Thermal simulations for integrated circuits

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 summaries 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.

Electronic systems work based on current and voltage signals.

Heat diffuses from the source following the above equation and solution in an homogeneous medium follows a Gaussian distribution.

[2] Although analytical solutions can only be found for specific and simple cases, they give a good insight to deal with more complex situations.

Analytical solutions for regular subsystems can also be combined to provide detailed descriptions of complex structures.

In Prof. Batty's work,[2] a Fourier series expansion to the temperature in the Laplace domain is introduced to find the solution to the linearized heat equation.

This procedure can be applied to a simple but nontrivial case: an homogeneous cube die made out of GaAs, L=300 um.

The top surface is discretized into smaller squares with index i=1...N. One of them is considered to be the source.

Next, by applying adiabatic boundary conditions at the lateral walls and fix temperature at the bottom (heat sink temperature), thermal impedance matrix equation is derived: Where the index

[2] The below figure shows the steady state temperature distribution of this analytical method for a cubic die, with dimensions 300 um.

A constant power source of 0.3W is applied over a central surface of dimension 0.1L x 0.1L.

As expected, the distribution decays as it approaches to the boundaries, its maximum is located at the center and almost reaches 400K Numerical solutions use a mesh of the structure to perform the simulation.

The finite-difference time-domain (FDTD) method is a robust and popular technique that consists in solving differential equations numerically as well as certain boundary conditions defined by the problem.

This is done by discretizing the space and time, and using finite differencing formulas, thus the partial differential equations that describe the physics of the problem can be solved numerically by computer programs.

The FEM is also a numerical scheme employed to solve engineering and mathematical problems described by differential equations as well as boundary conditions.

It discretizes the space into smaller elements for which basis functions are assigned to their nodes or edges.

Finally, a direct or iterative method is employed to solve the system of linear equations.

[3] The figure below shows the temperature distribution for the numerical solution case.

This solution shows very good agreement with the analytical case, its peak also reaches 390 K at the center.

The next figure shows a comparison of the peak temperature as a function of time for both methods.

The numerical methods such as FEM or FDM derive a matrix equation as shown in the previous section.

This method is based on the fact that a high-dimensional state vector belongs to a low-dimensional subspace [1].

Once the solution is obtained, the original vector is found by taking the product with V. The generation of heat is mainly produced by joule heating, this undesired effect has limited the performance of integrated circuits.

) applied over a single surface source on the top of a cubic die a peak increment of temperature in the order of 100 K was computed.

Such increase in temperature can affect the behavior of surrounding semiconductor devices.

That is why the heat dissipation is a relevant issue and must be considered for circuit designing.