Electromigration

Electromigration (EM) is the phenomenon of mass transportation due to momentum transfer from high current density. Such a mechanism can result in open or short circuit modes of failure. The mechanism also impacts both the design and manufacturing of metallization. For electromigration in a metal, the driving force of the net atomic flux consists of two forces: (1) the direct action of electrostatic field on the diffusion atom, electrostatic force, and (2) the momentum exchange between the moving electrons and the ionic atoms on the diffusion atom, electron wind force. It

can be expressed as

( ) (1-1)

where is the effective charge number, e is the electron charge, and E is the electric field. The effective charge is consisted of two terms, and . is positive and can be regarded as the nominal valence of the diffusion ion in the metal

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when the dynamic screening effect is ignored. When these positively charged metal ions are under the field effect, this so called ―direct force‖ draws atoms towards the

negative electrode. In the contrary, the wind force, is usually negative and represents the momentum effect from electron flow that pushes atoms towards the positive electrode. Generally, the electron wind force dominates and is found to be on the order of 10 for a good conductor like Ag, Al, Cu, Pb, Sn, etc [16]. can also be positive, but it was found only in transition elements with complex band structures where electron hole conduction plays a more important role [17]. The atomic flux is related to the electric field and thus the current density. The flux equation can then be

expressed as the following:

𝐽 𝐽 𝐽 (1-2) (1-3)

where C is atomic concentration, D is atomic diffusivity, k is Boltzmann’s constant, and T is temperature. is resistivity and j is current density. The flux is a function of temperature. As shown in the equation below, the atomic diffusivity is exponentially

dependent on temperature.

( ) (1-4)

where is diffusion coefficient, R is gas constant, and Q is activation energy of diffusion.

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For electromigration to occur, a nonvanishing divergence of atomic flux is a requirement. The divergence may be due to a temperature gradient or microstructural inhomogeneity. Since electromigration is cumulative, it affects the failure rate.

Therefore, the mean time to failure (MTTF) in the presence of electromigration is

1.3.1 Failure sites and flux divergence

Electromigration was first observed in Al metal interconnects. Less than 0.2%

Cu atoms were added to Al line to reduce the effect of electromigration [16]. Blech first developed a structure of a short Al or Cu strip in a base line of TiN to conduct electromigration tests as shown in Figure 1-2(a) [18,19]. Because Al or Cu has lower resistance, as electric field was applied on two ends of TiN line, electric current in TiN took a detour to go along the strip of Al or Cu. After electromigration test, a depleted region occurs at the cathode and an extrusion is observed at the anode.

Figure 1-2(b) is the scanning electron microscope (SEM) image of the morphology of

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a Cu strip tested for 99 h at 350 °C with a current density of 5 10⁵ A/cm². By

conservation of mass, both depletion and extrusion should have the same volume change. We can then calculate the drift velocity from the rate of depletion. In the past years, an impetus to study electromigration in very fine conductors has arisen from the development of very large-scale integrated circuits. The conductors are not only interested in small dimensions; they are often assembled into multilayered structure with a certain combination of conductors and insulators. This gives rise to electromigration problems which is distinctly different from simple single-level conductor line.

The metal layer is a two-dimensional (2D) conductor film that can be considered as an ensemble of grain boundaries and their intersections as illustrated in Figure 1-3.

Experimental observations have indicated that in most cases, mass depletion and accumulation initiate at grain boundary intersection such as triple junctions. The former would eventually lead to the formation of voids or cracks and the latter to hillocks or whiskers. The reason that the grain boundary intersections are likely to be the failure sites is that they often represent the spots where the mass flux would diverge the most. At the grain boundary intersections, there could be an abrupt change in grain size, which produces a change in the number of paths for mass movement;

there also could be a change in atomic diffusivity due to the change in grain boundary

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microstructure.

Recently damascene structure have been developed to form Cu interconnect. Cu material is used to replace Al due to its higher electrical conduction. Because Cu has higher melting temperature, its diffusion mechanism is surface diffusion instead of grain boundary diffusion [20]. As for solder joints, because it has lower melting temperature, the diffusion mechanism is lattice diffusion for most solders at a typical operation temperature of an electronic device around 100 °C. Table 1-1 lists the non-uniformity of the current density over a conductor line increases. Since the Joule heating is proportional to the square of the current density, the local temperature will also increase rapidly. The current crowding effect therefore plays dual roles here: both the elevated local density and temperature accelerate the electromigration process.

