Resistance extraction ■Prob| em formulation 口 A simple structure V OLOL R a TWO-terminal structure R= A-B It's a singler value +V. a Multi-terminal(port) structure R14 NXNR matrix ruaT I+ro F. RA2
11 ◼ Problem formulation A simple structure Two-terminal structure Multi-terminal (port) structure Resistance extraction L W H i V L L R i S HW = = = A B + - V i V R i = It’s a single R value NxN R matrix
Resistance extraction ■ EXtraction techniques L 口 Square counting R= R a Analytical approximate formula a For simple corner structure 02-d or 3-D numerical methods For multi-terminal structure current has irreqular distribution a Solve the steady current field for i under given bias voltages Set V1=1, others all zero R13 flowing-out current l R4↑1 lk R R23 R12 R4(/B4 Repeating it with different settings 12
12 ◼ Extraction techniques Square counting Analytical approximate formula ◼ For simple corner structure 2-D or 3-D numerical methods ◼ For multi-terminal structure; current has irregular distribution ◼ Solve the steady current field for i under given bias voltages ◼ Set V1 = 1, others all zero, Resistance extraction L R R W = 1 1 1 k k i R flowing-out current = Repeating it with different settings
Resistance extraction EXtraction techniques -numerical method D How to calculate the flowing-out current? Field solver a Field equation and boundary conditions Laplace equation inside conductor: auauau V·oⅴu=0 0 OX divergence E Boundary conditions: port surface I uk: u is known Normal component other surface: E is zero: current can not flow out 0 The BVP of Laplace equation becomes solvable k 13
13 ◼ Extraction techniques – numerical method How to calculate the flowing-out current ? Field equation and boundary conditions Resistance extraction Normal component is zero; current can not flow out Laplace equation inside conductor: = u 0 222 2 2 2 2 0 uuu u x y z = + + = divergence E Boundary conditions: uk port surface : u is known other surface: 0 n u E n = = The BVP of Laplace equation becomes solvable Field solver
Resistance extraction Numerical methods for resistance extraction a Methods for the BVP of elliptical Pde: Vu=O a Finite difference method i+1,j, k 211,jk a Derivative-> finite difference: ax2 (△x) Generate sparse matrix for ODE and PDE a Finite element method ∑x;9(t) a Express solution with local-support basis functions 1 a construct equation system with Collocation or galerkin method a Widely used for bVP of ode and pde a Boundary element method Only discretize the boundary, calculate boundary value a Generate dense matrix with fewer unknowns For elliptical PDE 14
14 ◼ Numerical methods for resistance extraction Methods for the BVP of elliptical PDE: Finite difference method ◼ Derivative -> finite difference: ◼ Generate sparse matrix; for ODE and PDE Finite element method ◼ Express solution with local-support basis functions ◼ construct equation system with Collocation or Galerkin method ◼ Widely used for BVP of ODE and PDE Boundary element method ◼ Only discretize the boundary, calculate boundary value ◼ Generate dense matrix with fewer unknowns Resistance extraction 2 = u 0 2 1, , , , 1, , 2 2 2 ( ) i j k i j k i j k u u u u x x + − − + → For elliptical PDE
Resistance extraction a Where are expensive numerical methods needed a Complex onchip interconnects rminal a Wire resistivity is not constant a Complex 3d geometry around VIas Terminas Terminal a Substrate coupling resistance in mixed-signal IC supply digital PORTS Current noise fk Eigs ate esiststra te ports o od sir mutated to commect exacted 15
15 ◼ Where are expensive numerical methods needed ? Complex onchip interconnects: ◼ Wire resistivity is not constant ◼ Complex 3D geometry around vias Substrate coupling resistance in mixed-signal IC Resistance extraction