Analytical modeling of orientation effects in random nanowire networks
AAnalytical modeling of orientationeffects in random nanowire networks
Milind Jagota ∗ and Isaac Scheinfeld † Stanford UniversityStanford, CA 94305 (Dated: November 6, 2019)Films made from random nanowire arrays are an attractive choice for electronics requiring flexibletransparent conductive films. However, thus far there has been no unified theory for predicting theirelectrical conductivity. In particular, the effects of orientation distribution on network conductivityremain poorly understood. We present a simplified analytical model for random nanowire networkelectrical conductivity that is the first to accurately capture the effects of arbitrary nanowire ori-entation distributions on conductivity. Our model is an upper bound and converges to the trueconductivity as nanowire density grows. The model replaces Monte Carlo sampling with an asymp-totically faster computation and in practice can be computed much more quickly than standardcomputational models. The success of our approximation provides novel theoretical insight intohow nanowire orientation affects electrical conductivity, illuminating directions for future research.
I. INTRODUCTION
Transparent conductive films are a crucial componentof touch screens and solar cells, among various other elec-tronics [1, 2]. One approach to making transparent con-ductive films that has been widely studied and deployedis to randomly disperse highly conductive nanowires intoa substrate. Films made in this way, using conductivematerial such as silver nanowires or carbon nanotubes,display competitive electrical and optical properties toalternatives, while being cheaper and more flexible thanthe performance standard Indium Tin Oxide [3–7]. Thelatter property is particularly valuable as flexible elec-tronics continue to become more mainstream in consumerdevices. However, despite the wide interest in applyingthem, there is no unified theory for predicting electri-cal properties of random nanowire networks, and manyobserved effects have not been fully characterized or ex-plained. As a result, the technology remains underdevel-oped, and there is undoubtedly still room for improve-ments in performance.The majority of results describing properties of randomnanowire networks have been experimental or via directcomputational simulation. Various studies have exper-imentally compared electrical properties of films usingdifferent conductive rods, such as silver nanowires andcarbon nanotubes [4, 5, 8]. Agreement between simu-lation and experimental observations of electrical prop-erties has also been well established for the classes ofrandom nanowire networks that are easiest to produceexperimentally [9]. More recently, computational mod-els have been used to maximize electrical performance ofrandom nanowire networks by varying the distributionsfrom which the networks are sampled [10–17]. Some ofthese results have been verified experimentally [18–21]. ∗ Corresponding author. Contact at [email protected] † [email protected] In particular, various computational studies have demon-strated that it is possible to improve electrical conductiv-ity of nanowire networks by controlling nanowire orienta-tion [10–16]. However, this effect is not well understoodand there is no simple framework to predict the result ofusing a specific, arbitrary orientation distribution.Recently, a number of analytical models have also beendeveloped to describe properties of random nanowire net-works, but none thus far have explained the effect ofnanowire orientation on electrical conductivity in fullgenerality [22–29]. Forro et al . proposed a model derivedassuming high nanowire density, so that potential dropacross nanowire networks can be assumed to be linear[25]. The model is accurate in the high density regimeand yields a closed-form expression. Benda et al . ob-tained a closed form expression for network conductivityby numerically fitting a physically interpretable form toMonte Carlo simulations, while Manning et al . developeda theoretical framework for analyzing both electrical andoptical performance of nanowire networks [26, 28]. How-ever, these models are developed under the assumptionof uniformly distributed wire orientation and do not gen-eralize in a clear manner to random orientation of anarbitrary distribution.In this work, we present the first analytical modelfor random nanowire network conductivity that accu-rately captures the effects of arbitrary distributions ofnanowire orientation. Our approximate model replacesMonte Carlo sampling with an asymptotically less ex-pensive computation and is empirically much faster thanstandard computational models. It approaches the lim-iting dependency of network conductivity on nanowiredensity, with small errors even at moderate nanowire den-sities. Furthermore, the structure of our approximationsprovides novel intuition for how orientation affects net-work conductivity as well as intuition for the behavior ofrandom nanowire networks in general. a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov II. MODEL CONSTRUCTIONA. Setting
