SISPAD SISPAD ’06 ’06 A 3-D Time-Dependent Green’s A 3-D Time-Dependent Green’s Function Approach to Modeling Function Approach to Modeling Electromagnetic Noise in On-Chip Electromagnetic Noise in On-Chip Interconnect Networks Interconnect Networks Zeynep Dilli, Neil Goldsman, Ak Zeynep Dilli, Neil Goldsman, Ak ı ı n Akt n Akt ü ü rk, rk, George Metze George Metze Dept. of Electrical and Computer Eng. Dept. of Electrical and Computer Eng. University of Maryland; University of Maryland; Laboratory for Physical Laboratory for Physical Sciences, Sciences, College Park, MD, USA College Park, MD, USA
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SISPAD ’06 A 3-D Time-Dependent Green’s Function Approach to Modeling Electromagnetic Noise in On-Chip Interconnect Networks Zeynep Dilli, Neil Goldsman,
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SISPAD SISPAD ’06’06
A 3-D Time-Dependent Green’s Function A 3-D Time-Dependent Green’s Function Approach to Modeling Electromagnetic Noise in Approach to Modeling Electromagnetic Noise in
•Objective: Investigate the response of a complex on-chip interconnect network to external RF interference, internal parasitic signals, or coupling between different regions•Full-chip electromagnetic simulation: Too computationally-intensive, but possible for small “unit cell”s:
•Simple seed structures of single and coupled interconnects
•We have developed a methodology to solve for the response of such a unit cell network to random inputs.
Model unit cells and combine them to form a networkModel unit cells and combine them to form a network– Simplified lumped element model: Uses resistors and Simplified lumped element model: Uses resistors and
capacitors (Unit cells marked with red boxes in the figure).capacitors (Unit cells marked with red boxes in the figure). Pick critical points as output nodes of interestPick critical points as output nodes of interest Solve for impulse responses to impulses at likely induction or Solve for impulse responses to impulses at likely induction or
injection pointsinjection points Use impulse responses to obtain outputs to general inputsUse impulse responses to obtain outputs to general inputs
The interconnect network is a linear The interconnect network is a linear time invariant system: Use Green’s time invariant system: Use Green’s Function responses to calculate the Function responses to calculate the output to any input distribution in output to any input distribution in space and time.space and time.
f x t f tWe can write these input components fi[t] as
Writing fi[t] as the sum of a series of time-impulses marching in time:
[x-xi]=
1, x=xi0, else
Define a unit impulse at point xi:
This yields a system impulse response:
Let an input f[x,t] be applied to the system. This input can be written as the superposition of time-varying input components fi[t]=f[xi,t] applied to each point xi:
[ ] [ , ] [ ] [ ]i i jj
f t f x t x x t t
Numerical Modeling: TheoryNumerical Modeling: Theory
•Full-wave electromagnetic solutions only possibly needed for small unit cells•The input values at discrete points in space and time can be selected randomly, depending on the characteristics of the interconnect network (coupling, etc.) and of the interference. Let
: [ , ]ij i jf x t
[ ] [ , ]ij i ji j
F t h x t t •Then we can calculate the response to any such random input distribution αij by only summation and time shifting•We can explore different random input distributions easily, more flexible than experimentation
On-chip interconnects on lossy On-chip interconnects on lossy substrates: capacitively and inductively substrates: capacitively and inductively coupled to each othercoupled to each other– Characterized with S-parameter Characterized with S-parameter
measurementsmeasurements– Equivalent circuit models found by Equivalent circuit models found by
• Developed an in-house network solver. • Inputs: A 2-D or 3-D lumped network; input waveforms with the
input locations indicated; locations that the user wishes to observe responses at.
• Outputs: Impulse responses at given output locations to impulses at given input locations; the composite output at given output locations to the input waveforms provided.
• Algorithm:
1. Read in network mesh structure, the input impulse locations, the output locations
2. Set up the KCL-based system of difference equations for the mesh
3. For each impulse location, stimulate the system with a unit impulse
1. Solve for the time evolution of the voltage profile across the network
2. Record the values at the set output points, creating impulse responses vs. time
4. Use the full input waveforms together with calculated impulse responses to compose the full output at the requested output locations.
An example three-metal-layer interconnect network representation. The An example three-metal-layer interconnect network representation. The connections are resistive and/or capacitive as required.connections are resistive and/or capacitive as required.
Vias: Vias: X X marks. Inputs: marks. Inputs: U U marks. Outputs: marks. Outputs: marks. marks.
•We are developing unit cells modeling physical interconnect structures:
•With appropriate unit cells, we can investigate the full networks of 3-D integrated chips•We plan to use EM modeling tools and S-parameter measurements and extraction•An integrated circuit layout featuring an interconnect layout designed for unit cell extraction has been sent for fabrication
•Example goal application: Determine which locations are most vulnerable for substrate and ground/VDD noise-sensitive subcircuits included in 3-D integrated system with different types of circuit networks on the individual layers (e.g. communication on top layer, data storage in the middle, data processing at the bottom…)
•A computationally efficient method to model and investigate the response of a complex on-chip interconnect network to external RF interference, internal parasitic signals, or coupling between different regions•Computational advantages:
•Can rapidly model the effect of many sources on the same network;•Impulse responses at only the desired points in the system need to be stored to calculate the output at those points for any input waveform; •The same unit cells can be recombined in different configurations; thus flexibility in the systems that can be investigated;•It is straightforward to expand the method to three-dimensional chip stacks as well as layers on a single chip.