Lifetime Analysis

As it was mentioned the classical mean time to failure of an interconnect is often evaluated by the well-known Black’s equation where 𝐴 is a constant and 𝑛 is a model parameter which are obtained empirically. Yet, determining the current density exponent is a crital task which may vary for interconnects with different electrical, thermal and mechanical loads. We propose to determine a true mean time to failure based on the atomic flux divergences. In other words, the parameters in Black’s model can be determined readily by the following method.

Here we build a novel systematic basis for lifetime modeling based the fundamental definition of lifetime. As it was shown in the previous Section, an interconnect may experience different aging stages during its lifetime. Thus, based on the general failure or aging mechanism, a generic lifetime expression can be stated as (Fig. 1.4):

𝑡 = max 𝑡 + 𝑡 + 𝑡 (4.27)

In order to find 𝑡 , in (4.27), we decompose the expressions to: 𝑡 , 𝑡 and 𝑡 .

Regarding 𝑡 , it can be obtained when the stress reaches the critical value via equating as follows:

𝜎(𝑡) = 𝜎 (4.28)

While a closed form for 𝑡 can be found based on the strongest term in the series in (4.3), classical numerical methods can be applied to solve (4.28) in which mathematical complexity is very straightforward and not challenging. For 𝑡 , depending on the void type and its motion velocity (e.g. (1.14)) and the visibility criteria (e.g. specific length or volume), the time to

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visibility can be obtained. Similarly, the late growth phase, 𝑡 can be obtained when void

is grown up to a critical length, 𝐿𝑐𝑟𝑖𝑡. Thus, having the speed of void presented in (1.14), the time for a void to cause a total failure can be computed.

Korhonen solved (1.20) which is similar to (4.3) without including thermomigration caused by Joule heating for straight line with various boundary conditions [29]. Based on solutions in [29] , authors in [102] using a simplified version of (4.27) where 𝑡 is ignored, propose a time to failure model for slit-like void for a semi-infinite line (i.e. one blocking end and long length) which is as follows: 𝑡 = 𝑡 + 𝑡 =𝜋 4 Ω𝑘𝑇 (𝑒𝑍𝜌𝑗) 𝐵𝐷 𝜎 + 𝑘𝑇 𝑒𝑍𝜌𝑗𝐷 𝐿 (4.29)

Indeed, 𝑡nuc in (4.29) is obtained by solving (4.28) for a semi-infinite line (i.e. one blocking end and long length). Also, 𝑡 is when void is grown up to a critical length, 𝐿_crit. Thus,

having the speed of void presented in (1.14), the time for a void to cause a total failure can be computed.

A more advanced work is done by the author in [103] where again 𝑡nuc is obtained by

solving (4.28) for a finite line (i.e. blocking boundaries).

𝑡 ≈ 𝐿 𝑘𝑇 2𝐷 𝐵𝛺𝑙𝑛 𝑒𝑍𝜌𝑗𝐿 2𝛺 𝜎 +𝑒𝑍𝜌𝑗𝐿_2𝛺 − 𝜎 (4.30)

In (4.30) 𝜎 is the residual stress due to thermal expansion. For the growth phases they employ a progressive model for resistance change based on the length and the speed of void growth, and resistivity of line and liner.

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It should be noted that the condition for saturation must be checked making sure whether a new steady state is achieved. The time needed to reach the saturated volume is as follows:

𝑡 = 𝑡 + 𝐿 𝑘𝑇

2𝐷 𝐵𝛺 1 +

2𝜎 𝛺

𝑒𝑍𝜌𝑗𝐿 (4.31)

As it can be seen, none of these models include thermomigration due to Joule heating as they all borrow Korhonen’s model in their core.

In addition to exclusion of thermal effects in the aforementioned models and similar ones [47, 31], extending them to complex structures and real-world interconnect networks is difficult due to many assumptions made during these models derivations.

To overcome such complexity, we propose a new approach for looking at lifetime. As it was discussed before, the key to determine the depletion or accumulation of atoms in a region is to look at the overall intake and outtake of atoms due the multiple fluxes. This quantity can be measured accurately by atomic flux divergence. Therefore, it can be stated that

MTF ∝ 1

∇ . 𝐽⃗ (4.32)

The individual flux divergences can be mathematically obtained as follows:

𝛻 . 𝐽 ⃗ = 𝛻𝐶 𝐶 + 𝐸 𝑘𝑇− 1 𝛻𝑇 𝑇 + 𝛻 𝑉 𝛻𝑉 . 𝐽 ⃗ (4.33) 𝛻. 𝐽 ⃗ = 𝛻𝐶 𝐶 + 𝐸 𝑘𝑇− 2 𝛻𝑇 𝑇 + 𝛻 𝑇 𝛻𝑇 . 𝐽 ⃗ (4.34) 𝛻. 𝐽 ⃗ = 𝛻𝐶 𝐶 + 𝐸 𝑘𝑇− 1 𝛻𝑇 𝑇 + 𝛻 𝜎 𝛻𝜎 . 𝐽 ⃗ (4.35) 𝛻. 𝐽 ⃗ = 𝐸 𝑘𝑇 𝛻𝑇 𝑇 + 𝛻 𝐶 𝛻𝐶 . 𝐽 ⃗ (4.36)

