Thermal Modeling of Fluid Cooled 3D ICs

Ayman Dorrah, Jeff Nicholls and Miaofei Mei

• Introduction/ Motivation

• Architecture

• Thermal Model

• Modeling of Solids

• Modeling of Fluids

• Solution Procedure

• Results

• Example results

• Comparison between BE and TR methods

• Conclusion

• References

Agenda:

• There is demand for ever increasing powerful integrated circuits, but interconnecting

the components on a 2D chip has poor scalability, so the semiconductor industry is

moving towards 3D IC stacks.

• One of the challenges of engineering 3D ICs is designing for heat dissipation.

• Heat sinks are incapable of cooling the middle of a stack.

• Liquid coolant flowing through microchannels in between the layers of ICs is a

scalable solution for the heat problem.

• A thermal model of the 3D IC stack and microchannel system is presented.

Introduction/ Motivation:

• A compact model based on the work of Sridhar et al.

• Spatial dimensions are coarsely discretized, leading to large speedups. Time is

discretized through finite differencing.

• The model is solved using Backward Euler and Trapezoidal Rule methods.

Architecture:

Thermal Modeling of solids: Conservation Law

Thermal Modeling of Solids: Equivalent Circuit

Thermal Modeling of Fluids: Conservation Law

Thermal Modeling of Fluids: Equivalent Circuit

Thermal Modeling of Fluids:

• Boundary Conditions Dirichlet BCs at Channel’s Inlet and Neumann BCs

everywhere else.

• C is a diagonal matrix and G is an asymmetric block-tridiagonalmatrix

• Input stack file (.stk) -> equivalent circuit -> G, C, B

• Solution computed using Backward Euler:

• ..and Trapezoidal Rule:

• With 400,000 nodes, 600 timesteps, solution takes ~ 7 mins

Solution Procedure (Solver):

TR Ringing

• Try to solve:

• Exact solution for nth time step:

• Numerical solution:

• For :

Numerical solution approaches zero but flips between positive and negative.

TR Ringing: Stability Analysis

• Smoothing attempts to correct ringing:

While (smoothing):

Calculate xn+1

Generate improved estimate for xn

Use improved estimate to recalculate xn+1

• Repeated for multiple timesteps after a change in the input

Smoothing

Smoothing

Example Results

• The test case is composed of the layers

(7 mm long by 10 mm wide)

• Initial temperature was 20 ◦C. The

active silicon was producing heat at a

constant 50 W/cm2. A hotspot produced

between at 50 W/cm2, 250 W/cm2 and

450 W/cm2 of heat. The microchannels

were 50 µm wide by 100 µm tall. The

incoming coolant was 26 ◦C. Coolant

velocities of 1.62 m/s, 0.81 m/s, and

0.405 m/s were simulated.

• The test case was simulated until t = 0.6

s with both BE and smoothed TR using a

timestep of Δt = 0.001 s.

Example Results

• Temperature over time in the middle of the die:

Example Results

Example Results

Example Results

Results: Comparison Between BE and TR methods

Results: Comparison Between BE and TR methods

Results: Comparison Between BE and TR methods

• Presented a compact model capable of efficiently simulating thermal effects in 3D

microchannel cooled ICs

• Model demonstrates/extends the applicability of the equivalent circuit approach

• Trapezoidal Rule suffers from “ringing” in simulations with fast transients

• Smoothed Trapezoidal Rule is a good way of reducing ringing and improving

simulation accuracy with minimal time overhead

Conclusion:

References:

Thanks! Questions?Email: engineering@ecf.utoronto.ca