A 3-D Time-Dependent Green’s Function Approach
to Modeling Electromagnetic Noise in On-Chip
Interconnect Networks
Zeynep Dilli, Neil Goldsman, Akın Aktürk,
George Metze
Dept. of Electrical and Computer Eng.
University of Maryland;
Laboratory for Physical Sciences,
College Park, MD, USA
SISPAD ’06
Introduction
•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 computationallyintensive, 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.
Sample unit cells for a
two-metal process
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Methodology




Model unit cells and combine them to form a network
– Simplified lumped element model: Uses resistors and capacitors
(Unit cells marked with red boxes in the figure).
Pick critical points as output nodes of interest
Solve for impulse responses to impulses at likely induction or
injection points
Use impulse responses to obtain outputs to general inputs
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Methodology

The interconnect network is a linear time
invariant system: Use Green’s Function
responses to calculate the output to any input
distribution in space and time.
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Numerical Modeling: Theory
f[x,t]
Define a unit impulse
at point xi:
[x-xi]=
1, x=xi
0, else
This yields a system impulse response:
[x-xi][t]
hi[x,t]
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:
f [ x, t ]   fi [t ]
i
We can write these input components fi[t] as
fi [t ]  f [ x, t ] [ x  xi ]
Writing fi[t] as the sum of a series of time-impulses marching in time:
f i [t ]   f [ x, t ] [ x  xi ] [t  t j ]
j
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Numerical Modeling: Theory
Let Fi[x,t] be the system’s response to this input
applied to xi:
fi [t]
Fi[x,t]
For a time-invariant system we can use the
impulse response to find Fi[x,t] :
Fi [ x, t ]   fi [t j ]hi [ x, t  t j ]
j
Then, since
f [ x, t ]   fi [t ]  F [ x, t ]   Fi [ x, t ]
i
i
 F [ x, t ]   f i [t j ]hi [ x, t  t j ]
i
j
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Computational Advantages
•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 : f [ xi , t j ]
 F [t ]    ij hi [ x, t  t j ]
i
j
•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
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Computational Cost
•Knowing impulse responses hi[x,t] and input fi[x,t], the
response is calculated by adding the output contribution
from each input time step:
Vout [t ]  Vout [t ]  fi [tn ]  hi [t  tn ]
•For tin temporal and Nin spatial input points and
impulse responses decaying in th timesteps, this is done
tinNinth times
•If tin<<th, for Nout output points, this costs NinNoutO(th)
•SPICE solves entire N-node network at each time step;
costing Nm for m>1
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Interconnect Network Modeling

On-chip interconnects on lossy substrates:
capacitively and inductively coupled to each other
– Characterized with S-parameter
measurements
– Equivalent circuit models found by parameterfitting
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Implementation: Interconnect Network Solver
• 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.
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Sample 3-D Network
Only 5x5x2 mesh shown for
simplicity. Not all vertical
connections shown.
All nodes on the same level
connected with an R//C to
their neighbors.
All nodes on lowest level are
connected with an R//C to
ground.
All nodes in intermediary
levels are connected with an
R//C to neighbors above and
below.
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 21x21x5 Mesh
Input points: (1,1,1) (bottom layer, southwest corner), (11,20,5) (near north edge
center, topmost layer).
Sample output points (5,5,1) (bottom layer, southwest of center); (11,11,3)
(center layer, exact center); (20,2,5) (top layer, southeast of center).
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 21x21x5 Mesh
Sample impulse response over all five layers:
Unit impulse at point (11,20,2).
The animation shows the
impulse response until t=6 nsec
with 1 nsec increments.
Note that this result is from a
network with all horizontal
connections resistive only.
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 21x21x5 Mesh
Sample impulse responses shown one layer at a time (horizontal connections resistive only.):
Layer 5
Layer 1
Impulse at (1,1,1)
Impulse at (11,20,5)
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 5x5x3 Mesh, SPICE Comparison
5x5x3 network; each element connected with R//C to six nearest neighbors.
Cadence SPECTRE takes 0.24 msec per timestep. Our code takes 0.012 msec.
UMCP Solver
SPECTRE
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 50x50x3 Non-uniform Mesh
An example three-metal-layer interconnect network representation.
The connections are resistive and/or capacitive as required.
Vias: X marks. Inputs: U marks. Outputs:  marks.
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 50x50x3 Non-Uniform Mesh
Sample impulse responses shown one layer at a time.
Layer 1
Impulse at (5,15,1)
Impulse at (25,25,3)
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 50x50x3 Non-Uniform Mesh
Sample impulse responses shown one layer at a time.
Layer 2
Impulse at (5,15,1)
Impulse at (25,25,3)
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 50x50x3 Non-Uniform Mesh
Sample impulse responses shown one layer at a time.
Layer 3
Impulse at (5,15,1)
Impulse at (25,25,3)
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Solver Results: 50x50x3 Non-uniform Mesh
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Current work
•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…)
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Conclusion
•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 threedimensional chip stacks as well as layers on a single chip.
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Impulse Response RC Network Solver Results
Input points: (10,1,1)
(bottom layer, south edge
center), (11,25,3) (near
north edge center,
topmost layer).
T=5
RC parameters from
Weisshaar et.al., 2002;
uniform two layer
network, unit cell size 10
μm.
Right: Level 1,
input at (10,1,1)
Top to bottom:
tstep=5, tstep=15,
tstep=25
T=15
T=25
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Impulse Response RC Network Solver Results
Level 2, input at (11,25,3)
Left: tstep=5, Below: tstep=15
SISPAD ’06, Dilli, Goldsman, Akturk, Metze
Impulse Response RC Network Solver Results
Level 3, input at (11,25,3)
Left: tstep=5, Below: tstep=15, 25
SISPAD ’06, Dilli, Goldsman, Akturk, Metze