Application of Monte Carlo method to the electron probe microanalysis of thin films

Materials Chemistry 4 (1979) 495 - 506 © C E N F O R S.R.b. - Printed in Italy

APPLICATION

OF MONTE

PROBE MICROANALYSIS

A. DESALVO** LAMEL

CARLO

METHOD

OF THIN

FILMS*

TO THE ELECTRON

a n d R. R O S A

- C.N.R., Istituto Chimico, Facolth di Ingegneria, Universith di Bologna -

Via C a s t a g n o l i 1 - 4 0 1 2 6 B O L O G N A

- Italy.

Abstract -- In the present paper the basic features of Monte Carlo m e t h o d as applied to the electron probe microanalysis are outlined. In particular, applications to a large variety of experimental situations are reviewed. The problems examined are as follow: i) a binary film on a substrate of a third element; ii) a ternary film without substrate; iii) a ternary film on a substrate of an element present in the film; iv) a multi-layer elemental film; v) a multi-layer compound film. The o u t p u t of the computer program consists of: a) spatial distribution of the penetrating electrons; b) depth distribution of the generated X-rays; c) spatial distribution of the deposited energy; d) spatial distribution of the electron-hole pairs created in semiconductors. By comparing X-rays intensities with experimental data, one is able to obtain both the unknown composition and the thickness of the film. In some particular instances, additional informations, such as an independent determination of the thickness or measurements of the X-rays intensities at different electron energies, may be required.

BASIS OF MONTE

CARLO

MODELS

The Monte Carlo method

( M . C . , f o r s h o r t ) c o n s i s t s e s s e n t i a l l y in

* Presented at the IV Scientific Meeting of the Italian Association for Crystal Growth (AICC), Parma, Italy, 26-28 February 1979.

496 modelling stochastic variations present (or assumed as such) in nature by using random numbers to make a choice among possible outcomes. By means of computer programs based on this technique the path of electrons through the target material is traced. Each trajectory is build u p as a sequence of successive scattering points; the sequence is terminated when the energy of the primary electron fails below some chosen value or the last scattering point lies out of the surface. From a large n u m b e r of simulated trajectories information about the spatial distribution of electrons, as well as their energy distribution is obtained in a straight-forward way. In principle by the M.C. a full event-byevent treatment between the incident electron and the atoms of the target is available, but actually every M.C. model contains simplifications which may exert influence on the final results I . Our M.C. computer program, named CARLONE, is based on the so called "single scattering model", developed by Murata et ai.2 and extended to compounds by Kyser and Murata a. The model is described in detail elsewhere 4' s; we recall here the main outlines for the reader's convenience. a) The structure of the target is ignored, i.e., the real matter is viewed as a random assembly of scattering centres. b) The electronic and nuclear interactions are taken as independent of each other. For the former the Bethe's continuous slowing down approximation is assumed 6, i.e., electrons lose their energy via a purely frictional force without affecting the direction of motion. The energy loss is given by:

(1)

dE ~ dz

=

21r e2 N Z E

In

3'E J

where Z is the atomic number, N the number of atoms per unit volume, 3' = 1.166 and J the mean ionization energy given by Duncumb et al. 7 . As regards interaction with nuclei, they are almost entirely responsible for the angular scattering, producing, however, negligible energy transfer because of the large mass difference.

497

c) The distance between two successive scattering events is given by the elastic mean free path: (2)

Ai = [N a T (Ei)]-', for the i-th event

where o T is the screened Rutherford total cross section: 7r Z (Z + 1) e 4 (3)

OT •

4fl(l +13)E2

1

t3 = - i - - ( l ' 1 2 ~ / a T r ) 2 (aTF is the Thomas-Fermi radius) is a parame, ter to account for the electrostatic screening of the nucleus by the orbital electrons s. The usual Z 2 factor is replaced by Z(Z + 1) to account for the contribution of electron-electron scattering 9. d) R a n d o m numbers are employed to select the angular deflection 0 from the scattering cross section after each event:

(4)

cos0=l

213R

1 +13-R

, 0
R is a random number. The azimuthal angle ~0 is in turn selected from 27r radians by another random n u m b e r R': (5)

~0=2~rR'

,

0
e) In a composite target the scattering atomic specie is determined in a random fashion, weighted according to the stoichiometric composition, through a third random number. f) If the target is composed of two or more layers of different composition, as the electron crosses the boundary, the physical parameters appropriate to the layer must be changed according to the new composition.

