Recently, genetic algorithms have been applied for ultrafast optical spectrometry in systems with several convoluted lifetimes. We apply these algorithms and compare the results with POSFIT (by Kirkegaard and Eldrup) and LT programme (by Kansy). The analysis was applied to three types of samples: molybdenum monocrystals, Czochralski-grown silicon with oxygen precipitates, Si with under-surface cavities obtained by He + H ion co- implantation. In all three tests, the genetic algorithm performs very well, in particular for short lifetimes. Further developments to model the resolution function in genetic algorithms are needed

Brusa, R.S.

Fisz, J. J.

Kansy, Jerzy

Karbowski, A.

Karwasz, G.P.

Acta Physica Polonica A

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Genetic algorithms for positron lifetime data  Author: A. Karbowski, J.J. Fisz, G.P. Karwasz, Jerzy Kansy, R.S. Brusa  Citation style: Karbowski A., Fisz J.J., Karwasz G.P., Kansy Jerzy, Brusa R.S. (2008). Genetic algorithms for positron lifetime data. "Acta Physica Polonica A" (Vol. 113, nr 5 (2008), s. 1365-1372).  Vol. 113 (2008) ACTA PHYSICA POLONICA A No. 5Proceedings of the 37th Polish Seminar on Positron Annihilation, La¸dek-Zdro´j 2007Genetic Algorithms for Positron LifetimeDataA. Karbowskia, J.J. Fisza, G.P. Karwasza,∗, J. Kansyband R.S. BrusacaInstitute of Physics, Nicolaus Copernicus UniversityGrudzia¸dzka 5, 87-100 Torun´, PolandbInstitute of Physics, Silesian UniversityBankowa 12, 40- 007 Katowice, PolandcDipartimento di Fisica, Universita´ di TrentoVia Sommarive 14, I-38050 Povo, Trento, ItalyRecently, genetic algorithms have been applied for ultrafast optical spec-trometry in systems with several convoluted lifetimes. We apply these algo-rithms and compare the results with POSFIT (by Kirkegaard and Eldrup)and LT programme (by Kansy). The analysis was applied to three types ofsamples: molybdenum monocrystals, Czochralski-grown silicon with oxygenprecipitates, Si with under-surface cavities obtained by He + H ion co- im-plantation. In all three tests, the genetic algorithm performs very well, inparticular for short lifetimes. Further developments to model the resolutionfunction in genetic algorithms are needed.PACS numbers: 71.60.+z, 78.70.Bj, 71.55.Cn1. IntroductionPositron annihilation techniques are a versatile tool for identifying the typeof defects in semiconductors, see e.g. [1]. Values of lifetimes for pure and defectedelements can be calculated with high precision, see e.g. [2] for Si. On the otherhand, numerous measurements [3–5] still differ in precise determination of theselifetimes. One reason for the discrepancies are differences in experimental setups,another can be details of the numerical procedures used for analysis of the lifetimespectra.Recently [6], genetic algorithms have been applied to the analysis of mul-ticomponent fluorescence spectra in complex samples. In this work we develop∗corresponding author; e-mail: karwasz@fizyka.umk.pl(1365)1366 A. Karbowski et al.procedures for applying these algorithms for positron lifetime spectroscopy. Ge-netic algorithms have been recently applied to numerous scientific subjects, rangingfrom catalyst design [7], protein structures [8] to search for extra-solar planets [9].As far as we know, no attempt has been to apply these algorithms for positronlifetime spectra. Therefore, we test this method for three cross-check cases:1) lifetime spectra in pure, defect free monocrystal molybdenum samples mea-sured with good (169 ps) resolution [10];2) lifetime spectra in high-purity Czochralski-grown silicon (Cz-Si) in whichoxygen precipitates were induced by thermal treating at high hydrostaticpressures; our previous analysis of these spectra [11] with POSFIT method[12] indicated presence of nano-voids with lifetimes in a broad range (0.5–1.1 ns)3) silicon samples in which nanocavities were formed by He+ and successiveH+ ion implantation [13]; these spectra were obtained with the use of depth-profiling (positron beam) technique, with about 250 ps resolution.Present method of analysis of lifetime spectra combines the genetic algorithm withmultiple linear regression (GA-MLR) method. For all three types of samples wecompare GA-MLR results with two other methods traditionally used in positronlifetime analysis, POSFIT [12] by Kirkegaard and Eldrup and