A calculation method of X-ray emitted intensity in multi-layer films is proposed in this paper. The method is based on the work developed by us: (1) a simplified physical model of electron scattering and Monte Carlo evaluations in a single medium and in multi-layer media and (2) the theories and the formulae for excitation, absorption and fluorescence of characteristic X-rays. The intensity ratio of X-rays for the known thickness films, Au/Cu/Si and Cr/Ni/Si, were calculated at 20, 25 and 30 keV. Calculated results are compared with experimental values of electron microprobe analysis for the multi-layer film specimens, and the correspondence is excellent. The work lays foundations for X-ray quantitative microanalysis of multi-layer specimens

Chen, Jia-guang

Ho, Yen-cai

Wang, Xin-lei

English

DigitalCommons@USU

Scanning Microscopy Volume 3 Number 3 Article 4 9-5-1989 A Calculation Method of X-Ray Emitted Intensity in Multi-Layer Films by Monte Carlo Simulation Yen-cai Ho Shanghai Institute of Ceramics Jia-guang Chen Shanghai Baoshan Steel Works Xin-lei Wang Shanghai Institute of Ceramics Follow this and additional works at: https://digitalcommons.usu.edu/microscopy  Part of the Biology Commons Recommended Citation Ho, Yen-cai; Chen, Jia-guang; and Wang, Xin-lei (1989) "A Calculation Method of X-Ray Emitted Intensity in Multi-Layer Films by Monte Carlo Simulation," Scanning Microscopy: Vol. 3 : No. 3 , Article 4. Available at: https://digitalcommons.usu.edu/microscopy/vol3/iss3/4 This Article is brought to you for free and open access by the Western Dairy Center at DigitalCommons@USU. It has been accepted for inclusion in Scanning Microscopy by an authorized administrator of DigitalCommons@USU. For more information, please contact digitalcommons@usu.edu. Scanning Microscopy, Vol. 3, No. 3, 1989 (Pages 725-729) 0891-7035/89$3.00+.00 Scanning Microscopy International, Chicago (AMF O'Hare), IL 60666 USA A CALCULATION METHOD OF X-RAY EMITTED INTENSITY IN MULTI-LAYER FILMS BY MONTE CARLO SIMULATION l * 2 l Yen-cai Ho , Jia-guang Chen and Xin-lei Wang ~Shanghai Institute of Ceramics, Academia Sinica Research Department, Shanghai Baoshan Steel Works (Received for publication April 18, 1989, and in revised form September 5, 1989) Abstract A calculation method of X-ray emitted intensity in multi-layer films is proposed in this paper. The method is based on the work developed by us: ( 1) a simp lifi ed physical model of electron scattering and Monte Carlo eval ua-tions in a single medium and in multi-layer media and (2) the theories and the formulae for excita -tion, absorption and fluorescence of characteris -tic X-rays . The intensity ratio of X-rays for the known thickness films, Au/Cu/Si and Cr/Ni/Si, were calculated at 20, 25 and 30 keV. Calculated results are compared with experimental values of electron microprobe analysis for the multi-layer film specimens, and the correspondence is excellent . The work lays foundations for X-ray quantitative microanalysis of multi-layer specimens . KEY WORDS: Multi-layer films, X-ray Quantitative microanalysis, Electron Scattering, Fluorescence correction, Monte Carlo simulation . *Address for correspondence: 865 Chang-ning Road, Shanghai 200050, China Phone No. 512990 725 Introduction In X-ray quantitative microanalysis ( XQMA), there are some special cases which are not con-sidered by the conventional ZAF procedure. These are XQMA for the specimens of microparticles with irregu lar shapes, and thin films or multi-layer films. In recent years, the theoretical and experi mental difficulties of XQMA for micro-particlfs)have already been solved by thfl) authors and others, ( SuIB)as Armstrong Some valuable contributio ns ' have already been made in research on XQMA of film specimens. Based on Monte Carlo calculations, using our simplified model of electron scattering, fluorescence and absorption formulae of X- rays, the intensity ratio values, k, of X-rays for multi-layer films Au/ Cu/Si and Cr/Ni/Si (on substrate Si) were eva luat ed at the various acceleration voltages. Monte Carlo calculations show good correspondence to the experimental results of