Numerical analysis of the electrical failure of a metallic nanowire mesh due to Joule heating
 Yuan Li^{1}Email author,
 Kaoru Tsuchiya^{1},
 Hironori Tohmyoh^{1} and
 Masumi Saka^{1}
DOI: 10.1186/1556276X8370
© Li et al.; licensee Springer. 2013
Received: 11 July 2013
Accepted: 23 August 2013
Published: 30 August 2013
Abstract
To precisely examine the electrical failure behavior of a metallic nanowire mesh induced by Joule heating (i.e., melting), a previously developed numerical method was modified with regard to the maximum temperature in the mesh and the electrical resistivity of the nanowire. A sample case of an Ag nanowire mesh under specific working conditions was analyzed with highly accurate numerical results. By monitoring the temperature in the mesh, the current required to trigger the melting of a mesh segment (i.e., the melting current) could be obtained. The melting process of a mesh equipped with a current source during actual operation was predicted on the basis of the obtained relationship between the melting current and the corresponding melting voltage in the numerical melting process. Local unstable and stable melting could be precisely identified for both the currentcontrolled and voltagecontrolled current sources in the present example.
Keywords
Electrical failure Joule heating Metallic nanowire mesh Stable/unstable meltingBackground
Given the remarkable physical properties of metallic nanowires, their successful assembly into both regular [1, 2] and random [3–5] networks can achieve three goals: high transparency, low electrical resistivity, and good flexibility. Therefore, these metallic nanowire networks offer promising alternatives to indium tin oxide (ITO) for possible application in optoelectronic devices, such as touch screens and solar cells. For example, high optical transmittance and electrical conductance have been reported for a flexible transparent Cu nanowire mesh (i.e., a regular network) [1]. In addition, an organic solar cell integrated with such a Cu nanowire mesh electrode has been shown to perform comparably to one using an ITO electrode [1]. Another study on a transparent conductive Ag nanowire mesh has also been shown to exhibit a similarly good performance [2].
As we all have known, when current flows through any electrically conductive material, some electrical energy is transformed into thermal energy, which means the occurrence of Joule heating [6]. Undoubtedly, this general knowledge also applies to individual metallic nanowire and the corresponding nanowire mesh, both of which are conductors. Due to the size effects on the nanoscale (e.g., the increase in electrical resistivity [7–9] and the decrease in both thermal conductivity [10–12] and melting point [13, 14]), the high current density and the substantial Joule heating induced in metallic nanowires may cause or accelerate electrical failure related to the phenomena of melting [15–17], electromigration [16, 18–21], and corrosion [22]. The size effects will definitely also degrade the electrical performance of the corresponding nanowire mesh and therefore reduce the reliability of meshbased devices. To prevent this problem, there is an urgent need to examine the electrical failure of a metallic nanowire mesh induced by Joule heating.
Unfortunately, in contrast with the numerous reports on electrical failure of individual metallic nanowires [15–21], little is currently known about the electrical failure of metallic nanowire mesh, which is expected to exhibit different characteristics because of its unique mesh structure. A recent and pioneering study [23] reported the electrical failure of an Ag nanowire random network due to Joule heating and offered possible solutions to the potential for electrical failure of a metallic nanowire mesh. In addition, a numerical method has also been proposed [24] which provided meaningful yet preliminary results regarding the electrical failure of a metallic nanowire mesh due to Joule heating.
The present work aims to clarify the electrical failure behavior of a metallic nanowire mesh induced by Joule heating. To that end, two vital modifications were proposed to the previously developed numerical method and compiled into a computation program. The first relies on the identification of the maximum temperature in the mesh, which relates to the criterion used to determine the melting of the mesh segment. The second modification relates to the resistivity of the metallic nanowire. As an example, the melting process of an Ag nanowire mesh was analyzed under specific working conditions. Numerical results allow monitoring of the temperature in the mesh under current stressing and determination of the current that triggers the melting of a mesh segment. Using the relationship between the melting current and the corresponding melting voltage, the electrical failure behavior of an Ag nanowire mesh system equipped with a current source can be predicted during actual operation.
