Design Parameters for Subwavelength Transparent Conductive

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Design Parameters for Subwavelength Transparent Conductive Nanolattices Juan J. Diaz Leon,†,‡ Eyal Feigenbaum,§ Nobuhiko P. Kobayashi,‡ T. Yong-Jin Han,† and Anna M. Hiszpanski*,† †

Materials Science Division and §National Ignition Facility, Lawrence Livermore National Laboratory, Livermore, California 94550, United States ‡ Baskin School of Engineering, University of California, Santa Cruz, California 95064, United States S Supporting Information *

ABSTRACT: Recent advancements with the directed assembly of block copolymers have enabled the fabrication over cm2 areas of highly ordered metal nanowire meshes, or nanolattices, which are of significant interest as transparent electrodes. Compared to randomly dispersed metal nanowire networks that have long been considered the most promising next-generation transparent electrode material, such ordered nanolattices represent a new design paradigm that is yet to be optimized. Here, through optical and electrical simulations, we explore the potential design parameters for such nanolattices as transparent conductive electrodes, elucidating relationships between the nanowire dimensions, defects, and the nanolattices’ conductivity and transmissivity. We find that having an ordered nanowire network significantly decreases the length of nanowires required to attain both high transmissivity and high conductivity, and we quantify the network’s tolerance to defects in relation to other design constraints. Furthermore, we explore how both optical and electrical anisotropy can be introduced to such nanolattices, opening an even broader materials design space and possible set of applications. KEYWORDS: transparent electrodes, metal nanowires, transparent conductive nanolattice, subwavelength periodicity, electrical and optical anisotropy, defect effects

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characterized by sheet resistance (Rs) less than 100 Ω/sq and transmittance across the visible regime greater than 90%.5,7 Rather than randomly dispersing nanowires to form irregular networks, several groups have shown that higher transparency, conductivity, and uniformity of these properties across larger areas can be attained if nanowires are dispersed to be macroscopically aligned along two orthogonal directions, thereby increasing the number of junctions between nanowires compared to randomly dispersed nanowires.8−10 These moreordered networks are fabricated by screening solutions through microfluidic channels to dictate nanowires’ orientation. For example, Kang et al.8 uniaxially aligned Ag nanowires by dragging a patterned polydimethylsolixane (PDMS) stamp with nanochannels across substrates with Ag nanowire solutions, and Cho et al.9 similarly induced uniaxial alignment by dragging a rod wrapped with μm-sized wires (which effectively act as microfluidic channels) across substrates with Ag nanowire solutions. By repeating the coating process orthogonally to the first direction, networks of orthogonally crossing nanowires can be created, increasing the number of junctions per wire compared to networks formed from randomly dispersed

INTRODUCTION Given its high transparency at visible wavelengths and relatively high conductivity, indium tin oxide (ITO) is the de facto standard transparent conductive electrode material widely used in optoelectronic devices from displays to solar cells.1−3 However, increased indium costs and ITO’s lack of mechanical flexibility have necessitated research into alternative transparent conductive materials, particularly to meet the needs of nextgeneration flexible and wearable electronics. Among the most promising alternative transparent conductive materials being explored are randomly dispersed networks of metallic nanowires, which are most often Ag or Cu.1−5 These networks are formed by synthesizing metal nanowires typically with sub-100 nm diameters (D) and varying lengths (L) that are then dispersed from solution, creating irregular percolating metal nanowire networks. The number of nanowires required to surpass the percolation limit in a random two-dimensional network scales inversely with the nanowires’ length squared (N ∝ 1/L2),6 and the fewer nanowires composing a network, the higher the network’s transmissivity. Thus, high aspect ratio nanowires (L/D), which are challenging to synthesize, are required to attain both high transmissivity and conductivity.4 For example, Mutiso et al. calculated that for a randomly dispersed nanowire network, nanowires with L/D > 400 are needed to attain high-performance transparent electrodes © XXXX American Chemical Society