We begin by presenting the setting in which we de-velop our model. We consider networks comprised of 1-dimensional nanowires (linear, widthless sticks) inside asquare space of unit length in each direction with periodicboundary conditions at the top and bottom. To simplifynotation, we assume nanowires have fixed length l , butour approach generalizes naturally to having a randomdistribution over wire length. Each nanowire is describedby an ( x, y ) coordinate pair and an angle θ , where thecoordinate pair represents the location of the wire centerand θ is the angle relative to the horizontal. The coor-dinates and the angle are sampled randomly, where allvalues are assumed independent and each nanowire in anetwork is assumed to be independent. We denote thesampling distributions of x, y, θ by X , Y , Θ, respectively.The primary electrical property of interest for randomnanowire networks is the sheet conductivity σ , which is arandom variable. Sheet conductivity transitions sharplyfrom being zero with overwhelming probability to be-ing greater than zero with overwhelming probability ata particular number of nanowires that is a function of l ,known as the percolation threshold [30]. The dimension-less quantity C N := N | l | (1)where N is the number of nanowires in an network and | l | indicates the wire length normalized by dividing bybox width, is often used as a normalized concentrationof nanowires because it allows direct comparison to thepercolation threshold [10]. We assume that our nanowirenetworks are well above the percolation threshold so thatthey are guaranteed to have conductivity greater thanzero. We focus on modeling the expected value of thesheet conductivity E σ , because the variance of sheet con-ductivity is typically small relative to the expected sheetconductivity for large N [9, 10].Figure 1 displays how the sheet conductivity is physi-cally defined, using a network sampled with nanowire po-sitions and orientations both distributed uniformly. Weplace electrodes at the left and right boundary of the net-work ( x = 0 and x = 1) and calculate the current when1 volt is applied. This current can then be used to cal-culate the sheet conductivity. In general, there are threesources of resistance in nanowire networks which deter-mine the conductivity along with the geometry. Thesethree sources are the resistance of wires themselves, theresistance at the junctions between two wires, and theresistance at the junctions between a wire and an elec-trode. In many real nanowire networks, the wire resis-tance is small compared to the resistance at junctions [9].We assume that this is the case and choose to ignore thewire resistance moving forward. However, our methodcan be generalized to account for wire resistance, and we FIG. 1. The sheet conductivity of a nanowire network is cal-culated by computing the current when 1 Volt is applied byelectrodes spanning the left and right border of the network.We assume that nanowire network resistance is dominatedby junction resistance and ignore wire resistance. The x di-rection is defined as the direction of current flow, and the y direction is perpendicular. discuss this in Section V. We set the resistance betweentwo wires to be a constant 1Ω and set the resistance be-tween a wire and an electrode to be a constant Ω. Theconductivity of a particular network is determined solelyby the ratio between these two quantities up to scaling.We expect the wire-wire resistance to be multiple ordersof magnitude larger than the wire-electrode resistance,and these quantities are thus reasonable.In this setting, σ can be calculated exactly for a partic-ular network from the symmetric ( N + 2) × ( N + 2) adja-cency matrix of the electrical network, which we denoteas A . The first N rows of this matrix each correspond toa single wire, while the last two rows correspond to theleft and right border electrode. An off-diagonal elementof the matrix is 1 if the two corresponding objects touch,and all diagonal elements are 0. From A , we can use thetwo resistance values to construct the Laplacian matrixof the nanowire network L , of the same shape as A . Thisis the matrix that, when multiplied by the vector of nodevoltages V , gives the vector of node net current flow J as given in Eq. (2), and is a linear function of A [31]. LV = J (2)We can then calculate the current flowing from the leftelectrode by setting the voltages at the left and rightelectrodes in the vector V and solving for the remainingvoltages. Dividing this current by the applied voltageyields the sheet conductivity [31]. B. Model definition
