Table Based Models Victor Bourenkov Computational Modelling Group - PowerPoint PPT Presentation

Table Based Models Victor Bourenkov Computational Modelling Group Tyndall National Institute, Cork, Ireland Kevin G. McCarthy Department of Electrical and Electronic Engineering University College Cork, Ireland

Outline • Table look-up models – Interpolation methods – Generation of data tables • SPICE implementation and performance • Further developments and critique of the approach • Summary/conclusion Table Based Models MOS-AK Grenoble 16.09.2005 2

Table look-up models i 0 1 2 3 4 5 6 V j 0 0.1 0.5 1.0 1.5 2.5 3.5 V DS GS Model setup 5 × 10 -7 6 × 10 -7 7 × 10 -7 8 × 10 -7 1 × 10 -6 2 × 10 -6 0 0.3 0.0 6 × 10 -5 7 × 10 -5 8 × 10 -5 9 × 10 -5 1 × 10 -4 2 × 10 -4 1 0.5 0.0 2 0.7 0.0 0.0014 0.0020 0.0023 0.0025 0.0028 0.0032 Tables of electrical 3 0.9 0.0 0.004 0.0082 0.0089 0.0094 0.010 0.011 characteristics 4 1.2 0.0 0.007 0.020 0.022 0.023 0.025 0.027 5 1.8 0.0 0.011 0.042 0.055 0.058 0.062 0.065 6 2.5 0.0 0.014 0.059 0.09 0.10 0.11 0.11 Search function 7 3.5 0.0 0.017 0.073 0.12 0.15 0.17 0.18 Table of I DS , ids[i][j] Interpolation routines Given bias values search for V DS =1.7 V, V GS =1.5 V nearest table entries i=4, j=4 Interface to simulator Interpolate I DS (V DS ,V GS ) Table Based Models MOS-AK Grenoble 16.09.2005 3

Interpolation method requirements • Compatible with the Newton-Raphson algorithm – Continuous – Preserve monotonicity of data (non-oscillatory) – Preferably C 1 smooth (continuous derivatives) or better • Accurate • Fast • Optimal memory usage • Easy to understand Table Based Models MOS-AK Grenoble 16.09.2005 4

Interpolation and approximation methods • Polynomial interpolation – Linear – Quadratic – Exponential • Variation diminishing B-spline approximation • Combined interpolations • Other interpolations – Spline interpolations – Variation diminishing interpolations (ENO) Table Based Models MOS-AK Grenoble 16.09.2005 5

Linear interpolation • Advantages – Computationally simple – Preserves monotonicity of data – Accuracy is easily controlled by table density • Disadvantages – Discontinuous first derivatives – Relatively large tables are needed for good accuracy − V V = + − i I ( V ) I ( I I 1 ) + − DS i i i V V + i 1 i Table Based Models MOS-AK Grenoble 16.09.2005 6

Quadratic interpolation • Advantages – More accurate than linear interpolation – Accuracy is easily controlled by table density – Control of derivative continuity • Disadvantages − − ( V V )( V V ) = + + i 1 i 2 L ( V ) – Not guaranteed to be − − 0 , i ( V V )( V V ) + + i 2 i i 1 i monotonic − − ( V V )( V V ) = + i i 2 L ( V ) − − 1 , i ( V V )( V V ) – Slower than linear interpolation + + + i 1 i 2 i 1 i − − ( V V )( V V ) = + i 1 i L ( V ) − − 2 , i ( V V )( V V ) + + + i 2 i i 2 i 1 = + + I ( V ) I L ( V ) I L ( V ) I L ( V ) + + DS i 0 , i i 1 1 , i i 2 2 , i Table Based Models MOS-AK Grenoble 16.09.2005 7

Exponential interpolation • Advantages – Preserves monotonicity of data – Very good fit to experimental data • Disadvantages – Computationally expensive – Discontinuous first derivatives − V V I + i i 1 ln( ) + − = V V I I ( V ) I e i 1 i i DS i Table Based Models MOS-AK Grenoble 16.09.2005 8

B-spline approximation • Advantages – Continuous first derivative – Preserves monotonicity of data – Accuracy is easily controlled by table density • Disadvantages – Slower than linear or quadratic interpolations – Approximation is not as accurate as interpolation = + + I ( V ) I B ( V ) I B ( V ) I B ( V ) + + + + DS i i , t i 1 i 1 , t i 2 i 2 , t Table Based Models MOS-AK Grenoble 16.09.2005 9

Combined interpolation • 1. Subthreshold region: exponential interpolation f EXP V ( ) 3 GS • 3. Strong inversion: linear 2 (quadratic) interpolation f LIN V ( ) GS • 2. Transition region: blending function [#] 1 − V V ( V , V ) µ = GS TH BS DS ( V ) ∆ GS = − µ + µ I ( V ) ( 1 ( V )) f ( V ) ( V ) f ( V ) DS GS GS EXP GS GS LIN GS [#] V.Bourenkov, K. G. McCarthy, A. Mathewson. ICMTS 2003 Table Based Models MOS-AK Grenoble 16.09.2005 10

