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As discussed in Section 2.2, the electromigration driving force acting on the metal atoms is caused by the contribution of two microscopic forces, namely "direct force" and "electron wind". The "direct force" refers to the direct action of the local electric field E on the migrating ions in the conducting metal line. The action of the local electric field on the metal atoms is related to the electric current density j by Ohm{'}s law [70] with
where σe is the electrical conductivity of the metal. Since the electric field has zero curl and is a conservative vector field, it can be expressed as a function of the electric potential Ve such that
The principle of charge conservation states that electric charge can neither be created nor destroyed at every point in the line. Considering a region with no charges, the charge continuity equation, which is related to the distribution of current density in the line at any time [70], reduces to
By applying the divergence theorem, together with equation (3.1) and (3.2), the distribution of the electric potential in the metal line is given by
| \[\begin{equation} \nabla \cdot \left(\sigma_\text{e}\nabla V_\text{e}\right)=0. \end{equation}\] | (3.4) |
Under the assumption of constant electrical conductivity along the line, equation (3.4) reduces to Laplace's equation for electric potential
The electric potential distribution in the metal interconnect line is subjected to the insulating surface boundary condition. If the space surrounding the metal line is taken to be non-conducting, charge would not be leaking out into it and the current density on the surface, with normal vector n, should vanish by obeying the following boundary condition [70]
Hence, by applying equation (3.5), equation (3.6) implies that the normal derivative of the electric potential vanishes on the surface as follows
A potential which satisfies Laplace{'}s equation and has a specific normal derivative on all boundaries is a uniquely defined electrostatic potential [70].
The temperature distribution within the metal line is determined by the solution of the heat equation. The heat equation is derived from the conservation law of heat energy for materials and Fourier's law [23]. The heat equation is a differential statement of thermal energy balance. The thermal energy of a metal line can change only if heat is conducted in/out through its surface boundary or heat is generated or adsorbed within the line [120]. The net outflow of thermal energy through the bounding surfaces of the line should therefore be balanced by its internally-generated heat and its ability to store some of this heat. This is stated by the law of conservation of heat energy for the metal line as follows
| \[\begin{equation} \nabla \cdot \vec q = q_\text{gen} - \frac{\partial Q}{\partial t}, \end{equation}\] | (3.8) |
where q is the heat flux, qgen is the rate of heat generation within the line, and Q is the thermal energy stored within the metal per unit volume. The change in thermal energy in time is related to the capacity of the line to store heat by raising its temperature T as follows
| \[\begin{equation} \frac{\partial Q}{\partial t} = \rho_\text{m}c_\text{p} \frac{\partial T}{\partial t}, \end{equation}\] | (3.9) |
where ρm is the metal mass density, and cp is the metal specific heat capacity. The net heat conducted out of the line implies a temperature gradient. When there exists a temperature gradient within the line, heat energy flows from regions of high temperature to regions of low temperature. Therefore, the rate of flow of heat energy through the line is related to the temperature gradient across it by Fourier's law
where km is the metal thermal conductivity. By substituting equation (3.9) and (3.10) in equation (3.8), the heat equation is rewritten as
| \[\begin{equation} \nabla^2 T -\cfrac{\rho_\text{m}c_\text{p}}{k_\text{t}} \frac{\partial T}{\partial t} = - \cfrac{q_\text{gen}}{k_\text{t}}. \end{equation}\] | (3.11) |
The temperature T at a given location in the line changes over time as heat spreads throughout the metal. The heat generated in the interconnect is related to Joule heating, which is described as the heat released in the conductor when an electric current passes through it and is given by
| \[\begin{equation} q_\text{gen}=\vec j \cdot \vec E = \sigma_\text{e}\vec E \cdot \vec E =\sigma_\text{e}\parallel{\vec{E}}\parallel^2= \sigma_\text{e}(\nabla V_\text{e})^2. \end{equation}\] | (3.12) |
Joule heating is included in the heat equation (3.11), and couples the electrical problem with the thermal problem as follows
| \[\begin{equation} \nabla^2 T -\cfrac{\rho_\text{m}c_\text{p}}{k_\text{t}} \frac{\partial T}{\partial t} = - \cfrac{\sigma_\text{e}}{k_\text{t}}(\nabla V_\text{e})^2. \end{equation}\] | (3.13) |
The temperature distribution in the line is determined by setting thermal boundary conditions to the metal line and taking into account the Joule heating effect. A Dirichlet boundary condition for the temperature specifies the values along the boundaries of the domain. The effect of Joule heating in the thermal problem is properly considered by including portions of the dielectric material surrounding the metal line. Both the electrical and thermal conductivities are assumed to be temperature dependent parameters in the forms
| \[\begin{equation} \sigma_\text{e}(T) = \cfrac{\sigma_\text{e,0}}{1+\alpha_\text{e}(T-T_\text{0})+\beta_\text{e}(T-T_\text{0})^2} \end{equation}\] | (3.14) |
and
| \[\begin{equation} k_\text{t}(T) = \cfrac{k_\text{t,0}}{1+\alpha_\text{t}(T-T_\text{0})+\beta_\text{t}(T-T_\text{0})^2} , \end{equation}\] | (3.15) |
where σe,0 and ke,0 are the conductivities at the reference temperature T0, αe and αt are the linear temperature coefficients, and βe and βt are the quadratic temperature coefficients [40].
By solving the non-linear system of equations (3.3),(3.5), and (3.13) which describe the electro-thermal model, it is possible to obtain the current density, the electric potential, and the temperature distributions in the interconnect line.