EPMA for the film specimens. This research provides a means for realizing XQMA of multi-layer film specimens . Principle Figure l demonstrates that an electron beam bombards a multi-l ayer film specime n on a thick su bstrate with normal incidence. We have derived a series of theories to treat the complex physical processes, suc h as electron scatteri ng and genera-tion, absorption and fluorescence of X-rays in a multi-lay er target. Electron scattering in multi-layer media 1) A simp l ified model of electron scatteri ng in a single medium. Elast4c 11)attering of elec-trons is calculated from ' : 8 P·E ctg-- = ( 1) 2 0.0072·Z I where Pis the impact parameter, P=P0 R2 • R is uniform random number in the range of O -1. If N is the scattering step number in the fixed step length model, the maximum collision parameter P0 may be expressed as follows: Yen-cai Ho et al. P = f(N) Z □. 4/E ( 2) o o where Z is the atomic number, and E is the primar y energy of incident electron~. In our model N is taken as 100, then f(N) approximates to 0.85. Inelastic scattering is represented by the Bethe equation: 4 dE = - d f's 2 e NA Z 1 1.166E E • A n J (3) in which NA is Avogadro's number, A is atomic Equivalent)single ---1--"d---'H.--- medium ,.: . (.A •. ,·•,.) •.j A ,_ ---~--------B (a) Substrate (b) Substrate Fig. 1. Electron scattering and X-ray excitation in multi-layer media. weight and J is the mean ionization potential. The reliability of this physicr~ §~delis con-firmed in our preceding papers ' • 2) Electron scattering in multi-layer media--'eguivalent single medium scattering' calculation method. Suppose that electron i comes from medium m-1 into medium m, the scattering paths of the electron in the two media are AD and DB respec-tively (Fig. 1a), the step length (mass length) of electron scattering on both sides of a boundary may be approximately considered as the one in the preceding medium, we have (4) where S 1 is the mean free path (in length units) of elecrron i in medium m-1, then DB= Pm-1Sm-1 - fm_,-AD Pm (5) After scattering many times, the energy of electron i at point Bin medium mis E . The scattering path of the electron from p8int B only relates to the direction cosine of the vector AB and energy E . So all of the scatterings of electron i b~fore point B may be regarded equiva-lently as those happening in medium m. Then the problem will be simplified. Based on the equal 726 step length model, the energies E1, E2, ... , Ek''' of the electron with the primary energy E at every step in medium m can be calculated By Bethe formula. By comparing E with E1 , E2 , ••• , Ek''''' the energy Ek which is ~he closest Ea E among the energies is obtained. Then k is regQrded as the step number of scattering before electron i enters medium m. The scattering path of electron i from k+1 step in medium mis continuously simulated by using the 'equal step length' model, until the electron enters another medium or stops in medium m due to loss of all its energy. Once an electron comes into another medium, the previous scattering history is considered to be equivalent to that one happens in the 'new' medium. This is the 'equivalent single medium scattering' concept. The dotted line in Fig. 1b represents the scattering track of an electron in the 'equivalent single medium'. Electron scatter-ing and excitation of X-ray in multi-layer media can be conveniently calculated by using the theory of 'equivalent single medium scattering ' and the 'equal step length' model. Calculation of X-ray emitted intensity in multi-layer films 1) General equation. Considering the absorp-tion and fluorescence of characteristic X-rays, the general equation of the emitted intensity of X-rays of element i from the nth layer film of multi-layer specimens: nth layer I . =Lil . f+( 'f ( P,Z)df.Z n.l n.i. J (n-1 ) th layer n n n .