The expected sheet conductivity E σ has most oftenbeen studied by direct sampling of nanowire networks [9–17]. This procedure involves numerous steps. For eachnetwork, N nanowires are sampled according to the dis-tributions X , Y , Θ. Then, the adjacency matrices A forthe networks are generated. From these matrices, obser-vations of the sheet conductivity can be calculated byapplying Kirchhoff’s Laws, which are then averaged toyield an estimate. We denote this empirical estimateby ˆ σ , defined in Eq. (3), where σ ( A i ) refers to the sheetconductivity of the network represented by the adjacencymatrix A i . ˆ σ = 1 M M (cid:88) i =1 σ ( A i ) (3)While this approach converges rapidly to E σ as thenumber of sampled networks M increases, it has a num-ber of drawbacks. First, it is slow: calculating the adja-cency matrix A from a list of wire coordinates and anglesrequires checking all pairs of nanowires for intersection,as well as computing a Cholesky decomposition of an N × N matrix. While there are methods to speed up bothof these steps, the procedure is still at least O ( M N ) andso collecting many samples for high conductivity films isslow. In addition, this sampling based procedure makesinterpretation of observed effects difficult, which limitsphysical intuition.An exact analytical model for the sheet conductivitywould fix these issues, but directly deriving an expressionfor E σ is very difficult even under the simplest distribu-tions X , Y , Θ. A common approximation for this type ofproblem is to move the expectation inside of the compli-cated function, as shown in Eq. (4). The right side ofthis equation is defined by treating E A as a weighted ad-jacency matrix; the Laplacian L is constructed from E A by the same linear relationship as for an ordinary adja-cency matrix A , and the sheet conductivity is calculatedby solving the same matrix equation involving L .E σ ( A ) ≈ σ (E A ) (4)However, this naive approach fails catastrophically forrandom nanowire networks. None of the spatial struc-ture of the networks is captured because all nanowiresare indistinguishable according to E A . Using σ (E A ) asa model results in a massive overestimate of the sheetconductivity that is not useful.To develop our analytical model, we modify the ap-proach of moving the expectation inside the function todirectly capture spatial structure of random nanowirenetworks. We first observe that σ is clearly invariant toreindexing the wires in a network and recalculating theadjacency matrix A accordingly. We choose to assume,without loss of generality, that the wires are always rein-dexed according to increasing x -coordinate. Specifically, define the random matrix A ∗ as A ∗ rank( i ) , rank( j ) = A ij (5)where the function rank( i ) gets the placement of x i in thelist of x -coordinates when sorted from smallest to largestand leaves the electrode indices fixed. Our approximatemodel σ ∗ is then defined in Eq. (6), where σ (E A ∗ ) isdefined in the same way as the right side of Eq. (4). σ ∗ := σ (E A ∗ ) (6)Under slightly more restrictive assumptions, we canprove that σ ∗ is greater than E σ for all X , Y , Θ usingJensen’s inequality; details are presented in Section II.C.Despite being an upper bound, σ ∗ is able to capture thedependency of conductivity on both wire concentrationand orientation distribution due to the choice of assumedwire permutation; E A ∗ encodes most of the spatial struc-ture of the networks. We illustrate this property of ourmodel in Figure 2 by plotting the values of a randomsorted adjacency matrix A ∗ as well as the values of E A ∗ under the same distributions. Due to the sorted orderthat is assumed, the matrices A ∗ for sampled randomnetworks are banded, because wires near in index arealso near in x -coordinate and therefore more likely to in-tersect. The expected adjacency matrix E A ∗ reproducesthis key property well.In the true system, nanowires that intersect are closein y -coordinate as well as in x -coordinate. We encodethis effect only with respect to x -coordinate and not y -coordinate, but our empirical results verify that ourmodel is useful regardless. This result has interestingimplications which we discuss in Section IV. C. Proof that σ ∗ is an upper bound We argue that σ ∗ is an upper bound on E σ underslightly more restrictive assumptions. Note that theLaplacian matrix L for a particular nanowire networkis a linear function of the sorted adjacency matrix A ∗ [31]. It satisfies the Kirchhoff current equation given inEq. (2), where V is the vector of voltages at each of the N + 2 objects and J , the net current flowing into eachnode, is zero at all nodes other than the electrode nodes.To reduce notation for units, we assume in this sectionthat V is made dimensionless by dividing each element by1V. L and J then both have units of inverse resistance.Under the assumed normalization of V , the sheet con-ductivity is equal to the current flowing out of the leftborder electrode (node N + 1) when we set the voltageat the left border to be 1V and the voltage at the rightborder to be 0V ( V N +1 = 1, V N +2 = 0). With thesevalues of V set, the Kirchhoff current equation is givenby Eq. (7). L N, N V = − L N,N +1 (7) FIG. 2. We plot the values of (a) the sorted adjacency ma-trix A ∗ of a random nanowire network, and (b) the expectedadjacency matrix E A ∗ for the distributions that the samplewas drawn from, with N = 500 and l = 0 .