498

EXAMPLES OF APPLICATION In the following we present a few representative examples of calculations, highlighting the applications of M.C. programs in electron probe microanalysis (EPMA). All the results shown were obtained by the CYBER 72/76 CDC-System of C.I.N.E.C.A.. In CARLONE the computer time for 10 kV electrons in Si, which penetrate at a maximum depth of the order of 1.2/am, is ~" 0.01 sec. per trajectory. Primary interaction volume

A ) Penetration of electrons The M.C. electron path calculation is directly visualizable by a computer-drawn projection of the trajectories of electron through the

0.1

Si

02

t

!

02

I

I

t

~

(11

I

I

al

I

I

0.2

pX Emg//cm2]

Fig. 1 -- Trajectories o f 100 electrons simulated by "CARLONE in a double layer o f A g (600 A thick) and A l (1000 fl{ thick) on Si. Incident enerh~y = 10 keV, angle o f incidence = 15".

499 target. The extent of the primary interaction volume is shown in Fig. 1, which refers to 20-keV electrons impinging on a double layer of Ag and AI on a substrate of Si at a beam incidence angle of 15 °. In a more quantitative fashion (see Fig. 2) the M.C. simulation enables one to obtain separately the distribution of i) the "final points", i.e., the point at which the electron energy has decreased to the preset value; ii) the m a x i m u m depth reached by each electron before coming to rest in the target; iii) the m a x i m u m penetration depth of back-scattered electrons.

i

"t'~ '

I

i

f---.\

/

X.

!

i

10 keV

6

ZI

o

b IP,/

:,

Ii1",I

~b!l

)\\ \".,

X

\,,,

07/'",,,...t, 0

0.5

1

z(~m) Fig. 2 - Depth distribution o f maximum penetration depth for absorbed electrons (dash-dot line) and back:scattered electrons (dashed line), and depth distribution of final points of absorbed electrons (continuous line) of lO-keV primary electrons at normal incidence in SL

500

B) Energy dissipation The distribution of energy loss in the target is easily obtained by the M.C. code by memorizing the amount of energy lost by each penetrating electron along its trajectory. Fig. 3a shows the energy dissipation profile for 10 keV-electrons in Si. This type of curves is of increasing interest for the study of semiconductors in the SEM or in connection with problems of resist exposure in modem electron device microfabrication I 0. Moreover the energy dissipation profile can furnish directly the depth distribution of the electron-hole pairs created, by taking into account the mean energy of pairs generation.

C) Distribution of X-ray signals As is known the generation of X-rays by electrons is a relatively inefficient process; however it is possible to simulate such a process by introducing in the code the probability of X-ray excitation as appropriate fractions of a photon to be excited for each electron scattering event. For the nl shell ionization this probability is expressible by means of the Qnl ionization cross section. One of the most widely used expression for Qnl is due to Worthington and Tomlin' 1 and is given, for the i-th event, by: (6)

Qnl E2nl = 6.51" 10 "14 Znl bnl

Unl

In

4Unl 1.65 + 2.35 exp(1 - U n l )

(cm 2 eV 2 )

where Ui = Ei/Enl, Enl is the binding energy of electrons in the nl shell; Znl is the number of electrons in the nl shell and bnl is a parameter. The number of photons generated by a'primary electron in a step length dS is proportional to Ira, where I m is the number of ions created of specie m:

501

(7)

Im=

Cm p N A Am

Qnml dS

where Cm, Am are the weight fraction and the atomic weight of the m-th element respectively, p is the mass density and NA is Avogadro's number. Summing the contribution for all steps, as well as that from all electron trajectories the code produces a distribution q~ (z) in depth of X-ray generation, as shown in Fig. 3b. When the specimen is comi

i

i

1

b)

~-~

lOk~t

4

~K.

2 1 i

.

