Electromigration has been studied extensively for simple interconnect segments with blocking boundary conditions at both ends. Stress evolution, void nucleation and failure conditions are well established for such structures. The compact model for simple structures is known as Blech length or Blech effect [20, 25]. In this work we consider copper (Cu) interconnects manufactured in dual damascene technology. We say that a connection is simple if it consists of a straight segment of Cu wire terminating at both ends at a via or a contact. An interconnect tree, also referred to as a net or a complex interconnect, consists of a Cu structure within one layer of metallization which terminates at diffusion barriers such as vias or contacts. The diffusion barriers allow electrons to pass through but block the movement of atoms. The finite element simulations are performed for direct current (DC) in bamboo-like dual damascene Cu interconnects with Ta liner, and SiNx capping embedded in a low-k dielectric.
Authors in [36] extend the Blech criterion (𝑗𝐿 ) from one segment to a multi-segment
tree. They showed that the maximum stress difference in an interconnect tree, Δ𝜎 , is given by the path with the greatest sum of segment 𝑗𝐿 products:
Δ𝜎 = 𝜌𝑒𝑍
∗
Ω (𝑗𝐿) (2.1)
37
(𝑗𝐿) = max
,
𝑗 𝐿 (2.2)
The effective 𝑗𝐿 product (𝑗𝐿 ) is then calculated and compared to the 𝑗𝐿 . Using their
method the failure due to the void saturation can be checked by comparing 𝑗𝐿 to 𝑗𝐿 . In
this method, Δ𝜎 is assumed to be equal to 2𝜎 where 𝜎 is the critical tensile stress for
a void to nucleate. This assumption can be valid for symmetric structures. However, in multi-
branch nets with arbitrary topology, 𝑗𝐿 may reach Δ𝜎 but void nucleation may not occur
because the stress at the cathode may be less than 𝜎 . Such situations occur for critical compressive stress for low-k dielectric breakage or extrusion. Although this model is very powerful, it does not capture the effect of segments adjacent to the path with the highest sum of the 𝑗𝐿 products, yet not belonging to it. In other words, while the max 𝑗𝐿 product remains the same, the entire stress distribution might be affected by the segments not on the path with the maximum sum of the 𝑗𝐿 products. For instance, consider two nets shown in Fig. 2.1; the only difference between these nets is the connectivity at node 𝑏 (or 𝑏 ′). In Fig 2.1 net I has a branch 𝑎 𝑎 which is connected to the net on the same layer. On the other hand, net II has an alternative configuration in which the segment 𝑎 ′𝑎 ′ is connected through a via below at 𝑏 ′. Based on the method developed in [36] and continued in [37], both configurations, have the
same 𝑗𝐿 = 𝑗𝐿 (the path with greatest summation of 𝑗𝐿 is marked by the broken line arrow).
This is, both structures should experience the same EM conditions. But the stress distributions on net I and net II are different. In other words, even though (𝑗𝐿) are the same in these structures, due to the other segments configuration (i.e. 𝑎 𝑎 in net I and 𝑎 ′𝑎 ′ in net II), the overall stress distribution is different.
38
Figure 2.1: A multi-segment net. 𝑗𝐿 = maxΣ𝑗𝐿 is the same when segment 𝑎 ′𝑎 ′ is connected
to 𝑏 ′ as a side branch directly in the same layer or when a1a2 is connected through a via to the lower layer. Red and green dots indicate a via above and via below, respectively. Hydrostatic stress along the broken line near the interface is shown at the bottom (in all figures arrows show electron flow j= 10 mA/um ).
39
Fig. 2.1 shows the hydrostatic stress along the broken line (i.e. the path with greatest Σ𝑗𝐿). In this particular example, analyzing stresses in different nodes and juxtaposing them with the corresponding nodes in the other structure demonstrates this disparity. The number of atoms accumulated in the anodes 𝑎 and 𝑎 of net I is greater than in the anodes 𝑎 ′ and 𝑎 ′of net II, since the only places for accumulation of atoms migrating from the cathode ends are 𝑎 and 𝑎 . On the other hand, in net II, atoms can be accumulated in 𝑎 ′,𝑎 ′, 𝑎 ′ and 𝑎 ′ . Therefore, the compressive stresses in nodes 𝑎 and 𝑎 are less than those of node 𝑎 ′ and 𝑎 ′ . Similarly, node 𝑏 is experiencing a large atom flux divergence since the segment 𝑏 𝑐 supplies more
atoms than what is taken by segments 𝑏 𝑎 and 𝑏 𝑎 . However, in net II, 𝑏 is experiencing
less atomic flux divergence since 𝑏 ′𝑎 ′ and 𝑏 ′𝑎 ′ also contribute in atom accumulation (i.e. atom suction). This disparity in stress is caused by the fact that the via located in 𝑏 in net I only supplies electrons but no atoms. Yet, node 𝑏 ′ is supplied with atoms by the wire segments 𝑏 ′𝑎 ′ and 𝑏 ′𝑎 ′. Likewise, cathode 𝑐 ′ of net II experiences less back stress (since atoms can distribute among more branches) and therefore supplies more atoms (i.e. more atoms are depleted) compared to cathode 𝑐 in net I. Thus, the tensile stress in cathode 𝑐 is less than that of cathode 𝑐 ′. These differences may result in different failure mechanisms. One structure may even be immortal and the other one may be mortal.
Another effort to model immortality in complex interconnect is presented in [38]. The authors propose a method to calculate effective current density based on atomic flux divergence at each via node. This method states that the effective atomic flux can be expressed as 𝑗 = (𝐹 𝐹 𝐹 )𝑗 where 𝐹 , 𝐹 and 𝐹 model the non-electrical effects such as length, width and wire segment interaction on the atomic flux of a lead. Then, effective current density divergence at each node is calculated by 𝑗 . = Σ𝑗 . The limitation of this model is that it
40
only explains the effect of the immediately adjacent segments and does not include the effect of other segments in the net, passive extensions, or their material properties.
We believe that the method presented in this work has broader and more accurate applications. It not only includes the non-electrical and interaction effects of adjacent segments but also explains the non-electrical effects of all connected segments as well as the effects of passive wire extensions.