A percolative approach to investigate electromigration failure in printed Ag structures

Printed microelectronics have the potential for broad application and ease of use. This has been a driving force for significant advances in the inks that are used to print conducting lines. Such advances include lower sintering temperatures and lower resistances.^1–4 For example, newer nanoparticle-based silver inks can be chemically sintered at room temperature using polyanionic compounds.^5,6 Despite these advances, very few industries outside of the photovoltaic industry have adopted silver inks for their metallization schemes and even fewer use drop-on-demand (DOD) printed inks in real-world applications. This limited adoption of DOD printed electronics can be attributed to the limited availability of commercial-scale DOD printers capable of handling large production volumes with high resolutions and the high cost of DOD-compatible nanoparticle inks. Reactive inks are a new approach to DOD printed electronics that are easy to synthesize and that often do not require high-temperature sintering.^7,8 Unlike traditional inks that print clusters of particles, reactive inks print chemical precursors that react to form a solid material.^9,10 These reactions can be initiated by elevated substrate temperatures (thermally), solvent or stabilizing agent evaporation (chemically), or by some increased catalytic activity of the substrate (kinetically).^9,11–17 Recent advances in silver,⁹ copper,¹⁰ and aluminum^13,18 reactive inks have brought the reaction temperatures of these inks to below 180 °C and to even room temperature for silver-diamine based inks.^7,19,20 Compared to silver paste and nanoparticle-based inks, these new reactive inks provide superior conductivities (that are close to bulk material values) at lower temperatures and significantly lower costs. Applications for reactive inks include printed electronics,⁹ stretchable electronics,⁸ photovoltaic metallization,²¹ and more.

The long-term performance of conducting lines that are printed with reactive Ag inks is currently unknown. This is because the most promising, low-temperature reactive ink chemistries for Ag, Cu, and Al are fairly new, with keystone publications having appeared only within the last few years.^{9,10,13,20,22} As a result, the relationships among processing parameters, printed physical structure, material properties, and long-term performance have yet to be studied in detail. Of particular concern is how the porous nature of silver that is printed from a reactive silver ink impacts its properties and reliability. Electromigration and Joule heating are two phenomena that contribute to early failure in electronic devices. For a high-quality, solid material, electromigration (EM) generates voids that, due to current crowding, lead to localized Joule heating and rapid device failure.²³ The failure time of metal lines due to electromigration is typically predicted by Black's equation, which correlates the lifetime with the current density and temperature during the electromigration.²⁴ Since reactive inks currently print porous materials with a large volume-fraction of voids,⁷ these materials should experience earlier failure than conducting elements fabricated using traditional methods (DC sputter coating, evaporation, electrochemical deposition, etc.). However, no studies have been published that detail the performance reliability of these printed reactive inks under a bias (temperature or current). Initial stress tests of printed silver reactive inks showed lifetimes that were quite random and that were independent of the current density. A deeper understanding of the relationship between morphology and failure mechanisms is needed before reactive silver inks (RSI) can replace more established particle-based inks and pastes.

In this work, we have studied the failure mechanisms of silver lines that were printed using a low-temperature, self-reducing, silver-diamine based ink. These studies demonstrated that Black's equation does not suitably account for morphology-induced current crowding when predicting the lifetimes of printed silver or other porous conducting elements. Overall, a model is needed that can take into account the effects of physical structure, mass-transport, and other degradation mechanisms that can impact the lifetimes of porous or “imperfect” printed electronic devices. In this work, we demonstrate that a percolation-based model (for predicting the electromigration failure time) is suitable for predicting the increase in resistance of porous conducting lines under a current bias.

The base silver ink (that was used for this study) was prepared following Lewis' reactive silver ink recipe.⁹ All of the chemicals were used as received: silver acetate (C₂H₃AgO₂, anhydrous 99%, Alfa Aesar); ammonium hydroxide (NH₄OH, 28–30 wt. %, ACS grade, BDH Chemicals); formic acid (CH₂O₂, ≥96%, ACS reagent grade, Sigma Aldrich); and ethanol (EtOH, C₂H₆O, 99.5 wt. %, Koptec). 1.0 g of silver acetate was dissolved in 2.5 ml ammonium hydroxide. The solution was then stirred for 2 min on a vortex mixer to dissolve the silver acetate. Next, 0.2 ml of formic acid was added in two steps with a quick stir at the end of each step. The ink was then allowed to sit for 12 h in the dark at room temperature before being filtered through a 450 nm nylon filter. Following this, the ink was stored at 4 °C (refrigerator temperature) until it was used.