Received: June 12, 2017 Accepted: September 13, 2017

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DOI: 10.1021/acsami.7b08446 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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ACS Applied Materials & Interfaces nanowires.8,9 However, while these networks have macroscopic ordering, nanowires in these networks still have no local ordering or periodicity in their spacing. Even more desirable than to increase the conductivity and mechanical robustness of these networks would be to pattern nanowires, thereby forming transparent conductive nanolattices with subwavelength periodicity that maximize nanowires’ interconnectivity.11,12 For example, Hsu et al. fabricated ordered Au nanowire arrays via electron beam lithography and measured their sheet resistance to be 7.2 Ω/sq and their transmission to be 95%.10 However, attaining transparent conductive nanolattices with nanowires that are less than 100 nm in diameter (as solution-synthesized nanowires are) is difficult with existing nanofabrication techniques, and furthermore, these nanofabrication techniques are not easily scalable to the large areas that applications require. Rolith Inc. (now Metamaterials Technologies Inc.) has developed a scalable method for fabricating metal meshes with high transmissivity (96%) and conductivity (5 Ω/sq) using their propriety rolling mask lithography technique.11,13 However, their metal wires are significantly larger (ca. 300 nm wide and between 300 and 500 nm thick), and thus, to attain such high transmissivity, a large mesh pitch of 30 μm is required. Such large wire spacing makes these metal mesh electrodes unsuitable for flexible organic electronic devices since organic semiconductors have short carrier diffusion lengths, commonly on the order of 100 nm or 2 orders of magnitude smaller than the pitch of Rolith’s wire meshes, and the total film thickness of organic devices is often less than the thickness of these 300−500 nm thick wires.14,15 For such applications, a thin metal nanowire mesh with periodicity on the order of ca. 100 nm is more desirable if it can also attain high transmissivity. Recently, Majewski et al.16 demonstrated that metal nanowire meshes with sub-100 nm diameter and periodicity can be fabricated over large cm2 areas by an entirely different approach using block copolymers. Specifically, applying shear forces to block copolymers induces their rearrangement into line-patterns orientated in the direction of applied shear stress with ca. 10−100 nm diameter and periodicity.17−19 Majewski et al. used laser-heating and the resulting local variations in thermal expansion to produce shear forces that cause the alignment of block copolymers.16 Block copolymer poly(styrene)-b-poly(2-vinylpyridine) (PS-P2VP) is the block copolymer of choice since the P2VP phase complexes with metal salts,20 thereby providing a means of directly creating metal nanowires from the shear-aligned block copolymer rearranged into lines. After metal salt infiltration, the block copolymer is plasma-etched away, leaving behind only metal nanowires. Repeating the process of making nanowires and shear-aligning orthogonally to the direction of nanowire alignment in the first layer results in the creation of the rectangular-shaped lattice patterns. Figure 1 shows our own platinum metal nanowire meshes,21 fabricated by adapting the techniques of Majewski et al.16 While the resistivity of platinum nanowires produced in such a manner is initially high due to grain boundaries,20 Majewski et al. found that with sintering, the Pt nanowires’ resistivity approaches that of bulk platinum.16 Furthermore, a single array of platinum nanowires has transparency of ∼90% across the visible regime,16 which suggests that orthogonally crossing arrays of such block copolymer-derived metal nanowires may be promising as transparent conductive nanolattices (TCNs) with further optimization.

Figure 1. Scanning electron microscopic images of (a) isotropic square and (b) anisotropic rectangular Pt nanowire meshes with sub-100 nm periodicity fabricated using the sequential and layer-by-layer shear alignment of block copolymer films of poly(styrene)-b-poly(2vinylpyridine), PS-P2VP.

In addition to providing a highly scalable means of producing over large areas TCNs with subwavelength periodicity, the use of block copolymers also provides significant tunability over TCN parameters. Specifically, the choice of metal salt dictates the type of metal nanowire produced,20 and the dimensions (height and width) of the nanowires and their periodicity are dictated by the degree of polymerization of the block copolymer.21 However, guidelines for designing subwavelength TCNs as transparent electrodes have not been previously explored as such structures have only recently become capable of being fabricated over large areas, specifically by using block copolymers. Experimentally exploring this parameter space to elucidate design guidelines for TCNs is challenging since each new set of nanowire dimensions requires the synthesis of a new block copolymer having the appropriate degree of polymerization and reoptimization of the processing conditions to align the block copolymer. To help guide experimental efforts, we performed optical and electrical simulations exploring the TCN parameter space and elucidating design rules for their use as transparent conductive electrodes. Specifically, we explore how the dimensions and periodicity of TCNs affect their transmissivity and conductivity. We furthermore derive relationships for how defects in the nanowire lattice, which are common in block copolymer-derived patterns, affect the design and performance constraints. Finally, we present how structural anisotropy, which can be introduced by using block copolymers with differing degrees of polymerization, presents unique opportunities to engineer anisotropy into the electrical and optical properties of TCNs.

