Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://danp.sinp.msu.ru/LIIWM/Munteanu.pdf Äàòà èçìåíåíèÿ: Fri Aug 1 04:19:06 2014 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:00:04 2016 Êîäèðîâêà: Windows-1251

Nuclear Instruments and Methods in Physics Research B 322 (2014) 2-6

Contents lists available at ScienceDirect

Nuclear Instruments and Methods in Physics Research B
journal homepage: www.els e vier.com/locate/nimb

Behavioral modeling of SRIM tables for numerical simulation
S. Martinie 1, T. Saad-Saoud, S. Moindjie, D. Munteanu, J.L. Autran
Aix-Marseille University and CNRS, IM2NP, UMR 7334 - Faculté des Sciences - Service 142, Avenue Escadrille Normandie Niémen, F-13397 Marseille Cedex 20, France

article

info

abstract
This work describes a simple way to implement SRIM stopping power and range tabulated data in the form of fast and continuous numerical functions for intensive simulation. We provide here the methodology of this behavioral modeling as well as the details of the implementation and some numerical examples for ions in silicon target. Developed functions have been successfully tested and used for the simulation of soft errors in microelectronics circuits. Ó 2013 Elsevier B.V. All rights reserved.

Article history: Received 19 November 2013 Available online 22 January 2014 Keywords: SRIM Stopping power Stopping range Behavioral modeling Numerical simulation

1. Introduction Stopping and Range of Ions in Matter or SRIM is a collection of software packages developed by Ziegler and Biersack [1-3] that calculate the stopping and range of ions into matter. SRIM is a reference and extremely popular program in the radiation effects community; it is based on a Monte Carlo simulation method, namely the binary collision approximation with a random selection of the impact parameter of the next colliding ion. Among all functionalities of the developed packages, SRIM includes quick calculations that produce tables of stopping powers, range and straggling distributions for any ion at any energy (in the range 10 eV-2 GeV) and in any elemental target. More elaborate calculations include targets with complex multi-layer configurations [1]. SRIM tables of stopping power, linear energy transfer (LET) and projected range (R) versus particle energy are very useful for the computation of particle transport in a wide range of simulation applications which do not directly involve a specific ion transport code. Although a special independent executable program (SRIM Module.exe) can be used as a subroutine for Windows applications, for other operating systems or simulation frameworks, libraries of fast table lookup and interpolation algorithms are generally necessary to embed SRIM data in the simulation flow. Another alternative, described here, is to use very accurate fitting functions that can be parameterized as a function of the ion parameters, i.e. the atomic number, mass (amu), and kinetic energy. In the following, we detail how SRIM tabulated data can be described in the form of fast and continuous numerical functions. The approach is
Corresponding author. Tel.: +33 413594627; fax: +33 491288531.
E-mail addresses: sebastien.martinie@cea.fr (S. Martinie), jean-luc.autran@univamu.fr (J.L. Autran). 1 Permanent address: CEA-LETI, 17 avenue des Martyrs, F-38054 Grenoble Cedex 9, France. 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.12.023

particularly well adapted for intensive simulation that requires very frequent LET and R calculations for a wide variety of ions, for example for the simulation of single-event effects in microelectronics circuits. We illustrate in the following the details of the implementation of the proposed solution in such a particular application framework. 2. Numerical model and approach The behavioral modeling of SRIM data has been performed on the basis of power polynomial fitting functions as a function of particle energy (E) for a given ion, a given target and a given energy range. This later has been arbitrarily fixed to a relatively wide interval [1 keV, 1 GeV] that should be suitable for many applications. But nothing prevents a user to choose another energy interval more adapted to its application. After exploring and evaluating a large collection of fitting functions, we definitively adopt the following expressions for modeling the linear energy transfer given by SRIM tables in the defined energy range:

LETðEÞ ? 10A AðEÞ ?
8 X i?1

ðE Þ

ð1Þ ð2Þ

ai  sinðbi Á log10 ðEÞþ ci Þ

where ai, bi and ci correspond to a set of 8 Â 3 = 24 real coefficients characteristics of a given ion for a given target material. Similarly, for the projected range versus energy in the same energy domain, we adopt the following modeling:

RðEÞ ? 10B BðEÞ ? p10

ðE Þ 9 X þ pi Â?log10 ðEÞ i? 1

ð3Þ
10Ài

ð4Þ


S. Martinie et al. / Nuclear Instruments and Methods in Physics Research B 322 (2014) 2-6 Table 1 Fitting coefficients p1 to p10 corresponding to Eqs. (3) and (4) for the behavioral modeling of the LET versus ion energy. Ion
1 1H 2 4 He 3 6 Li 4 8 Be 5 10 B 6 12 C 7 14 N 8 16 O 9 18 F 10 20 Ne 11 22 Na 12 24 Mg 13 26 Al 14 28 Si 15 28 P

3

p1 À0.000072 0.000230 À0.000228 0.000285 0.000505 0.000408 0.000354 0.000037 À0.000045 À0.000022 À0.000085 À0.000132 À0.000162 À0.000305 À0.000224

p2 0.000313 À0.001294 0.000661 À0.001407 À0.002682 À0.002376 À0.002357 À0.000902 À0.000707 À0.000764 À0.000196 À0.000110 0.000257 0.000815 0.000467

p

3

p

4

p

5

p

6

p

7

p

8

p

9

p

10

0.000801 À0.001047 0.002999 À0.000455 À0.000231 0.000552 0.001813 0.003432 0.004194 0.003949 0.003569 0.004453 0.003401 0.004645 0.003608

À0.005735 0.015509 À0.005415 0.011422 0.019618 0.017223 0.015679 0.003016 0.002069 0.001845 À0.003571 À0.005289 À0.007000 À0.013059 À0.007899

0.004161 À0.005743 À0.025006 À0.016504 À0.023021 À0.025489 À0.031779 À0.032759 À0.035774 À0.032257 À0.027971 À0.032196 À0.021882 À0.023557 À0.019252

0.016050 À0.086562 0.004855 À0.029281 À0.036910 À0.027506 À0.015370 0.024607 0.024161 0.029588 0.053294 0.063200 0.061737 0.081572 0.056598

À0.058306 0.074551 0.149170 0.140768 0.139100 0.140963 0.145143 0.140068 0.143401 0.121298 0.111525 0.114376 0.078688 0.066732 0.056851

0.172295 0.364402 0.157453 0.118997 0.076552 0.033676 À0.011943 À0.081166 À0.085871 À0.118845 À0.178127 À0.207351 À0.196060 À0.208296 À0.167203

1.508268 0.937028 0.727124 0.655164 0.668135 0.642289 0.636229 0.628646 0.615726 0.644585 0.645318 0.647418 0.693492 0.721947 0.731879

1.212035 0.546634 0.420795 0.337527 0.228816 0.194686 0.161748 0.164015 0.132176 0.145051 0.163270 0.164492 0.120705 0.047830 0.019093

where coefficients pi corresponds to a set of 10 real coefficients for each given ion. Considering Eqs. (1)-(4), the proposed solution then requires to determine a unique set of 34 reals per given ion and per given target to analytically describe both LET(E) and R(E) relations in the full energy range [1 keV, 1 GeV]. In order to calculate such coefficients, we developed a dedicated MatlabÒ macro2 that minimizes the numerical error between SRIM data and analytical representations. Briefly, this script creates a data structure for the fit using the MatlabÒ ``fitoptions'' function [4] that specifies information like weights for the data, fitting types (``sin8'' for Eq. (2), ``poly9'' for Eq. (4)), fitting methods (``NonlinearLeastSquares'') and other options for the fitting algorithm [4]. SRIM data are loaded into the structure directly from the reading and pre-processing of SRIM output files and, as the result, the script generates a text file with the 34 real fitting coefficients plus the evaluation of the minimum, maximum and averaged errors on the specified energy domain for each ion. The program can directly process several SRIM files in a single execution sequence, then directly providing a matrix of coefficients for a collection of ions and/or targets. The resulting set of coefficients can be directly copy and paste to create numerical functions in any programming language on the basis of Eqs. (1)-(4). 3. Results and discussion We illustrate in the following the results of this behavioral modeling of SRIM tables for fifteen ions ranging from 1 H to 28 P in 1 15 a silicon target. This particular case is important for microelectronics since it corresponds to secondary ions that be produced from nuclear reactions between atmospheric neutrons and silicon [5], possibly inducing single event effects in circuits. Numerical simulation of such effects requires very frequent computation of stopping power and range of produced ions, fully justifying the present approach. Tables 1 and 2, respectively gives the pi and ai, bi, and ci coefficients for the LET and the range of all these ions in silicon, as obtained from the processing of the corresponding SRIM output files (SRIM version 2012) with the MatlabÒ program (R2012a release). These coefficient values have then been implemented in C++ functions to generate continuous numerical data. Fig. 1 illustrates the comparison of both tabulated and modeled data for a few ions, demonstrating the excellent fitting of the SRIM values
2 This macro is available for free download at the following URL: http://www.natural-radiation.net (top menu ``Modeling/Simulation'', section ``SRIM Table Modeling'').