£. exp (- JJ P. d csc'f'), m=1 mm m (6) nth layer where AI . f and J( 1) 1 lfn( f.nZ )d f.nZ are n.1. n- ayer the emitted fluorescence intensity and the generated intensity of characteristic X-rays, respectively, and fi-1 exp(- J.J p d csc 'f') is the absorption term;~~ ~ andmJ.Jm ~re the distri-bution in depth ofnX-rQy~, the ~ensity of nth (or mth) layer medium, and the mass attenuation coe -fficient of mth layer medium respectively, and'/' is the X-ray take-off angle. The p (pZ) function can be easily evaluated by using tRe electron scattering model in multi-layer media mentioned above, and the excited pro-bability of characteristic X-rays is obtained, according to Bethe as: lnU u 1 -2-E k. l in which Q~ 1 is the ionization k or 1 shell electrons, Ek 1 is ionization potential, qk 1 ·1s a (7) cross-section for the minimum constant, and U is overvoltage (E/ Ek 1) . · 2) Formulae for Fluorescence of Characteris-tic X-ray s in multi-layer films. One of the most difficult problems in XQMA for multi-layer films is the calculation of the fluorescence Calculation of X-ray Inte nsity in Mult i -lay er Films contribution. Castaing ' s equation can be use d to estimat 2)the fluor esce nce correction in a bulk t ar get . Cox e t al. dev e loped a correction pro-cedure to ca lculat e the fluor er c,nce produc ed in a thin coating on a s ubstra t e 3 . In Cox's formula, th e mass t hi ck ness of t he film is assumed to be much le ss th an that of exci ted X-ra y depth in a targ et . So Cox's equation is s uitabl e for th e very thin film. In thi s pap er, formulae are pr esen ted to ca l culat e th e fluor esce nc e inten s ity from an element excited by the characteri s t ic X-rays emit ted from another element in multi-layer films. The formulae are applicable whether the l ayer is thin or th ic k, and are extended to include multi-la ye r multi-component film s . Figure 2 shows a multi-layer speci men which co nsis t s of film 1,2, ... , m, . .. , n, .... As we know, the distribution of the X- ray photons exc it ed by incident e le ct ron s is sy mmetrf9a t 2~o the incident direction of the e l ectron beam ' , i . e . the direction of Z axis. So we s uppose th e charac -teristic X-r ays of eleme nt j in film mare emitted from mid-poi nt ~ of film m to sim plif y t he calculations. Fluorescence X- rays of th e film are also conside r ed to be emitted from the mid-point of the fi lm. (9) Then we derive th e equations for calcula-ting the fluoresc ence intensity of eleme nt i in film n, A I . f . , pro duced by the characteristic X-ray of el~ffie~t j in film m. I t is El ectro n beam d m n-d n z Fig. 2. Schematic diag r am of ca l cu l ati ng fluor escence correctio n for multi-layer fi l ms. expressed as follows ( the derivation is in r e ference (9 )) : '!T" . (2 J sin p 0 . r e- Usec 'f _ e - ( U+V)sec 'f ] d <f ( 8 ) 727 in which I . is the int ensity of the character-istic X-ra~ s J emi tt ed fr om e l ement j in film m, d n 2 d m U = JJm.j Pm 2 + Jlm+l ,j f m+l dm+l+. · · · · + ).ln- 1, j fn - 1 dn-1 and ( 9 ) ( 10) (11) In the above -mentioned equations, U i s the mass attenuation coefficient of film n for i radiation, U . is the m.a.c of film radiation, d is the thi~KMess of the nth layer film, and( 12, c8nve r sion effic i ency facto r, £ , is giv en by £ = C . n.l i ~ ,lJn. j r. -1 l r. l . W( i) ( 12 ) where C is the wei ght fraction of element i in film n,n.i ~ 1 . i s the m. a . c of e l ement i for j radiation inn fi l m n, r the abso rption edge j ump ratio of element i, an~ W( i ) the fluorescence yie ld of i radiation . Equation (8) is still applicable when film n is above film m. If m=n, i. e. there are element i and element j in the same layer film, the form of equation (8) i s : 'Ir (2 J sin If 0 d n -vsec 'f . - 2-( 1-e ) d<f ( 13) If layer mi s ver y thick, e . g . it is a sub-s tr ate, t he assumption that j radiat ion conce ntr a-tes on the mid- point of film m ( t he s ubstrate) is unreasonable. For th is case, j radiation is ass umed to concentrate on the mid-po in t of the mean exci t a tion depth, d in the s ubst ra te . d ca n be ca l culat ed from r ~f~r ences (5,6). 1/2 d s . x in equa t io n ( 10 ) will be replaced with d m s . x Analytical Result s Result s of experim en t and ca l cu l ation In order to exami ne the reliabi l ity of this ca l cu lation method, we did t he experiment of EPMA for the carefully prepar ed mul ti-layer films (Fig .3 ) Au/Cu/Si and Cr/ Ni/Si (o n substrate Si) at 20 kV, 25kV, 30kV. The X-r ay i ntensity ratios, k of t he characteri stic X-ra y intensity emitte d from every Yen-cai Ho et al. Multi-layer El ement Calculat ed k val ues Measu red k values bv M. C method with EPMA 20 keV 25 keV 30 keV 20 keV 25 keV 30 keV Au 0.4110 0 . 