1B).
In contrast to this approach, the present invention utilizes “percolation-doping” of a polycrystalline film, preferably a single layer graphene film as described above, using conductive nanowires or nanotubes. In this approach, the RS of graphene network is reduced, not by increasing carrier density, as in chemical doping, but by opening up new conduction channels by bridging the HGBs in the SLG with nanowires or nanotubes.
Thus, the present disclosure relates to hybrid TCMs including a polycrystalline film (e.g., a poly-graphene film) that is “percolation doped” with conductive nanostructures (e.g., metallic NWs). As noted above, HGBs dominate the resistance in SLG. In hybrid structures with appropriate densities of AgNWs, the NWs bridge the HGBs, providing a percolating transport path for the electrons and therefore lower the sheet resistance. An experimentally calibrated, comprehensive numerical model for electron transport in poly-graphene was used to determine that the high resistivity of pure poly-graphene films reflects an intrinsic percolation bottleneck, in which electrons are periodically trapped in domains formed by high-resistance grain boundaries (“GBs”). FIG. 5 illustrates the percolation of charge carriers through a poly-graphene film including high-resistance and low-resistance GBs, while FIG. 6 illustrates the percolation of charge carriers through a poly-graphene film including high-resistance and low-resistance GBs where the poly-graphene film has been “percolation doped” with metallic nanowires.
As shown in FIG. 6, the density of the NWs may be below a percolation threshold, so that the NWs themselves do not form a percolating network across the hybrid TCM between electrical contacts. If the average length of the NWs is larger than the average grain size, <Lgrain>, the NWs will cross the GBs of the poly-graphene with probability approaching one. Where a NW intersects a high-resistance GB, that GB will no longer inhibit current conduction (as illustrated by the dashed lines in FIG. 6), so that the effective percentage of high-resistance GBs, PGB, is reduced. Given the exponential dependence of conductivity on PGB (see FIG. 4), even a modest percolation-doping with NWs can dramatically decrease the sheet resistance RS of poly-graphene films. Thus, the density of the nanowires can be used to control the properties of the hybrid material.
As used in the present disclosure, “percolation doping” refers to the inclusion and/or addition of conductive nanostructures in a polycrystalline film, at a density below a percolation threshold, to improve conductivity by creating conducting pathways that bridge high-resistance GBs in the polycrystalline film. The continuity of the polycrystalline film in such a hybrid TCM ensures vertical current collection free from current crowding, while the relatively low density of conductive nanostructures ensures that the high transmittance of the polycrystalline film is not compromised. As further described below, a hybrid TCM including a polycrystalline film that is percolation doped with conductive nanostructures simultaneously achieves both low sheet resistance (e.g., RS<20Ω/□) and high transmittance (e.g., T>90%). Its performance with respect to these properties is comparable to, or better than, ITO, while it is free of the negative attributes of ITO discussed herein. Although the illustrative embodiments described below are described primarily with reference to a poly-graphene film, it is contemplated that any polycrystalline material may similarly benefit from percolation doping with conductive nanostructures. Examples of other polycrystalline films for forming transparent conducting electrodes include transparent conductive oxides (TCO) which are doped oxides of tin, zinc, cadmium and their alloys, such as ITO. However, the utility of percolation doping in a two-dimensional (2D) polycrystalline material by a one-dimensional (1D) random nanostructure extends beyond TCE. Any large scale, polycrystalline 2D material such as MoS2, silicene, germanane etc can benefit from the teachings according to the present invention.
Graphene-silver nanowires hybrid transparent conducting electrodes are formed as follows. Silver nanowires of number density Di (Di varies roughly from 2.0×106 to 4.8×106), diameters 70 nm-110 nm, and length of 20-60 μm were uniformly dispersed on to RCA cleaned quartz or transparent and flexible substrate (PET) to form a nano-net structure. The structure is dried for 12 hours in nitrogen ambient maintained within a glove box. Surfactants and residue on substrate/nanonet surface are cleaned by dipping in methanol for 30 minutes and dried by blowing with dry nitrogen gas. The sample is then annealed at 180° C. for 1 hour in forming gas with 40 sccm flow rate (For samples with a bendable substrate (PET), the process step are performed at 150° C. temperature). Independently, in another process, AZ1518 photoresist is spin-coated on one side of a copper foil containing graphene on its surface. The graphene on the other side of the copper foil is plasma etched for 10 seconds, using oxygen plasma at 100 Watt power, 50 mtorr O2 pressure, and 50 sccm flow rate. Then the copper foil is etched in copper etchant to get a graphene/AZ1518 stack floating on the etchant.