LT by one of presentauthors [14].2. Numerical methodThe POSFIT package [12] the most widely used in lifetime measurements isbased on the convolution method, i.e. constructs the trial function from expectedguessed lifetimes and the expected instrument function. The LT allows analy-sis of con-tinuous distributions of lifetimes and is based on the de-convolutionapproach [14, 15].The GA-MLR optimizer has been designed [6] for the nonlinear least-squaresproblems, in which the model function are linear combinations of nonlinear func-tions. GA optimizes the nonlinear parameters and the linear parameters are cal-culated from multiple linear regression. In general, optimizing by GA consists innarrowing the range of the variability of the fitted parameters in a way in whichbiological populations are selected, i.e. by employing two genetic operators —crossover and mutation. GA-MLR is an intuitive optimization approach and itexploits all advantages of the genetic algorithm technique. The MLR method isembedded in the GA optimizer and linear and nonlinear model parameters areoptimized in parallel.In the GA-MLR optimization approach, the GA optimizer provides a trialpopulation of vectors q (i.e., a trial population of the complete sets of the nonlinearparameters in Fk(q, x) (k = 1, . . . ,M). Each trial vector q returns the correspond-ing χ2(q, s) function of vector s of the combination parameters ak standing at theGenetic Algorithms for Positron Lifetime Data 1367corresponding function Fk(q, x), namelyχ2(q, s) =1vfN∑i=11σ2i[yi −M∑k=1akFk(q, x)]2, (1)where vf is the number of freedom.All ak can be determined from the MLR method, i.e. from the minimum-value condition of χ2(q, s) with respect to s, that is∂χ2(q, s)∂s= 0. (2)This condition leads to the following set of equations:M∑k=1[N∑i=1Fj(q, xi)Fk(q, xi)σ2i]ak −N∑i=1yiFj(q, xi)σ2i= 0, (3)where j = 1, . . . ,M , which is equivalent to the matrix equationAs = c, (4)where s is a column vector of the amplitudes ak and ,Ajk =N∑i=1Fk(q, xi)Fj(q, xi)σ2i, cj =N∑i=1yiFj(q, xi)σ2i. (5)A formal solution to Eq. (4) readss = A−1c. (6)Substituting the obtained values of the amplitudes ak into Eq. (1) andrepeating this procedure over all trial vectors q, the trial set of the values ofχ2(q, s) is obtained. As temporarily the “best” vectors, q and s in the populationare considered to be those that correspond to the lowest χ2 value in the set of allχ2(q, s) obtained. The GA optimizer keeps for further reproduction those vectorsq in the population that, together with the corresponding vectors s (estimatedfrom the MLR method), return better χ2(q, s) values. The less “fit” vectors qare replaced by new randomly generated individuals. This procedure is repeatediteratively χ2 until reaches a predefined tolerance or the number of iterations(generations) reaches its predefined maximum. Finally, one obtains the coordinatesof a probable minimum value χ2(q, s) and its coordinates qm and sm in the spaceof the allowed values of q and s.This method of optimization is similar both in LT [14] and GA-MLR, withthe difference that LT uses the double methodology, i.e. the non-linear parametersby the MINSQ procedure [14] and the linear ones outside MINSQ from Eqs. (5).In LT the values of the initial parameters (vector q) are postulated and for thesevalues optimum parameters of ai (vector s) are chosen from Eq. (6). For s obtainedin this way MINSQ calculates new values of q. Then, from Eq. (6) new optimumvalues of s are calculated until a satisfactory value of χ2 is obtained. In the secondstage all parameters (i.e. both linear and non-linear) are treated as non-linear, andfitted by MINSQ.1368 A. Karbowski et al.The implementation of GA-MLR developed for fluorescence spectroscopy [6]to positron lifetime spectra finds an essential difference with POSFIT and LT asfar as the resolution function is considered. The GA-MLR program, working withmuch slower signals assumes the spectrometer resolution function infinitely narrow.We mimic the resolution function in the GA-MLR program using a “laser pulse”option, in the first approximation as a single Gaussian, with an adjustable position,width, and amplitude. The convolution with the experimental spectrum is doneoutside the GA-MLR program. Obviously, this does not correspond strictly tothe parameters of the resolution function as defined