Instytut Technologii Elektronowej Al.Lotnik

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Extraction of the EKV model parametersselected
aspects of the underlying optimization task
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Outline

• Task specification
• Comparison of parameter extraction methods
• Random sampling
• Local minimization starting from randomly
  selected point
• Evolutionary algorithm
• Evolutionary algorithm local minimization
• Comparison of results
• Evolutionary algorithm local minimization for
  disturbed I-V data
• Summary
• Future work

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Task specification

• Given
• MOSFET model EKV
• MOSFET reference electrical characteristics I-V
• measured
• simulated numerically
• generated using this or another compact model
• Information about model parameters (approx.
  values, ranges)
• Objective (well known)
• determine model parameters in order to obtain an
  optimum matching of model and reference
  characteristics

A set of extraction meth.
set of initial parameters
Parameter extraction
MOSFET model equations
parameters
MOSFET electr. characteristics
set of final parameters
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Task specification

• A set of parameters selected for evaluation of
  extraction methods
• VTO, nominal threshold voltage range (0.0 ..
  2.0)
• GAMMA, body effect factor range (0.0 .. 2.0)
• PHI, bulk Fermi potential range (0.0 .. 2.0)
• KP, transconductance parameter range (0.0 ..
  0.001)
• THETA, mobility degradation coeff. range (0.0
  .. 0.2)
• UCRIT, longitudinal critical field range (106
  .. 108)processed in logarithmic scale

Mean Squared Error (mse) function is used to
evaluate a quality of the set of parameters
For the purpose of calculations in optimization
procedures all the parameters are reduced to a
common domain (0.0 .. 1.0). Before putting them
into the EKV model they are transformed into the
original domains. Scaling of the parameters
balances optimization process for parameters of
different range (e.g. VTO vs UCRIT).
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Task specification
Reference data generated by the EKV model with
the different sets of parameters, e.g.
Reference data different numbers of points in
the range of

• VTO 0.647
• GAMMA 0.78
• PHI 0.93
• KP 4.304e-05
• THETA 0.026
• UCRIT 4.0e6

VGS in a range (0.0, 5.0) VDS in a range (0.0,
5.0) VBS in a range (-5.0, 0.0)
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Comparison of parameter extraction methods

• The following extraction methods have been
  considered
• Random sampling
• Local minimization starting from randomly
  selected point
• Evolutionary algorithm
• The best point of evolutionary algorithm local
  minimization

• Methods of results presentation
• "Tornado" projection of mse function in
  multi-dimensional space on a 2-D plane (par,mse)
  each point of the "tornado" represents result of
  a single extraction sequence execution (single
  local minimum or "plateau" of mse ?)
• Histogram of mse (logarithmic scale) its
  location and shape illustrate properties of the
  extraction sequence distribution of sampled
  and/or extracted points, degree of conglomeration
  of resulting points, convergence of method
• Comparison of I-V curves generated by the model
  under consideration with reference data this is
  the most popular metod of fitting estimation

Result The best point generated by the
extraction sequence
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Random sampling
Sampling of parameters according to uniform
distribution. Population size 2500 points. No
correlation of sampled parameter with mse
(exception KP, where a visible correlation was
obtained).
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Local minimization startingfrom randomly
selected point
Nelder-Mead's (NM) algorithm (J.A. Nelder,R.
Mead, A simplex method for function
minimization,The Computer Journal,pp.308313,
1965) A direct search of mse minimumthe
(n1)-vertices simplex in n-D space creeps
through the domain, and is subjected to the
following operations

• reflection
• expansion

• contraction
• shrinking

Stops at local minimum or "plateau" of objective
function. The method is non-gradient, easy to
implement
Nelder-Mead simplex search over the Himmelblau's
function. http//en.wikipedia.org/wiki/Nelder-Mead
_method Himmelblau's function is a multi-modal
function, used to test the performance of
optimization algorithms f(x, y) (x2y-11)2
(xy2-7)2
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Local minimization starting from
randomlyselected point
Starting point selected randomly according to
uniform distribution. Better quality of
optimization results. Significant correlation of
extracteded parameters VTO, KP, THETA with mse.
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Evolutionary algorithm (EA)
Evolutionary algorithm Def. Evolutionary
algorithms (EAs) are population-based
metaheuristic optimization algorithms that use
biology-inspired mechanisms in order to refine a
set of solution candidates iteratively, namely
mutation, crossover, natural selection, and the
fact, that individuals of better fitness have
more children. The advantage of evolutionary
algorithms compared to other optimization methods
is their black box character that makes only
few assumptions about the underlying objective
functions. Furthermore, the definition of
objective functions usually requires lesser
insight to the structure of the problem space
than the manual construction of an admissible
heuristic. EAs therefore perform consistently
well in many different problem categories.
Thomas Weise, "Global Optimization Algorithms
Theory and Application", 2nd ed.,
http//www.it-weise.de/projects/book.pdf Jaroslaw
Arabas, "Lecture notes on evolutionary
computation", 2nd ed., WNT, Warszawa, 2004 (in
Polish)
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Evolutionary algorithm (EA)
Population size 15 individuals, number of
generations 250. First population selected
randomly according to uniform distribution. Result
s quality intermediate. Weak correlation of
extracted parameters with mse.
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The best point of EA local minimization
The best point of EA becomes a starting point for
NM method. The best quality of optimization
results. Two-mode histogram. Significant
correlation of extracteded parameters VTO, GAMMA,
KP, THETA with mse.
local optima
global optimum
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The best point of EA local minimization
500 executions of EA NM tasks
start best worst
0.0 1e-3 2e-3 3e-3
0.0 1e-4 2e-3
0.0 1e-4 2e-3
ID ID
0 1 2 3 4 5
0 1 2 3 4 5
0 1 2 3 4 5
VDS VDS VDS
0.0 5e-6 1e-5 1.5e-5
0.0 5e-6 1e-5 1.5e-5
0.0 1e-4 2e-4
0 1 2 3 4 5
0 1 2 3 4 5
0 1 2 3 4 5
VGS VGS VGS
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Comparison of results
EA
Evolutionary algorithm
number of points (rel.) 0.0 0.2 0.4 0.6
0.8 1.0 1.2
random
Nelder-Mead
R
NM
EA NM
EANM
-20 -15 -10 -5 log10mse
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Comparison of results