A mixture of 0.5 M tin (II) chloride solution in deionized water mixed 1:1 by volume with 0.5 M HCl was used as a sensitizing adhesion promoter¹⁴ to keep the silver samples from peeling off from the glass substrate. The substrates were dipped in this solution for 300 s and dried using N₂. Silver reactive ink lines and contact pads were printed using a MicroFab Jetlab II micro-dispensing inkjet printer. Drop volume, velocity, and quality were observed using a horizontal camera and strobe light. The Jetlab II was equipped with an MJ-ATP-01 piezoelectric-driven printer head with a 60 μm orifice coated with a diamond-like coating to reduce wetting. The mean measured droplet size was 40 μm when printed with rise and echo driving voltages of ±25 V and dwell and echo times of 5 μs and 5 μs, respectively.

Single-line test structures were printed using 1:1 EtOH:Ag ink on the glass substrates, at 78 °C. Each of the printed lines had a length of 5 mm with a pitch of 25 μm between droplets and a measured droplet diameter of 100 μm. A Dektak XT stylus (12.5 μm stylus tip diameter) profilometer was used to collect cross-section profiles across each sample. The cross-section area of each sample was calculated by integrating the area under the measured profile.

The structures were then tested for reliability on a probe station (MC Systems, Inc., Model 8832) equipped with a temperature-controlled hot stage. The samples were tested at an elevated temperature of 94 °C to accelerate the failure process. Since early failure modes occurred at the interface between the probe and the contact pad instead of at the Ag line, conductive silver paint (Colloidal silver liquid, Ted Pella, Inc.) was applied on the contact pads in order to increase the contact surface area and to decrease the contact resistance between the probe and the contact pads to avoid the failure at their interface. The samples were tested using two probes in contact with opposite pads. A constant current was applied across the probes using an Agilent DC power supply (N5752A) and the voltage drop between the probes was measured every 1 s using a Keysight digital multimeter (34461A) until an open occurred in the line. Several different current values were applied to study the failures under different current stresses.

Typically, the failure time of a metal line can be predicted by using Black's equation²⁴ 

where A_B is a constant, J is the current density in A/cm², n is the current-density exponent, E_a is the activation energy in eV, k_B is Boltzmann's constant, and T is the substrate temperature in Kelvin. From Black's equation, n can be extracted by fixing the test temperature and varying the current density.^25,26

The printed Ag line has a porous microstructure. A representative cross-section secondary electron microscopy (SEM) image of an as-printed Ag line is shown in Figure 1(a). The micrograph shows that the printed Ag film is porous and not solid. Due to the porous microstructure, only the Ag component (in the printed line) carries the current. As a result, the actual current-carrying cross-section area is equal to the cross-section area of Ag (A_Ag) if the pores are taken out from the printed Ag sample and the sample is 100% dense (0% porosity) as a result. A_Ag is calculated by dividing the volume of silver used in each printed Ag line by the line length. Each Ag line was printed using the same known amount of silver. Therefore, A_Ag is 152 μm², and it has the same value for all of the samples (the calculation of A_Ag is shown in the supplementary material). Next, the current density (J) is obtained by dividing the current (I) by A_Ag (to take into the account of current-carrying cross-section area). Six samples were tested using the same temperature but different current densities, until they failed. Their failure times (⁠$tf$⁠) are plotted on a log-log scale, as the function of current density (J), as shown in Figure 1(b). For this study, the failure time is defined as the time that Ag line breaks. According to Eq. (1), $tf$ vs. J should be a linear function on the log-log plot and the current exponent n can be estimated by calculating the slope. However, it is evident from Figure 1(b) that the experimental results do not fall on a straight line as would be predicted by Black's equation.

We suspect that the erratic lifetimes of the RSI printed Ag lines is likely due to the porous microstructure of the lines, and that Black's equation cannot be used for these material systems because the equation assumes a solid cross-section for calculating the current density. This assumption breaks down for porous conductors because it does not account for current crowding in such porous materials.^27–29 With this in mind, it is clear that failure models that account for porosity and morphology must be used when working with porous printed conductors.

Percolation theory has been used to study the conductivity of carbon nanotube thin film networks.³⁰ Similarly, to take into the account of the porosity of our printed sample, the conduction of this porous structure can be described by a standard (unbiased) percolation model.^31,32 The line can be viewed as a random network of insulators (voids) and conductive resistors (silver).^31,32 All of the resistors are assumed to have the same resistance, in order to simplify the model. Based on percolation theory, the resistivity (ρ) is a function of the void fraction, and is given by the following equation:³³ 

where ρ₀ is the pre-factor and is proportional to the metal's bulk resistivity, and m is the percolation coefficient. The resistivity (ρ) of a printed Ag line is obtained from the measured resistance, the measured area, and the line length. f_V is the fraction of voids in the printed sample.