The single layer graphene sheet protected with photoresist (AZ1518) layer is wet transferred on to a quartz/nano-net or PET/nano-net substrate, cleaned twice using DI water, cleaned twice in dilute hydrochloric acid (5% by volume in DI) for 15 minutes, and then cleaned twice again in DI. It then is dried for 12 hours, preferably in nitrogen glove box. After drying, the photoresist is stripped in hot acetone at 90° C. for 3 hours, followed by an acetone rinse and then a methanol rinse, and is blown dry gently with nitrogen.
The quartz/AgNW-nanonet/graphene stack is annealed in forming gas at 300° C. for 1 hour with 40 sccm flow rate, then cooled to 50° C. or below in forming gas before removing from furnace. For PET/AgNW-nanonet/graphene hybrid bendable TCE, the annealing temperature is 150° C. (all other parameters remaining the same).
For CTLM device fabrication, a standard photolithography process, development, metallization and lift-off is used. The quartz/AgNW-nanonet/graphene is dipped in chlorobenzene solution for 5 minutes before the photolithography. No chlorobenzene treatment is used for PET/AgNW-nanonet/graphene. Contact electrodes for electrical measurements: Ti (1 nm)/Pd (30 nm)/Au (20 nm) are e-beam evaporated with a rate of 0.7 Å/sec, 1 Å/sec, and 1 Å/sec respectively. Finally, after the lift-off, the devices are cleaned in hot acetone at 90° C. for 2 hours, rinsed in acetone and methanol and blow dried with nitrogen.
The hybrid Graphene-AgNW TCE shows excellent chemical and optical stability over four months of time (examined after four months of storage in glove box and couple of days exposure to air before the measurements), and has a stabilized sheet resistance of 13Ω/ with 88% optical transmittance at 550 nm wavelength. The hybrid Graphene-AgNW TCE on PET substrate shows excellent mechanical stability over bending. Pure graphene resistance changed by ˜20% with bending radius ranging from 8.3 mm to 14.6 mm, while hybrid graphene-AgNW changed its resistance much less.
To better understand the resistivity of poly-graphene, a process model was used to produce representative structures, an electrical model was used to compute sheet resistances, and an optical model was used to compute transmittances. First, polycrystalline graphene samples were synthetically generated using Voronoi tessellation. In this algorithm, input parameters (e.g., the pattern and number of seed sites) were used to control statistical features of the resulting Voronoi cells (e.g., shape, size. statistical distributions, etc.). This approach allowed the generation of a wide variety of film morphologies characteristic of various deposition conditions and captured the universal features of carrier transport in poly-graphene films, independent of the details of film deposition. Five types of microstructures with increasing complexity were used to represent grain-size distributions in poly-graphene films: uniform square grains as a reference structure (“square,” illustrated in FIG. 1A), uniform hexagonal grains to approximate poly-graphene films produced by a seeded growth method (“hex1” illustrated in FIG. 1B), perturbed hexagonal grains with Gaussian size distribution typical of poly-graphene films produced by a seeded growth method (“hex2” illustrated in FIG. 1C), random grains with normal size distribution to approximate CVD poly-graphene films (“rand1” illustrated in FIG. 1D), and random grains with log-normal size distribution typical of CVD poly-graphene (“rand2” illustrated in FIG. 1E). Corresponding grain size distributions for the “hex2”, “rand1” and “rand2” microstructures are illustrated in FIG. 2A, FIG. 2B, and FIG. 2C, respectively. In each of the five types of microstructures, the average grain size (i.e., the average grain diameter. <Lgrain> was ˜5 μm (consistent with reported values). Several hundred samples of the foregoing microstructures were synthetically generated for statistical study of the transport characteristics of poly-graphene films.
Two important electrical parameters for polycrystalline films are the resistances of the grains (i.e., inter-grain resistances) and of the GBs intra-grain resistances). In poly-graphene, it is experimentally observed that ratio between these resistances typically range from ˜1 to ˜30. Although the GBs may exhibit a distribution of resistances (as a function of misorientation between neighboring grains), for simplicity, each GB is classified as either a high-resistance GB or a low-resistance GB in the present disclosure. In the illustrative microstructures shown in FIG. 1A-E, approximately half of the GBs are classified as high-resistance GBs (lines of heavier weight), and approximately half of the GBs are classified as low-resistance GBs (lines of lighter weight). In other words, the percentage of high-resistance GBs, PGB, is about 50% in these microstructures.
