Monte Carlo model of electron elastic scattering in solids

surface science ELSEVIER

Surface Science 351 (1996) 303 308

Monte Carlo model of electron elastic scattering in solids R. C h a k a r o v a * Department of Reactor Physics, Chalmers University of Technology, S-41296 GOteborg, Sweden Received 10 July 1995; accepted for publication 7 December 1995

Abstract

An analog Monte Carlo model is presented simulating electron elastic scattering in solids and capable to treat problems of quantitative AES and XPS. Its analytical and numerical verification is described. The model is used to investigate the dependence of the elastic reflection coefficient on the inelastic mfp. It was found that the slope decreases with increasing inelastic mfp. The single elastic scattering coefficient is shown to be proportional to the total, not to the inelastic, mean free path. An analytical expression of the single elastic scattering coefficient is derived and supported by Monte Carlo calculations.

Keywords: Electron-solid interactions; Electron-solid scattering - elastic

1. Introduction

Electrons penetrating a medium scatter elastically or inelastically and change their direction or lose energy if an ionization or excitation event takes place. Investigation of elastic scattering effects only requires an adequate description of electron elastic scattering and inclusion of total inelastic interaction data without treatment of each inelastic scattering type. Each electron can be followed directly from one interaction point to another for energies less than a few keV. The method is classified as analog Monte Carlo method. Standard Monte Carlo packages for low energy electron transport are not available and calculations are performed by user developed codes. It is essential to verify the code in each particular case and to establish its accuracy and

* E-mail: [email protected]. 0039-6028/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII 0 0 3 9 - 6 0 2 8 ( 9 5 ) 0 1 2 9 7 - 4

proper working. Analytical, numerical or experimental results may serve as a test as well as other Monte Carlo calculations. A Monte Carlo algorithm, proposed by Jablonski about ten years ago [,1], has been applied extensively to study elastic scattering effects associated with problems of quantitative AES and XPS [2]. Single and multiple scattering methods were derived to determine the inelastic mean free path from elastic backscattering data. Recently some contradiction arose between the Monte Carlo calculations [-1] and results obtained within the transport approximation [-3]. The purpose of this article is to present an alternative Monte Carlo elastic scattering model with its analytical and numerical verification as well as its application to examine results obtained earlier [,,1,4,5]. Angular distributions of elastically backscattered electrons will be compared as well as the reflection coefficient dependence on the inelastic mean free path (mfp). Multiple and single backscattering will be analysed.

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A more detailed description of the model and other applications can be found elsewhere [6,7].

escape from the medium. The current contribution of a scored electron is equal to unity.

2. Monte Carlo elastic scattering model

3. Analytical test of the Monte Carlo model

The Monte Carlo algorithm proposed by Jablonski [ 1] contains the following features: elastic and inelastic scattering are taken into account separately. First the electron trajectory through the medium is constructed assuming that the medium has elastic scattering properties only. The probability is then calculated for an electron to pass the simulated total path inside the medium without energy loss. This probability represents the electron contribution to the scored quantities. The electron history is terminated if it escapes from the medium or its path exceeds a certain conventional length. Decoupling of spatial and angular transport from energy loss processes is the essence of the Tougaard and Sigmund theory [8]. Thus the algorithm may be considered as a particular application of this theory. The Monte Carlo model to be presented here simulates the transport of the elastic scattered electrons rigorously. The trajectory of each electron is constructed as a series of successive interaction events. The distance, S, between (i-1)th and ith interaction is sampled from an exponential distribution governed by the total mean free path, (mfp), •t:

A semi-infinite medium was considered which scatters the electrons isotropically elastically or absorbs them. The case was used to test the Monte Carlo model since the albedo problem has an analytical solution by Chandrasekhar's theory [9]. The analytical flux, q~, and current, J, of the backscattered particles are expressed as a function of the escape angle, #:

S = - 2 t log R,

(1)

where 1/2t = 1/28 + 1/2in, )L~ and 2m are the elastic and inelastic mfp, respectively, and R denotes a random number uniformly distributed in an interval [0, 1]. The interaction type is decided by comparison of another random number R with the elastic scattering probability P~ = 2t/L~. In case of elastic collision, (R ~ P~), the electron history is terminated. Electrons suffering successive elastic scatterings are followed until they

coo H(~)~(m) ~(~)=5-m ~+m ' J(#) = #~(#),

(2)

where #o is the cosine of the incident angle, H(#) is the Chandrasekhar's H-function tabulated in [9] and coo is the single scattering albedo. According to the definitions, coo represents the fraction of the elastically scattered radiation, while 1 - coo is the fraction transformed into other forms of energy. The total half-space current albedo, c~, is given by: = 1 -

,Ja - O~o/-/(m).