• Random sampling reference
• Nelder-Mead
• "funnels" on "tornado" charts are noticeable
  they indicate that local optmization algorithm
  has tried to find optimum
• improved quality of fitting
• Evolutionary algorithm
• average results are better than for random
  sampling however EA has not found absolute
  optimum but located neighbourhoods of local
  optima
• none of the parameters have been priviliged
• EA NM
• correlation of parameters and mse due to
  filtering of the starting points by EA
• the most pronounced conglomeration of the
  parameters ("funnels") extracted by the
  optimization method the method detects optimum
  values of the parameters, hence
• the method detects single global optimum

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EA NM for disturbed I-V data
Measurement data are always burdened by errors.
In order to investigate this effect, the
reference data were generated in such a way, that
voltages as well as currents are independently
randomly disturbed

1. A set ofinput voltages was generated on a
   rectangular grid.
2. Voltages were disturbed using a random variable
   of normal distribution(mean value 0, std dev.
   0.1, 0.5)
3. Drain current of the EKV model was calculated
   using the disturbed voltages
4. Calculated currents were disturbed using a random
   variable of normal distribution(mean value 0,
   std dev. 0.1, 0.5)

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EA NM for disturbed I-V data
mse values obtained for a set of 2197 measurement
points with disturbed data (0.5) there are no
"funnels" characteristic for objective function
with non-disturbed reference data difficulties in
obtaining true values of parameters,
particularlyPHI and UCRIT
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EA NM for disturbed I-V data
0.0 5e-5 1e-4 1.5e-4 2e-4
0.0 1e-5 2e-5 3e-5 4e-5 5e-5
ID
0 1 2 3 4
0 1 2 3 4
VDS
VGS
Results of parameter extraction for disturbed
reference data(std dev. of error 0.5)
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Summary

• A combination of the search method in the
  multi-dimensional space of parameters (e.g. EA
  algorithm) with the local optimization method
  (e.g. NM) seems to be the reliable and efficient
  way to find the unique set of the EKV model
  parameters minimizing the misfit between the
  experimental (disturbed ?) and model I-V data
• The approach is supposed to overcome the problem
  of mutual dependence of parameters, which makes
  questionable the task of their extraction by
  means of optimization
• The proposed approach allows to evaluate any set
  of parameter extraction methods1,2 particularly
  important is a question Where is the extracted
  point located in the enabled space of parameters
  ?
• The approach allows to evaluate a shape of
  objective function and acceptable boundaries of
  parameter ranges
• The approach is valid for a wide class of models
  and objective functions
• 1 M.Bucher, C.Lallement, C.C.Enz, An Efficient
  Parameter Extraction Methodology for the EKV MOST
  Model, Proc.1996 IEEE International Conference on
  Microelectronic Test Structures, Vol.9,
  pp.145-150, 1996
• 2 C.C.Enz, F.Krummenacher, E.A.Vittoz, An
  Analytical MOS Model Valid in All Regions of
  Operation and Dedicated to Low-Voltage and
  Low-Current Applications, Analog Integrated
  Circuits and Signal Processing, 8, pp.83-114
  (1995)

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Future work

• Implementation of a set of local methods for
  fitting of experimental/simulated and model I-V
  characteristics
• Analysis of a "quality" of a starting
  approximation generated by the set of local
  methods
• Project "Extraction of semicondutor devices
  parameters based on global optimization methods
  and compact models" submitted for financing by
  Polish Ministry of Science and Higher Education
• Implementation of EKV3.0 (other MOSFET models ?)
• Implementation of BJT model

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Thank you
Acknowledgments
The authors would like to express a gratitude to
Dr.Wladek Grabinskiand to Prof.Matthias Bucher
for a code of the EKV model as well asfor
support and interest in this work.
Jaroslaw Arabas J.Arabas_at_ise.pw.edu.pl Lukasz
Bartnik lbartnik_at_elka.pw.edu.pl Slawomir
Szostak S.Szostak_at_imio.pw.edu.pl Daniel
Tomaszewski dtomasz_at_ite.waw.pl