Miniaturizing components has always been a primary goal in the semiconductor industry because it cuts production cost and lets companies build smaller computers and other devices. Miniaturization, however, has increased dissipated power per unit area and made it a key limiting factor in integrated circuit performance. Temperature increase becomes relevant for relatively small-cross-sections wires, where it may affect normal semiconductor behavior. Besides, since the generation of heat is proportional to the frequency of operation for switching circuits, fast computers have larger heat generation than slow ones, an undesired effect for chips manufacturers. This article summarizes physical concepts that describe the generation and conduction of heat in an integrated circuit, and presents numerical methods that model heat transfer from a macroscopic point of view.
Generation and transfer of heat
Fourier's law At macroscopic level, Fourier's law states a relation between the transmitted heat per unit time per unit area and the gradient of temperature:
q = − κ ∇ T {\displaystyle q=-\kappa \nabla T}
Where κ {\displaystyle \kappa } is the thermal conductivity, [W·m−1 K−1].
Joule heating Electronic systems work based on current and voltage signals. Current is the flow of charged particles through the material and these particles (electrons or holes), interact with the lattice of the crystal losing its energy which is released in form of heat. Joule Heating is a predominant mechanism for heat generation in integrated circuits and is an undesired effect in most of the cases. For an ohmic material, it has the form:
Q = j 2 ρ {\displaystyle Q=j^{2}\rho }
Where j {\displaystyle j} is the current density in [A·m−2], ρ {\displaystyle \rho } is the specific electric resistivity in [ Ω {\displaystyle {\Omega }} ·m] and Q {\displaystyle Q} is the generated heat per unit volume in [W·m−3].
Heat-transfer equation The governing equation of the physics of the heat transfer problem relates the flux of heat in space, its variation in time and the generation of power by the following expression:
∇ ( κ ( T ) ∇ T ) + g = ρ C ∂ T ∂ t {\displaystyle \nabla \left(\kappa \left(T\right)\nabla T\right)+g=\rho C{\frac {\partial T}{\partial t}}}
Where κ {\displaystyle \kappa } is the thermal conductivity, ρ {\displaystyle \rho } is the density of the medium, C {\displaystyle C} is the specific heat, α = κ ρ C {\displaystyle \alpha ={\frac {\kappa }{\rho C}}} , the thermal diffusivity and g {\displaystyle g} is the rate of heat generation per unit volume. Heat diffuses from the source following the above equation and solution in an homogeneous medium follows a Gaussian distribution.
Techniques to solve heat equation
Kirchhoff transformation To get rid of the temperature dependence of κ {\displaystyle \kappa } , Kirchhoff transformation can be performed
θ = T s + 1 κ s ∫ T s T κ ( T ) d T {\displaystyle \theta =T_{s}+{\frac {1}{\kappa _{s}}}\int _{T_{s}}^{T}\kappa (T)dT}
where κ s = κ ( T s ) {\displaystyle \kappa _{s}=\kappa \left(T_{s}\right)} and T s {\displaystyle T_{s}} is the heat sink temperature. When applying this transformation, the heat equation becomes:
α ∇ 2 θ + α κ s g = ∂ θ ∂ t {\displaystyle \alpha \nabla ^{2}\theta +{\frac {\alpha }{\kappa _{s}}}g={\frac {\partial \theta }{\partial t}}}
where α = κ ρ C {\displaystyle \alpha ={\frac {\kappa }{\rho C}}} is called the diffusivity, which also depends on the temperature. To completely linearize the equation, a second transformation is employed:
α s τ = ∫ 0 t α ( θ ) d t {\displaystyle \alpha _{s}\tau =\int _{0}^{t}\alpha (\theta )dt}
yielding the expression:
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