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### 2 Physics

#### 2.1  Historic Outline of Electromigration

Electromigration was first discovered by the french physicist M. Gerardin in 1861 [104]. In the systematic studies of electromigration were carried out by W. Seith and H. Wever [120] showing the correlation between the direction of the current flow and the material transport. In the electromigration was recognized as one of the main failure phenomena leading to the development of different failure criteria and physically based models [143].

##### 2.1.1  Black’s Equation

While working for Motorola in the , James R. Black was involved in the understanding of the “cracked stripe” problem [95]. This phenomenon was found to be electromigration induced. He carried out a systematic investigation and derived a model for the failure time prediction [10]. The model is based on the concept that a failure criterion is reached, if a structure specific mass is transported away.

is the mass transported per time. This leads to the inverse proportionality of the TTF to the rate of mass transport.

This rate is modeled by the proportionality

with being the conducting electron concentration, the impulse transferred from the electrons to the atoms while scattering, and the density of thermal activated atoms. The first two variables and are proportional to the current density and the density of activated atoms is modeled by the Arrhenius law [137]

with the activation energy , the Boltzmann constant and the temperature . The combination of leads to Black’s equation.

The constant comprises the material properties as well as the geometry and must be empirically determined [11, 12, 13].
The inverse dependence is a special case which can be extended to the generalized Black’s equation

by substituting the exponent 2 by a second parameter called the current exponent.
The correct value for this parameter was extensively debated [77, 93]. According to Clement [26] lifetime models can be roughly classified into two groups. Void growth models, where the failure is triggered by the growth of a void over a critical size and nucleation models, where the failure is triggered by the stress build-up in the structure exceeding a critical value. For the void growth model the current exponent is found to be 1, because the mass transported is proportional to the current resulting in an inverse relation to the TTF [126]. Models based on the nucleation show an exponent of 2 as in the original Black equation due to the stress induced back flow flux [5, 91].

For the prediction of the TTF the generalized Black equation is used to extrapolate the results from accelerated test conditions with increased currents and elevated temperatures to normal operation conditions. In this application the current exponent is a second fitting parameter beside the parameter . Measurements reported in the literature show values for ranging form slightly greater than one to six [107]. Values above two are explained by the improper treatment of the Joule heating. Values in the range between one and two are interpreted as a failure based on a mixture of the two models, where a void is first nucleated and followed by a growth phase.

##### 2.1.2  Blech Effect

Ian A. Blech from the Technion in Israel carried out a study, where he deposited golden islands onto a refractory underlay made out of titanium nitride. In his experiments he stressed the film with high current densities [14]. Due to the much higher resistivity of the underlayer the current mainly passed through the gold in the gold covered regions.
By observing the movement of the islands he discovered a length dependent behavior.
For long islands the edge, where the electrons pass into the gold, moved in the direction of the stripes with the velocity

where is the self diffusion coefficient, the effective valance, the unit charge, the specific electrical resistance, and the current density. At the other side of the island extrusions were formed. For islands short enough the movement of the ends was not observed. For islands in between, the side, where the electrons entered, also moved with but stopped after a certain time, when a critical length was reached. At the other end no extrusion was formed. Blech discovered a critical value for the product of current density and the length of the islands under which electromigration does not occur. This finding leads to the concept of the Blech length as a critical value for a given current density.

The explanation for this phenomenon was found in the fact that different densities of mass in the island lead to a mechanical back stress working against the electromigration force. This compression stress has to be below the critical value of extrusion forming. While the islands are in steady state, the back flux induced by the stress gradient is totally compensating the EM flux. This back flux is proportional to the gradient of the tensile stress. Therefore, the maximum stress divided by the island length is proportional to the back flux. With the fact that the EM flux is proportional to the current the following can be deduced.

By taking a critical stress value into account a critical product follows [125, 142].
As the stress build-up due to electromigration in microelectronic structures is highly depending on the surrounding materials, on the physical design and on the fabrication process, the Blech length or product can hardly be pre-determined. Therefore, this concept was never seriously considered [95].