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Considering EM, TM and SM, the total flux divergence can be written as follows (E /kT ≫ 1,2):: ∇ . 𝐽⃗ ≈ 𝐸 𝑘𝑇 ∇𝑇 𝑇 + ∇ 𝑉 𝛻𝑉 . 𝐽 ⃗ + 𝐸 𝑘𝑇 ∇𝑇 𝑇 + ∇ 𝑇 𝛻𝑇 . 𝐽 ⃗ + 𝐸 𝑘𝑇 ∇𝑇 𝑇 + 𝛻 𝜎 𝛻𝜎 . 𝐽 ⃗ (4.37)

Plugging the fluxes due to EM, TM and SM and rewriting (31) quantitatively from (2) yields:

MTF ∝ 1

𝐷(α𝑗 + 𝛽𝑗 + 𝛾𝑗 )≈ 𝐴𝑗 𝑒 (4.38)

where α, 𝛽 and 𝛾 are determined by the parameters in experiments or equivalently the ones in previous equations. The rightmost-hand side equation is the well-known Black’s equation where 𝐴 is a pre-exponential constant factor and 𝑛 is an empirically found parameter. As it can be seen there is an intuitive agreement between flux divergence with Black’s equation where the current density exponent 𝑛 and constant 𝐴 are traditionally determined empirically. In other words, flux divergence analysis can enrich Black’s empirical equation by enlightening the model parameters. Empirical extractions of 𝑛 also usually set it to be between 2 to 4 – depending on which terms are more dominant [27].

Material Effects: Microstructures and Interfaces: Interconnect aging is a diffusive process and thus very sensitive to diffusion coefficients and constants. In the above calculation diffusivity is assumed to be temperature dependent described as follows:

𝐷 = 𝐷 𝑒 (4.39)

where D is the self-diffusion coefficient of the conductor material and E is the activation energy for diffusion [10]. It should be mentioned that the more accurate model for activation energy depends on hydrostatic stress, vacancy formation and diffusion energy. Nevertheless,

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this dependency is found to be weak and a constant activation energy is widely used by researchers.

Korhonen [29] discusses the effect of different parameters such as stress and temperature on diffusion coefficient while solving (1.14) under various settings. His numerical results turned out to be not too different from the analytic solutions with a constant 𝐷. Similar

observations were made when solving (4.3) in which TM is included. Fig. 4.1 shows the solutions of (4.3) when diffusion depends on the parameters mentioned above versus the case where diffusion coefficient is constant. As it can be seen the solutions for both cases are matching quite well.

Furthermore, the rate of mass transport and void growth in dual damascene process largely depend on lattice microstructure such as grain boundaries and interfaces. The fastest path for atoms to migrate is at the capping layer interface. The effective diffusivity in copper interconnects, 𝐷eff, can be written in terms of diffusions and dimensions of bulk, grain boundaries, liner interface and capping surface as follows:

(a) (b)

Figure 4.1: Stress evolution using (4.3) for the cases where (a) D is costant (b) D is not constant.

100 𝐷 = 𝐷 𝑛 + 𝐷 𝛿 1 𝑑+ 𝐷 𝛿 1 ℎ+ 2 𝑤 + 𝐷 𝛿 1 ℎ (4.40)

where the subscripts correspond to the diffusion pathway, 𝐷b, 𝐷gb, 𝐷 , and 𝐷c are diffusion constants, 𝛿gb, 𝛿l and 𝛿c denote the effective thicknesses, d is the grain size, and 𝑛b is the fraction of atoms diffusing through the bulk. Also, 𝑤 and ℎ are the width and thickness of a

wire [91].

Fig. 4.2. demonstrates how a passive dummy via with no electric current can affect the lifetime of a wire through diffusion. The proposed systematic way of computing lifetime explained above can distinguish between the structures shown in Fig 4.2. The structure shown in Fig 4.2 (b) differs from that in Fig 4.2 (a) in that it has one dummy via above the active wire. Even though it does not carry current, it affects the EM process by slowing down the atom and vacancy fast track at the interface of the copper and capping layers. Note that the effective diffusivity of interconnect is affected and can be modeled using (4.40). A huge slack in the interface by introducing more dummy vias can lead to a very long time to failure and therefore the wire might practically be considered immortal [104]. This is observed in finite element simulation [20] as the steady state for net I is reached 5 times earlier than that for net II (Fig 4.2).

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In addition, impurity or bulk imperfection which usually appear in a form of pre-existing voids in capping interface or dislocations, can be included similarly. These embryo voids are mainly produced by the thermal mismatch between copper and the surrounding dielectric and can significantly affect the electromigration lifetime [105].

In short, to include the microstructural effect of lattice, detailed lifetime analysis may be needed. Fig. 4.3 shows detailed experiments including microstructure effects. In these experiments, different diffusion coefficients are used in different regions within the lattice.

Figure 4.2: Current and stress distribution in two segments. (a) a simple interconnect (b) the same interconnect segment with a dummy via in the middle. The via does not carry current but changes the bulk effective diffusion which prolongs the time to reach steady state. Yet, the effect is not visible in steady state. (J=1mA/um2_).

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Figure 4.3: Stress evolution and void nucleation in bamboo-like interconnect obtained using COMSOL Multiphysics. DCu-SiN_cap > DCu-Ta_liner (left), DCu-SiN_cap = DCu-Ta_liner(right).

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