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RESULTS AND DISCUSSION Effects of Nanowire Dimensions in TCNs. First, we evaluated how changes in the TCNs’ dimensions affect their performance as transparent conductive electrodes. While the dimensions and periodicity of nanowire networks formed by the shear alignment of block copolymer can be tuned along different directions of the lattice and while defects within the network are to be expected, for simplicity we first evaluated the optoelectronic properties of defect-free, isotropic TCNs (i.e., perfect nanowire networks with comparable periodicity and wire dimensions in both the x- and y-direction). Furthermore, we limited the lattice spacing to sub-100 nm as this is the approximate upper-limit of spacing achievable using block copolymers. Like others modeling randomly dispersed networks of nanowires,7,22,23 we assumed that nanowires interpenetrate (i.e., junctions between wires are in-plane). B

DOI: 10.1021/acsami.7b08446 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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values for the size-dependent resistivity of Cambrios’ wellcharacterized Ag nanowires.27 To scale the expected resistivity appropriately for the dimensions and square-shape of nanowires under consideration, we fitted these values using models by Steinhoegl, Dingle, and Fuchs (see Methods for details). Furthermore, for simplicity of our initial calculations and to focus on the effects of defects and anisotropy on the electrical properties of the nanowire meshes, we assume that there is no junction resistance between crossing wires. While junction resistance between silver nanowires deposited from solution can be quite high due to their polymer coating,28 all organics are plasma-etched and sintered away in the block copolymerbased fabrication scheme, which we anticipate should yield low junction−junction resistance. Figure 2b depicts a schematic of the simulation of a TCN where nanowires are represented in gray and electrodes are placed at both ends of the wires along the x-axis. Superimposed with blue arrows is a current propagation map based on the path of least resistance, where the size of the arrow is proportional to the magnitude of the current. The direction of the arrow at each junction represents the vector addition of currents from the top, bottom, and right wires. If the nanowire network has no defects, as is shown in Figure 2b, then when a voltage difference is applied across the electrodes, current propagates directly along the nanowires aligned in the x direction as this is the shortest and least resistive path connecting the electrodes. We will call the nanowires aligned along the x-axis the “direct” network. For such a defect-free nanowire network, the wires aligned along the y-axis do not carry any current, and the sheet resistance is dictated by the parallel resistance of the nanowires forming the direct network. To evaluate the optical properties of TCNs, we simulated the transmission spectra of various Ag nanowire meshes at normal incidence using Comsol (see Figure S1). Most of the spectra show a characteristic surface plasmon resonance (SPR) absorption peak, though the magnitude and wavelength of this peak change depending on the dimensions29 of the Ag nanowires and periodicity 30,31 of the mesh. To ease comparisons of the transmissivity of different nanowire meshes,

To evaluate the electrical properties of TCNs, we developed an analytical model using a resistor network. Figure 2a depicts a

Figure 2. Schematics showing (a) the defining parameters of a defectfree TCN and (b) the model used to calculate the sheet resistance of a defect-free TCN with current propagation mapped in blue arrows on top of the nanowires.