##### 2.1.3  Korhonen’s Model

The coupling of the stress development with the vacancy dynamics was first introduced by Korhonen et al. [82]. They consider a thin narrow interconnect line deposited on a silicon oxide substrate covered by a dielectric passivation layer of infinite length. At moderate temperatures the electromigration induced flux is mainly flowing along the grain boundaries. Therefore the effective diffusion coefficient for the whole interconnect can be calculated by

where is the grain boundary width, is the grain size, and is the grain boundary diffusion coefficient. However, in certain cases, like bamboo structures, the grain boundaries are nearly perpendicular to the line and the simplification of zero bulk flux from (2.9) is not applicable. A diffusion flux arises due to a difference in the chemical potential in the interconnect, which can be written [29, 57] by

where is the atomic volume and is the tensile stress. For thermal equilibrium of the vacancies must be set to zero. The atomic flux including the electromigration induced flux is then represented by

For the equilibrium concentration of the vacancies in the presence of stress Korhonen et al. [82] derived the equation

which leads to the vacancy flux

and the conservation law for the vacancies

where the added by Clement [26] represents a sink or source term. This model was extensively studied and presented in [26, 27, 28].

Under the assumption of a low vacancy concentration compared to the atomic concentration and climbing grain boundary dislocations consuming the net flux divergence, Korhonen et al. [82] developed a PDE for the build-up stress.

is the applicable mechanical modulus for an aluminium line in a confined silicon matrix ranging form 0.5 to 0.75 times the Young modulus [82].

These equations were generalized for 3D calculations, allowing to simulate 3D structures which are more complex then straight interconnect lines. This formulation was used to investigate structures with grain boundaries explicitly given [118].

##### 2.1.4  Kirchheim’s Fluxes

In the model presented in the previous chapter the stress is only included in the calculation of the equilibrium concentration of the vacancies. This was argued with the stress being in local equilibrium with the vacancy concentration. Kirchheim [76] added an extra driving force due to a gradient of the stress leading to the driving force approach allowing the stress and the concentration being out of equilibrium.

is a relaxation factor. Tan and Roy [139] showed that the typical order of the stress induced flux and the electromigration induced flux are of the same order and therefore the flux driven by the stress gradient can not be neglected. An additional important physically induced flux is the flux due to a temperature gradient. This flux arises from the non-isothermal second Fick law [74].

stands for the heat of transport, which relates the temperature to the chemical energy by [114]

For the calculation of the stress in the structure, the volume change due to the sink/source has to be taken into account by [100]

The state of the art model is entirely based on the physical phenomena and its mathematical description was developed by Korhonen et al. [82] and the extension of Clement’a [28] and Kirchheim’s [74] work.

##### 2.1.5  Towards the State of the Art Model

The equations of Kirchheim are derived for one-dimensional problem sets and are therefore only capable to model straight lines. In these lines the values of the concentration, the stress etc. are assumed to be the same in every cross section. The extension of this equations by Sarychev et al. [116] to three-dimensional problems, while also taking the inelastic strain due to mass displacement into account, allows the simulation of EM in complex interconnect structures. Furthermore, the directional dependence of the diffusion was included by introducing a diffusion coefficient tensor. This constitutes a self-consistent model connecting the transport dynamics of the vacancies and the stress build-up in the conducting materials under EM.

Some extensions are required for interfaces like grain boundaries and interfaces to surrounding isolation or dielectric materials regarding, but not limited to, different diffusion coefficients. For instance, the segregation model of Lau et al. [90] was applied [22] to self diffusion and EM, furthermore, the Fischer model [42] was implemented to include a possible accumulation behavior of grain boundaries [4, 34].

The diffusion coefficients obeying an Arrhenius law were extended by a second dependence on the stress [136]. Also the equilibrium concentration of the vacancies was extended by an Arrhenius law [22, 55]. The full mathematics of the model PDE and the connecting quantities are presented in detail in Chapter 3.

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