2. The expectedadjacency matrix captures the banded structure of the sam-pled matrices A ∗ . Note that E A ∗ has a small maximum valuebecause even if two nanowires have no x -separation, the prob-ability of them intersecting is small when l is small. We use the notation B i : j,k : l to refer to the submatrix of B from rows i to j and columns k to l . A single indexindicates taking a single row or column.We proceed by adding two minor assumptions. We firstassume that for fixed distributions X , Y , Θ, the numberof nanowires crossing the left electrode is a constant in-teger M . For the high density networks we study, thevariance of this quantity is small with respect to its ex-pected value and does not cause much variance in sheetconductivity. Second, we assume that the M nanowiresthat cross the left border are the first M indices in A ∗ .Under the sorting that is used for A ∗ , this is the mostlikely set of M wires to cross the left border, and the vari-ance of these indices also does not cause much variancein sheet conductivity. This assumption can be viewed asa definition of sheet conductivity where we attach ourleft electrode to the leftmost M wires based on center location, as opposed to based on left endpoint location.Under these assumptions, the sheet conductivity isgiven by Eq. (8), where R ew is the wire-electrode resis-tance in Ohms and E M is the N dimensional vector thatis 1 in the first M elements and 0 otherwise. σ = 1 R ew M (cid:88) i =1 (1 − V i )= 1 R ew ( M − M (cid:88) i =1 V i )= 1 R ew ( M − E TM V )= 1 R ew ( M + ( E TM ( L N, N ) − L N,N +1 ) (8)The inverse of L N, N exists when the network is con-nected, which is true because we assume that our net-works are well above the percolation threshold.Since the first M nanowires cross our left measurementelectrode, L N,N +1 is given by L N,N +1 = − R ew E M (9)We can use this value to write another expression for σ in Eq. (10). σ = 1 R ew ( M − R ew E TM ( L N, N ) − E M ) (10)Since M is assumed to be constant, the only random-ness in σ comes from L N, N . Since this matrix is posi-tive definite when the network is connected and is a linearfunction of A ∗ , σ is a concave function of A ∗ . Jensen’s in-equality then tells us that for all X , Y , Θ, σ ∗ is an upperbound on E σ as shown in Eq. (11).E σ ( A ) = E σ ( A ∗ ) ≤ σ (E A ∗ ) = σ ∗ (11) III. MODEL COMPUTATIONA. Methods for computing E A ∗ The approximate model σ ∗ is useful because we can di-rectly compute E A ∗ in a wide variety of circumstances.This eliminates the need for Monte Carlo sampling of net-works and solving a linear system of equations for eachsample. Here we present a method for computing E A ∗ when X , Y are the uniform distribution, which is an as-sumption used throughout the literature. This procedureis applicable for any orientation distribution Θ that canbe parameterized by a vector α .Recall that A ∗ is the sorted adjacency matrix of a ran-dom nanowire network and has size N + 2 × N + 2. Thefirst N indices correspond to a nanowire, sorted by in-creasing x -coordinate, while indices N + 1 and N + 2correspond to the left and right border electrode. Theelements of the expected adjacency matrix E A ∗ are thusthe probability of intersection between the objects of in-dices i, j . To compute the matrix, we thus need to com-pute the probability of intersection between every pairof wires, conditioned on the rank of the x -coordinate ofeach wire. We also need to calculate the probability ofintersection between each wire and the border electrodes,conditioned on the rank of the x -coordinate of the wire.Because the matrix is symmetric, we only need to do sofor i > j , and we only need to do the calculation for asingle border electrode because the probabilities for theother border electrode are symmetric.We will first calculate the probability of intersectionbetween any two nanowires. Denote a wire as w =( x, y, θ ) and let w ∗ i , w ∗ j be the i th and j th wire accordingto the sorted order based on x -coordinate. The desiredprobability is then denoted by P( w ∗ i ∩ w ∗ j ). The event of w ∗ i intersecting w ∗ j is a deterministic function of the dif-ference in x coordinates, the difference in y coordinates,and the angles of the two wires. Under our independenceassumptions, we can thus calculate E A ∗ ij by calculatingthe distributions of x ∗ i − x ∗ j and y ∗ i − y ∗ j and then using theknown distributions of θ i and θ j . For brevity, we define x ij = x ∗ i − x ∗ j (12) y ij = y ∗ i − y ∗ j (13)We will first analyze randomness solely in y ij by com-puting the intersection probability conditioned on x ij ,denoted by P( w ∗ i ∩ w ∗ j | x ij ). This is the probabilitythat two wires intersect if we know the difference in x -coordinates between them. For any pair of wires w ∗ i , w ∗ j with x -separation x ij and angles θ i , θ j , we can define thehorizontal range of overlap b as the length of the intervalof x -coordinates that both wires lie in. For particular val-ues of b, θ i , θ j , there is an interval of y ij values for which w i and w j will cross. We denote the length of this inter-val of by h . We illustrate these quantities with examplenanowire pair configurations in Figure 3.Now observe that the distribution of y ij is identical forall i (cid:54) = j . Furthermore, because of our use of periodicboundary conditions, since Y is the uniform distribution, y ij is in fact the uniform distribution in the range [0 , w i and w j conditioned on x ij is given by the conditional expec-tation of h : P( w ∗ i ∩ w ∗ j | x ij ) = E[ h | x ij ] (14)We can calculate this conditional expectation by observ-ing that b and h can be calculated from x ij and θ i , θ j , asin Eq. (15, 16). Here, ( f ) + is defined to be max { f, } . b ( x ij , θ i , θ j ) = min (( l/
2) (cos θ i + cos θ j ) − x ij ) + l cos θ i l cos θ j (15) h ( x ij , θ i , θ j ) = b ( x ij , θ i , θ j ) | tan θ i − tan θ j | (16) FIG. 3. If two nanowires have known x -separation of x ij andangles θ i and θ j , we can calculate the length of the horizontalregion of overlap b . We visualize this quantity in the casewhere the horizontal region of one nanowire is contained inthe region of the other (a) as well as the case where this is nottrue (b). We can then use this quantity to calculate the lengthof the range of y -separations h for which the two nanowireswould intersect as visualized in (c). In Eq.(15), the latter two cases correspond to the situa-tion when the interval of x -coordinates that one wire liesin is contained by the interval of x -coordinates that theother lies in, as in Figure 3a. The first case is taken whenthis situation does not occur, as in Figure 3b.The conditional expectation is then given by integrat-ing out θ i and θ j drawn independently from Θ:P( w ∗ i ∩ w ∗ j | x ij ) = (cid:90) h ( x ij , θ i , θ j ) p ( θ i ) p ( θ j ) d ( θ i θ j ) (17)Since we have assumed the wires are sorted by x -coordinate, the difference x ij is the difference in orderstatistics i and j from the distribution X . Because X isthe uniform distribution, x ij follows the Beta distribu-tion with parameters i − j and N − i + j + 1, if i > j [32]. However, for the networks with large N which westudy, these distributions become strongly concentratedat their mean, which is i − jN +1 . We thus assume that x ij is equal to its expected value, and empirically observe noloss in accuracy. This yields a formula for the probabilityof intersection between any two nanowires:P( w ∗ i ∩ w ∗ j ) = P( w ∗ i ∩ w ∗ j | x ij = i − jN + 1 ) (18)A similar argument can be used to calculate the prob-ability that any wire crosses the left border electrode, de-noted by e . Observe that w ∗ i intersects e if and only if( l/
2) cos θ i ≥ x ∗ i . Assuming that x ∗ i equals its expectedvalue of iN +1 , the desired probability is then given byEq. (19), where θ ∼ Θ.P( x ∗ i ∩ e ) = P(cos θ ≥ il ( N + 1) ) (19)We can therefore calculate every element of E A ∗ for any N and any orientation distribution Θ, assuming wire po-sitions are uniform.To use these expressions efficiently, we numericallycompute the integral in Eq. (17) over a grid of valuesfor x ij and the parameters α of the orientation distri-bution Θ. We then fit a polynomial to the probabilityvalues on these grid points to obtain an expression forP( w ∗ i ∩ w ∗ j ) that is extremely rapid to use. We furtherdescribe the speed of our method in the next subsection. B. Analysis of computational speed
