Numer erica ical l Charact cteriz erization ation of Multi - - PowerPoint PPT Presentation

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

Numer erica ical l Charact cteriz erization ation of Multi ti-Di Diel electri ectric c Green’s Function for 3-D Capa pacit itance nce Extracti raction

• n wit

ith Fl Floatin ing g Rando dom m Walk Algo gori rithm thm

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

2

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

3

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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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

5

3-D D FRW W Alg lgor

• rithm

ithm for Cap apacitan acitance ce Ex Extr traction action

 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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3-D FR FRW W Algo gorit rithm hm for r Cap apac acit itan ance ce Extracti raction

• n

 For capacitance problem, set master conductor with 1 volt,

• ther 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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3-D D FRW Al Algorit

• rithm

hm for Ca Capacit itan ance ce Extraction ction

• Fig. Transition domain’s

PDF pre-computed

Gi

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3-D D FRW W Alg lgor

• rithm

ithm for Cap apacitan acitance ce Ex Extr traction action

 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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3-D D FRW W Alg lgor

• rithm

ithm for Cap apacitan acitance ce Ex Extr traction action

 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)

3-D D FRW W Alg lgor

• rithm

ithm for Cap apacitan acitance ce Ex Extr traction action

11

 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

3-D D FRW W Alg lgor

• rithm

ithm for Cap apacitan acitance ce Ex Extr traction action

 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

12

3-D D FRW W Alg lgor

• rithm

ithm for Cap apacitan acitance ce Ex Extr traction action

 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

13

3-D D FRW W Alg lgor

• rithm

ithm for Cap apacitan acitance ce Ex Extr traction action

 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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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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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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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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Numerical characterization of multi-layer Green’s functions

 Coefficient of inner grids and continuous

condition to avoid mismatch of numeric error

• rder

 (a) use normal 7 point scheme  (b) eq(12)  (c) u0: eq(13)

 And the coefficient on interface

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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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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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FDM & FRW’s numerical result

PDF Distribution solved by FDM

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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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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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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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Thank you Q&A

The END