in POSFIT or LT programsbut, as proved by present results, it works rather well, in particular with shortlifetimes.3. ExperimentalSpectra of Mo and Cz-Si were obtained at Trento laboratory with a fast-fastcoincidence spectrometer with about 169 ps resolution [10]. The total countingrate was about 1.5–2.0 × 106 for Mo and 1.1 × 106 for Si samples; for the lattercase the statistics was improved by performing several independent runs with thesame sample [11].Data on He+H implanted Si coming from the spectrometer at UniversittBundeswehr in Mnchen was used with about 230 ps resolution [13]. This spec-trometer is based on the beam setup and uses the bunching pulse as the startsignal. Positrons probe different depths of the material, according to their im-plantation energy. The total counting rate for these measurements was about1.2× 106.Samples of Cz-Si were obtained from the same wafer with initial oxygen con-centration 1.1× 1018 cm−3. All samples underwent a double “oxygen-nucleation”treatment, first at 450◦C for 20 h, then 20 h at 650◦C and an additional treatmentunder high pressure at 870–950◦C. Details of the thermal treatments are given inRef. [11].Implanted samples were obtained from p-type (100) Si (1.7–2.5 Ω cm),Czochralski-grown (about 6.6 × 1017 cm−3 oxygen content) wafers, co-implantedat room temperature with He+ ions at 30 keV with a dose of 1 × 1016 ions/cm2and successively with H+ ions at 24 keV with a dose of 1×1016 ions/cm2. Detailsof the thermal treatment are given in Ref. [10].4. ResultsFor molybdenum samples all three programs give similar τ1 values (seeTable I), slightly overestimating the theoretical value for the defect-free metal(119 ps); the values from GA are closest to the theoretical value. On the otherhand, POSFIT performs the best as far as the source background contribution isconcerns — this package has no problems in finding correctly the 383± 0.5 ps life-time for kapton with the exact 4% contribution. Both GA-MLR and LT programstend to overestimate the τ2 lifetime (compensating it by a lower I2 intensity).Genetic Algorithms for Positron Lifetime Data 1369TABLE IThe comparison of the lifetimescalculated for molybdenum inthe following programs: POS-FIT (first row for every sample),LT (second row) and GA-MLR(third row) method. FWHM =170 ps, the source contribution isestimated at 11% — 382 ps, GA-MLR shift — 6 channels.Sample τ1 [ps] τ2 [ps] I2 [%]M001 119.3 382.5 4.4119.8 408.7 3.6119.6 478.2 2.6M002 122.2 383.3 3.5122.7 433.1 2.8121.5 430.7 2.3M003 122.1 382.7 3.4123.1 457.0 2.4122.6 487.7 2.5M004 120.1 383.5 4.8120.3 394.5 3.7118.6 406.0 3.0In Table II we show lifetimes for Cz-Si samples. LT and GA-MLR programsgive similar results, in general not much different also from POSFIT analysis.All programs give values of τ1 ranging from about 226 to 231 ps. This is higherthan the theoretical value of 220 ps for non-defected Si [2]. This difference should,probably, be attributed to presence of oxygen-decorated defects. Both LT and GA-MLR give similar values for τ2, somewhat higher than POSFIT. In compensation,POSFIT gives higher intensity I2 of the second lifetime.For He–H implanted Si, in spite of the poorer statistics, the two packagesPOSFIT and LT agree pretty well (see Table III). For near-to-surface implanta-tion depth of positrons all three programs show shortening of the τ1 lifetime, belowthe value for pure Si. This is to be attributed to out-diffusing positrons towardssample surface. All three packages agree also well in determining the long life-time component — both its value and intensity. GA-MLR fails only when the I2intensity becomes weak.5. DiscussionGenerally, the new GA-MLR method performs somewhat poorer than tradi-tional packages (POSFIT and LT) for simple cases, like measurements with high1370 A. Karbowski et al.TABLE IIThe lifetimes calculated for the sam-ples of Czochralski-grown silicon.POSFIT (first row for each sam-ple), LT (second row) and GA- MLR(third row) FWHM = 165 ps, thesource contribution 11% — 382 ps,for GA-MLR “laser pulse” Gaussianshifted by –123 ps.Sample τ1 [ps] τ2 [ps] I2 [%]71A0001 231.6 487.7 5.9227.4 1178.6 0.5228.3 1197.8 2.7W2E0001 222.8 425.7 13.6226.5 606.1 0.1221.2 1100.0 2.3W540005 230.3 459.9 8.6228.3 597.3 1.6227.8 798.2 1.7W77A0002 227.1 861.0 10.3228.6 858.2 1.3228.9 899.6 4.3statistics in defect-free