(3)

In terms of the Monte Carlo method, coo is equal to the elastic scattering probability, i.e. coo = Po. The distances, S, are expressed in mfp units. Monte Carlo calculations were performed for normally incident electrons. The elastic scattering probability was varied from 0.1 to 0.95. The obtained and the analytical angular distributions of the electron flux and current for Pe = 0.5 are compared in Fig. 1. The deviations do not exceed the statistical error which is less than 1%. Fig. 2 illustrates the agreement of the analytical total current albedo with the Monte Carlo results. Fig. 2a shows the albedo as a function of the elastic scattering probability. Fig: 2b presents the albedo as a function of the ratio of the inelastic mfp to the transport mfp, 2in/,~tr • P e / ( 1 - Pc), using the fact that the elastic mfp is equal to the transport mfp for isotropically scattering medium. This unit transformation makes

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(a)

Fig. 1. Angular flux and current distribution of electrons reflected from isotropically scattering medium for elastic scattering probability 0.5, comparison between analytical and Monte Carlo data.

Fig. 2b directly comparable and consistent with Fig. 3 from Ref. [3].

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4. N u m e r i c a l calculations

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Carlo

o

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Numerical test of the Monte Carlo model was performed by comparison of Monte Carlo results with those obtained by S N method which is a standard method in reactor physics, see e.g. Ref. [ 10]. It uses Gauss quadrature for integration over the angle and difference between ordinates to express the spatial derivative of the flux. The transport equation is solved by iteration. Angular distributions of elastically reflected electrons have been calculated by Monte Carlo and SN methods for 1 keV electrons normally incident on aluminium, copper and gold when using one and the same cross section data [ 11 ]. The elastic scattering was described by Riley's differential cross sections [12] and the inelastic mfp was taken from Ref. [13]. The agreement between the Monte Carlo and SN results was very good, which is illustrated by the case for gold in Fig. 3. The SN calculations are estimated to be about five times faster than the

0.0

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; k i / / Xtr (b)

Fig. 2. Total half space albedo for isotropically scattering medium, comparison between analytical and Monte Carlo data: (a) albedo as a function of elastic scattering probability, (b) albedo as a function of the ratio of inelastic to transport mfp.

Monte Carlo ones for results with less than 1% statistical error. Figs. 4a-4c show comparisons between the Monte Carlo results obtained and the corresponding ones published in Refs. [ 4 ] and [5], respectively. Ref. [4-] provides distributions based on non-relativistic cross sections. The histograms 1 and 2 in Fig. 4c show the non-relativistic and

1L Chakarova/Surface Science 351 (1996) 303-308

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Fig. 3. Angular distribution of 1 keV electrons reflected elastically from gold. Comparison between Monte Carlo and SN method [ 11 ].

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COPPER 1 keV

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relativistic calculations from [,5]. The observed data deviations are considered as acceptable. They may be related to differences in the cross sections input. (Self computed cross sections are used in [,1,4,5] which are not explicitly available.) Further Monte Carlo calculations have been performed for 2.2 keV electrons normally incident on copper. The elastic reflection coefficient, l/e, was obtained with a statistical error less than 1% when varying the inelastic mfp from 5 to 60 A. The results (box-symbols) are presented in Fig. 5. It is seen that the slope decreases with the increasing inelastic mfp. T h e solid line in the figure is the data fit within 1% to the polynomial: r/e = A 2 i q-

B~,

(4)

where A = 1.0298 x 10 -3 [.4 -1] and B= - 1 . 3 8 6 x 10 - 6 I - A - l ] , i.e. A is positive and B is negative. Thus the behaviour of the reflection coefficient function supports the conclusions in [,3]. The results taken from [1] (star-symbols)

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Fig. 4. Angular distribution of 1 keV electrons reflected elastically from: (a) aluminium, comparison with data from [4]; (b) copper, comparison with data from [4]; (c) gold, comparison with data from [5]. Histogram 1 - non-relativistic data [5], histogram 2 - relativistic data [5].

O.O O

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R. Chakarova/SurfaceScience351 (1996) 303-308

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Elastic reflection of 2.2 kev electrons from copper

Elastic reflection of 2.2 kev electrons from copper 0.05

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lin~,

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J j/"

._e ._o

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o 0.04Present

work .~'j/ ," / Z,d/"

._u 0.01 //

Od

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i 50

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Fig. 5. Elastic reflection coefficient of 2.2 keV electrons backscattered from copper as function of the inelastic mfp. C o m p a r i s o n with data from Ref. [ 1 ] .

are shown for comparison. It has been proved that the differences can not be related to the features of the Monte Carlo algorithm used by Jablonski. They might be related to the cross section calculated in Ref. [, 1] within the first Born approximation that has limited application to the low energy electron transport. The shape of the calibration curve for copper, 1 keV electrons, presented in a later publication [,2b] is in agreement with the reflection coefficient function in Fig. 5. The single scattering model draws attention because it is very simple and because the elastic peak is dominated by a single large-angle elastic scattering event. It is discussed in Refs. r 1, 3, 14]. Here in order to investigate the elastic reflection in detail, single backscattered electrons were scored explicitly during the Monte Carlo simulation. In this way the single scattering reflection coefficient, q , , was obtained together with the multiple scattering one, 7°. The complete results are presented in Fig. 6. It is seen that the dependence of the single scattering coefficient on the inelastic mfp is not linear which invalidates the expression suggested [ 1 ]: q~s = Nc%fr2i.,

0

60

(5)

/

//

...... -~.......... ,.....

e ....

