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

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 on wit ith Fl Floatin ing g Rando dom m Walk Algo gori rithm thm Hao Zhuang 1, 2 , Wenjian Yu 1 *, Gang Hu 1 , Zuochang Ye 3 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

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) 4

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 orithm 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 6

3-D FR FRW W Algo gorit rithm hm for r Cap apac acit itan ance ce Extracti raction on  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. 7

3-D D FRW Al Algorit orithm hm for Ca Capacit itan ance ce Extraction ction Gi Fig. Transition domain’s PDF pre-computed 8

3-D D FRW W Alg lgor orithm 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 9

3-D D FRW W Alg lgor orithm 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  Lost efficiency in 5~10 layers of dielectrics walk stops frequently approaching dielectric interface increase hops!  Interface is really a problem Only equation we can use analytically 10

3-D D FRW W Alg lgor orithm 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 11

3-D D FRW W Alg lgor orithm ithm for Cap apacitan acitance ce Ex Extr traction action  The modified FRW in this paper Gaussian (Algo2) Surface  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 12

3-D D FRW W Alg lgor orithm ithm for Cap apacitan acitance ce Ex Extr traction action  The modified FRW in this paper Gaussian (Algo2) Surface  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 13

3-D D FRW W Alg lgor orithm ithm for Cap apacitan acitance ce Ex Extr traction action  The modified FRW in this paper Gaussian (Algo2) Surface  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 Store them in GFTs Question: How can we get the probability for transition? 14

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 15

Numerical characterization of multi- layer Green’s functions  Problem Formulation  Free charge space  Interface with continuous condition  Use Finite Difference method 16

Numerical characterization of multi- layer Green’s functions Inner grids  Matrix Formulation Boundary points Boundary condition  Potential value at inner grids Points reside at interface 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 17

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) u 0 : eq(13)  And the coefficient on interface 18

Numerical characterization of multi-layer Green’s functions  The situation when walk hits the interface requires interface in the middle layer of domain 19

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 20

FDM & FRW’s numerical result PDF Distribution solved by FDM 21

FDM & FRW Numerical Results The efficiency of FDM  Comparison with the same solver utilized by CAPEM* 4X Speedups * M. P . Desai, “The Capacitance Extraction Tool,” http://www.ee.iitb.ac.in/~microel/download. 22

FDM & FRW’s Numerical Results FRW results Compared to Algo1 41 wires in the 3 layers Placed in the brown zone h  The3 layers belongs to 5 The3 layers belongs to 9 layers layers without thin dielectrics without thin dielectrics 2.1X Speedups 3.5X Speedups  Increase only 6MB memory overhead 23

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 24

The END Thank you Q&A