Temperature on the IBM Power Grid Benchmark Examples

• IBM has placed a set of SPICE-format power grid example circuits here.
• The power grids are based on real designs and come with current sources at the substrate level (cells?) and voltage sources at the top level (bumps?).
• Power grid sizes vary from 30638 nodes (ibmpg1) to 1670494 nodes (ibmpg6).
• TDA has converted several power grids to IEEE481 standard parasitic extraction format (SPEF) by assuming a hypothetical metal interconnect stack to provide resistor width and thickness and thermal coupling to substrate.
• Ibmpg5 has a single VSS net and four VDD nets, each covering one quadrant of the chip, so this design was chosen to compute Ember™'s scaling with number of nodes.  Nets from ibmpg4 and ibmpg6 are included for comparison.
• For details of the ibmpg5 design, see Scaling.
• Temperature colour scale: 0.25 < ∆T < 6.0 C. For a clearer impression of the lower temperature range, see 3-D Substrate Temperature.

Ember™ temperatures vs. implied current density rule temperatures

• Foundry rms current density rules are intended to protect interconnect from high temperatures due to Joule self-heating without the need for a detailed temperature simulation.
• Rms current density rules are equivalent to the T∞ term in Ember™'s expression for temperature T(x).
• The benefits of using rms current density rules are:
  • easy to do
  • (that's the only benefit)
• Ember™ includes topology effects on temperature that are not addressed by the density rules:
  • finite metal segment length
  • thermal conduction between layers and down to the substrate through vias and contacts
  • thermal injection from vias
  • thermal coupling to locally variable substrate temperature
  • thermal coupling between nets (where specified, e.g. by extracting couping capacitance)
• These topology effect completely alter the temperature predictions of the current density rule (Tinf), resulting in Ember™ maximum resistor temperature predictions (Tmax).  See the graph below.

The detail below shows Ember™'s temperature solution at the same location.

Note that the high temperature region, through which significant heat is conducting, includes many more resistors than the high current region. Heat takes a different path, seeking cold, than current takes, seeking ground.

Clearly, temperature and self-heating of interconnects are not highly correlated in the power grid.  Why not?

The detail below shows Ember™'s dc current solution at a presumed bump, with a maximum current in the via indicated by the red arrow.  Only three paths exiting the bump carry significant current.