One of the significant advantages of our method isthat it replaces Monte Carlo sampling with an asymptoti-cally faster computation. Sampling-based models, whichare the most common approaches for studying randomnanowire networks, have two major components. First, anumber of nanowire networks M are sampled by directlysampling ( x, y, θ ) for each of N nanowires, and a collec-tion of M adjacency matrices are calculated. Second, theKirchhoff current equation is solved for each adjacencymatrix to collect M observations of sheet conductivity,and these observations are then averaged. The first ofthese steps has complexity O ( M N ). Within all net-works, each of the N nanowires must be compared witha fixed fraction of all other nanowires for intersectionto compute the adjacency matrix A . The second step,meanwhile, has complexity O ( M N ), which is the costof solving M linear systems of equations each involving N variables. While the second step has larger complex-ity, both steps require significant amounts of time and sospeeding up either is beneficial.Our model σ ∗ delivers a large asymptotic improvementto the first step and delivers a large constant factor im-provement to the second step. Recall that the probabilityof intersection between two wires under our model de-pends only on the expected x -separation between them.As a result, we only need to directly compute two rowsof E A ∗ in order to produce the entire matrix. Thisis because the expected x -separation between the wires w ∗ i and w ∗ j is determined completely by the quantity | i − j | . Equivalently, if we ignore the rows represent-ing electrodes, then all diagonals of E A ∗ are constant.We therefore must compute the first row of E A ∗ to ob-tain the probability of interaction between every pair ofnanowires, and also must compute the last row of E A ∗ to obtain the probability of interaction between everynanowire and an electrode. Therefore, the cost of com-puting E A ∗ is O ( N ) with constant proportional to thetime it takes to compute P( w ∗ i ∩ w ∗ j ). We must still solve asingle Kirchhoff current equation, and this step is O ( N ).Numerically integrating to compute each evaluation ofP( w ∗ i ∩ w ∗ j ) is in practice quite slow. We therefore pre-compute this function for a grid of values of x ij as well theparameters α of the orientation distribution Θ, and thenfit a polynomial to the computed values. A polynomial FIG. 4. Normalized conductivity as a function of normalizedwire concentration is shown for both the true empirical meanand our approximate analytical model, on a linear scale (a)and a double log scale (b). The model follows an exact powerlaw relationship and corresponds to the high density behaviorof the true conductivity. We emphasize that the model is notobtained by fitting to the true conductivity. fit is in practice quite accurate because the probabilityin question is smooth as a function of the parameters ofinterest. This step makes computation of P( w ∗ i ∩ w ∗ j )extremely rapid, but the precomputation cost is expo-nential in the number of parameters of the orientationdistribution. For the majority of interesting cases, theorientation distribution can be parameterized in one ortwo parameters, and this complexity is thus not signifi-cant compared to other steps.In total, our method has a small precomputation cost,but replaces the O ( M N ) complexity of Monte Carlosampling with an asymptotically faster O ( N ) computa-tion. It also reduces the cost of solving linear systems bya factor of M , the number of samples that are collected ina sampling based approach. In our implementation, thisallowed the model σ ∗ to be evaluated about 100 timesfaster than direct Monte Carlo sampling. IV. EMPIRICAL TESTS AND DISCUSSION
We examine the effectiveness of σ ∗ in modeling depen-dency of network conductivity on both nanowire densityand orientation distribution. We assume, as in the previ-ous section, that X , Y are both the uniform distribution.We implemented direct sampling of σ under this assump-tion and our previously stated setting, while allowing thedistribution Θ to be arbitrary. We use our implementa-tion of direct sampling of σ as a baseline comparison forall tests, estimating E σ with ˆ σ with M = 30. Through-out these experiments, we use l = 0 .
1. Larger values of M reduce the noise of ˆ σ , while smaller values of l reducefinite size error in ˆ σ and σ ∗ . The chosen values of M and l were found empirically to be sufficient to largelyeliminate these errors; σ ∗ and ˆ σ do not change much forhigher M or lower l , as long as C N is fixed. A. Dependence on nanowire density
We first assume that Θ is the isotropic distribution(uniform in [ − ,