TCN without defects and identifies its defining parameters. The periodicity of nanowires aligned in the x- and y-directions is denoted by px and py, respectively, and the resistivity of the wires (which is dependent on the nanowires’ dimensions) oriented in the x- and y-direction is denoted as rx and ry, respectively. When the dimensions of nanowires are greater than approximately 100 nm,24 then the nanowires’ resistivity approaches the bulk metal’s resistivity, and the nanowires’ electrical resistance scales inversely with wires’ cross-sectional area. However, it is important to note that the resistivity of the nanowires (rx and ry) can increase considerably with decreasing nanowires’ dimensions, particularly below ca. 100 nm. Specifically, as the dimensions of a nanowire decrease to a size comparable to the electron mean free path, grain size and surface roughness (which are impacted by the metal and its processing) play an increasing role in determining electrical resistivity of the nanowire.24−26 We are currently performing separate studies to evaluate the resistivity of metal nanowires fabricated from block copolymer templates. However, for the purposes of this study, we chose to use experimentally derived

Figure 3. Illustrative examples of how the sheet resistance (left axis, black circle) and transmission (right axis, blue squares) of defect-free Ag nanowire meshes change with increasing nanowire width for TCNs with (a) pitch = 100 nm and thickness = 10 nm, (b) pitch = 50 nm and thickness = 10 nm, and (c) pitch = 100 nm and thickness = 50 nm. The dashed red line represents the performance criterion (90% visible transmission) for transparent conductive electrodes. C

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Figure 4a depicts a TCN with defects and identifies its defining parameters with quasirandomly placed defects or

we averaged their simulated transmission spectra over visible wavelengths from 380 to 760 nm to attain a single transmission value. With these tools in hand, we evaluated how changing nanowires’ width, thickness, and periodicity each affect the sheet resistance and visible transmission of the TCN. Figure 3 depicts these values for three different cases of Ag nanowires’ pitch and thickness as a function of varying nanowire width. Though the conductivity and transmission criteria for transparent conductive electrodes vary depending on the application, the applications with the most demanding criteria (i.e., displays and solar cells) require transmission 90% or higher and sheet resistance below 100 Ω/sq,1,3 and we have thus set these two values as the benchmark for a “successful” TCN. In Figure 3, this benchmark is depicted as the dashed red line and thus any TCN with both sheet resistance and visible transmission above the red line meets the criteria. Figure 3a depicts an Ag TCN with fixed nanowire thickness of 10 nm, fixed spacing between nanowires (i.e., pitch) of 100 nm, and varying nanowire width between 5 and 95 nm. The calculated sheet resistance and transmission, depicted with black circles and blue squares, respectively, for these various TCN geometries are shown as a function of the nanowire width. From this figure, we observe an inverse relationship between the sheet resistance and transmission with increasing nanowire width, as expected. However, the transmission appears much more sensitive to the nanowires’ dimensions than sheet resistance. The sheet resistance of the mesh quickly decreases below 100 Ω/sq as the nanowire width surpasses 12 nm, but the transmission remains above 90% only up to a nanowire width of 27 nm. In Figure 3b, we maintain a nanowire thickness of 10 nm but decrease the spacing between nanowires by half to 50 nm. Then, we again vary the width of the Ag nanowires and calculate the expected sheet resistance and visible transmission. With more nanowires in the lattice (i.e., greater redundancy in the network), the lower limit on the nanowire width required to attain a sheet resistance of 100 Ω/sq decreases to 7 nm. However, the upper limit on the nanowire width to maintain at least 90% transmission also decreases to 16 nm. The nanowire dimensions required to attain sheet resistance less than 100 Ω/ sq and transmissivity higher than 90% are readily accessible by block copolymers.16 For example, using different molecular weights of PS-P2VP, we have been able to fabricate metal nanowires with approximately 10 to 25 nm widths, 8 to 50 nm heights or thicknesses, and 37 to 64 nm periodicity.21 Finally, in Figure 3c, we return to a periodicity of 100 nm and instead increase the nanowires’ thickness 5 times their original value to 50 nm. With such thicker nanowires, the transmission rapidly falls below 90% when the nanowire width increases beyond 8 nm. Realistically, fabricating nanowires that have less than 8 nm width would be challenging, and so, to reduce this constraint on nanowires’ widths, nanowires’ thickness should be less than 50 nm. Effect of Defects in TCNs. While we previously assumed that the TCNs are free of defects, block copolymer-derived nanowires have relatively high defect densities of approximately ca. 1−10 defects/μm2, where defects are classified as breaks in a nanowire.16,19,21 Thus, the effects of these defects must be evaluated to determine the maximum defect tolerance for TCNs. To our knowledge, there have not been previous studies regarding how current propagates in any sized periodic networks when randomized defects are present.

Figure 4. Schematics showing (a) the defining parameters of a TCN with defects (i.e., disconnects represented by effectively infinite resistivity) and (b) the model used to calculate the sheet resistance of a TCN with defects with current propagation mapped in blue arrows on top of the nanowires.

breaks in the nanowires. As before, px and py denote the pitch of nanowires aligned in the x- and y-directions, respectively, and rx and ry denote the resistivity of the wires traveling in the xand y-direction, respectively. However, we have now introduced two additional variables for TCNs with defects: lx and ly. These variables represent the average correlation length of nanowires aligned in the x- and y-directions, respectively (i.e., average distance between defects or average nanowire length). Similarly to Figure 2b, Figure 4b depicts an example resistor network formed from 8 wires crossing 8 wires (8 × 8) but with quasirandomly placed defects represented in light red. While current propagates solely along the wires in the x-direction when the TCN is defect-free (refer to Figure 2b), the nanowires aligned along the y-axis become essential to current propagation when defects are present. These nanowires interconnect broken wires along the direction of current propagation and help overcome defects. We were next interested to determine quantitatively the impact of these defects on the networks’ sheet resistance. To calculate this quantity and to broaden the applicability of this work to any-sized wire lattices, we chose to make the network dimensionless. To accomplish this, we normalized all measurements of length in the TCN by pxthe periodicity of wires aligned in the x-direction. Specifically, we denote dimensionless correlation lengths of wires aligned in the xand y-directions as Lx = lx/px and Ly = ly/px, and likewise, we denote the dimensionless periodicity of wires aligned in the xand y-directions as Px = px/px and Py = py/px. As an illustrative example, in Figure 4b, Lx = Ly = 6, which means that on average a wire aligned in one direction intersects six perpendicular wires before encountering a defect or breaking. Furthermore, we made the resistivity of nanowires in the x- and y-directionrx and rydimensionless by normalizing them by rx, thereby making rx always equal to 1. The calculated dimensionless sheet resistance of the network will be called Rnetwork. An ideal defectfree network yields Rnetwork = 1 by definition. Any network with defects will yield a Rnetwork greater than 1, where the number indicates how many times more resistive the network is than the defect-free case. To illustrate this point, if calculations on a network with defects yields Rnetwork = 2, this indicates that the sheet resistance of this network is two times greater than the sheet resistance of the ideal, defect-free case. D

DOI: 10.1021/acsami.7b08446 ACS Appl. Mater. Interfaces XXXX, XXX, XXX−XXX

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ACS Applied Materials & Interfaces To quantify the effects of defects, we modeled a 50 × 50 square (Px = Py = 1) network of wires with dimensionless resistivity equal to 1. For this network, we varied the dimensionless correlation length of the wires in both the x and y directionsLx and Lyfrom 1 to 50. For a 50 × 50 network of wires, Lx = Ly = 50 physically corresponds to wires within the lattice having no breaks or defects whatsoever; the calculated Rnetwork in this case is 1. At the other extreme, Lx = Ly = 1 physically corresponds to the wires breaking after every intersection with wires from the perpendicular direction; without any continuous paths for current propagation, the calculated Rnetwork in this case is infinity. Figure 5 is a color map depicting Rnetwork for a 50 × 50 network having Lx and Ly between 3 and 21. In this specific

To put these numbers in perspective, for a TCN having ca. 10 nm-wide nanowires with 40 nm periodicity, which is comparable to what we and others16 have produced with block copolymers, nanowires in the x- and y-direction would only have to be ca. 360 and 240 nm, respectively, to achieve sheet resistances only 1.4× higher than that of a perfect defect-free network. The L/D aspect ratio required of nanowires to achieve such high electrical conductivity thus only needs to be between 24 and 36significantly lower than the L/D aspect ratios in excess of 400 that are required when nanowires are randomly dispersed.7 Nanowires could also be 320 nm in both x- and ydirections to achieve a similar sheet resistance, yielding a L/D requirement of only 32. Increasing the correlation length in the x- direction to 600 nm (L/D = 60) with a correlation length of just 120 nm in the y-direction (L/D = 12) would decrease the sheet resistance to 1.2× its original value. Compared to the ultrahigh aspect ratio Ag nanowires (L/D > 400) that are typically required to attain decent conductivity and transmissivity when nanowires are randomly deposited forming a disordered array, these values are quite attainable and demonstrate the large improvements in performance and easing of material constraints when nanowires are ordered into periodic lattices with built-in redundancy. When building a nanowire network with two different layers of shear-aligned block copolymer nanowires, it might be beneficial to orient the nanowires with less defects along the direction of current propagation. Effect of Anisotropy in TCNs. In addition to scalable fabrication, using block copolymers in a layer-by-layer manner to form TCNs provides a unique opportunity to easily introduce anisotropy into the lattice structure, as experimentally demonstrated by Majewski et al.16 Specifically, the dimensions and periodicity of the nanowire arrays are dictated by the ratio and molecular weight of the two blocks of the block copolymer, and the direction of the nanowires in each layer is dictated by the shear alignment direction. Thus, by forming lattices in a layer-by-layer manner using different block copolymers, lattices having varying periodicity or wire dimensions along the x- and y-axis can be formed. Such structural anisotropy can potentially translate to anisotropy in the materials’ optical and electrical properties, which we are interested to understand and evaluate. Anisotropy in TCNs can be modified in two ways: either by changing the periodicity of wires in the y-direction relative to the x-direction or by changing their resistivity. We first focus on how changing the periodicity of the wires affects the electrical properties of the wire lattice, as illustrated in the top of Figure 6a. The periodicity of nanowires can be increased from approximately 20 to 100 nm by increasing the degree of polymerization of the block copolymers used to fabricate the nanowire mesh. For simplicity, we assumed that the wires in the x- and y-direction must have the same average length (i.e., Lx = Ly = L), and we increased the spacing of wires aligned in the ydirection (i.e., increased Py), calculating how long wires must be to achieve a specific Rnetwork. The bottom of Figure 6a shows the required dimensionless correlation length, L, for a given lattice periodicity and network resistance. Interestingly, there is a linear relationship between increasing periodicity of y-axisoriented wires and increasing the required correlation length. By increasing the spacing between y-oriented wires, we reduce the redundancy built into the network and therefore require that current carrying wires aligned along the x-axis have fewer defects or higher correlation lengths. Furthermore, the slope of this linear relationship between correlation length and lattice

Figure 5. Calculated dimensionless resistance of a nanowire network, Rnetwork, as a function of the average dimensionless length of wires in the network, Lx and Ly, which is inversely correlated with the number of defects or breaks along a wire.

plot, the defect distribution is kept fixed to an initial quasirandom uniform distribution, and the number of defects is reduced randomly (see Methods for details). Though we calculated Rnetwork for correlation lengths as high as 50 (which corresponds to the perfect, defect-free case where Rnetwork = 1), we found that Rnetwork reaches close to that value (ca. 1.1) even at dimensionless correlation lengths as low as 21 for the wires aligned in the direction of applied voltage (x-axis), and we thus choose to only show this reduced variable space. Comparing how Rnetwork varies along the x- and y-axis in Figure 5, we see that increasing the correlation length of wires aligned in the xdirection (i.e., direction of applied voltage) has a much greater effect in decreasing Rnetwork than increasing the correlation length of wires aligned in the y-direction perpendicular to the direction of applied voltage. However, the wires aligned in the y-direction do play a critical role in reducing Rnetwork when defects are present; we see that for a given correlation length of wires in the x-direction, Rnetwork decreases rapidly with the presence of even short wires in the y-direction (i.e., wires having dimensionless correlation lengths as low as 5). Indeed, high correlation values are not needed for wires in the x- and ydirections to obtain low values of Rnetwork; once the dimensionless correlation factors for wires in the x- and ydirections are above 8 and 6, respectively, Rnetwork does not increase significantly. E