Methods
Numerical model
The wire between two adjacent mesh nodes is called a mesh segment. The segment between node (i − 1, j) and node (i, j) is denoted by ${\mathrm{S}}_{\left(\mathrm{i},\mathrm{j}\right)}^{\mathrm{L}}$, and the segment between (i, j) and (i + 1, j) is denoted by ${\mathrm{S}}_{\left(\mathrm{i},\mathrm{j}\right)}^{\mathrm{R}}$. Similarly, the segment between node (i, j − 1) and (i, j) is denoted by ${\mathrm{S}}_{\left(\mathrm{i},\mathrm{j}\right)}^{\mathrm{D}}$, and the segment between (i, j) and (i, j + 1) is denoted by ${\mathrm{S}}_{\left(\mathrm{i},\mathrm{j}\right)}^{\mathrm{U}}$. Here, the letters L, R, D, and U denote the relative positions of the adjacent nodes (i.e., (i − 1, j), (i + 1, j), (i, j − 1) and (i, j + 1)) to node (i, j), meaning left, right, down, and up, respectively. The corresponding number of mesh segments is M(N − 1) + N(M − 1).
Fundamentals of governing equations
The melting behavior of a metallic nanowire mesh can be treated as an electrothermal problem. To simplify this problem, the following assumptions are made: (1) the material of the metallic nanowire is electrically and thermally homogeneous and isotropic, (2) the material properties of the metallic nanowire are temperature independent, and (3) the effects of electromigration and corrosion are neglected.
where T is the temperature and λ is the thermal conductivity of the nanowire. It should be noted that the effect of thermal conduction to the underlying substrate of the mesh is ignored here for simplicity.
The current density, temperature, and heat flux in the other mesh segments connected to node (i, j) can be obtained in a similar manner.
in which the subscript indicates the mesh segment connected to node (i, j) and A is the crosssectional area of the wire. Considering Equations 1, 5, and 6 for any mesh node (i, j), a system of linear equations can be constructed to obtain the relationship between ϕ and I_{external} for any mesh node. Once ϕ is obtained for every node by solving the system of linear equations, the current density in any mesh segment can readily be calculated using Equation 1.
Considering Equations 4, 7, and 8 for any mesh node, a system of linear equations can be constructed to obtain the relationship between T and Q_{external} for any mesh node. Once T is obtained for every node by solving the system of linear equations, the temperature at any location on any mesh segment can be calculated using Equation 3.
The current density and temperature in any mesh segment can be obtained using the previously described analysis for the electrothermal problem in a metallic nanowire mesh. This calculation will provide valuable information for the investigation of the melting behavior of a metallic nanowire mesh.
Computational procedure
Initially, the input current I is gradually increased in uniform increments, ΔI, and the corresponding temperature profile of the mesh is monitored. To cause the mesh segment to melt one at a time, ΔI must be properly tuned. When the temperature in a given mesh segment reaches the melting point T_{m} of the nanowire itself, the corresponding mesh segment melts and breaks with an arbitrary small force generated in actual operation such as a vibration. This temperature is considered the maximum temperature, T_{max}, of the mesh. The electrical failure is believed to occur at the mesh segment. Here, the following two critical modifications have been made to the previously developed numerical method [24]. First, instead of using the temperature in the center of a mesh segment to approximate the T_{max}, five points uniformly distributed along each segment are monitored to determine whether the temperature reaches T_{m} and melting occurs. If the temperature in a segment reaches T_{m} before the temperature at a mesh node, then the mesh segment melts and breaks. However, if the temperature of a mesh node reaches T_{m} first, then the adjacent segments connected to the node melt simultaneously and break. Second, the temperature dependence of the resistivity is ignored for simplification; thus, the resistivity of the metallic nanowire at the melting point, not the resistivity of the metallic nanowire at room temperature (R.T.), is employed during the simulation to approximate real conditions. The input current of the mesh triggering the melting of the mesh segment and the corresponding voltage of the mesh (i.e., the difference in the electrical potential between the input and the output) are recorded as the melting current I_{m} and the melting voltage V_{m}, respectively. The corresponding resistance R of the mesh can be calculated by dividing V_{m} by I_{m}.