Reference work

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/" .-/)~

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e ~ O.02-

,f

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Fig. 6. Single and multiple elastic scattering reflection coeffid e n t s as function of the inelastic mfp for 2.2 keV electrons and copper.

where N is the number of scattering centres per unit volume and aeff is an effective cross section. The proper way to derive an analytical formula for qes follows below. In terms of statistical theory, the particle transport can be considered as a stochastic process of Poisson which is of Markoff's type [,,15]. This implies that a penetrating particle generates series of mutually independent events. The probability of a certain "chain" event equals the product of the probabilities of each step. Here in the particular case of interest, the probability for an electron to be elastically backscattered at angle 0 after interaction in a layer dz is given by (see Fig. 7): Ptot = PcolP~PoPosc.

(6)

Here

is the probability of an electron travelling a distance z without interaction multiplied by the probability of having an interaction in dz. Po is the probability of elastic scattering used above.

Po= 2 ~ s i n 0 d 0

N2~

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Incident electron

with increasing inelastic mfp. The fitting function is a polynomial, qe = A2i + B22, where A is positive and B is negative. The single elastic scattering coefficient, q¢~, is shown to be p r o p o r t i o n a l to the total mfp. The analytical expression derived is general, as far as no energy or material assumptions were included. The analysis is supported by M o n t e Carlo calculations for 2.2 keV electrons normally incident on copper.

Scattered electron

-.... \ \

I I

Acknowledgements

Fig. 7. Single elastic scattering model.

is the probability that the electron is backscattered at an angle 0, and P e s o = e x p ( z s e c 0/At) is the probability of escaping from the m e d i u m without interaction. The contribution of elastically forward scattered electrons is neglected. The single elastic reflection coefficient is obtained after inserting the probabilities defined above in Eq. (6) and performing the integration over z

q~s =

i

i P dz dO = 2tNO-eff,

(7)

~/2 0 where Gff contains the integration over 0. It is seen that the single reflection coefficient is p r o p o r tional to the total and not to the inelastic mfp. For 2i>>2~, 2 t ~ 2 ¢ = const, i.e. the single reflection coefficient is independent of the inelastic mfp which tendency is seen in Fig. 6.

5. Conclusions The analog M o n t e Carlo model presented simulates successfully electron elastic scattering in solids. It is tested analytically and numerically and can serve as a b e n c h m a r k model for other M o n t e Carlo calculations associated with problems of quantitative AES and XPS. The dependence of the elastic reflection coefficient, ~e, on the inelastic mfp, 2i, is investigated for the case of 2.2 keV electrons normally incident on copper. It was found that the slope decreases

I a m grateful to Profs. N.G. SjOstrand, I. Pfizsit and A. Prinja for the helpful discussions.

References [1] A. Jablonski, Surf. Sci. 151 (1985) 166. [2] (a) A. Jablonski, Surf. Interface Anal. 14 (1989) 659; (b) A. Jablonski and C.J. Powell, Surf. Interface Anal. 20 (1993) 771. [3] V.M. Dwyer, J. Vac. Sci. Technol. A 12(5) (1994) 2680. f4] A. Jablonski, Phys. Rev. B 39 (1989) 61. [5] A. Jablonski, Phys. Rev. B 43 (1991) 7546. [6] R. Chakarova, Detailed Monte Carlo simulation of Electron Elastic Scattering, CTH-RF-103, 1994. [7] I. P~izsit and R. Chakarova, Phys. Rev. B 50 (1994) 13953. [8] S. Tougaard and P. Sigmund, Phys. Rev. B 25 (1982) 4452. [9] S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960). [10] G.I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold Co., New York 1970); also N.G. Sj6strand, Ann. Nucl. Energy 7 (1980) 435. [11] N.G. Sj6strand, Numerical Study of Electron Scattering, CTH-RF-102, 1994, Surf. Interface Anal. 23 (1995) 785. [12] M. Riley, G.J. MacCallum and F. Biggs, At. Data Nucl. Data Tables 15 (1975) 443. [13] S. Tanuma, C. Powel and D. Penn, Surf. Interface Anal. 17 (1991) 911. [14] V.M. Dwyer, Surf. Interface Anal. 20 (1993) 513; and A. Jablonski, B. Lesiak and G. Gergely, Phys. Scr. 39 (1989) 363. [15] L. Jfinossy, A. R6nyi and J. Acz61,Acta Matem. 1 (1950) 209; also J. Wood, Computational methods in Reactor Shielding (Pergamon, Oxford, 1982) p. 277; A. Profio, Radiation Shielding and Dosimetry (WileyInterscience, New York, 1979) pp. 183-192.