sispad ’06 a 3-d time-dependent green’s function approach to modeling electromagnetic noise in...

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

On-Chip Interconnect NetworksOn-Chip Interconnect Networks

Zeynep Dilli, Neil Goldsman, AkZeynep Dilli, Neil Goldsman, Akıın Aktn Aktüürk, 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 Sciences, Laboratory for Physical Sciences,

College Park, MD, USACollege Park, MD, USA

SISPAD ’06, Dilli, Goldsman, Akturk, MetzeSISPAD ’06, Dilli, Goldsman, Akturk, Metze

•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.

Sample unit cells for a two-metal process

IntroductionIntroduction


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

MethodologyMethodology


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.

MethodologyMethodology


[ ] [ , ] [ ]i if t f x t x x

[ , ] [ ]ii

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

f[x,t]

[x-xi][t] hi[x,t]


fi [t] Fi[x,t]

[ , ] [ ] [ , ]i i j i jj

F x t f t h x t t

[ , ] [ ] [ , ]i j i ji j

F x t f t h x t t

Let Fi[x,t] be the system’s response to this input applied to xi:

For a time-invariant system we can use the impulse response to find Fi[x,t] :

Then, since

[ , ] [ ] [ , ] [ , ]i ii i

f x t f t F x t F x 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

Computational AdvantagesComputational Advantages


•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:

[ ] [ ] [ ] [ ]out out i n i nV t V t f t h t t

•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

NinNout(th)•SPICE solves entire N-node network at each time step; costing Nm for m>1

Computational CostComputational Cost


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

parameter-fittingparameter-fitting

Interconnect Network ModelingInterconnect Network Modeling


• 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.

Implementation: Interconnect Network SolverImplementation: Interconnect Network Solver


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.

Sample 3-D NetworkSample 3-D Network


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).

Solver Results: 21x21x5 MeshSolver 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.



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)



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.

Solver Results: 5x5x3 Mesh, SPICE ComparisonSolver Results: 5x5x3 Mesh, SPICE Comparison

UMCP Solver

SPECTRE


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.

Solver Results: 50x50x3 Non-uniform MeshSolver Results: 50x50x3 Non-uniform Mesh


Sample impulse responses shown one layer at a time.

Layer 1


Solver Results: 50x50x3 Non-Uniform MeshSolver Results: 50x50x3 Non-Uniform Mesh



Layer 2





Layer 3




Solver Results: 50x50x3 Non-uniform MeshSolver Results: 50x50x3 Non-uniform Mesh


Current workCurrent 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…)


•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.

ConclusionConclusion


Impulse Response RC Network Solver ResultsImpulse Response RC Network Solver Results

Input points: (10,1,1) (bottom layer, south edge center), (11,25,3) (near north edge center, topmost layer).

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=5

T=15

T=25



Level 2, input at (11,25,3)

Left: tstep=5, Below: tstep=15



Level 3, input at (11,25,3)

Left: tstep=5, Below: tstep=15, 25

sispad ’06 a 3-d time-dependent green’s function approach to modeling electromagnetic noise in...

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t h i x

writing f i t

varying input components

unit impulse

metze f i tf

impulse responses

input fx

timeinvariant system