Electromigration

Electromigration is the transport of material caused by the gradual movement of the ions in a conductor due to the momentum transfer between conducting electrons and diffusing metal atoms. The effect is important in applications where high direct current densities are used, such as in microelectronics and related structures. As the structure size in electronics such as integrated circuits (ICs) decreases, the practical significance of this effect increases.

Electromigration is the transport of material caused by the gradual movement of the ions in a conductor due to the momentum transfer between conducting electrons and diffusing metal atoms. The effect is important in applications where high direct current densities are used, such as in microelectronics and related structures. As the structure size in electronics such as integrated circuits (ICs) decreases, the practical significance of this effect increases. F_{res}=F_e-F_p=q\cdot (Z_e-Z_p) \cdot E = q\cdot Z^*\cdot E=q\cdot Z^*\cdot j\cdot \rho where q is the electric charge of the ions, Z_e and Z_p the valences corresponding to the electrostatic and wind force respectively, Z^* the so-called effective valence of the material, j the current density, and \rho the resistivity of the material. Electromigration occurs when some of the momentum of a moving electron is transferred to a nearby activated ion. This causes the ion to move from its original position. Over time this force knocks a significant number of atoms far from their original positions. A break or gap can develop in the conducting material, preventing the flow of electricity. In narrow interconnect conductors, such as those linking transistors and other components in integrated circuits, this is known as a void or internal failure (open circuit). Electromigration can also cause the atoms of a conductor to pile up and drift toward other nearby conductors, creating an unintended electrical connection known as a hillock failure or whisker failure (short circuit). Both of these situations can lead to a malfunction of the circuit. where N is the number of steps in the bunch, and the proportionality coefficient has the dimension of length. This scaling law has been confirmed by numerous experimental studies. In 2018, Toktarbaiuly et al. reported electromigration-induced step bunching on vicinal W(110) surfaces. Their study revealed that step bunching occurred for both step-up and step-down current directions at the same temperature, T = 1500°C, with distinct size-scaling exponents depending on the current direction. More recently, Usov et al. (2020) demonstrated that electromigration-induced step bunching is not limited to silicon surfaces but can also occur on dielectric surfaces, such as sapphire (Al₂O₃(0001)). This study suggests that the fundamental mechanism of step bunching on W(110), Al₂O₃(0001), and Si(111) follows similar principles. Moreover, annealing W(110) offcut in the [001] direction with an up-step current produced a morphology where the bunch edges formed zigzag segments meeting at right angles. \frac{\partial N}{\partial t} + \nabla\cdot\vec J = 0 where N(\vec x, t) is the atom concentration at the point with a coordinates \vec x=(x, y, z) at the moment of time t , and J is the total atomic flux at this location. The total atomic flux J is a combination of the fluxes caused by the different atom migration forces. The major forces are induced by the electric current, and by the gradients of temperature, mechanical stress and concentration. \vec J = \vec J_c + \vec J_T + \vec J_\sigma + \vec J_N . To define the fluxes mentioned above: \vec J_c = \frac{NeZD\rho}{kT}\,\vec j . Here e is the electron charge, eZ is the effective charge of the migrating atom, \rho the resistivity of the conductor where atom migration takes place, \vec j is the local current density, k is the Boltzmann constant, T is the absolute temperature. D(\vec x, t) is the time and position dependent atom diffusivity. \vec J_T = -\frac{NDQ}{kT^2}\nabla T .  We use Q the heat of thermal diffusion. \vec J_\sigma = \frac{ND\Omega}{kT}\nabla\! H , here \Omega=1/N_0 is the atomic volume and N_0 is initial atomic concentration, H=(\sigma_{11}+\sigma_{22}+\sigma_{33})/3 is the hydrostatic stress and \sigma_{11},\sigma_{22},\sigma_{33} are the components of principal stress. \vec J_N = -D\,\nabla\! N . Assuming a vacancy mechanism for atom diffusion we can express D as a function of the hydrostatic stress D = D_0\exp\left(\tfrac{\Omega H - E_A}{kT}\right) where E_A is the effective activation energy of the thermal diffusion of metal atoms. The vacancy concentration represents availability of empty lattice sites, which might be occupied by a migrating atom. ==Electromigration-aware design== ===Electromigration reliability of a wire (Black's equation)=== At the end of the 1960s J. R. Black developed an empirical model to estimate the MTTF (mean time to failure) of a wire, taking electromigration into consideration. Since then, the formula has gained popularity in the semiconductor industry: \text{MTTF} = \frac{A}{J^n} \exp{\left(\frac{E_\text{a}}{k T}\right)} . Here A is a constant based on the cross-sectional area of the interconnect, J is the current density, E_\text{a} is the activation energy (e.g. 0.7 eV for grain boundary diffusion in aluminum), k is the Boltzmann constant, T is the temperature in kelvins, and n a scaling factor (usually set to 2 according to Black). The temperature of the conductor appears in the exponent, i.e. it strongly affects the MTTF of the interconnect. For an interconnect of a given construction to remain reliable as the temperature rises, the current density within the conductor must be reduced. However, as interconnect technology advances at the nanometer scale, the validity of Black's equation becomes increasingly questionable. Electromigrated nanogaps are gaps formed in metallic bridges formed by the process of electromigration. A nanosized contact formed by electromigration acts like a waveguide for electrons. The nanocontact essentially acts like a one-dimensional wire with a conductance of G = 2\,e^2\!/h . The current in a wire is the velocity of the electrons multiplied by the charge and number per unit length, \,I = veN/L\ or \ G=veN/LV . This gives a conductance of G=ve^2\!N/LE . In nanoscale bridges the conductance falls in discrete steps of multiples of the quantum conductance G = 2\,e^2\!/h . Electromigrated nanogaps have been proposed for use as electrodes in molecular scale electronics and as quantum tunneling sensors. Researchers have used feedback controlled electromigration to investigate the magnetoresistance of a quantum spin valve. ==See also== Kirkendall effect Sealing current ==References== Semiconductor electromigration in-depth at DWPG.Com Modeling of electromigration process with void formation at UniPro R&D site DoITPoMS Teaching and Learning Package- "Electromigration"

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