The electrical interconnect network is solved for voltage, but Ember™'s thermal network is solved for coefficients of the temperature formula T(x) on every element.
Closed-Form Temperature Trajectories Capture Key Thermal Behaviour
Ember™'s closed-form solution captures the complex shape of the temperature trajectory below with only 9 nodes! Temperatures calculated based on rms current density (the "T∞ formula" values on the graph) cannot model real heat conduction in complex interconnect. Notice that simply varying the material properties (e.g. from Si02 dielectric to low-k SiLK™) can shift the location of the hottest interconnect segment.
The reference temperature Tref for the resistor is the weighted average of the temperatures of all the resistors to which it is capacitively coupled, plus the local substrate temperature, which forms the boundary condition. A self-consistent solution for a design with multiple, coupled networks is found by iteration between nets.
Ember™ Transforms Electric Circuit Elements to Solve the Heat Equation
Electromigration analysis decomposes the interconnect into a network of π equivalent circuits representing (a) resistors and (b) vias. Ember™'s replaces electrical component values with transformed component values that express the closed-form solution to the heat equation.
Closed-Form Solution to the Heat Equation
The time-independent heat equation on a resistor can be solved in closed form, under reasonable assumptions:
Ember™ uses circuit analysis and closed-form models to forecast interconnect temperature rises due to Joule self-heating.
Electrical-Thermal Analogy
Ember™ exploits the well-known mathematical analogy between electrical circuit quantities and thermal quantities, re-using the electric circuit topology for thermal purposes.