13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Simulation Method for Electrical Breakdown of a Metallic Nanowire Mesh 2.1 Simulation Model and Basic Assumptions Fig. 1 Schematic illustration of a metallic nanowire mesh (m×n) A metallic nanowire mesh with size of m×n is schematically illustrated in Fig. 1, which is a uniform rectangular grid of wires with m rows (horizontal wires) and n columns (vertical wires). As shown in Fig.1b, the pitch size is p, the wire width and thickness are w and b, respectively. Due to the complex nature of electro-thermal for a metallic wire mesh, the following assumptions are made: (1) The material of metallic mesh is homogeneous and isotropic; (2) Material properties of metallic mesh are temperature independent; (3) The effect of electromigration is neglected for simplicity. 2.2 Fundamentals of Governing Equations A mesh node is an intersection of each row and column. Let node (i, j) be the intersection of the (i+1)th row and the (j+1)th column, where i=0, …, m-1; j=0, …, n-1. The wire between two adjacent mesh nodes is called a mesh segment. For the present mesh with size of m×n, the numbers of mesh nodes and mesh segments are m×n and m(n-1)+n(m-1), respectively. The segment between node (i, j-1) and node (i, j) is denoted as si,l j, and the segment between (i, j) and (i, j+1) is denoted as si,r j. Similarly, the segment between node (i-1, j) and (i, j) is denoted as si,d j, and the segment between (i, j) and (i+1, j) is denoted as si,u j. Here, the letters of l, r, d and u stand for the relative positions of the adjacent nodes (i.e., (i, j-1), (i, j+1), (i-1, j), (i+1, j)) to node (i, j), i.e., left, right, down, up, respectively. For any mesh segment, the current can be calculated using Ohm’s law. By considering the node (i, j), the current flowing between four adjacent nodes and it can be obtained as below Si,l j = Si,l j =−1 (i,j)− (i,j−1) (i,j)− (i,j−1) Si,r j = Si,r j =−1 (i,j+1)− (i,j) (i,j+1)− (i,j) Si,d j = Si,d j =−1 (i,j)− (i−1,j) (i,j)− (i−1,j)
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