Fast Capacitance Extraction in Multilayer, Conformal or Embedded Dielectric using Hybrid Boundary El

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Fast Capacitance Extraction in Multilayer,
Conformal or Embedded Dielectric using Hybrid
Boundary Element Method

• Ying Zhou and Weiping Shi Zhuo Li
• Dept. of ECE, Pextra Corporation
• Texas AM University (now with
  IBM)
• 
  

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Outline

• Motivation
• Equivalent Charge Method and Multilayer Greens
  Function Method
• HybCap Method
• Experimental Results
• Conclusion

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Motivation For This Work

• As the technology advances, inter-layer
  dielectrics become more complex
• More layers of dielectrics, etch stops.
• Conformal dielectrics, shields.
• Embedded dielectric and air gaps.
• Previous BEM algorithms for parasitic extraction
  is weak in dealing with the increasing complexity
  of dielectrics
• We propose an effective method to address this
  challenge

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Complex Multilayer Dielectric
Conformal dielectric
M7
M6
Planar dielectric
M5
M4
Conductor
M3
Embedded dielectric
M2
M1
substrate
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Previous Works on Capacitance Extraction

• Library look-up table method
• 2D/2.5D solver
• 3D field solver
• Boundary Element Method (BEM)
• FastCap, pFFT, IES3, HiCap, PHiCap
• Finite Difference/Element Method (FDM/FEM )
• Raphael, HFSS
• Other methods QuickCap (Stochastic)

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BEM on Multilayer Dielectrics

• Equivalent Charge Method
• FastCap
• Fast multipole acceleration
• Kernel dependent (for 1/r kernel only)
• Both planar and conformal dielectrics can be
  handled
• PHiCap
• Hierarchical refinement, fast multipole
  acceleration
• Sparse transformation and preconditioning
• Kernel independent
• Both planar and conformal dielectric can be
  handled
• Multilayer Greens Function Method
• IES3
• Multilayer Greens function is implemented in a
  kernel independent solver
• Only planar dielectric can be handled

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Outline

• Motivation
• Equivalent Charge Method and Multilayer Greens
  Function Method
• HybCap Algorithm
• Experimental Result
• Conclusion

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BEM Flow

• Partition conductor surfaces into n small panels
  A1,, An.
• Assume charge qi on each panel Ai.
• We have linear system Gqv, where q(q1,,qn) is
  the vector of unknown charges, v(v1,,vn) is the
  vector of known panel potential, and G is Greens
  function.

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What is G?

• G(i,j) is the potential on point ri due to the
  unit charge on point rj in the free space

Observation point ri
Linear system
GFqv
Source point rj
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Multilayer Greens Function

• G is the potential on point ri due to the unit
  charge on point rj in the multilayer dielectrics

Observation point ri
?
Linear system
GMqv
?
Requirement Kernel Independent Solver
?
Source point rj
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Multilayer Greens Function Method
Partition the surface into small panels, AiAn
Compute the potential coefficient matrix GM(i,j)
between panel Ai and Aj
Solve the linear system GMqv, q is the vector of
unknown charges
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Equivalent Charge Method
Observation point ri
Observation point ri
Source point rj
Source point rj
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Equivalent Charge Method

• expresses the potential at point ri due tothe
  unit charge at point rj in free space ,and
  boundary condition ?a Ea ?b Ebmust be
  satisfied.
• The linear system is

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Equivalent Charge Method Flow
Partition the surface into small panels, AiAn
Compute the potential coefficient matrix GF (i,j)
between panel Ai and Aj ,Eij between the panel
and the panel on interfaces
Solve the linear system GFqv, q is the vector of
unknown charges
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Current Boundary Element Methods (BEM)
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Outline

• Motivation
• Equivalent Charge Method and Multilayer Greens
  Function Method
• HybCap Method
• Experimental Results
• Conclusion

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HybCap Algorithm
Partition Complex multilayer dielectric into
planar dielectrics and non-planar dielectrics
Partition the surface into small panels, AiAn
Compute potential coefficient matrix GM(i,j)
between panel Ai and Aj for planar dielectric
Compute electric coefficient matrix E(i,j)
between panel Ai and Aj with GM(i,j) for
non-planar dielectric
Solve the linear system
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Continued

• The linear system
• Note
• For a system with nc conductor panels, ne
  non-planar dielectric interface panels, the
  dimension of matrix G is ncne

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Correctness

• Based on the previous analysis, it is sufficient
  to prove that MGMECM ECM

Potential due to charges on Non-planar dielectric
interfaces
Potential due to charges on planar dielectric
interfaces
Potential due to charges on conductor
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Example Step 1


Conductor
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Example Step 2
Use multilayer Greens function GM(i,j) to
compute potential coefficients between conductor
surfaces.
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Example Step 3
Build electric coefficient matrix for non-planar
dielectrics usingthe multilayer Greens function
GM
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Example Step 4

• Use the kernel independent solver to
  hierarchically build linear system
• Use wavelet method to transform dense matrix to
  sparse matrix
• Use incomplete LU factorization as preconditioner
  to speed up the algorithm
• Solve the linear system withiterative method

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Reflective Bounary Wall and Ground Plane

• Reflective Boundary Wall
• Boundary Condition
• Ground Plane
• Multilayer Greens function can automatically
  handle it

Boundary Wall E0
?a ?G(r , r ?)/?na 0 ?a 1.0 ?b 0.0
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HybCap Advantage

• Employ ECM to model the interfaces of non-planar
  dielectrics
• Employ MGM to model the planar dielectrics and
  the ground plane
• A smaller linear system compared toECM method
  alone is built by multilayer Greens function
• Use the kernel independent solver to solve
• the whole system such as PHiCap which has a
  good preconditioner to speed up the algorithm

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Outline

• Motivation
• Equivalent Charge Method and Multilayer Greens
  Function Method
• HybCap Method
• Experimental Results
• Conclusion

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Experimental Environment

• All experiments are executed on an Intel 3.2GHz
  Linux machine with 8 GB memory

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Planar Dielectric
Air e 1.0
M4
M3
1 ?m
0.4 ?m
M2
M1
substrate
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Result
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48 Conductors
Air e 1.0
M8
2 ?m
M7
M6
M5
0.8 ?m
M4
M3
M2
M1
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Result
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Embedded Dielectric, Reflective Boundary Wall and
Substrate
Air e 1.0
M4
1 ?m
M3
M2
0.4 ?m
M1
Embedded Dielectric
Reflect Wall
Reflect Wall
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Result
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Conclusion

• An efficient hybrid algorithm (HybCap) for
  capacitance extraction is proposed that combines
  equivalent charge method and multilayer Greens
  function method.
• The algorithm can deal with complex dielectric
  structures, ground plane and reflective boundary
  walls.
• The algorithm is accurate and consumes much less
  memory and achieves significant speedup over
  previous best methods.

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Thank You !