Resistance extraction ■Prob| em formulation 口Asmp| e structure L L R s HW a TWo-terminal structure R A-B It's a single R value a Multi-terminal(port) structure R14 NxN R matrix R23 R12 R24 , 2 11
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 W a Analytical approximate formula u For simple corner structure a2-D or 3-D numerical methods For multi-terminal structure; current has irregular distribution u Solve the steady current field for i under given bias voltages a Set v=1. others all zero R13 flowing-out current l1k R14 Ru R12 R34 r, 3 R24 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 a How to calculate the flowing-out current Field solver a Field equation and boundary conditions Laplace equation inside conductor: d-u dudu 0 u 0 2 divergence E Boundary conditions port surface uk: u is known Normal component other surface: E is zero: current can 0 not flow out 0 The BvP of Laplace equation becomes solvable Rk/a 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 21k+-1k a Derivative->finite difference (△x) Generate sparse matrix; for OdE and PDE 口 Finite element method ∑x;9(t) a EXpress solution with local-support basis functions 1 a construct equation system with 1.0 Collocation or galerkin method Widely used for BVP of ODE and PDe .oK 2 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 Where are expensive numerical methods needed? a Complex onchip interconnects Terminal a Wire resistivity is not constant Complex 3d geometry around Vlas Ie a Substrate coupling resistance in mixed-signal IC PORTS Current noise flow Fig. 1. Current is injected into the substrate and flows to other parts Fig. 2. Substrate ports (nodes) created to connect extm of the circuit substate resistive network to the circuit. 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