Jarosław Arabas 1 , Łukasz Bartnik 1 , Sławomir Szostak 2 , Daniel Tomaszewski 3 1 Institute of Electronic Systems 2 Institute of Microelectronics and Optoelectronics 3 Institute of Electron Technology. http://www.ise.pw.edu.pl http://www.imio.pw.edu.pl http://www.ite.waw.pl. - PowerPoint PPT Presentation
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MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009 Extraction of the EKV model parameters: selected aspects of the underlying optimization task Jarosław Arabas 1 , Łukasz Bartnik 1 , Sławomir Szostak 2 , Daniel Tomaszewski 3 1 Institute of Electronic Systems 2 Institute of Microelectronics and Optoelectronics 3 Institute of Electron Technology Warsaw University of Technology: http:// www.ise.pw.edu.pl http:// www.imio.pw.edu.pl http://www.ite.waw.pl
Transcript
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
Extraction of the EKV model parameters:selected aspects of the underlying
optimization task
Jarosław Arabas1, Łukasz Bartnik1, Sławomir Szostak2, Daniel Tomaszewski3
1 Institute of Electronic Systems2 Institute of Microelectronics and Optoelectronics
3 Institute of Electron Technology
Warsaw University of Technology:
http://www.ise.pw.edu.plhttp://www.imio.pw.edu.pl
http://www.ite.waw.pl
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
Outline
1. Task specification
2. Comparison of parameter extraction methods
a) Random sampling
b) Local minimization starting from randomly selected point
c) Evolutionary algorithm
d) Evolutionary algorithm & local minimization
3. Comparison of results
4. Evolutionary algorithm & local minimization for disturbed I-V data
5. Summary
6. Future work
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
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).
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
Reference data generated by the EKV model with the different sets of parameters, e.g.:
Task specification
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 = 4.0e+6
VGS in a range (0.0, 5.0)
VDS in a range (0.0, 5.0)
VBS in a range (−5.0, 0.0)
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
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
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
Random sampling
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PHI KP
THETA log10UCRIT
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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MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
Nelder-Mead's (NM) algorithm (J.A. Nelder,R. Mead, A simplex method for function minimization,The Computer Journal,pp.308–313, 1965)
A direct search of mse minimum:the (n+1)-vertices simplex in n-D space creeps through the domain, and is subjected to the following operations:
Local minimization startingfrom randomly selected point
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) = (x2+y-11)2 + (x+y2-7)2
• reflection
• expansion
Stops at local minimum or "plateau" of objective function.
The method is non-gradient, easy to implement
• contraction
• shrinking
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
Local minimization starting from randomlyselected point
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Starting point selected randomly according to uniform distribution.
Better quality of optimization results.
Significant correlation of extracteded parameters VTO, KP, THETA with mse.
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
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
Jarosław Arabas, "Lecture notes on evolutionary computation", 2nd ed., WNT, Warszawa, 2004 (in Polish)
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
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Population size: 15 individuals, number of generations: 250.
First population selected randomly according to uniform distribution.
Results quality: intermediate.
Weak correlation of extracted parameters with mse.
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Evolutionary algorithm (EA)
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
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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
globaloptimum
The best point of EA & local minimization
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
The best point of EA & local minimization
start best worst
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500 executions of EA & NM tasks
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
Comparison of results
randomNelder-Mead
Evolutionary algorithm
EA + NMNM
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MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
Comparison of resultsRandom 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
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
EA & NM for disturbed I-V dataMeasurement 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%)
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
EA & NM for disturbed I-V data
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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, particularly:PHI and UCRIT
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
EA & NM for disturbed I-V data
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Results of parameter extraction for disturbed reference data(std dev. of error: 0.5%)
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
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)
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
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
MOS-AK meeting IHP, Frankfurt(Oder), 3rd April 2009
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.
