13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Thus, considering coming and going of atoms within the microstructure unit, the atomic flux divergence at the line end, AFD* gen|end is expressed by Eq. (4). where Dx=Z *e ρj x- κΩ/N0(∂N/∂x), Dy=Z *e ρj y- κΩ/N0(∂N/∂y). AFD* gen|end expresses the amount of flux divergence at line end and represents the number of atoms decreasing per unit volume and unit time. Using the governing parameter of EM damage, numerical simulation of atomic density distribution in interconnect is performed under some kinds of input current density, j, at a certain substrate temperature, Ts. The line to be evaluated is two-dimensionally divided into elements and building up process of atomic density distribution is simulated by changing the atomic density of each elements based on the parameter. The boundary condition with respect to temperature is given on both line ends and that with respect to current density is given on via position. Atomic flow is insulated around the metal line. The end-parameter AFD* gen|end is used in elements at cathode and anode ends and on via and AFD* gen is used for elements except both line ends. The computational procedure is shown in Fig. 3. At first, the distributions of current density and temperature are calculated by two-dimensional FE analysis. The governing parameters are calculated in each element from the analysis results and the film characteristics. Next, the atomic density related to θ, N*, is calculated based on the value of the governing parameter. The atomic density in each element N is calculated by averaging N* among all θ’s value. By the repetitive calculation, the atomic density distribution in the line grows with time. The iteration is performed until the atomic density reaches a critical atomic density for damage initiation N* min or holds a steady state. If atomic density becomes steady state without reaching N* min, the input current density given in the simulation would be less than jth. 3. Evaluation We evaluated four line structures as shown in Fig.4. Sample 1 has no reservoir at both ends of line. Sample 2 has two reservoirs located on both vias. Sample 3 has a reservoir located only on the cathode via. And Sample 4 has a reservoir located only on the anode via. In each sample, the reservoir having shorter length was evaluated. After the simulation with current density smaller than the threshold, a steady state distribution of atomic density should be got without reaching critical atomic density N* min. The smallest value of the atomic density N* in all elements at steady state is plotted against supposed current density j. From an intersection point of the line of the smallest atomic density N* and the critical density, the threshold current density is evaluated. ⎥ ⎦ ⎤ ⎭ ⎬ ⎫ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + Ω − − Ω + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ Ω ⎟⎟− ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ Ω ∂ − ⎩ ⎨ ⎧ + − ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + Ω − − Ω − = y T D x T D kT Q N N N T y N D x N D kT N y N x N N d D D kT Q N N N T C N d AFD y x T T gb y x y x T T gb gb 1 ) / ( 1 / 4 3 6 cos 6 sin ) / ( exp 3 2 0 0 2 2 2 2 0 0 * 2 end * gen σ κ κ κ π β β σ κ π (4)
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