SISPAD ’06 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; University of Maryland; Laboratory for Physical Sciences, Laboratory for Physical Sciences, College Park, MD, USA

SISPAD ’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 processIntroduction

SISPAD ’06, Dilli, Goldsman, Akturk, Metze Model unit cells and combine them to form a network 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 Pick critical points as output nodes of interest Solve for impulse responses to impulses at likely induction or injection points Solve for impulse responses to impulses at likely induction or injection points Use impulse responses to obtain outputs to general inputs Use impulse responses to obtain outputs to general inputsMethodology

SISPAD ’06, Dilli, Goldsman, Akturk, Metze 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. 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.Methodology

SISPAD ’06, Dilli, Goldsman, Akturk, Metze We can write these input components f i [t] as Writing f i [t] as the sum of a series of time-impulses marching in time:  [x-x i ]= 1, x=x i 0, else Define a unit impulse at point x i : 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 f i [t]=f[x i,t] applied to each point x i : Numerical Modeling: Theory f[x,t]  [x-x i ]  [t] h i [x,t]

SISPAD ’06, Dilli, Goldsman, Akturk, Metze f i [t]F i [x,t] Let F i [x,t] be the system’s response to this input applied to x i : For a time-invariant system we can use the impulse response to find F i [x,t] : Then, since Numerical Modeling: Theory

SISPAD ’06, Dilli, Goldsman, Akturk, Metze 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 Then we can calculate the response to any such random input distribution α ij by only summation and time shifting We can explore different random input distributions easily, more flexible than experimentation Computational Advantages

SISPAD ’06, Dilli, Goldsman, Akturk, Metze Knowing impulse responses h i [x,t] and input f i [x,t], the response is calculated by adding the output contribution from each input time step: For t in temporal and N in spatial input points and impulse responses decaying in t h timesteps, this is done t in N in t h times If t in <<t h, for N out output points, this costs N in N out O (t h ) SPICE solves entire N-node network at each time step; costing N m for m >1 Computational Cost

SISPAD ’06, Dilli, Goldsman, Akturk, Metze On-chip interconnects on lossy substrates: capacitively and inductively coupled to each other On-chip interconnects on lossy substrates: capacitively and inductively coupled to each other –Characterized with S-parameter measurements –Equivalent circuit models found by parameter- fitting Interconnect Network Modeling

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

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

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

SISPAD ’06, Dilli, Goldsman, Akturk, Metze 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. Solver Results: 21x21x5 Mesh

SISPAD ’06, Dilli, Goldsman, Akturk, Metze 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) Solver Results: 21x21x5 Mesh

SISPAD ’06, Dilli, Goldsman, Akturk, Metze 5x5x3 network; each element connected with R//C to six nearest neighbors. Cadence SPECTRE takes 0.24 msec per timestep. Our code takes msec. Solver Results: 5x5x3 Mesh, SPICE Comparison UMCP Solver SPECTRE

SISPAD ’06, Dilli, Goldsman, Akturk, Metze 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. Vias: X marks. Inputs: U marks. Outputs:  marks. Solver Results: 50x50x3 Non-uniform Mesh

SISPAD ’06, Dilli, Goldsman, Akturk, Metze Sample impulse responses shown one layer at a time. Layer 1 Impulse at (5,15,1) Impulse at (25,25,3) Solver Results: 50x50x3 Non-Uniform Mesh

SISPAD ’06, Dilli, Goldsman, Akturk, Metze Sample impulse responses shown one layer at a time. Layer 2 Impulse at (5,15,1) Impulse at (25,25,3) Solver Results: 50x50x3 Non-Uniform Mesh

SISPAD ’06, Dilli, Goldsman, Akturk, Metze Sample impulse responses shown one layer at a time. Layer 3 Impulse at (5,15,1) Impulse at (25,25,3) Solver Results: 50x50x3 Non-Uniform Mesh

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 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.Conclusion