90] degrees) and explore the depen-dence of σ ∗ on normalized concentration C N . Figure4 shows a comparison between normalized σ ∗ and ˆ σ asa function of C N , starting just above the percolationthreshold, on both linear and double log scales. Con-ductivities are normalized by multiplying by junction re-sistance 1Ω so that they are unitless. It is a known resultthat E σ can be approximated as a power law function ofthe distance of normalized concentration from the per-colation threshold, with an exponent of around 1.75 atmedium densities which moves close to 2 at high densities[9]. Our estimate ˆ σ matches these known relationships;the growth pattern of conductivity in log-space becomeslinear as the subtraction of the percolation threshold be-comes negligible. Our model σ ∗ , however, displays a per-fect power law dependence on C N , with an exponent thatmatches the asymptotic exponent of E σ . Near the per-colation threshold, the error is large, as we have assumednanowire density above this threshold in developing ourmodel. However, the error in log space approaches zeroas concentration grows, and the model can thus be inter-preted as the limiting behavior of E σ at high concentra-tions.While σ ∗ is less precise than other recent models forpredicting dependency of conductivity on concentrationat small nanowire densities, the result that our approachyields the correct limiting behavior is theoretically inter-esting. By using E A ∗ , our model directly encodes clus-tering of nanowire only in the x -direction. However, thisis sufficient information to capture asymptotic behavior,and, as we next show, capture the effect of varying ori-entation distribution. FIG. 5. Nanowire networks drawn from (a) Θ , with α = 45and (b) Θ , with α = 60 are shown. In Θ , all nanowires haveorientation at ± α degrees from the horizontal ( x -direction).In Θ , nanowire orientation is distributed uniformly in [ − α, α ]degrees from the horizontal. B. Dependence on orientation distribution
Our model σ ∗ is particularly valuable because it is ableto predict the effect of arbitrary orientation distributionson sheet conductivity. The problem of optimizing ori-entation distribution in random nanowire networks hasbeen studied numerous times via computational models,but there is no unified understanding of the observed ef-fects [10, 11, 13, 14, 16].We consider two families of distributions for Θ, each ofwhich is described by a single parameter. For each family,we demonstrate that σ ∗ accurately captures the effect ofvarying the distribution parameter on conductivity. Thefirst family Θ ( α ) is given by p ( θ ) = (cid:40) θ = α θ = − α (20)for all 0 < α <
90. All probability mass is concentratedat ± α degrees from horizontal. The second family Θ ( α ) FIG. 6. Normalized conductivity is shown as a function ofdistribution parameter α for both the true empirical conduc-tivity and the approximate analytical model, for both (a) Θ and (b) Θ , with C N = 50. In both families, the shapes of thetwo curves match well and the optimal values (vertical lines)are close. The model also captures the fact that a gain overisotropic conductivity (horizontal lines) can only be achievedin Θ . is given by p ( θ ) = (cid:40) α | θ | < α − α, α ]degrees. Figure 5 shows a sample network from a singledistribution within each family. These two families werepreviously studied, and it was found that while a con-ductivity gain over isotropic networks could be achievedwithin Θ , no gain could be achieved within Θ [10].Figure 6 shows a comparison between σ ∗ and ˆ σ for de-termining the relationship between distribution parame-ter α and normalized conductivity for both Θ and Θ .The normalized concentration is fixed at 50 in both cases.Within both families, the shape of the curve matcheswell, and the optimal values are within a few degrees of each other. Moreover, the predictions from σ ∗ match E σ in that a gain over isotropic orientation is attainable inΘ but not Θ .To the best of our knowledge, our model is the firstto accurately reproduce the effects of orientation distri-bution on sheet conductivity without relying on MonteCarlo sampling in any capacity. These results indicatethat orientation effects can be modeled by analyzingtheir effects on network connectivity in a single direc-tion, as our model A ∗ only takes into account positionsof nanowires in the x -direction. V. CONCLUSION
We developed an approximate analytical model forsheet conductivity of random nanowire networks thatcondenses a large amount of their structure through aspecific choice of nanowire permutation. We showed thatthis model is an upper bound and matches the asymp-totic dependency of the true sheet conductivity on wireconcentration. We also demonstrated that the model ac-curately captures the effects of orientation on nanowirenetwork conductivity, a result that has limited theoreti-cal explanation in the literature. Our model is the first toaccurately capture the effects of arbitrary orientation dis-tributions on network conductivity, and replaces MonteCarlo sampling with an asymptotically faster computa-tion. These results and the structure of the model wedeveloped provide novel theoretical intuition about ran-dom nanowire network conductivity. Namely, our resultsdemonstrate that network connectivity in the directionof current flow is the key factor in determining the de-pendence of conductivity on wire density and orientationdistribution, because our model only encodes connectiv-ity information in the x -direction.The most pressing direction for future research is torelax our assumption of zero wire resistance, as recentwork has indicated that the junction resistance in silvernanowire networks can be reduced to a comparable mag-nitude as the wire resistance [18]. This could be done,for example, by using an approximate function to cal-culate sheet conductivity based on E A ∗ in the presenceof wire resistance. Various recent analytical models forrandom nanowire network conductivity have successfullyused approximations about the number of nanowires thata given nanowire will intersect [25, 29]. Rather than us-ing approximations derived in the setting of uniform wireorientation, these models could instead use approxima-tions obtained from E A ∗ for an arbitrary orientation dis-tribution. The success of these existing models indicatesthat they would likely function as accurate approximatefunctions to calculate sheet conductivity given the infor-mation in E A ∗ . [1] A. Kumar and C. Zhou, ACS nano , 11 (2010).[2] R. G. Gordon, MRS bulletin , 52 (2000).[3] S. Choi, S. I. Han, D. Kim, T. Hyeon, and D.-H. Kim,Chemical Society Reviews , 1566 (2019).[4] D. S. Hecht, L. Hu, and G. Irvin, Advanced materials ,1482 (2011).[5] L. Hu, H. Wu, and Y. Cui, MRS bulletin , 760 (2011).[6] J.-Y. Lee, S. T. Connor, Y. Cui, and P. Peumans, Nanoletters , 689 (2008).[7] A. Teymouri, S. Pillai, Z. Ouyang, X. Hao, F. Liu,C. Yan, and M. A. Green, ACS applied materials & in-terfaces , 34093 (2017).[8] M. Marus, A. Hubarevich, H. Wang, A. Stsiapanau,A. Smirnov, X. W. Sun, and W. Fan, Optics express ,26794 (2015).[9] R. M. Mutiso, M. C. Sherrott, A. R. Rathmell, B. J.Wiley, and K. I. Winey, ACS nano , 7654 (2013).[10] M. Jagota and N. Tansu, Scientific reports , 10219(2015).[11] A. Behnam, J. Guo, and A. Ural, Journal of AppliedPhysics , 044313 (2007).[12] A. Behnam and A. Ural, Physical Review B , 125432(2007).[13] F. Du, J. E. Fischer, and K. I. Winey, Physical ReviewB , 121404 (2005).[14] S. I. White, B. A. DiDonna, M. Mu, T. C. Lubensky, andK. I. Winey, Physical Review B , 024301 (2009).[15] Y. Y. Tarasevich, I. V. Vodolazskaya, A. V. Eserkepov,V. A. Goltseva, P. G. Selin, and N. I. Lebovka, Journalof Applied Physics , 145106 (2018).[16] N. Pimparkar, C. Kocabas, S. J. Kang, J. Rogers, andM. A. Alam, IEEE electron device letters , 593 (2007).[17] J. Hicks, J. Li, C. Ying, and A. Ural, Journal of Applied Physics , 204309 (2018).[18] A. T. Bellew, H. G. Manning, C. Gomes da Rocha, M. S.Ferreira, and J. J. Boland, ACS nano , 11422 (2015).[19] T. Ackermann, R. Neuhaus, and S. Roth, Scientific re-ports , 34289 (2016).[20] M. Marus, A. Hubarevich, R. J. W. Lim, H. Huang,A. Smirnov, H. Wang, W. Fan, and X. W. Sun, Opti-cal Materials Express , 1105 (2017).[21] F. Wu, Z. Li, F. Ye, X. Zhao, T. Zhang, and X. Yang,Journal of Materials Chemistry C , 11074 (2016).[22] M. Marus, A. Hubarevich, H. Wang, Y. Mukha,A. Smirnov, H. Huang, W. Fan, and X. W. Sun, ThinSolid Films , 140 (2017).[23] D. Kim and J. Nam, Journal of Applied Physics ,215104 (2018).[24] Y. Y. Tarasevich, I. V. Vodolazskaya, A. V. Eserkepov,and R. K. Akhunzhanov, Journal of Applied Physics ,134902 (2019).[25] C. Forr´o, L. Demk´o, S. Weydert, J. V¨or¨os, and K. Ty-brandt, ACS nano , 11080 (2018).[26] H. G. Manning, C. G. da Rocha, C. OCallaghan, M. S.Ferreira, and J. J. Boland, Scientific reports , 1 (2019).[27] A. Ponzoni, Applied Physics Letters , 153105 (2019).[28] R. Benda, E. Canc`es, and B. Lebental, Journal of AppliedPhysics , 044306 (2019).[29] A. Kumar, N. Vidhyadhiraja, and G. U. Kulkarni, Jour-nal of Applied Physics , 045101 (2017).[30] D. Stauffer and A. Aharony, Introduction to percolationtheory (Taylor & Francis, 2018).[31] D. J. Klein and M. Randi´c, Journal of mathematicalchemistry , 81 (1993).[32] H. Weisberg, The Annals of Mathematical Statistics42