2.1.1 1960s: Black's Equation

In 1966 Motorola, IBM, Texas Instruments, and Fairchild independently observed "cracked stripe" problems in aluminium interconnects in ICs. At this time, electromigration was recognized to be one of the main reasons for IC failure which threatened the microelectronics industry. While working for Motorola, J. R. Black was involved in the investigation of the origin of electromigration failure in aluminum conductors. Based on the work of Huntington and Grone [82], he derived a simple semi-empirical model for the time to failure prediction of aluminum film conductors stressed at high current densities and temperatures [12,13,14]. He assumed an entirely random motion of the electrons, both before and after collisions with ions under influence of the electric field. The electrons are accelerated and impart all of their full momentum to the ions during nearly elastic collisions. The rate of mass transport R_m by a momentum exchange between electrons and thermally activated ions may be expressed as

\[ \begin{equation} R_\text{m} = F P N_\text{e} N_\text{a} \Sigma, \end{equation}\] (2.1)

where F is a constant, P is the electron momentum, N_e is the number of electrons passing through a unit volume of metal per second, N_a is the number of activated ions available per cm³, and Σ is the ionic scattering cross section. The momentum picked up by a conducting electron when falling through an electric field E over its mean free path p_f, with an average velocity v, is

\[\begin{equation} P = eE\cfrac{p_\text{f}}{v}=e\rho j \cfrac{p_\text{f}}{v}, \end{equation}\] (2.2)

where e is the electron charge, ρ is the volume resistivity of the metal, and j is the current density. The number of electrons per seconds available for striking the activated ions is related to j by

\[\begin{equation} N_\text{e} = n_\text{e} v_\text{d}=\cfrac{j}{e}, \end{equation}\] (2.3)

where n_e is the electrons density, and v_d is the drift velocity which perturbs slightly the average velocity v. The equation for the number of activated metal ions which overcome a potential barrier E_a in the metal lattice follows the Arrhenius law

\[\begin{equation} N_\text{a} = F_\text{1} \ \text{exp}\left(-\cfrac{E_\text{a}}{k_\text{B}T}\right), \end{equation}\] (2.4)

where F₁ contains the diffusion constants, E_a is the activation energy, k_B is the Boltzmann constant, and T is the film temperature. Since the MTTF of a metal conductor is related to the rate of mass transfer and the conducting cross section by

\[\begin{equation} R_\text{m} = \cfrac{F_\text{2}wt}{MTTF}, \end{equation}\] (2.5)

where F₂ is a constant, w is the conductor width, and t is the film thickness, equation (2.1) may be rewritten by substituting equations (2.2)-(2.5). By consolidation of the constants, the Black's equation is obtained as follows

\[\begin{equation} \cfrac{1}{MTTF} =Aj^{2} \ \text{exp}\left(-\cfrac{E_\text{a}}{k_\text{B}T}\right). \end{equation}\] (2.6)

The constant A contains several physical properties such as the metal resistivity, the electron free path, the average velocity, the effective ionic scattering cross section for electrons, and the metal self-diffusion jump frequency factor. Based upon the above theory, experiments were designed and carried out to confirm that the failure rate of a group of identical aluminum stripes, which were exposed to the same electromigration stressing, is proportional to the inverse square of the current density. Furthermore, Black's equation can be an important tool for the design and the fabrication of highly reliable metal conductors for specific stress conditions of temperature and current density, as described in Chapter 1.