6 Finite Element Discretization of the Basic Equation

We are going to demonstrate the numerical handling of the basic continuum equation of the Sarychev model. For the sake of simplicity we assume unpassivated interconnects where the stress relaxation to the hydrostatic state is faster than vacancy diffusion, and where the vacancy concentration in the initial phase of electromigration stressing does not deviate from the equilibrium concentration. In that case the vacancy dynamics is described by a three-dimensional Korhonen-type equation,

$$\frac{\partial C_V}{\partial t}=D_V\Bigl(\Delta C_V+\frac{Z^*e}{kT}\nabla\varphi\nabla C_V\Bigr)-\frac{C_V-C_V^{eq}}{\tau},$$ (249)

with the weak formulation,

$$\int_T\frac{\partial C_V}{\partial t}N_j (\mathbf{x}) d\Omega=-D... ...ma+D_V\frac{Z^*e}{kT}\int_T \nabla C_V \nabla \varphi N_j(\mathbf{x}) d\Omega-$$

$$-\int_T \frac{C_V-C_V^{eq}}{\tau}N_j(\mathbf{x}) d\Omega.$$ (250)

Figure 4.4: Electrical potential distribution [mV]

[fillstyle=slopes,slopesteps=10000,slopecolors=0 0.0 0.0 1.0 1000 0.0 0.0 1.0 3000 0.0 1.0 0.0 5000 0.0 1.0 0.0 6000 0.85 0.917 0.160 6200 0.85 0.917 0.160 9000 1.0 0.0 0.0 7,gradangle=90,swapaxes=true,linestyle=none](-3.0,0.4)(2.0,0.0) $\parbox{5cm}{ \vspace*{2.2cm} $70.0$\ [6mm] $60.0$\ [6mm] $50.0$\ [6mm] $40.0$\ [6mm] $30.0$\ [6mm] }$

Figure 4.5: Vacancy distribution [cm$^{-3}$].

[fillstyle=slopes,slopesteps=10000,slopecolors=0 0.0 0.0 1.0 1000 0.0 0.0 1.0 3000 0.0 1.0 0.0 5000 0.0 1.0 0.0 6000 0.85 0.917 0.160 6200 0.85 0.917 0.160 9000 1.0 0.0 0.0 7,gradangle=90,swapaxes=true,linestyle=none](-3.0,0.4)(2.0,0.0) $\parbox{5cm}{ \vspace*{2.2cm} $7.6 \cdot 10^{15}$\ [6mm] $6.5 \c... ... 10^{15}$\ [6mm] $4.2 \cdot 10^{15}$\ [6mm] $3.0 \cdot 10^{15}$\ [6mm] }$

By introducing the spatial discretization of the vacancy concentration $C_V$ and the electrical potential $\varphi$ with the linear basis functions $N_i(\mathbf{x})$,

$$C_V=\sum_{i=1}^4 C_{V,i} N_i(\mathbf{x}),\quad \varphi=\sum_{i=1}^4 \varphi_i N_i(\mathbf{x}),$$ (251)

and backward Euler scheme for the time discretization, we obtain,

$$\sum_{i=1}^4\Bigl(\int_T N_i(\mathbf{x}) N_j(\mathbf{x}) d\Omega... ...l(\int_T \nabla N_i(\mathbf{x}) \nabla N_j(\mathbf{x}) d\Omega\Bigr)C_{V,i}^n+$$

$$+D_V\frac{Z^*e}{kT}\sum_{i=1,p=1}^4\Bigl(\int_T \nabla N_i(\mathbf{x}) N_p(\mathbf{x}) \nabla N_j(\mathbf{x}) d\Omega\Bigr)C_{V,i}^n\varphi_{i}^n-$$

$$-\frac{1}{\tau}\Bigl(\sum_{i=1}^4\Bigl(\int_TN_i(\mathbf{x})N_j(\mathbf{x}) d\Omega\Bigr)C_{V,i}^n-C_V^{eq}\int_TN_j(\mathbf{x}) d\Omega\Bigr).$$ (252)

The construction of the nucleus matrix follows the scheme described in Section 3.8. The fully discretisatised Sarychev model includes also the finite element discretization of equations (4.6)-(4.11).

Already a simple electromigration model described by (4.39) enables some insight in the electromigration caused material transport phenomena taking place in the copper interconnect layout. In Figure 4.4 the interconnect via with barrier layer is displayed. Here we can also see the electrical potential distribution which determines the electromigration driving force intensity and direction. The vacancy concentration distribution, obtained by solving equation (4.39), shows a peak value at the bottom of the via (Figure 4.5). This peak concentration region indicates a probable location of void nucleation.