The floating random walk (FRW) algorithm has several advantages for extracting interconnect capacitance. However, for multi-layer dielectrics in VLSI technology, the efficiency of FRW algorithm would be degraded due to the frequent stop of walk at dielectric interface. In this paper, an approach is proposed to calculate multi-dielectric Green's function, which is utilized to enable hops across dielectric interface in the FRW. Numerical results show that the proposed approach is about 4X faster than an existing method, and brings several times speedup to the FRW-based capacitance extraction for actual multi-dielectric interconnect structures.
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RWCap ASCION2011
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 Greenâs Function for 3-D Capacitance Extraction with Floating Random Walk Algorithm
2. Outline
īBackground
ī3-D Floating Random Walk Algorithm for Capacitance Extraction
īNumerical characterization of multi-layer Greenâs functions by FDM
īFDM & FRWâs Numerical Results
īConclusions
2
3. Background
īField Solver on Capacitance Extraction based on
īDiscretization-based method (like FastCap):
īfast and accurate
īnot scalable to large structure due to
īthe large demand of computational time or
īthe bottleneck of memory usage.
īDiscretization-free method
īlike Floating Random Walk Algorithm (FRW) in this paper
īAdvantages:
īlower memory usage
īmore scalability for large structures and
ītunable accuracy
īFRW algorithm evolved to commercial capacitance solvers like QuickCap of Magma Inc.
īRecent advances for variation-aware capacitance extraction [ICCAD09] by MIT
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4. Backgrounds
īChallenges
īLittle literature reveals the algorithm details of the 3-D FRW for multi-dielectric capacitance extraction.
īCAPEM is a FRW solver to deal with these problems, but not published and only binary code available.
īRecently, weâve developed FRW to handle multi-dielectric structure, by sphere transition domain to go across dielectrics interface [another article in ASICONâ12].
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
īBackground
ī3-D Floating Random Walk Algorithm for Capacitance Extraction
īNumerical characterization of multi-layer Greenâs functions
īFDM & FRWâs Numerical Results
īConclusions
5
6. 3-D FRW Algorithm for Capacitance Extraction
īFundamental formula is potential calculation,
is the electric potential on point r, S is a closed surface surrounding r. is called the Greenâs function,
īRecursion to express
īCan be solved by Monte Carlo (MC) Integration
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7. 3-D FRW Algorithm for Capacitance Extraction
īFor 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
īTransform (3),obtain
is weight function.
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8. 3-D FRW Algorithm for Capacitance Extraction
Fig. Transition domainâs PDF pre-computed
Gi
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9. 3-D FRW Algorithm for Capacitance Extraction
īIt is a homogeneous case in last slide. To my best of knowledge, the analytical equation for transition domain with dielectrics is not available.
īRecently, The FRW weâve 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
īLost efficiency in 5~10 layers of dielectrics
īInterface is really a problem
Gaussian Surface
walk stops frequently approaching dielectric interface
increase hops!
Only equation we can use analytically
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īIt is a homogeneous case in last slide. To my best of knowledge, the analytical equation for transition domain with dielectrics is not available.
īRecently, The FRW weâve 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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īThe modified FRW in this paper (Algo2)
īPre-characterize the transition domain by Greenâs Function (GF) to obtain transition probability
īand store them in GF Tables
īto aid random walk to cross the interface
12. 3-D FRW Algorithm for Capacitance Extraction
īThe modified FRW in this paper (Algo2)
īPre-characterize the transition domain by Greenâs Function (GF) to obtain transition probability
īand store them in GF Tables
īto aid random walk to cross the interface
īFinite Set V.S infinite online walk
ī Mismatch?
Store them in GFTs
Gaussian Surface
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13. 3-D FRW Algorithm for Capacitance Extraction
īThe modified FRW in this paper (Algo2)
īPre-characterize the transition domain by Greenâs Function (GF) to obtain transition probability
īand store them in GF Tables
īto aid random walk to cross the interface
īMismatch? Shrink the size of domain
īTrade-off between memory & speed
Store them in GFTs
Gaussian Surface
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14. 3-D FRW Algorithm for Capacitance Extraction
īThe modified FRW in this paper (Algo2)
īPre-characterize the transition domain by Greenâs Function (GF) to obtain transition probability
īand store them in GF Tables
īto aid random walk to cross the interface
īMismatch? Shrink the size of domain
īTrade-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
īBackground
ī3-D Floating Random Walk Algorithm for Capacitance Extraction
īNumerical characterization of multi-layer Greenâs functions
īFDM & FRWâs Numerical Results
īConclusions
15
16. Numerical characterization of multi-layer Greenâs functions
īProblem Formulation
īFree charge space
īInterface with continuous condition
īUse Finite Difference method
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17. Numerical characterization of multi-layer Greenâs functions
īMatrix Formulation
īPotential value at inner grids
īThe k-th gridâs potential by multiple a vector with 1 in k-th position and 0 (otherwise)
īEliminate 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 Greenâs functions
īCoefficient of inner grids and continuous condition to avoid mismatch of numeric error order
ī(a) use normal 7 point scheme
ī(b) eq(12)
ī(c) u0: eq(13)
īAnd the coefficient on interface
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19. Numerical characterization of multi-layer Greenâs functions
īThe situation when walk hits the interface requires interface in the middle layer of domain
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20. Outline
īBackground
ī3-D Floating Random Walk Algorithm for Capacitance Extraction
īNumerical characterization of multi-layer Greenâs functions
īFDM & FRWâs Numerical Results
īConclusions
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21. FDM & FRWâs numerical result PDF Distribution solved by FDM
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22. FDM & FRW Numerical Results The efficiency of FDM
īComparison with the same solver utilized by CAPEM*
* M. P. Desai, âThe Capacitance Extraction Tool,â http://www.ee.iitb.ac.in/~microel/download.
4X Speedups
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23. FDM & FRWâs Numerical Results FRW results Compared to Algo1
īThe3 layers belongs to 5 layers without thin dielectrics
2.1X Speedups
h
The3 layers belongs to 9 layers without thin dielectrics
3.5X Speedups
īIncrease only 6MB memory overhead
41 wires in the 3 layers
Placed in the brown zone
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24. Conclusions
īBy using pre-computed 2-layer Greenâs function for cube transition domain will accelerate FRW in multi-dielectric cases around 2X~4X
īOur generator is faster than CAPEMâs
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