Thus obtaining an accurate current crowding density distribution is necessary in determining the flux divergence [20].

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Current crowding phenomenon is an even more serious issue in flip-chip solder joints. Figure 1-4(a) demonstrates the unique line to bump geometry of a flip-chip solder bump joining an interconnect line on the chip side (top) and a conducting trace on the substrate side (bottom) [17,21,22,23]. Because the cross section of the line on the chip side is about two orders of magnitude smaller than that of a solder joint, the current density changes significantly from the metal line to the solder as current enters the solder joint. The change leads to the current crowding at the entrance into the solder bump, thus resulting in the change of magnitude of current density from 10⁵ A/cm² to 10⁴ A/cm² at the current crowding region in a typical Al line to solder bump structure. Figure 1-4(b) is a 2D simulation of current distribution in a solder joint.

Note that this current crowding phenomenon leads to non-uniform current distribution inside a solder joint. The current density at the current crowding region is one order of magnitude higher than the average current density at the center of the solder joint.

Since the drift velocity is proportional to the current density and non-uniform temperature distribution inside a solder joint due to local Joule heating effect (see Section 1.3.3), electromigration-induced damage occurs near the contact between the on-chip line and the bump; voids formation for the bumps with electrons downward and hillock or whisker for the bumps with electrons upward. Therefore, current crowding effect plays a crucial role in the flip-chip solder joints under

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electromigration. Consequently, electromigration damage occurs near the contact between the line and the bump; voids induced from the damage can propagate along the interface due to the non-uniform current distribution [24].

In those flip-chip solder joints using a thin film UBM, the current crowding leads to a pancake-type void across the entire cathode contact [25,26]. Figure 1-5(a) time increases, pancake-type voids propagate across the top of solder joints, resulting in open failure.

1.3.3 Joule Heating Effect

When the current flow passes through a conductor, the heat generated due to the electrons vibrating the atoms in the conductor. This is so called Joule heating effect.

The heating power can be describe as:

𝑃 𝐼²𝑅 ² 𝑉 (1-6)

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where P is heating power, I is applied current, R is resistance of the conductor, j is current density, is resistivity of the conductor, and V is volume of the conductor.

Thus, the heating is controlled by two factors: first is the applied current, and the other is the resistance of the conductor.

When the current applied to a flip-chip solder joint is relatively high, the whole system will generate a huge amount of heat resulting from the conducting traces and solder bumps. Generally, the total length of the Al trace can reach few meters, which the effective resistance is approximately few ohms. In contrast, the resistances of the solder bumps and the Cu trace on the substrate are relatively low, typically in the order of few or tens of milliohms. Therefore, the major contribution for Joule heating in flip-chip solder joints comes from the Al trace [27-29]. As a result, the temperature in the bumps during accelerated testing is likely to be much higher than that of the ambient because of the Joule heating. Moreover, the current crowding effect will cause the local high current density; in other words, there will be a local Joule heating

in the solder joints and result in a non-unformed temperature distribution. Chiu et al reported the ―hot spot‖ exists inside the solder bumps at the current crowding region

[29,30]. The combination of the Joule heating of Al interconnects on the chip side and the non-uniform current distribution will lead to a temperature gradient across the solder joints. Consequently, Joule heating effect induced temperature increase in the

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flip-chip solder joints under electromigration significantly affects the analysis of failure time.

1.3.4 Mean time to failure

Electromigration requires a nonvanishing divergence of atomic flux. Since electromigration is cumulative, it affects the failure rate. In statistical study, the test samples should be stressed at the same current and temperature conditions. Then, the failure times or lifetimes can be recorded and plot by Weibull or normal distribution.

In Weibull distribution, the time of 63.2% of reliability is denoted as the mean-time-to-failure (MTTF) [31]. In 1969, James R. Black explained the MTTF in

the presence of electromigration is given by the equation [32]:

( ) (1-5)

where A is a constant, j is the current density, n is a model parameter, Q is the activation energy, k is the Boltzmann’s constant, and T is the average temperature.

There are four of parameters: j, n, Q, and T needed to be examined and analyzed.

However, the current crowding effect and the Joule heating effect play important roles under electromigration in flip-chip solder joints. To further consider about these two

effects in MTTF analysis, the modified Black’s equation becomes:

( ) * _{( +∆ )}+ (1-7)