A drift-diffusion formulation may be used to describe electronic transport through the microstructures described above: J=(σ·∇(Fn/q), where J is the current density in A/m, (σ is the sheet conductivity, and Fn is the electrochemical potential). This drift-diffusion formulation is appropriate because the average grain size (˜5 μm) is much larger than the typical mean-free path (hundreds of nanometers). Assuming charge current is conserved (i.e., no recombination-generation), ∇·J=0 may be solved.
Within the bulk of a poly-graphene grain, σ=σ0. The theoretical lower limit of sheet resistance is 30Ω/□, which occurs when only acoustic deformation potential scattering is present. A low-resistance GB may characterized as being perfectly transparent to charge carriers (i.e., σGB(lo)≡σ0). A high-resistance GB may be characterized by transport energy gap (EG) below which charge carriers are perfectly reflected, i.e., σGB(hi)<σ0. With these three conductivities, σ0, σGBlo, σGBhi, the transport problem is fully defined. The foregoing model of high and low resistance GBs leads to a maze-like morphology landscape through which a charge carrier injected from one contact travels to the other contact (as further described below with reference to FIG. 5), thereby transforming electronic transport in poly-graphene into a percolation problem.
For each of the microstructures discussed above, the finite difference method (“FDM”) was used to calculate electronic transport properties (with each grain having about 200 nodes). The input parameters used for the FDM calculations were the sheet resistance within the grains (Rlo≈30Ω/□) and the sheet resistance across high-resistance GBs (Rhi≈63Rlo). The FDM results were compared to a simple “one-node model” in which each grain was represented by only one node. In the one node model, each grain is represented by a single “super-node” of a network. These grains are separated from each other by the intergrain sheet resistances, which could be either high-resistance grain boundaries (RHGB), or low-resistance grain boundaries (RLGB). A single AgNW bridges multiple grains. For the illustrative microstructure of FIG. 1A, the one-node model is shown schematically in FIG. 1F, where the sheet resistance across each low-resistance GB and the sheet resistance across each high-resistance GB are denoted as Rlo and Rhi, respectively. The high-resistance and low-resistance GBs in this one-node model can be easily measured. See also Figure S6 in the Supporting Information of Chen et al.
The normalized sheet conductance, G/σ0, for each of the microstructures discussed above is plotted in FIG. 3 as a function of the sample length, Lc, for three different percentages of high-resistance GBs (PGB=20%, 50%, and 80%). The sample width was fixed at seven times the average grain size, after which the width dependence of the electronic transport properties disappears (as one of ordinary skill would expect for large area films). The inset of FIG. 3 shows the dependence of the conductance exponent, n, on the sample length, i.e., G∝Lc, for the three different percentages of high-resistance GBs. If the sample length is smaller than about ten times the average grain size, the conductance exponent becomes significantly larger than −1.0, indicating a nonlinear dependence on sample length. Thus, as compared to a longer sample, there is a higher probability in a shorter sample that low-resistance GBs and grains form a percolation path (i.e., a continuous network) between the electrical contacts. For large area poly-graphene (e.g., square meters, as may be used in PV applications), however, the conductance exponent approaches −1.0 with increasing length, regardless of the percentage of high-resistance GBs. As can be seen in FIG. 3, the grain shape and grain size distributions have little effect on the conductance of poly-graphene. Rather, it is the average size of the grains, <Lgrain>, and the percentage of high-resistance GBs, PGB, that dictate the overall electronic transport properties of poly-graphene.
The dependence of the normalized conductivity, σ/σ0, on the percentage of high-resistance GBs, PGB, for a relatively long sample (e.g., Lc≈100×<Lgrain>) is plotted in FIG. 4. As can be seen in FIG. 4, even small increases in the percentage of high-resistance GBs may result in dramatic suppressions of conductivity. This result may be understood with reference to FIG. 5, which interprets the resistance of a poly-graphene film as a percolation problem defined by high-resistance GBs (lines of heavier weight) and low-resistance GBs (lines of lighter weight). The percolation threshold for a Voronoi tessellation is 0.667-0.68, while the percolation threshold of a hexagonal lattice is 0.6527. Therefore, regardless of the specific form of the GB distribution, when the percentage of high-resistance GB approaches ˜0.66 (i.e., PGB≈66%), charge carriers traveling through a poly-graphene film between a pair of electrical contacts must cross one or more high-resistance GBs (the movement of the charge carriers is indicated by arrows in FIG. 5). This percolation bottleneck suppresses the conductivity of the poly-graphene film exponentially.
To confirm this percolation analysis quantitatively, the numerical results from the FDM simulation discussed above were interpreted using the generalized effective media (“GEM’”) theory. The GEM equation is given by: