A Green FunctionBased Parasitic Extraction Method for Inhomogeneous Substrate Layers

1
A Green Function-Based Parasitic Extraction
Method for Inhomogeneous Substrate Layers

• Chenggang Xu, Ranjit Gharpurey
• Terri S. Fiez, and Kartikeya Mayaram
• School of EECS, Oregon State University,
  Corvallis, OR
• Dept. of EECS, University of Michigan, Ann
  Arbor, MI

2
Outline

• Introduction and motivation
• Two-problem approach for inhomogeneous substrate
  layers
• Efficient calculation methods
• Numerical examples
• Conclusions

3
Substrate Noise Coupling Problem

• Reduced isolation between analog and digital
  circuitry
• Increasing integration for system on a chip (SOC)
• Increasing operating frequency

4
Methods for Reducing SubstrateNoise Coupling

• Modify process technology (SOI, triple wells,
  etc.)
• Design noise tolerant circuits by including a
  substrate model in circuit simulations

Substrate Model
5
Numerical Methods for Substrate Parasitic
Extraction

• Volume element methods (finite difference or
  finite element) computationally expensive but
  versatile
• Large number of nodes
• Suitable for complex structures
• Boundary element methods computationally more
  efficient but typically limited to multilayered
  homogeneous substrates

Boundary element methods (BEM)
Volume element methods(VEM)
6
Green Function for Layer-Wise Homogeneous
Substrates

• The Green function represents the potential at a
  field point induced by a unit
  point charge at a source point

a, b, d substrate dimensions in x-, y-, and
z-directions u, l superscripts for upper and
lower solutions j, k subscripts for field and
source layers function of layer
thicknesses and permittivities
Source point
Field point
A. M. Niknejad, et al., Trans. CAD, 1998
7
Coefficient-of-Potential and Impedance Matrices

• Potential as an integration of Green function
• Discretization of above integration gives
  
• 
  
• where
• Coefficient-of-potential
  matrix Impedance matrix
• Vector of panel charge
  Vector of panel current
• Vector of panel potential
  
• Capacitance and resistance calculation from
  and

for capacitance extraction
for resistance extraction
8
Inhomogeneous Substrate Layers

• Homogeneous approximation not valid for sinkers,
  wells, and trenches
• Difficult to derive an analytical Green function
  for complex substrate structures

Cross section of a BiCMOS process with
approximate resistivities
9
Combined FEM/BEM Method

• Basic ideas
• Split substrate intotwo regions
• BEM for homogeneous region
• FEM for inhomogeneous region
• Reduced volume for FEM
• Reduced matrix size andCPU cost
• Possible problems
• FEM for inhomogeneous region
• Large number of nodes
• BEM meshing of interface
• Large number of panels

Example 1024µm1024µm6µm inhomogeneous region
uniform meshing (1µm) of Nodes from FEM 6M
of panels from BEM1M
E. Schrik and N.R. van der Meijs, Proc. 39th
DAC, 2002
10
Lateral Inhomogenity Is Local

• Small percentage of chip area for sinkers, wells,
  and trenches
• Combined FEM/BEM methods ignore local
  inhomogenity
• Possible ways to reduce problem size
• Reduce volume for FEM meshing
• Constrain volume meshing to local regions
• Reduce area for BEM meshing
• Constrain area for BEM meshing tosurfaces of
  local regions

11
Two-Problem Approach
Two contacts in a substrate with one local
inhomogeneous region
Companion region of permittivity
Homogenous substrate of permittivity
Original problem local region permittivity
R. Gharpurey, Ph.D. Dissertation, UC Berkeley,
1995
12
Two-Problem Approach - Implementation

• Problem 1
• Insert sockets for plugging in parasitic
  circuit of companion region
• Virtual contacts for meshinglocal surfaces
• Extract the parasitics forreal and virtual
  contacts
• Problem 2
• Grid representation of companion region
• Complete network obtained by plugging network
  from Problem 2 into network for Problem 1

13
Challenges with the Two-Problem Approach

• Previous work limited to horizontal planar
  contacts 1, 2
• No formulation available for vertical planar
  contacts
• Need formulation for vertical 2-D contacts
• A vertical 2-D contact is a special case of a 3-D
  contact with one of its x- and y-dimensions set
  to zero

1 R. Gharpurey, Ph.D. Dissertation, UC
Berkeley, 1995 2 A. M. Niknejad, et al., Trans.
CAD, 1998
14
Efficient Calculation of Coefficient of Potential
Matrix

• Equations for matrix entry calculation are in the
  form of infinite series
• Efficient calculation of truncated series by
  discrete transforms
• Equations for 3D contacts and horizontal 2D
  contacts cast into summation of discrete cosine
  transforms (DCT)
• Equations related to vertical 2D contacts require
  additional discrete sine transforms (DST) and
  mixed discrete sine-cosine transforms (MDSCT)

15
Lumping Parasitics to Real Contacts for Fast
Circuit Simulation

• Plugging Network 2 into Network 1 results in
  substrate network with many internal nodes
• Circuit simulation cost increases
• Lump parasitics to surface contacts before
  including in circuit simulation

nx ny nz 10 results in 1600 new nodes
16
Numerical Examples for Verification

• Two identical contacts (10µm10µm) separated by a
  trench (L-region)
• Extract resistances for the equivalent circuit
  model

Two contacts and a trench
Equivalent circuit model
17
Lightly and Heavily Doped Substrates

• Substrates are layer-wise uniform except for
  sinkers, wells, and trenches
• Thickness and resistivity of each layer
  determined from process doping profile

18
Comparison between EPICand ATLAS

• EPIC, a Green function-based BEM software we've
  developed for the Extraction of Parasitics in
  ICs.
• Equation solved Poisson equation
• Two-Problem approach implemented in EPIC
• ATLAS, a FDM-based device simulator from SILVACO
• Equation solved
• Poisson Equation
• The continuity equation with transport models

19
Comparison on Accuracy

• EPIC shows good agreement with ATLAS for both
  heavily and lightly substrates

Lightly Doped Substrate
Heavily Doped Substrate
20
Comparison on Computation Cost

• EPIC cost increases with trench depth
• More virtual contacts
• More companion nodes
• ATLAS cost less sensitive to trench depth
• Large number of nodes less sensitive to trench
  depth
• EPIC orders of magnitude faster than device
  simulation

CPU cost as a function of trench depth for the
heavily doped substrate
21
Resistances as a Function of Trench Length and
Depth

• An increase in trench length or depth improves
  isolation

Schematic of current flow path
Resistances as a function of trench depth
22
Trench Isolation for Different Contact Separations

• Trench isolation more effective for small
  separations
• Normalized resistances approach unity for large
  separations

Resistance values normalized to those without
trench
23
Conclusions

• Presented a two-problem approach for Green
  function based parasitic extraction with
  inhomogeneous substrate layers
• Computational efficiency improved by
• Efficient calculation of equations using DCT, DST
  and MDSCT
• Lumping parasitics of local region to surface
  contacts
• Numerical accuracy and efficiency verified with
  three-dimensional device simulation (ATLAS)
• Accuracy to within 10 of ATLAS
• Orders of magnitude speedup over ATLAS