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Extraction of the EKV model parameters: selected aspects of the underlying optimization task

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Jarosław Arabas1, Łukasz Bartnik1, Sławomir Szostak2, Daniel Tomaszewski3

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

Warsaw University of Technology:

Extraction of the EKV model parameters:selected aspects of the underlying optimization task

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

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

MOSFET model equations

Parameter extraction

parameters

MOSFET electr. characteristics

set of final parameters

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).

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 = 4.0e+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)

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

log10mse

log10mse

log10mse

log10mse

log10mse

VTO

GAMMA

PHI

KP

number of points

THETA

log10UCRIT

log10mse

Random samplingSampling 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).

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.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:

- 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) = (x2+y-11)2 + (x+y2-7)2

log10mse

log10mse

log10mse

log10mse

log10mse

VTO

GAMMA

PHI

KP

number of points

THETA

log10UCRIT

log10mse

Local minimization starting from randomlyselected pointStarting point selected randomly according to uniform distribution.

Better quality of optimization results.

Significant correlation of extracteded parameters VTO, KP, THETA with mse.

Evolutionary algorithm (EA)

Evolutionary algorithm

Def. Evolutionary algorithms (EAs) arepopulation-based metaheuristic optimization algorithms that use biology-inspired mechanisms in order to refinea set of solution candidates iteratively, namelymutation, crossover, natural selection, and the fact, that individuals of better fitness have more children.

The advantage of evolutionary algorithms compared to other optimization methods istheir “black box” character that makes only few assumptions about the underlying objectivefunctions. Furthermore, the definition of objective functions usually requires lesser insight tothe 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)

log10mse

log10mse

log10mse

log10mse

log10mse

VTO

GAMMA

PHI

KP

number of points

THETA

log10UCRIT

log10mse

Evolutionary algorithm (EA)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.

log10mse

log10mse

log10mse

log10mse

log10mse

VTO

GAMMA

PHI

KP

number of points

THETA

log10UCRIT

log10mse

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

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

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

EA+NM

-20 -15 -10 -5

log10mse

- 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

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:

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

log10mse

log10mse

log10mse

log10mse

log10mse

VTO

GAMMA

PHI

KP

THETA

log10UCRIT

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, particularly:PHI and UCRIT

EA & NM for disturbed I-V data

0.0 1e-5 2e-5 3e-5 4e-5 5e-5

0.0 5e-5 1e-4 1.5e-4 2e-4

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%)

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)

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

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.

Jarosław Arabas [email protected]Łukasz Bartnik [email protected]ławomir Szostak [email protected] Daniel Tomaszewski [email protected]

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