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changing the material from which they are made (which is dictated by the choice of metal salt used to complex with the P2VP block).20 For simplicity, we again assumed that the wires in the x- and y-direction must have the same average length (i.e., Lx = Ly = L), and we increased the resistivity of wires aligned in the y-direction (i.e., increased Ry), calculating how long wires must be to achieve a specific Rnetwork. The results of our calculations are shown in the bottom plot of Figure 6b. Again, we observe a linear relationship between increasing the resistivity of y-axis-oriented wires, Ry, and increasing the required correlation length, L. Interestingly, the slopes in Figure 6b are smaller than those in Figure 6a, which suggests that increasing the resistivity of wires oriented in the y-direction is less harmful to the overall conductivity of the network than reducing the number of wires oriented in the y-direction. These results again emphasize the benefits of a fabrication scheme where ordered nanowire networks can be created since this order creates more redundancy in the network and makes it more defect-tolerant when the distribution of defects is random. We next evaluated how anisotropy in the Ag nanowires’ dimensions may affect the optical properties of TCNs. While the electrical effects of anisotropy could be studied assuming a dimensionless wire lattice, optical effects are sensitive to nanostructures’ dimensions and thus must be considered. To evaluate the effects of anisotropy on the optical properties of lattices, we focused on TCNs with fixed periodicity and thicknesses of 50 and 10 nm, respectively, as shown in Figure 7a. We used tabulated data from Palik32 for the index of refraction of the Ag wires in our simulations. We kept the width of the wires oriented along the x-axis (width 1 in Figure 7a) fixed at 10 nm and varied the width of wires oriented in the ydirection (width 2) between 5 and 45 nm. Figure 7b shows the polarization-specific transmission spectra derived from Comsol simulations for four specific cases where the width of the yoriented wires was set to 5 nm (gray traces), 15 nm (blue traces), 30 nm (red traces), and 45 nm (green traces). T∥ and T⊥ denote the transmission of light parallel and perpendicular to the incident H-field. From Figure 7b, we observe that T∥ remains relatively constant regardless of wavelength and decreases with increasing nanowire width, as may be expected. However, T⊥ shows much greater variance. Specifically, we observe a decrease in the T⊥ transmission at increasing wavelengths with increasing width of the y-oriented nanowires.

Figure 6. Anisotropy can be introduced into the wire lattice by changing (a) wires’ periodicity or pitch or (b) wires’ resistivity (which can be accomplished by using different wire dimensions or different material). (Top) Cartoon schematics of anisotropic wire lattices and (bottom) relationships showing how the lengths of wires must increase with increasing anisotropy of wires in the y-direction.

periodicity increases as lower sheet resistance (Rnetwork) is required. Physically, this indicates that increasingly longer wires or fewer defects are required for a given increase in periodicity as one sets higher demands on the electrical performance and redundancy of the network. These derived relationships provide helpful constraints when designing wire networks with specific electrical properties in mind. For example, given a correlation length and a network resistance limit value, we can decide how much redundancy (i.e., the minimum spacing Py) is needed to maintain the network resistance value below the resistance limit, and combine that with the optical properties of the TCN. In addition to changing the relative periodicity of wires in the x- and y-directions, yet another means of introducing anisotropy to the network is by using nanowires with different resistivities along the two axes. The resistivity of nanowires derived from the PS-P2VP block copolymer can be easily tuned either by changing the wires’ dimensions (which are dictated by the degree of polymerization of the P2VP block) or by

Figure 7. Anisotropic TCNs show a different plasmonic response for different polarizations, making them attractive as wavelength-specific polarizers. (a) Picture representing anisotropic TCNs and how the width was varied normally or in parallel with the incoming H-field. (b) Parallel and perpendicular transmission of the anisotropic TCN represented in (a) for when the variable width is 5, 15, 30, and 45 nm. (c) Polarization efficiency (PE) of the structured shown in (a). F

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our current work, the wire junction resistance was not taken into account (kept to values 6 orders of magnitude lower than the wire resistance in the x direction). The blue arrows are quiver plots representing the net magnitude and direction of current propagation at every node. Both surface roughness and grain size depend on the fabrication process and the material deposited. In the simulations shown above, we use the reported diameter-dependent resistivity values of Cambrios’ cylindrical Ag nanowires,27 and to account for our nanowires’ rectangular geometry, these resistivity values were then fitted and modified to rectangular wires using24−26

This decrease can be attributed to energy coupling to surface plasmon resonance modes in the structures.33 This structural anisotropy and difference in surface plasmon resonance can be used as the basis for highly wavelength-specific polarizers. Polarization efficiency, PE, is calculated as ⎛ T − T⊥ ⎞1/2 ⎟⎟ PE = ⎜⎜ ⎝ T + T⊥ ⎠

We calculated the polarization efficiency at each structures’ T⊥ minimum wavelength, thereby maximizing the polarization efficiency. Figure 7c summarizes the polarization efficiencies of the anisotropic TCN used in this example. As the width is increased from 10 to 45 nm, the wavelength of the polarizer redshifts from ca. 370 to 570 nm, and its polarization efficiency increases, as well, approaching 1.