Other interpolations • Cubic spline interpolation – Smooth first derivatives – May oscillate, computationally expensive • Bicubic interpolation (in 2D) – Monotonic, continuous first derivatives – Complex implementation for 3D • Essentially Non-Oscillatory approximation # – Monotonic, continuous first derivatives – Complex implementation [#] B. Yang, B. McGaughy. DAC 2004 Table Based Models MOS-AK Grenoble 16.09.2005 11

3D Interpolation • MOSFET is a four-terminal device • Device characteristics are functions of three relative voltages • Three-dimensional tables to store measured data • Three-dimensional interpolation Table Based Models MOS-AK Grenoble 16.09.2005 12

Extrapolation • “Phantom vertices” method – Linear extrapolation in strong inversion – Exponential extrapolation in weak inversion Table Based Models MOS-AK Grenoble 16.09.2005 13

Generation of data tables (I) • Measurements • Device simulations • Analytical compact models Measure DC currents for different bias conditions D G B I D I B + + - V V V V G S B D S - - + Table Based Models MOS-AK Grenoble 16.09.2005 14

Generation of data tables (II) Extraction of terminal charges From analytical model From transient analysis (QS) # From DC and s-parameter 1 dQ dQ dv = + = = i cond ( v ) [ i ( t ( v )) i ( t ( v ))] i cap ( v ) 1 2 measurements (NQS) @ 2 dt dv dt 1 ∫ v = + 0 = − Q ( v ) Q i ( u ) du i cap ( v ) [ i ( t ( v )) i ( t ( v ))] 0 cap 1 2 v 2 [#] G. Schrom, A. Stach, S. Selberherr. Microelectronics Jornal, 1998. [@] M. F. Barciela et al . IEEE Tran. On Microwave Theory and Technics, 2000. Table Based Models MOS-AK Grenoble 16.09.2005 15

Channel geometry scaling • Inter-table interpolation • Linear interpolation in W dimension • Quadratic interpolation in L dimension Table Based Models MOS-AK Grenoble 16.09.2005 16

Outline • Table look-up models – Interpolation methods – Generation of data tables • SPICE implementation and performance • Further developments and critique of the approach • Summary/conclusion Table Based Models MOS-AK Grenoble 16.09.2005 17

SPICE implementation Setup Initial operating point Main device model routines Load DEV.c Solve linear DEVpar.c matrix equations DEVmpar.c DEVsetup.c Convergence ? DEVload.c Yes DEVacload.c Increment time DEVcvtest.c DEVask.c End of time interval ? DEVmask.c Yes Output [#] V.Bourenkov, K. G. McCarthy, A. Mathewson. Electrosoft V (2001) Table Based Models MOS-AK Grenoble 16.09.2005 18

Performance Model run time per transistor 90 80 70 Table model: linear 1 60 Time ( µ s) Table model: linear 2 50 Table model: quadratic 40 Table model: B-spline 30 BSIM 3v3.2.2 20 10 0 Total simulation time 18 16 14 Simulation time (s) Table model: linear 1 12 Table model: linear 2 10 Table model: quadratic 8 Table model: B-spline 6 BSIM 3v3.2.2 4 2 0 Table Based Models MOS-AK Grenoble 16.09.2005 19

Performance • Accuracy and table size • Memory requirements 1600 1400 Memory Usage, kbytes 1200 Table Model, 1000 14x26x8 Table Model, 800 11x16x8 BSIM3v3.2.2 600 400 200 0 1 8 (4) 17 (7) Number of MOSFETs (and unique geometries) Table Based Models MOS-AK Grenoble 16.09.2005 20

Circuit simulation results (I) • Data tables generated from BSIM3v3.2.2 • Analyses: – CMOS inverter, DC analysis – Ring oscillator, transient analysis – Op-amp, DC and frequency response Table Based Models MOS-AK Grenoble 16.09.2005 21

Circuit simulation results (II) Table Based Models MOS-AK Grenoble 16.09.2005 22

Interpolation of derivatives Table Based Models MOS-AK Grenoble 16.09.2005 23

Outline • Table look-up models – Interpolation methods – Generation of data tables • SPICE implementation and performance • Further developments and critique of the approach • Summary/conclusion Table Based Models MOS-AK Grenoble 16.09.2005 24

Further developments •“Context aware” interpolation • Subcircuit level table models • Hybrid table/analytical approach • Temperature scaling • Noise modelling Table Based Models MOS-AK Grenoble 16.09.2005 25

Subcircuit modelling •Sub-circuits can be represented by table models 3 I 3 (V IN ) I 1 (V IN ) I 2 (V IN ) 1 2 0 Table Based Models MOS-AK Grenoble 16.09.2005 26

Table Model: The good points • Models for new devices can be implemented quickly • Less time-consuming parameter extraction • Fewer errors in implementation • Controllable accuracy – Density of table data and interpolation method • Measurement-based model – no need to change model equations Table Based Models MOS-AK Grenoble 16.09.2005 27