2.2 Electromigration Driving Force

In a simple picture, electromigration can be described as the atomic migration caused by the contributions of two microscopic forces on mobile defects (2.1). The microscopic forces arise from the influence of the local electric field and the resulting electron transport in the conductor.

Figure 2.1: The two contributions of the electromigration driving force.

The electric field force F_direct, often referred to as the "direct force", is caused by the direct action of the external macroscopic field E on the charge of the migrating metal ions [142]. The electrostatic force is the result of the interaction between ionic atoms, with valence Z, and the applied electric field, because the valence electrons responsible for electrical conductivity are no longer bound to the atomic nuclei. Furthermore, it takes into account the electrostatic screening s_e from the surrounding electrons [87]. The electrostatic force is given by

\[\begin{equation} \vec{F}_\text{direct}= Z(1-s_\text{e}) |e|\vec{E}=Z^*_\text{direct} |e|\vec{E}, \end{equation}\] (2.12)

where F^*_direct is the direct valence related to the nominal valence of the metallic ion when shielding processes are absent.

The second microscopic force responsible for atomic transport is a consequence of the "electron wind", described in Section 2.1. The semi-classical ballistic model of scattering, proposed independently by Fiks [55] and Huntington and Grone [82], is useful in describing the interaction between conducting electrons, carrying the electric current, and the migrating ions. The conducting electrons, driven by the external field, are scattered by point defects, and the resulting momentum transfer per unit time from scattering electrons to the impurities leads to the so-called "wind force" F_wind as

\[\begin{equation} \vec{F}_\text{wind}= \cfrac{n_\text{e}\rho_\text{d}m_\text{0}}{n_\text{d}\rho m^*}|e|\vec{E}=Z^*_\text{wind} |e|\vec{E}, \end{equation}\] (2.13)

where n_e is the conduction electrons density, ρ is the resistivity of the metal, n_d is the defects density, ρ_d is the defects resistivity contribution, m₀ is the mass of the free electrons, and m_* is the effective electron mass. The wind valence Z^*_wind is related to the magnitude and the direction of the momentum exchange between conducting electrons and point defects. Quantum-mechanical approaches of the electron wind effect were provided by Bosvieux and Friedel [18], Kummar and Sorbello [97], Sham [137], and Schaich [134].

By summing equations (2.12) and (2.13), the total electromigration driving force F_EM acting on a metal ion can be written as

\[\begin{equation} \vec{F}_\text{em}=\vec{F}_\text{direct}+\vec{F}_\text{wind} =(Z^*_\text{wind}+Z^*_\text{wind})|e|\vec{E}=Z^*|e|\vec{E}, \end{equation}\] (2.14)

where Z_* is referred to as the effective valence (or effective charge number) and represents a parameter that comprises the quantum-mechanical effects of the electromigration phenomenon. It describes the ion-electron interaction which can be both theoretically calculated as well as experimentally measured [40]. The sign of the effective valence determines the nature of the transport mechanism. A negative value indicates atomic transport in the direction of the electron flow, i.e. direction opposite to the current flow. Furthermore, the impact of the contribution of the two aforementioned forces in electromigration can be determined from the variation of the effective valence as a function of temperature [149].