5.3.1 Geometries and Model Parameters

The geometries used in the comparative study are shown in (5.16). The reference interconnect structure is a simple straight copper line without a via ((5.16(a))). The second geometry is proposed by adding a 90° angle in the line ((5.16(b))). In both interconnect structures, the network of grain boundaries is indicated. By assuming the interconnect width of 80nm, the grain boundaries can be assumed to be distributed in a near bamboo formation, as it could be expected for copper interconnects with a grain size comparable to line width (Section 1.4.5). Furthermore, the triple point, generated from the intersection between two grains and the material interfaces, becomes important when the location of void nucleation in the structure must be found. This is illustrated in the zoomed-in detail view of (5.16(a)). Material interfaces and grain boundaries of the structure have to be supplied with an appropriately fine mesh. This is necessary to provide sufficient accuracy for the results along the triple point regions.

Figure 5.16: Schematic view of the (a) linear and (b) L-shaped geometry used in this analysis. The zoomed-in detail view of a triple point depicts the mesh density employed in the numerical calculations. The arrows show the direction of the current density. The red circles S₁ and S₂ represent the spots where peak values of vacancy concentration and stress are extracted.

Since the scope of this work is to investigate the impact of current crowding and microstructure during the void nucleation mechanism induced by electromigration in the described geometries, the simulation procedure of void nucleation described in Section 4.3.2 is applied to study the electromigration failure development in these structures. In particular, the models derived in Section 3.2.1 have been used to describe the vacancy dynamics in the presence of grain boundaries and material interfaces. The derivation of these models is based on the segregation model [101] and the approach applied in this work consists of the expression for the vacancy annihilation/recombination term, as presented in equations (3.41) and (3.43) [25]. The models parameters for electro-thermal and solid mechanics models are the same of those presented for copper in table 5.1, while the parameters for the vacancy dynamics model are summarized in table 5.4. Electromigration simulations are performed at a temperature T of 573K and current density j of 4MA/cm².

Table 5.4: Materials parameters for the vacancy dynamics model [40, 25].

Model	Parameter	Cu
Vacancy Dynamics	Dv,0b [cm2 s-1]	0.52
	Dv,0gb [cm2 s-1]	0.52 102
	Dv,0i [cm2 s-1]	0.52 103
	Eab [eV]	0.89
	Eagb [eV]	0.7
	Eai [eV]	0.5
	Z*	-5
	Q* [J]	1.2 10-20
	f	0.4
	Ωa [cm3]	1.18 10-23
	Cv,0 [cm-3]	1 1016
	τvb [s]	1
	τvgb [s]	1
	τvi [s]	1
	ωr [m4 s-1]	5 10-33
	ωt [m s-1]	5 10-11
	δ [m-9]	1