2382 0 . 1582 0.3874 0 . 2253 0 . 1487 Cu 0.1852 0 . 1415 0.0986 0 . 1979 0.1428 0 . 1035 Au/Cu/Si Si 0.1890 0 . 2612 0 . 2983 0 . 1893 0.2422 0.2835 Cr 0.0627 0 . 0374 0.0327 0 .0561 0.0 341 0 .0 234 Ni 0.2688 0 .1 517 0.0939 0.234 5 0.1399 0.0930 Cr/ Ni/ Si Si 0.3869 0 .4448 0.4742 0 . 3953 0 .44 51 0.4702 Table 1. Compariso n between the calculated k values using Monte Carlo method and the experimental results with EPMA for t he multi-layer spec~mens Au/ Cu/ Si and 2 Cr/ Ni/ Si. Thick~ess_of every thin 2layer: Au, 0.1 5 mg/c m; Cu, 0 . 099 mg/ cm; Cr, 0 . 034 mg/c m; N1, 0.117 ~g/c m. X- ray l ines : Au, L« ; Cu, K~ ; Cr, Koi Ni, Koi . layer to the intensity from s tandard specimens. Agreement of the calculated values with the experimental result s is fairly good . ( table l ) Sample preparation and thickness determination in multi -la yer specimens . The multi-la ye r film spe ci mens are prepared wit h vacuum evaporator (Mode l: JEE-4X) . Let us explain the preparing cour se with the example Au/ Cu/S i films. Si substrate was finely finished, then Cu and Au films were deposited with vacuum eva poration method, and th e single-layer films Au/Si, Cu/Si (o n s ubstrate Si) were deposited s imultaneou s ly for meas uring thickness. The thickness of the s tandard s pecimen, was determined by using th e nucl ear back-scattering method and s hown in table 1 . Since ~uclear backsca tt ering method ca n directly give th e mass thickness of specimens , the ca l culated error res ult ed from the differ ence between the f i lm density and theoretical den sity may be avoided. EPMA ex periment of multi-lay e r film specimens . The characteristic X-ray intensities Au-Lo1 , Cu-K~, Cr-K~, emitted from eac h layer film and s ubs trate were determined at 20kV, 25kV, 30kV. The experimental conditio ns ~g.EPMA are as follow s : probe current, 2*10 A; probe diameter, 5 ,um. References 1. Arms trong, J.T. ( 1978 ) . Method of quantitati ve ana l ysis of indi vi dual microparticles with electron beam instrument, Scanning Electron Microsc . 1978; 1:455 - 467. 2 . Cast aing R. (1951). A method for application of elec tron beams to local chemica l and crystallo-graphica l analysis, Ph.D. thesis. Univer si t y of Paris, Fr ance. 3 . Cox M. G.C. , Love G., and Scott V. D. (1979). A cha r ac teristic fluorescent correction for elec-tr on probe microanalysis of thin coa tin gs, J. Phys. D: Appl. Phy ., 12, 1441-1451. 4. Cur genven L., Duncumb P. (1971). Simulati on of elec tr on tr ajec t ories in a so lid t arge t by a simple Monte Carlo t ech nique, Tub e Investment Resea r ch Report 303. 728 Electron Au-L., Cu-K,, Si -K,1 Au Cu (a) Substrate (Si) Cr-K., Ni-Ko\ Si -K" ( b ) Substrate (S i ) Fig . 3 . Schematic diagram of X-r ay exci tat io n in multi-la yer films Au/ Cu/S i and Cr/Ni/Si . 5 . Ho Y.C., Huang Y. H. ( 1982 ) . A s imple method for el ectro n probe det ermination of thickness of t hin films by Monte Carlo simulatio n, Scanning Electron Microsc. 1982, II: 559- 562. 6 . Ho Y.C., Huang Y.H. ( 1982 ) . Calculation of tra nsmission coefficients of electrons and t he depth of X-ra y production in solids by Monte Carlo simulation, J. Physics ( in Chinese ) , 11, 537- 539 . 7 . Ho Y,C., and Cao L.Q. ( 1984 ) , Calculation of the X-ra y spatial re so lution by Monte Carlo s imulation, Kexue Tong baa (A Monthl y Journal of Science ) , Bei j in g, 29, 1447- 1451. Calculations of X-ray intensity in multi-layer films 8. Ho. Y.C. (1987). A calc ulation method for quantitative X-ray microanalysis for micro-particle s pecimens by Monte Carlo si mulation, Scanning Microscopy, 1, 943-950. 9. Ho Y.C., Hu Y.H.,Chen J.G.,Wang X.L. (1988), An equation of fluorescence correction for XQMA of multi-la yer films, Kexue Tongbao, Beijing. 33, 711-714. 