Our measurements show that the measured cross-section areas are different from sample-to-sample, even though the lines were printed using the same amount of Ag ink. This is an indication of the variation in porosity from sample-to-sample and even within the same sample. The porosity can be estimated by calculating the fraction of the voids in the sample (f_V)

where V_Ag and A_Ag are the volume and cross-section area if the printed Ag sample was 100% dense (0% porosity), V_measured and A_measured are the measured volume and measured cross-section area of the printed Ag sample as determined by profilometry, and l is the length of the printed Ag sample.

The ρ and f_V of several printed Ag samples were determined, to validate the percolation model on the printed Ag sample. The resulting ρ is plotted as a function of f_V and can be fitted by Eq. (2), as shown in Figure 2. The values of $ρo$ and m were calculated to be 1.51 × 10⁻⁵ Ω cm, and 0.91, respectively. Using a least-squares model, the adjusted R-square of fit was 0.97. Based on this fit, we conclude that the conduction pathways of the printed reactive inks can be described by the standard percolation model.

Given that the conduction occurs via a percolation process, one would expect that the failure should also occur by such a process. When failure occurs by the standard percolative model, the damage would be uniform and dependent on a single test parameter.^31,32 In the case of electromigration, the failure occurs due to void formation and coalescence during the current stressing.²³ The percolative model for predicting the electromigration failure has to be biased given that the failure is nonhomogeneous, and it depends on the position and time.^31,32 Monte-Carlo simulations have been used to describe the evolution of resistance for biased percolation.^31,32 The scaling of the resistance as a function of time (⁠$R(t)$⁠) is given by

where $R0$ is the initial resistance, $γ$ is a statistical variation factor which does not have physical meaning but is due to statistical variation during data collection, $t$ is time under bias, $tf$ is the failure time, and $μ$ is the percolation parameter. $μ$ is dependent on the current density and the temperature applied on the sample.^31,32

Two printed samples were tested under current densities of 2.63 MA/cm² and 3.62 MA/cm². Both samples were tested at 94 °C until an open-circuit was measured (i.e., failure). Their resistance as a function of time is plotted in Figure 3. Given that their initial resistance and failure times are known, the $γ$ and $μ$ can be extracted by fitting the R vs. t curves using Eq. (4). A typical fit using Eq. (4) is shown in Figure 3. The extracted results and detailed parameters from the samples are listed in Table I. The results show that $μ$ has the same value (of 0.012) when the samples have the same current density (of 2.63 MA/cm²). However, $μ$ had a value of 0.048 when the sample was tested under a different current (of 3.62 MA/cm²). This result is consistent with the fact that $μ$ depends on the current density and temperature in the percolation model. The deviation of R from the fitting model between 2000 and 2500 s is probably due to the evolution of the voids/pores in the printed Ag line. It has been proposed³⁴ that the resistance change depends on the void shape and the change in void volume, ΔR/R= f·ΔV/V.³⁴ The volumes and shapes of the local pores/voids in the porous Ag line are more likely to change due to current-induced mass transportation. This will cause a deviation of the R from the model, since the model does not take into account the effects of changes in the local voids.

The percolative model can be further modified to incorporate the void fraction. In the percolative model, R₀, which is the initial resistance of the sample, can be related to the void fraction using Eq. (2). This can be expressed as

Hence, the void fraction can be incorporated into Eq. (4), by using Eq. (5)

Inspection of Eq. (6) shows that the porosity has a direct impact on the time dependent resistance during biasing at high current levels. In addition, the EM failure-time and the reliability of printed lines can be predicted using percolation theory. Future work will focus on the use of Eq. (6) to experimentally demonstrate that for similar structures (i.e., similar metals and with similar porosity) that are undergoing EM degradation, the exponents m and $μ$ can be accurately extracted from a small population of samples.

Ag metallization structures were printed by using reactive Ag ink. The reliability of the printed single-line Ag structure was tested through EM. The test results were first analyzed using Black's equation. However, J vs. t did not follow Black's equation. SEM analysis showed that the printed Ag lines are porous. These results indicated that it is inappropriate to apply Black's equation to porous structures. In contrast, the conduction in the porous Ag line was successfully described by using percolation theory. Therefore, we expected that the percolative model should be able to describe the failures of the porous printed Ag lines. Our analysis shows that R vs. t can be fitted well by the percolative model. A dependence of $μ$ on current density is observed in the experiment. This further supports the hypothesis that the percolative model is valid for describing failures of porous printed Ag lines. A model that links the physical structure and mass-transport to the lifetime is obtained by modifying the percolative model.

The supplementary material shows the detailed calculation for cross-section area of Ag (A_Ag) if the pores are moved from the printed Ag sample and the sample is 100% dense (0% porosity) as a result.

The authors thank N. David Theodore in NXP Semiconductors N.V. for his great help in offering suggestion on preparing manuscript. This work was partially supported by the National Science Foundation (C. Ying, Grant No. DMR-0902277) to whom the authors are greatly indebted.