by Eqs. (1)-(4). In order to quantify the accuracy of these numerical representations, Fig. 2 shows the averaged error estimated by the MatlabÒ fitting routines for both LET and range in silicon of the different ions defined in Tables 1 and 2. The averaged error is less than 1.2% for LET and less than 1% for range for the selected ions, which confirms the very good accuracy of the proposed approach. This later has been also successfully tested for silicon dioxide and cupper target (data not shown) with numerical errors in the same order of magnitude, around or below the percent. Finally, to validate our approach with a practical case, we implemented and used the developed numerical functions into the TIARA code for the simulation of soft errors induced by atmospheric radiation in static memories (SRAMs). TIARA (Tool suIte for rAdiation Raliability Assessment) is a general-purpose Monte Carlo program written in C++ that simulates the interaction of several particles (neutrons, protons, alpha-particles and heavy ions) with various architectures of electronic circuits [6-8]. The code was initially designed to directly import SRIM data and to linearly extrapolate range and LET values of secondary ions from SRIM tabulated values. In a recent study [9], the code was used to evaluate the soft error rate (SER) induced by atmospheric neutrons in three generations of SRAM circuits, respectively manufactured in 130 nm, 65 nm and 40 nm technologies. Fig. 3 shows a comparison of the SER values evaluated using the initial release of the TIARA code and a new version implementing the numerical functions for both LET's and ranges proposed in the present work. For both simulations, we considered 500 Â 106 of incident neutrons distributed in energy following the reference atmospheric spectrum at ground level given in [10]. The SER results are found strictly equivalent for both versions of the code; the small differences are only attributable to numerical fluctuations usually observed in the Monte Carlo simulation method from a simulation run to another. 4. Conclusion In summary, we proposed in this work a behavioral modeling of SRIM tabulated data in the form of fast and continuous power polynomial fitting functions for the stopping power and projected range versus particle energy. A dedicated automatic procedure has been also developed under MatlabÒ to calculate the polynomial coefficients by minimizing the error between the data to model and the numerical representation. Typical accuracy below the percent has been obtained in the range 1 keV-1 GeV for a wide variety of ions and targets. Finally, we successfully tested and implemented the proposed solution for the case of secondary ions