Subsequently, the crosssectional area of the melted mesh segment is set at a very small value to approximate a crosssectional area of zero. The pathway of the current and heat in the mesh will be correspondingly renewed. By increasing the input current gradually, the current that triggers the subsequent melting of the mesh segment can be determined.
By repeating the aforementioned process until the mesh opens, the relationship between I_{m} and V_{m} can be determined throughout the melting process.
Results and discussion
Numerical model of an Ag nanowire mesh
Physical properties of an Ag nanowire
In addition, the following working conditions are specified in the present study. The external current flows into the mesh from node (0, 0) and flows out of the mesh from node (9, 0), which means that node (0, 0) has an external input current and node (9, 0) has an external output current (see Figure 4). For all the other nodes, there is no external input or output current. A constant electrical potential is assigned to node (9, 9). The temperature of the boundary nodes ((i, 0), (0, j), (i, 9), (9, j) in which i, j = 0,…, 9) is set at room temperature of 300 K. For all of the other nodes, there is no any external input or output heat energy.
Using the developed computational program, the temperature in the Ag nanowire mesh can be monitored, allowing for determination of the melting current. The input current, I, is increased with a ΔI value of 0.001 mA to cause the mesh segments to melt one at a time if possible. The corresponding melting current and melting voltage (i.e., the difference in electrical potential between node (0, 0) and node (9, 0)) are recorded as melting current I_{m} and melting voltage V_{m}, respectively. Using the relationship between I_{m} and V_{m}, the variation in mesh resistance R throughout the melting process could be calculated.
Numerical analysis of the failure behavior of the mesh
Subsequently, the mesh structure undergoes a process of the consecutive melting of large numbers of individual nanowires. During the melting of the mesh as shown in Figure 5a, the variation in I_{m} and V_{m} of the mesh exhibits the repetition of three different trends: (I) both I_{m} and V_{m} decrease, (II) both I_{m} and V_{m} increase, and (III) I_{m} decreases while V_{m} increases. The solidline arrows in Figure 5c,d indicate these three trends. Such repetition of zigzag pattern as shown in Figure 5a can be explained in detail as below. After one mesh segment is melted, the electrical pathway in the mesh is changed so that the mesh resistance increases, and therefore Joule heating increases. In one case, the maximum temperature of the mesh may be far beyond the melting point of the wire, which means the present current is much higher than that for the subsequent wire melting. To precisely obtain the melting current for the subsequent wire melting (i.e., the current when the maximum temperature of the mesh properly reaches the melting point), the input current has to be decreased, which means the decrease of melting current. In another case, the maximum temperature of the mesh is still lower than the melting point of the wire. To make further melting, the input current has to be increased to make the maximum temperature rise up to the melting point, which implies the increase of melting current. The irregular alternation of these two cases leads to the zigzag pattern of the relationship between I_{m} and V_{m} during the melting process of the mesh. Moreover, it is thought that if the pitch size of the mesh is smaller, the extent of zigzag pattern will be mitigated. In an extreme case, when the pitch size is zero which makes the mesh transit to thin film, the present zigzag pattern will be diminished and the relationship between I_{m} and V_{m} will become smooth.
Finally, the mesh becomes open when two segments, marked by red cross symbols in Figure 7b, melt. Obviously, the broken mesh segments are sufficient to eliminate the continuous electrical pathway across the mesh. At this time, the total number of melted mesh segments is 99, which is slightly more than half the total number (180) of mesh segments in the intact Ag nanowire mesh.
The results of this work differ with those previously reported [24] in the following ways: First, the melting current is reduced by half, and the range of the melting voltage is increased, which can be attributed to the inclusion of ρ_{m}. Second, any unreasonable drop in the melting current due to a possible numerical error has been removed. Third, throughout the melting process, the mesh remains symmetric regardless of the number of segments that melt, as shown in Figure 7. These results suggest a dramatic increase in the accuracy of numerical results, supporting the feasibility of the present modified numerical method.
