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Next: 1.3.3 EM Lifetime Extrapolation Up: 1.3 Empirical and Semi-Empirical Previous: 1.3.1 Black's Equation

1.3.2 Resistance Increase and Void Growth

EM experiments normally use a given resistance increase as failure criterion. For a void spanning the entire cross sectional area of a line, the total resistance of a damascene line, as shown in Figure 1.5, is given by

$\displaystyle R = \frac{\symElecRes_{barrier}\Delta l}{A_{barrier}} + \frac{\symElecRes_{Cu}(\symL - \Delta l)}{A_{Cu}},$ (1.8)

where $ \symElecRes_{barrier}$ is the electrical resistivity and $ A_{barrier}$ is the cross sectional area of the barrier layer, $ \symElecRes_{Cu}$ is the electrical resistivity and $ A_{Cu}$ is the cross sectional area of the copper line, $ \symL$ is the line length, and $ \Delta l$ is the length of the void. Thus, this equation relates resistance increase with void size. In turn, void size is connected to mass transport, which is expressed in the form [57]

$\displaystyle \JA = \CA\symDriftVel = \CA\frac{\symIntThickness}{h}\frac{\DA}{\kB\T}\Z\ee\symElecRes\symCurrDens,$ (1.9)

where $ \CA$ is the atomic concentration, $ \symDriftVel = \Delta l/\Delta t$ is the drift velocity, $ \symIntThickness$ is the width of the interface which controls mass transport, $ h$ is the line height, $ \DA$ is the atomic diffusivity of the interface which dominates the EM transport, $ \Z$ is the effective valence, $ \ee$ is the elementary charge, $ \symElecRes$ is the electrical resistivity, and $ \symCurrDens$ is the current density.

Figure 1.5: Void growth in a single-damascene copper interconnect.

The atomic flux can also be written as [22]

$\displaystyle \JA = \frac{\CA\Delta V}{A_{Cu}\Delta t},$ (1.10)

where $ \Delta V$ is the change of the void volume in a time $ \Delta t$. Combining (1.9) and (1.10) yields

$\displaystyle \Delta t = \frac{h\kB\T}{\symIntThickness\DA\Z\ee\symElecRes\symCurrDens A_{Cu}}\Delta V,$ (1.11)

so that a given test time $ t$ is related to the void volume $ V_{void}$,

$\displaystyle t = \frac{h\kB\T}{\symIntThickness\DA\Z\ee\symElecRes\symCurrDens A_{Cu}}V_{void}.$ (1.12)

Equation (1.12) is commonly used for estimation of the void growth time. In turn, a critical void volume is related to the line resistance according to (1.8). Thus, the time to failure for a given resistance increase criterion can be determined.

An interesting application of such an approach was performed by Hauschildt et al. [20,22]. For a void spanning the entire line width, $ w$, and assuming a rectangular void shape, the void volume becomes $ V_{void}=wA_{void}$, and (1.12) yields

$\displaystyle t = \frac{wh\kB\T}{\symIntThickness\DA\Z\ee\symElecRes\symCurrDens A_{Cu}}A_{void},$ (1.13)

where $ A_{void}$ is the area of the void, which can be measured using SEM pictures. EM tests were then carried out and stopped after a given test time (called t-based by the authors) or after a certain resistance increase (R-based). In both cases, the void area mean value and standard deviation were determined. For the R-based tests the mean time to failure $ \symMTF$ can be determined which, in turn, can be used for the t-based experiments. Finally, using the void area statistical distribution of both tests and applying (1.13) a new distribution is obtained following [20,22]

$\displaystyle t = \frac{A^{R-based}}{A^{t-based}}\symMTF.$ (1.14)

Hence, once the statistics of the void area distribution for both test types is known, the distribution of the EM lifetimes can be estimated.

Equation (1.9) is also commonly used to extract the diffusivity from void drift experiments [57]. The drift velocity of the void boundary in Figure 1.5 is

$\displaystyle \symDriftVel = \frac{\symIntThickness}{h}\frac{\Z\DA}{\kB\T}\ee\symElecRes\symCurrDens = \frac{\Delta l}{\Delta t},$ (1.15)

which for a given test time $ t_{test}$ with a corresponding void length $ l_{void}$ can be rearranged as

$\displaystyle \Z\DA = \frac{h\kB\T}{\symIntThickness\ee\symElecRes\symCurrDens}\frac{l_{void}}{t_{test}}.$ (1.16)

Since $ \DA$ can be expressed by an Arrhenius relationship, the activation energy for diffusion can be extracted from a plot $ \ln(\Z\DA)$ versus $ 1/\kB\T$ [38].

next up previous contents
Next: 1.3.3 EM Lifetime Extrapolation Up: 1.3 Empirical and Semi-Empirical Previous: 1.3.1 Black's Equation

R. L. de Orio: Electromigration Modeling and Simulation