Table 2 Fitting coefficients (a1, b1, c1) to (a10, b10, c10) corresponding to Eqs. (1), (2) for the behavioral modeling of the range versus ion energy. Ion a1 b1 c1 1.956220 0.323238 À2.247633 1.343970 0.515700 À2.445174 0.807533 1.205272 0.903794 2.218317 1.329730 0.725164 4.086886 0.776971 0.684197 0.872859 0.916953 0.486781 0.714216 0.790150 1.175368 0.709179 0.731914 1.133574 0.765294 0.682119 0.998941 0.617390 0.574873 1.022313 0.878029 0.686509 0.850637 0.952986 0.666869 0.726381 0.938132 0.628930 0.795827 1.056648 0.633748 0.888116 1.093973 0.633742 0.850780 a2 b2 c2 0.827775 0.850600 1.823258 1.161264 1.028412 1.035452 0.652014 0.580013 À2.720591 1.571908 1.430394 À2.488038 3.649270 0.725456 3.729080 0.533693 0.744836 2.758709 0.132630 1.382951 À0.142722 0.188903 1.236427 0.040294 0.177953 1.310367 0.212229 0.358670 1.107210 0.369183 0.158675 1.271184 À0.191131 0.134501 1.511807 À0.118447 0.173064 1.280560 À0.211745 0.026705 2.515204 À0.688891 0.137012 1.265217 À1.005518 a3 b3 c3 0.097947 2.794768 À2.611099 0.189282 1.803855 À3.027577 0.110944 2.121241 3.344983 0.390005 0.535122 3.275295 0.010275 2.725142 À0.912435 0.007681 2.678857 À1.355789 0.032608 3.754745 0.917596 0.031090 3.540927 0.455165 0.038004 3.804492 0.329324 0.036321 3.802612 0.129087 0.023183 3.993324 0.199708 0.029698 3.844329 0.071261 0.021520 3.903995 0.065100 0.026059 3.774124 0.104702 0.016655 3.801339 0.200337 a4 b4 c4 0.148241 3.495576 À0.118307 0.001706 3.932618 À0.152822 0.036488 3.623005 À0.192994 0.012631 3.848398 0.756920 0.017128 5.262824 1.836107 0.027130 3.842144 1.085322 0.007397 2.517784 À1.908563 0.011392 2.411203 À1.348325 0.010038 2.530653 À2.918651 0.023972 2.535185 À2.683918 0.026452 5.811342 0.712431 0.063492 5.805575 0.401430 0.004522 5.057902 0.951061 0.152847 1.273392 À0.918549 0.012632 5.068163 À0.379640 a5 b5 c5 0.086313 3.772755 2.745605 0.008809 5.006213 2.256803 0.021734 4.845533 2.313348 0.017044 4.906220 1.935856 0.026199 3.893906 1.346917 0.012006 5.170452 1.773573 0.018223 5.242180 1.423563 0.014088 4.667325 0.903043 0.031058 6.917037 À3.391312 0.019304 4.860104 1.399990 0.025415 6.022104 3.615855 0.060289 5.978005 3.350518 0.008647 3.113949 À1.419880 0.014583 5.068415 0.121050 0.006995 6.336489 0.962197 a6 b6 c6 0.001234 6.184156 À1.591242 0.005177 5.685879 À1.623921 1.776972 6.115976 À1.296632 0.005777 6.234944 À1.438986 0.010496 6.657418 À2.459317 0.007783 6.624791 À2.377272 0.015762 6.674428 À3.010529 0.003956 7.234530 À3.165647 0.006750 4.448031 2.838901 0.039577 6.875954 1.620334 0.019122 2.598439 À0.293170 0.963900 11.377166 1.509906 0.004948 7.660582 2.476814 0.008291 6.294246 1.149604 0.007014 2.515705 À1.031704 a7 b7 c7 0.001723 7.263390 0.886562 0.001628 7.087200 1.013650 1.768770 6.120922 1.842873 0.002627 7.500284 1.015901 0.010002 7.739144 À0.167626 0.007509 7.943302 À0.204121 0.001915 13.774456 1.123507 0.003201 7.884317 À0.348409 0.028665 7.222415 À0.633313 0.036053 7.069492 À1.817266 0.093455 10.800094 2.689499 0.055281 11.067628 À1.309931 0.004149 6.996557 0.625648 0.004048 7.667861 2.023986 0.002935 7.604467 2.021591 a8 b8 c8 0.000541 8.850405 À3.689021 0.000281 8.438661 À3.124702 0.001840 8.691919 À2.605091 0.001935 8.837898 À2.877071 0.006215 8.276188 2.634144 0.005588 8.608565 2.452943 0.009392 7.150627 À0.277629 0.001512 11.363786 3.508969 0.005273 11.388164 2.976188 0.004802 11.408358 2.236381 0.092681 10.767664 À0.415540 0.909693 11.395236 À1.650419 0.002828 8.708039 À1.803427 0.002263 8.834262 À2.358316 0.001869 8.869279 À2.623060

4

1 1

H

2 4

He

3 6

Li

S. Martinie et al. / Nuclear Instruments and Methods in Physics Research B 322 (2014) 2-6

4 8

Be

5 10

B

6 12

C

7 14

N

8 16

O

9 18

F

10 20

Ne

11 22

Na

12 24

Mg

13 26

Al

14 28

Si

15 28

P


S. Martinie et al. / Nuclear Instruments and Methods in Physics Research B 322 (2014) 2-6

5

0.6 SRIM data Numerical model LET (MeV/(mg/cm )) 0.5
2

Proton in Silicon 900 800 700 Range (mm) SRIM data Numerical model

0.4 0.3 0.2 0.1

600 500 400 300 200 100

0 0 2 4 6 8 10 Energy (MeV) 1.6 1.4 LET (MeV/(mg/cm ))
2

0 0 100 200 300 400 500 600 Alpha in Silicon 90 80 70 Range (mm) 60 50 40 30 20 10 0 0 20 40 60 80 100 0 100 200 300 400 500 600 Energy (MeV) Energy (MeV) Silicon in Silicon 450 SRIM data Numerical model 400 350 SRIM data Numerical model Energy (MeV) SRIM data Numerical model