Prediction of the electrical failure behavior of the mesh equipped with current source
Achieving an immediate decrease in the current or voltage during practical experiments is known to be difficult due to the limited properties of current sources. Therefore, one cannot reproduce the abovementioned zigzag pattern of I_{m} and V_{m} observed in the numerical melting process in actual experiments. Considering a system composed of an Ag nanowire mesh and a current source, the electrical failure behavior of the mesh in actual experiments could be predicted using the aforementioned numerical results. Two common modes of current sources, a currentcontrolled current source (CCCS) and a voltagecontrolled current source (VCCS), are discussed below.
Similarly, for the VCCS mode, the relationship between I_{m} and V_{m} of the mesh in a real experiment can be predicted as indicated in Figure 8b by the dottedline arrows. The repetition of the vertical decline stage is marked by a green dottedline arrow pointing downward, and the diagonal ascent stage is marked by a green dottedline arrow pointing up and to the right. The vertical decline stage indicates the simultaneous melting of several mesh segments at a constant voltage. This local unstable melting is similar to the local unstable melting that occurs in the CCCS mode. When compared to the curve of I_{m} vs. V_{m} during numerically simulated melting, there is a jump (e.g., from point P_{C} to point P_{D} in the enlarged part of Figure 8b). The reason for this jump is that in real experiments, it is difficult to decrease the voltage immediately, just as it is difficult to decrease the current immediately. Therefore, it is difficult to reproduce the region to the left side of the vertical decline stage (i.e., the decrease in voltage and its subsequent increase), which is marked by a green dashed rectangle in the enlarged part of Figure 8b. The diagonal ascent stage indicates that an increase in the voltage is necessary for further melting. This stable melting is also similar to the stable melting that occurs in the CCCS mode. However, no global unstable melting occurs as in the CCCS mode due to the decrease in Joule heating, which is caused by the increase in the mesh resistance that accompanies the melting of the mesh segments.
To fully understand the unique melting behavior of a metallic nanowire mesh, the melting behavior of an individual nanowire itself is summarized for comparison as follows: For both the CCCS and VCCS modes, once the maximum temperature in the nanowire reaches T_{m}, the nanowire melts and breaks. This behavior has been used to cut metallic nanowires at desired locations [15, 17].
The predicted stable and unstable melting in the Ag nanowire mesh equipped with a current source is only an example. In the present case, the thermal conduction to the underlying substrate of the mesh is ignored. According to the above analyses, it could be speculated that the melting current I_{m} and the corresponding melting voltage V_{m} will increase if the effect of the underlying substrate is taken into account. The reason is the thermal conduction to substrate can effectively mitigate the temperature rise. However, as thermal conduction to the substrate is a global effect, the mesh itself including all mesh segments will be affected. Therefore, the overall zigzag behavior of the mesh and the predicted stable/unstable melting may not be changed largely.
Moreover, it should be noted that there are some important parameters including boundary conditions (e.g., thermal conduction to substrate), mesh structure, electromigration, and corrosion, all of which will make a great effect on the electrical failure behavior of metallic nanowire mesh due to Joule heating. The present study just provides a basis for investigating the reliability of metallic nanowire mesh.
Conclusions
With a modified effective computational method in terms of the maximum temperature in the mesh and the electrical resistivity, the electrical failure of a metallic nanowire mesh due to Joule heating (i.e., melting) was investigated. As an example, the melting process of an Ag nanowire mesh under specific working conditions was analyzed via monitoring of the temperature in the mesh and determining the melting current that triggers the melting of a mesh segment. Using the asobtained relationship between the melting current and the corresponding melting voltage during the melting process, the real melting behavior of a mesh system equipped with a current source could be predicted. The corresponding numerical results indicate with high accuracy that local unstable and stable melting can be identified in both currentcontrolled and voltagecontrolled current sources in the present example.
Abbreviations
 ITO:

Indium tin oxide
 CCCS:

Currentcontrolled current source
 VCCS:

Voltagecontrolled current source.
Declarations
Acknowledgements
This work was supported by the Tohoku Leading Women’s Jump Up Project for 2013 (J130000264) from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan.
Authors’ Affiliations
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