This document presents a 3-D floating random walk algorithm for capacitance extraction, highlighting its advantages over discretization-based methods in terms of memory usage and scalability. The authors detail the numerical characterization of multi-layer Green's functions and present numerical results showing significant speed improvements. The conclusions emphasize that utilizing pre-computed Green's functions can accelerate the capacitance extraction process in multi-dielectric structures.

1. Hao Zhuang1, 2, Wenjian Yu1*, Gang Hu1, Zuochang Ye3
1 Department of Computer Science and Technology, 3 Institute of Microelectronics, Tsinghua University, Beijing, China
2 School of Electronics Engineering and Computer Science,
Peking University, Beijing, China
Speaker: Hao Zhuang
Numerical Characterization of Multi-Dielectric Green’s Function for 3-D Capacitance Extraction with Floating Random Walk Algorithm

2. Outline
Background
3-D Floating Random Walk Algorithm for Capacitance Extraction
Numerical characterization of multi-layer Green’s functions by FDM
FDM & FRW’s Numerical Results
Conclusions
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3. Background
Field Solver on Capacitance Extraction based on
Discretization-based method (like FastCap):
fast and accurate
not scalable to large structure due to
the large demand of computational time or
the bottleneck of memory usage.
Discretization-free method
like Floating Random Walk Algorithm (FRW) in this paper
Advantages:
lower memory usage
more scalability for large structures and
tunable accuracy
FRW algorithm evolved to commercial capacitance solvers like QuickCap of Magma Inc.
Recent advances for variation-aware capacitance extraction [ICCAD09] by MIT
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4. Backgrounds
Challenges
Little literature reveals the algorithm details of the 3-D FRW for multi-dielectric capacitance extraction.
CAPEM is a FRW solver to deal with these problems, but not published and only binary code available.
Recently, we’ve developed FRW to handle multi-dielectric structure, by sphere transition domain to go across dielectrics interface [another article in ASICON’12].
However, extraction of VLSI interconnects embedded in 5~10 layers of dielectrics, the efficiency would be largely lost. (see later in the talk)
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5. Outline
Background
3-D Floating Random Walk Algorithm for Capacitance Extraction
Numerical characterization of multi-layer Green’s functions
FDM & FRW’s Numerical Results
Conclusions
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6. 3-D FRW Algorithm for Capacitance Extraction
Fundamental formula is potential calculation,
is the electric potential on point r, S is a closed surface surrounding r. is called the Green’s function,
Recursion to express
Can be solved by Monte Carlo (MC) Integration
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7. 3-D FRW Algorithm for Capacitance Extraction
For capacitance problem, set master conductor with 1 volt, other with 0 volt, calculate the charge accumulated in conductors,
Gi is the Gaussian surface containing only master conductor inside. D(r) is the field displacement in r, F(r) is dielectric constant at r, n(r) is normal vector at r from Gaussian surface
Transform (3),obtain
is weight function.
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8. 3-D FRW Algorithm for Capacitance Extraction
Fig. Transition domain’s PDF pre-computed
Gi
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9. 3-D FRW Algorithm for Capacitance Extraction
It is a homogeneous case in last slide. To my best of knowledge, the analytical equation for transition domain with dielectrics is not available.
Recently, The FRW we’ve developed handles multi-dielectric structure, by introducing sphere transition domain when hitting interface. (Algo1)
Gaussian Surface
Only equation we can use analytically
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10. 3-D FRW Algorithm for Capacitance Extraction
Lost efficiency in 5~10 layers of dielectrics
Interface is really a problem
Gaussian Surface
walk stops frequently approaching dielectric interface
increase hops!
Only equation we can use analytically
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It is a homogeneous case in last slide. To my best of knowledge, the analytical equation for transition domain with dielectrics is not available.
Recently, The FRW we’ve developed handles multi-dielectric structure, by introducing sphere transition domain when hitting interface. (Algo1)

11. 3-D FRW Algorithm for Capacitance Extraction
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The modified FRW in this paper (Algo2)
Pre-characterize the transition domain by Green’s Function (GF) to obtain transition probability
and store them in GF Tables
to aid random walk to cross the interface

12. 3-D FRW Algorithm for Capacitance Extraction
The modified FRW in this paper (Algo2)
Pre-characterize the transition domain by Green’s Function (GF) to obtain transition probability
and store them in GF Tables
to aid random walk to cross the interface
Finite Set V.S infinite online walk
 Mismatch?
Store them in GFTs
Gaussian Surface
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13. 3-D FRW Algorithm for Capacitance Extraction
The modified FRW in this paper (Algo2)
Pre-characterize the transition domain by Green’s Function (GF) to obtain transition probability
and store them in GF Tables
to aid random walk to cross the interface
Mismatch? Shrink the size of domain
Trade-off between memory & speed
Store them in GFTs
Gaussian Surface
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14. 3-D FRW Algorithm for Capacitance Extraction
The modified FRW in this paper (Algo2)
Pre-characterize the transition domain by Green’s Function (GF) to obtain transition probability
and store them in GF Tables
to aid random walk to cross the interface
Mismatch? Shrink the size of domain
Trade-off between memory & speed
Q
Question: How can we get the probability for transition?
Store them in GFTs
Gaussian Surface
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15. Outline
Background
3-D Floating Random Walk Algorithm for Capacitance Extraction
Numerical characterization of multi-layer Green’s functions
FDM & FRW’s Numerical Results
Conclusions
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16. Numerical characterization of multi-layer Green’s functions
Problem Formulation
Free charge space
Interface with continuous condition
Use Finite Difference method
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17. Numerical characterization of multi-layer Green’s functions
Matrix Formulation
Potential value at inner grids
The k-th grid’s potential by multiple a vector with 1 in k-th position and 0 (otherwise)
Eliminate the boundary condition vector, This is the transition probability we want! It describe the relation between center point and boundary points
Inner grids
Boundary points
Points reside at interface grids
Boundary condition
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18. Numerical characterization of multi-layer Green’s functions
Coefficient of inner grids and continuous condition to avoid mismatch of numeric error order
(a) use normal 7 point scheme
(b) eq(12)
(c) u0: eq(13)
And the coefficient on interface
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19. Numerical characterization of multi-layer Green’s functions
The situation when walk hits the interface requires interface in the middle layer of domain
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20. Outline
Background
3-D Floating Random Walk Algorithm for Capacitance Extraction
Numerical characterization of multi-layer Green’s functions
FDM & FRW’s Numerical Results
Conclusions
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21. FDM & FRW’s numerical result PDF Distribution solved by FDM
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22. FDM & FRW Numerical Results The efficiency of FDM
Comparison with the same solver utilized by CAPEM*
* M. P. Desai, “The Capacitance Extraction Tool,” http://www.ee.iitb.ac.in/~microel/download.
4X Speedups
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23. FDM & FRW’s Numerical Results FRW results Compared to Algo1
The3 layers belongs to 5 layers without thin dielectrics
2.1X Speedups
h
The3 layers belongs to 9 layers without thin dielectrics
3.5X Speedups
Increase only 6MB memory overhead
41 wires in the 3 layers
Placed in the brown zone
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24. Conclusions
By using pre-computed 2-layer Green’s function for cube transition domain will accelerate FRW in multi-dielectric cases around 2X~4X
Our generator is faster than CAPEM’s
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25. Thank you
Q&A
The END