SRIM data Numerical model

1.2 1 0.8 0.6 0.4 0.2 0

14 12 LET (MeV/(mg/cm2)) 10 8 6 4

Range (÷m) 0 100 200 300 400 500 600

300 250 200 150 100

2 0 Energy (MeV)

50 0 0 100 200 300 400 500 600 Energy (MeV)

Fig. 1. Comparison between SRIM data and numerical values (LET and range) calculated using the approximation functions for protons, alphas and silicon ions in silicon target as a function of ion energy.

10 Error on LET

10 Error on Range

Error (%)

0.1

Error (%)

1

1

0.1

0.01 0 2 4 6 8 10 12 14 Atomic number

0.01 0 2 4 6 8 10 12 14 Atomic number

Fig. 2. Numerical averaged error between SRIM data and values calculated using the approximation functions for both LET and range in silicon as a function of the atomic number of the ions defined in Tables 1 and 2.


6

S. Martinie et al. / Nuclear Instruments and Methods in Physics Research B 322 (2014) 2-6

500 SRIM Model 400 SER (FIT/Mbit)

References
[1] J.F. Ziegler, J.P. Biersack, M.D. Ziegler, SRIM - The Stopping and Range of Ions in Matter, SRIM Co., 2008. ISBN 0-9654207-1-X. [2] J.F. Ziegler, M.D. Ziegler, J.P. Biersack, SRIM - the stopping and range of ions in matter (2010), Nucl. Instrum. Methods Phys. Res. Sect. B 268 (11-12) (2010) 1818-1823, http://dx.doi.org/10.1016/j.nimb.2010.02.091. [3] http://www.srim.org. [4] Curve Fitting ToolboxTM User's Guide 2012, The MathWorks Inc. [5] S. Serre, S. Semikh, S. Uznanski, J.L. Autran, D. Munteanu, G. Gasiot, P. Roche, Geant4 analysis of n-Si nuclear reactions from different sources of neutrons and its implication on soft-error rate, IEEE Trans. Nucl. Sci. 59 (4) (2012) 714- 722, http://dx.doi.org/10.1109/TNS.2012.2189018. [6] S. Uznanski, Monte-Carlo simulation and contribution to understanding of Single Event Upset (SEU) mechanisms in CMOS technologies down to 20nm technological node (Ph.D. thesis), Aix-Marseille University, 2011. [7] S. Uznanski, G. Gasiot, P. Roche, C. Tavernier, J.-L. Autran, Single event upset and multiple cell upset modeling in commercial bulk 65 nm CMOS SRAMs and Flip-Flops, IEEE Trans. Nucl. Sci. 57 (4) (2010) 1876-1883, http://dx.doi.org/ 10.1109/TNS.2010.2051039. [8] S. Uznanski, G. Gasiot, P. Roche, J.L. Autran, Combining GEANT4 and TIARA for neutron soft error rate prediction of 65 nm Flip-Flops, IEEE Trans. Nucl. Sci. 58 (6) (2011) 2599-2606, http://dx.doi.org/10.1109/TNS.2011.2170853. [9] J.L. Autran, S. Serre, D. Munteanu, S. Martinie, S. Sauze, S. Uznanski, G. Gasiot, P. Roche, Real-Time Soft-Error Testing of 40nm SRAMs, in: IEEE Proceedings of the International Reliability Physics Symposium IRPS, (2012) 3C.5.1-3C.5.9, http://dx.doi.org/10.1109/IRPS.2012.6241814. [10] JEDEC Standard Measurement and Reporting of Alpha Particles and Terrestrial Cosmic Ray-Induced Soft Errors in Semiconductor Devices, JESD89 Arlington, VA: JEDEC Solid State Technology Association. Available online: http:// www.jedec.org/download/search/JESD89A.pdf.

300

200

100

0
130nm 65nm 40nm

Technology node
Fig. 3. Soft error rate (SER) values computed by the TIARA simulation code for three generation of SRAM memories (see details in [9]) when directly considering SRIM tabulated data (SRIM) or using the approximation functions developed within this work (model).

(ranging from protons to silicon ions) produced from nuclear reactions between neutrons and the silicon bulk material of microelectronics circuits.