5.1 Electromigration in a Nutshell

Electromigration is the transport of material caused by the gradual movement of the atoms in a conductor due to the momentum transfer between conducting electrons and diffusing metal atoms. All work in this field was pioneered by James R. Black [85], who established the basis. Accordingly, Black's semi-empirical equation, developed at the end of 1960s, is given by:

$$MTF=\frac{A}{j^{n}}\exp{}\Big( \frac{E_{a}}{\mathrm{k_B}T} \Big).$$ (5.1)

It describes the median time to failure (MTF) caused by electromigration, where $A$ is a pre-exponential constant based on the cross-sectional area of the interconnect, $E_{a}$ is the activation energy, $\mathrm{k_B}{}$ is Boltzmann's constant, $T$ is the temperature, $j$ is the current density, and $n$ is the so-called current density exponent. The problem with this formula is, that the activation energy $E_{a}$ and the current density exponent $n$ have to be determined experimentally and therefore its validity is limited to specific test configurations. In order to understand the phenomenon of electromigration, a more physical approach based on the main diffusion scenes within interconnect structures must be applied.

Figure 5.1: Figure 5.1(a) gives a schematic overview of mass transport of metal atoms along different diffusion paths in a typical Cu interconnect line. Figure 5.1(b) shows an failure of a copper conductive strip due to electromigration, viewed with a scanning electron microscope (SEM). The passivation layer was removed using reactive ion etching and hydrofluoric acid [86].

The mechanism of electromigration can be explained by the interaction of two counteracting forces:

• activated, positively charged metal ions suffer a force to the cathode direction,
• the electrons, while moving to the anode, transmit an impulse to the metal ions.

Because of a shielding effect of the conduction electrons on metal ions, generally the first force is small. This means, in a metal lattice a mass flux of ions exists, which is directed parallel to the so-called electron wind. The mass flow takes place in the form of material transport along interfaces, as grain boundaries and surfaces, and by volume diffusion. In Cu interconnects, grain boundary and interface diffusion are the dominating transport mechanisms at normal temperature operating conditions [87].

A schematic overview of different diffusion paths is shown in Figure 5.1(a). Local mass flux divergences cause the formation, growth, and movement of voids and hillocks. Especially the movement of electromigration induced voids is very challenging to model, since the simulation domain and, therefore, also the spatial discretization changes its structure after every time step. Voids can grow and shrink, they can move within Cu lines, aggregate, and split. These changes have to be considered during a time dependent electromigration simulation where primarily the interface between a void and the surrounding environment has to be tracked carefully.

The following is mostly focused on modeling the movement of voids within an arbitrary interconnect structure and the development of a powerful mesh adaptation technique which allows to use a diffuse interface model for the description of the metal void interface.