Theoretical consideration of electromigration damage around a right-angled corner in a passivated line composed of dissimilar metals

Abstract

Electromigration (EM) evaluation near contact corners in interconnection structures composed of dissimilar materials in particular is becoming increasingly important. A theoretical analysis of the atomic density distribution around a right-angled corner in a passivated line composed of dissimilar metals is performed. The 2D structure considered, including the corner, is assumed to consist of dissimilar single-crystal metals with uniform passivation and line widths. While the atomic density distribution in the passivated 1D straight line has been reported previously, the atomic density distribution in a passivated 2D structure that contains a right-angled corner composed of two dissimilar metals has not been studied. This work clarifies the atomic density distribution in the 2D structure under equilibrium conditions for the two fluxes, which are the atomic flux due to the electron wind force and the flux due to the back flow caused by the atomic density gradient in EM, and presents the predicted locations of EM damage in the form of voids and hillocks. In addition, this paper proposes countermeasures to increase the threshold current density for EM, which contributes to enhancement of the reliability of the metal line against EM. The selection of suitable values for several material properties are discussed from the viewpoint of preventing initiation of EM damage by increasing the EM threshold current density. Suitable material combinations to increase the threshold current density are also discussed.

Appendices

Appendix A

The details of the solution procedure for obtaining Eqs. (8–11) are given herein. By referring Eq. (2), we have the following equations in the radial direction:

$$\frac{{\partial N_{1} }}{\partial r} = - \frac{{N_{01} |Z_{1}^{*} |e\rho_{1} }}{{\kappa_{1} \Omega_{1} }}j_{\rm r1} ,$$

(12)

and

$$\frac{{\partial N_{2} }}{\partial r} = - \frac{{N_{02} |Z_{2}^{*} |e\rho_{2} }}{{\kappa_{2} \Omega_{2} }}j_{\rm r2} .$$

(13)

By substituting Eqs. (4) and (6) into Eqs. (12) and (13), respectively, and integrating Eqs. (12) and (13) with respect to r, N ₁ and N ₂ are obtained for each of the materials as expressed in Eqs. (8–11).

Similarly, if we consider the circumferential direction, we have the following equations:

$$\frac{1}{r}\frac{{\partial N_{1} }}{\partial \theta } = - \frac{{N_{01} |Z_{1}^{*} |e\rho_{1} }}{{\kappa_{1} \Omega_{1} }}j_{\theta 1} ,$$

(14)

and

$$\frac{1}{r}\frac{{\partial N_{2} }}{\partial \theta } = - \frac{{N_{02} |Z_{2}^{*} |e\rho_{2} }}{{\kappa_{2} \Omega_{2} }}j_{\theta 2}$$

(15)

By substituting Eqs. (5) and (7) into Eqs. (14) and (15), respectively, and integrating Eqs. (14) and (15) with respect to θ, N ₁ and N ₂ are obtained for each of the materials as expressed in Eqs. (8–11), which are same as the case with Eqs. (12) and (13).

Appendix B

In this work, the distributions of N in the two materials are represented as shown in Figs. 4 and 5. The treated structure composed of dissimilar metals is geometrically asymmetric about the interface, with θ ₁ = π/2 and θ ₂ = 3π/2, as shown in Fig. 3. If the structure that is geometrically symmetric about the interface with θ ₁ = 3π/4 and θ ₂ = 3π/2, as shown in Fig. 6, is considered, the singularity parameter is geometrically determined to be ξ = 2/3 without dependence on ρ ₁/ρ ₂ (Saka and Zhao 2012). The atomic densities N ₁ and N ₂ at the interface (θ ₁ = 3π/4) in the geometrically symmetric structure are independent of r and are constant.

Fig. 6

Right-angled metal line that is geometrically symmetric about the interface and is composed of dissimilar metals with θ ₁ = 3π/4 and θ ₂ = 3π/2