metals. As far the main lifetime component (i.e. the shortone) is well identified by all three methods, the GA- MLR overestimates the sourcelifetime.On the other hand, once fixed the source component, using the well knownlifetime in kapton, GA-MLR performs even better than POSFIT, producing littlescatter between different samples in Cz-Si. GA-MLR has again some problemswith the long components, which is not a surprise taking into account not fullycorrect implementation of the resolution function in the present version of theprogram.A weak point of the present application of GA-MLR is a simplified treat-ment of the resolution function. As the present version of the program is just amodification of the code applied for the fluorescence spectroscopy, the convolu-tion of spectra with the resolution function is different in LT (and POSFIT) vs.GA-MLR. In GA-MLR program we mimic the resolution function by the “laserpulses”. Obviously, this pulses approach works well just at the beginning of thespectra and does not correspond precisely to the convolution. Moreover, both theshift of the “laser” pulse as well its intensity must be carefully chosen.From comparison of all three methods, a conclusion on the further needfor development is clear. POSFIT performs very well but when the lifetimes showGenetic Algorithms for Positron Lifetime Data 1371TABLE IIIThe lifetimes τ1 calculated for thesamples of He+H implanted Si.POSFIT (first row for every sam-ple), LT (second row) = 230 ps,GA-MLR (third row). Resolu-tion function for POSFIT andLT 230 ps, for GA “laser pulse”shifted by –123 ps. “Sample” la-bel gives the implantation energyof positrons.Sample τ1 [ps] τ2 [ps] I2 [%]0.5 keV 140.7 436.3 90.4142.0 442.1 90.7145.1 447.2 85.82.5 keV 211.5 452.1 90.1212.0 456.7 92.0210.0 450.1 93.55 keV 223.3 488.5 63.5223.6 477.0 69.9222.7 481.7 38.510 keV 223.5 492.6 17.8224.5 484.1 22.1227.1 471.9 54.2pretty distinguishable values and strong intensities. The LT program shows a widerange of the convergence, i.e. allows working with unknown starting parameters.GA-MLR is particularly efficient in guessing short lifetimes. Future developmentsshould combine these strong features of all three methods.AckowledgmentsTwo of us (A.K. and G.K.) acknowledge support from Rector’s UMK grants:No. 354F/2007 and 778F/2006.References[1] G.P. Karwasz, A. Zecca, R.S. Brusa, D. Pliszka, J. Alloys Comp. 382, 244 (2004).[2] M. Hakala, M.J. Puska, R.M. Nieminen, Phys. Rev. B 57, 7621 (1998).[3] I. Itoh, H. Murakami, A. Kinshita, Appl. Phys. Lett. 63, 2798 (1993).[4] S. Yang, Y. Li, Q. Ma, H. Liu, Q. Hao, Y. Li, S. Niu, H. Li, J. Cryst. Growth276, 43 (2005).[5] P. Banerji, Appl. Surf. Sci. 253, 5129 (2007).1372 A. Karbowski et al.[6] J.J. Fisz, J. Phys. Chem. A 110, 12977 (2006).[7] U. Rodemerck, M. Baerns, M. Holena, Appl. Surf. Sci. 223, 168 (2004).[8] J.N. Cawse, M. Naerns, M. Holena, J. Chem. Inf. Comput. Sci. 44, 143 (2004).[9] K. Gozdziewski , A. J. Maciejewski, C. Migaszewski, Astrophys. J. 657, 546(2007).[10] R.S. Brusa, W. Deng, G.P. Karwasz, A. Zecca, D. Pliszka, Appl. Phys. Lett. 79,1492 (2001).[11] G.P. Karwasz, R.S. Brusa, A. Misiuk, A. Zecca, Acta Phys. Pol. A 95, 575 (1999).[12] P. Kirkegaard, M. Eldrup, Comput. Phys. Commun. 3, 240 (1972).[13] R.S. Brusa, C. Macchi, S. Mariazzi, G.P. Karwasz, W. Egger, P. Sperr, G. Kgel,Appl. Phys. Lett. 88, 011920 (2006).[14] J. Kansy, Nucl. Instrum. Methods Phys. Res. A 374, 235 (1996).[15] J. Kansy, Mater. Sci. Forum 652, 363 (2001).

Genetic algorithms for positron lifetime data

Fisz, J.

Karwasz, G.

Kansy, J.

Brusa, R.

Biblioteka Nauki - repozytorium artykuÅÃ³w

Genetic Algorithms for Positron Lifetime Data

A. Karbowski

J.J. Fisz

G.P. Karwasz

J. Kansy

R.S. Brusa

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https://rebus.us.edu.pl/bitstream/20.500.12128/8319/1/Karbowski_Genetic_algorithms_for_positron_lifetime.pdf

Genetic algorithms for positron lifetime data

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Repozytorium Uniwersytetu Śląskiego RE-BUŚ

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Genetic algorithms for positron lifetime data

Abstract

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Repozytorium Uniwersytetu Śląskiego RE-BUŚ

Biblioteka Nauki - repozytorium artykuÅÃ³w

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