⎡ U 1 ⎢ ρ = ρ0 ⎢C(1 − p) lmfp + S 3 ⎣ α=

■

1 3

−

α 2

(

+ α 2 − ln 1 +

1 α

)

⎤ ⎥ ⎥, ⎦

lmfp

R d 1−R

where ρ is the resistivity of a nanowire having a rectangular crosssection, ρ0 is the bulk resistivity of the material, C is a constant, p is the electron specular reflectance off of the nanowire surface, U is the perimeter, S is the surface area, lmfp is the mean free path of an electron inside the material, d is the smallest nanowire dimension, and R is the electron grain boundary reflectivity. For the simulations above, ρ0 = 1.81 · 10−8 Ωm (from Cambrios27), lmfp = 2 · 10−8 m, C = 0.25, P = 0.3, and R = 0.3.

CONCLUSIONS Increasingly advanced processing of block copolymers has allowed for the fabrication over large cm2-areas of subwavelength metal nanowire meshes with tunable materials, geometry, and dimensions.16,18,19,21 Using optical and electrical simulations, we have explored this wide parameter space to evaluate how such subwavelength metal nanowire meshes (i.e., TCNs) may perform as transparent electrodes. We found that the ordered nature of the nanowires significantly decreases the average nanowire length required to maintain high conductivity, increasing defect tolerance. Specifically, for an Ag TCN with ca. 40 nm periodicity, nanowires need only be ca. 300 nm to achieve sheet resistances only 1.4× higher than that of a perfect defect-free network. These values are quite attainable and demonstrate the large improvements in performance and easing of material constraints when nanowires are ordered into periodic lattices with built-in electrical redundancy. We furthermore evaluated and extracted relationships for how introducing anisotropy by changing nanowires’ periodicity or resistivity affects both electrical and optical performance. Having explored the TCNs’ parameter design space using numerical calculations, these design rules can help guide future fabrication and experimental studies.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.7b08446. Simulated transmission spectra for various Ag nanolattices (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Anna M. Hiszpanski: 0000-0002-2705-3263 Notes

The authors declare no competing financial interest.

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METHODS

ACKNOWLEDGMENTS The authors thank Carla L. Watson and Timothy D. Yee for fabrication and characterization of the nanowire meshes motivating this work. This work was funded by the Laboratory Directed Research and Development (LDRD) program at Lawrence Livermore National Laboratory (16-LW-041). Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration under Contract DE-AC52-07NA27344, LLNL-JRNL732399.

Optical Properties. Transmission of TCNs was simulated using Comsol in the 250−1000 nm wavelength range using full wave calculations at normal incidence. The simulated nanowire structure was surrounded by air at all interfaces. Silver’s refractive index and extinction coefficient were taken from Palik.32 Comparatively, simulations with nanowires on top of each otherrather than on the same z-planewere done to study the possibility of a different optical response due to this inhomogeneity. The optical response of both in-plane and off-plane structures was comparable. Electrical Properties. The TCN was modeled as a resistor network in Matlab. Specifically, the network is assumed to be made of nanowires oriented in the x- and y-directions interconnected through a junction resistance or an additional junction “wire.” This model provides tunability of the wire resistance in the x and y directions, junction resistance, input and output electrodes, pitch in the x and y directions, and network size. Defects representing breaks in a wire were added to the network with a uniform spatial distribution using a Mersenne Twister quasirandom number generator;34 the defects were assumed to have a resistance 6 orders of magnitude higher than the resistance of wires oriented in the x-direction. The sheet resistance was calculated by extracting the resistivity of a 50 × 50 nanowire network (the ratio changed for anisotropic spacing) and dividing by the number of wires propagating in the same direction as the current. In

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