2.2.5 Heat Sinks and Sources

For the simulation of self-heating, the generated heat is modeled through JOULE's power loss. Hence, each interconnect line with an electrical burden represents a distributed heat source which is modeled via an additional heat source term ${H_{\mathrm{th}}}$ in the local heat conduction equation (2.9).

As initial conditions, the ambient temperature at $T_0 = T(t=t_0)$ is chosen. Typical initial values for $T_0$ are 300K for room temperature and 330K or 350K for heated device structures which have already reached their stationary operational conditions.

An ideal heat sink provides a constant temperature at a certain part of the device structure. Therefore, the boundary condition for the temperature can be modeled by a DIRICHLET^2.23 boundary condition

$$\forall {\mathbf{{x}}}\in \partial \Omega_1: T({\mathbf{{x}}}) {=}T_0,$$ (2.94)

where $\Omega$ represents the simulation domain and $\Omega_1 \in\Omega$ a part of the simulation. $\partial\Omega_1$ is the corresponding boundary related to the ideal heat sink. This represents a very good assumption if actively cooled heat sinks are considered.

Adiabatic boundary conditions can be used, if only a single part of a device is sufficient to describe the device behavior due to symmetry of the device structure [94]. The adiabatic boundary condition can be expressed by homogeneous NEUMANN^2.24 boundary conditions

$$\forall {\mathbf{{x}}}\in \partial \Omega_2: {{\tilde{\lambda}}} ... ...}}) {=}{{{\mathbf{{q}}}_{\mathrm{th}}}}_{,0}({\mathbf{{x}}}) {=}{\mathbf{{0}}}.$$ (2.95)

External heat sources, for instance, at boundaries of the simulation domains $\partial\Omega_3$ , require inhomogeneous NEUMANN boundary conditions described by

$$\forall {\mathbf{{x}}}\in \partial \Omega_3: {{\tilde{\lambda}}} ... ...hbf{{x}}}) {=} {{{\mathbf{{q}}}_{\mathrm{th}}}}_{,0}({\mathbf{{x}}}),% \neq 0,$$ (2.96)

where ${{{\mathbf{{q}}}_{\mathrm{th}}}}_{,0}({\mathbf{{x}}})$ is the externally applied heat flux density, this might be for example a heat generating power line or an active cooling element.

Applying additional (fast) heat diffusion paths influences the original heat flux distribution and might result in new and sometimes unwanted heating effects at their surrounding material environments. Therefore, a rigorous investigation of the thermal influence has to be included for modern chip design because even cooling a semiconductor device structure may cause additional mechanical stress as has been outlined in [95].