10. Kyser OF, Murata K. (1976). Application of Monte Carlo simulation to electron micro-probe analysis of thin films on substrates, in use of Monte Carlo Calculations in Electron Probe Microanalysis and Scanni ng Electron Microscopy, K.F. Heinrich, D.E. Newbury and H. Yakowitz, eds., NBS, Washington DC, Spec. Publ. 460, 129-138. 11. Love, G., Cox M.G.C., Scott V. D. (1977) . A simple Monte Carlo method for sim ulating electron-so lid interactions and its applica-tion to electron probe microanalysis, J. Phys. D, 10, 7-23. 12. Reed SJB. ( 1975 ) . Electron microprobe Analysis, Cambridge University Press, London, 215, 265. Discussion with Reviewers G. Laudron: You have used a simplified Rutherford equation for elastic scattering and the Bet he continuous energy l oss law for inelastic scat -tering. Have you investigated the effect of introducing more appropriate cross sections to describe electron scattering? Authors: In the energy range of microprobe analysis, the continuous energy loss approxima-tion and use of Rutherford equation are accep-table. But for lower beam energies, the Ruther-ford equation is a very poor approximation, any calculations attempting to achieve absolute accuracy will have to adopt these more precise theories such as the exact Mott elastic cross-section or the partial wave expansion cross -section. J.A. Small: Why did the authors use so old a version of the Monte Carlo program 7 If the version used was dictated by the XT comput er, do the authors intend to use a larger computer with an updated Monte Carlo program? Authors: For the problem of interest in this paper, the differences between the calculated results using the Monte Carlo program based on the simplified model to be run on XT computer and the updated Monte Carlo Program base d on a stricter calculation method are very small. So it is not necessary to use a larger computer in these cases. G. Laudron: Can you describe any developments in th e method of quantitative analysis of multi-la yer films for non normal incident geometrie s? Would the calculation become more accurate? Authors: The method of quantitativ e analysis of multi-layer for non normal incident geometries meri t s attention. Under this circumstance the X-ra y intensity emitted from multi-layer can be properly increased, i.e. the accuracy of EPMA 729 experime nt is raised, since th e action range of incident electrons in multi-layer films becomes larger than that for normal incidence. This method and that of normal incidence all need to be developed in the theoretical calculation and the experimental technique. First, the physical model describing electron scatteri ng should be improved. Second, and even more importantly, the sensitivity of examining X-ray intensity should be raised to the full for XQMA of very thin multi-layer specimens . P.J. Statham: For some combination s of film thicknesses and incident electron energy, your simulations would be ent irely inappropriate. In particular, if the step length were of the same order as film thick ness in any one film, choice of suc h a "multipl e scattering" simulation would be i nvalid . Therefore, have you any recommenda-tion for how the validity can be checked before one commenced a full sim ul ation? Authors: This is one of the important problems for Monte Carlo simulation in XQMA of film specime ns. According to film thicknesses and incident electron energy, one ought to select a suitab l e step length so that there is enough scattering number in films to decrease the statistical error of Monte Carlo calculation . If the thickness in any one film is very thin, in order to obtain enough scattering number in the film both points need to be considered: ( 1) one should adopt the electron beam with lower energy to enable the average free path of electron scattering to be small enoug h, and (2 ) increase the number of simulated electrons. 

A Calculation Method of X-Ray Emitted Intensity in Multi-Layer Films by Monte Carlo Simulation

https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=2031&context=microscopy

A Calculation Method of X-Ray Emitted Intensity in Multi-Layer Films by Monte Carlo Simulation

Abstract

Similar works

Full text

Available Versions

DigitalCommons@USU