.

a)

! ,=, W

0

I

0

05

1

z(pm) Fig. 3 - Depth distribution o f (a) energy dissipation and (b) SiKa X-ray radiation, calculated for lO-keV primary electrons at normal incidence in SL

502 posed by one or more films on a substrate of different materials, the (z) for either the film(s) or the substrate are calculated, as in Fig. 4 for the case of Ag on A1 on Si. I

S

I

I

2 AgLv¢

I

I

SiKo¢

AIKv¢ Q,

e

o

o

0.05

O.lO

o.15

0.20

0.25

0.30

pZ [mg/cm2] Fig. 4 - Depth of AgLo~ AIKO~ and SiK~ X-ray radiation produced in the multilayer specimen of Fig. 1.

Application to the field of EPMA The M.C. technique has been employed successfully not only to investigate the physical foundation of electron scattering in solids, but also for practical purposes such as the quantitative analysis in the EPMA. Comparing the M.C. with other approaches, it appears at once the wide range of situations not handled by the more usual theoretical procedures, based on various approximations. By the same com-

503 puter program one can easily vary incident energies, incident angles, kinds of target materials as well as geometrical configurations of the specimen.

A) Binary films The • (pz) function can be used to calculate the emitted X-ray intensity via the Laplace transform of each ,b (p z)

(8)

I E = f o '°Zmax~b (p z) exp ( - X p z) d p z

where X = # cosec 4,/a being the X-ray mass attenuation coefficient and ~k the take-off angle. PZma x is the maximum depth at which a primary electron produces X-rays; for thin films p Zma x = p t, where p t is the film thickness. The use of the units of mass distance has the advantage of requiring no knowledge of the mass density p of a c o m p o u n d target. By the same program the X-ray intensity I0 emitted by standards of elemental or compound samples is also calculated and memorized, so that the intensity ratio k = IE/I0 for each characteristic X-ray line from every element of the film represents the final o u t p u t of the simulation. Kyser and Murata 3 worked out a graphical procedure to determine the composition CA and cB = 1 - cA as well as the mass thickness pt for a binary AxBy film either freestanding or supported on a substrate which does not contain the elements A or B. Essentially the problem is that to determine tb¢o unknowns, p t and CA, with two knowns, the esperimental values kA and k B. In this case instead of the rather tedious graphical procedure, we supplied CARLONE with a numerical method of a convergence to obtain the unique solution for composition and mass thickness. For example weappliedsuccessfullyCARLONE intheanalysisfor TiOx (x ~- 2) antireflection coatings for silicon solar cells 12. Notice that when an element of the film, e.g. B, is present also in the substrate, kB is equal to 1 and insensitive to variations of its concentration in the film. To solve this kind of problem an indepen-

504 dent determination of the film thickness is necessary. Another approach consists in reducing, if possible, the incident energy in such a way as the primary interaction volume remains confined in the film.

r

~

a)

r

4

J°

koV

4 keV

1

c) lOkeV

8 keV

0

ZrO,

o

0

0.06

0.10

0

0.05

0.10

Fig. 5 - Depth distribution o f ZrL~ X-ray radiation produced in a double layer o f Zr02 (0.07657 mg/crn 2 thick) and ZrO (0.0589 mg/cm 2 thick) on Zr, by electrons at normal incidence with different initial energies.

Fig. 5 shows how the reduction of the incident energy below 5 keV allows to carry out the analysis only in the first film of ZrO~ while at higher energies both ZrO2 and ZrO are involved in X-ray emission.

505

B) Ternary films We e x t e n d e d K y s e r and Murata's p r o c e d u r e for the b i n a r y case to t e r n a r y B R L ' s ( b o r o n - r i c h layer), g r o w n o n Si, an alloy o f B-O-Si. T h e films, some h u n d r e d A thick, were a n a l y z e d with and w i t h o u t rem o v i n g the u n d e r l y i n g silicon substrate s' 13. When the substrate was present, we assumed the thickness f o u n d for the same f r e e s t a n d i n g film. Tab. 1 shows the results o b t a i n e d in b o t h cases, i.e., w i t h and Table 1 - Composition of BRL's on silicon substrate and freestanding obtained by Monte Carlo simulation of electron probe experimental data. Nuclear backscattering results are shown for comparison (Armigliato et al.s ).

BRL's without substrate Film No.

BRL-2

BRL-3

Thin film approximation

Monte Carlo

BRL's on substrate Monte Carlo

CB

71.8

68.3

72.5

74.8

co

9.9

11.2

6.5

7.1

csi

18.3

20.5

21.0

18.1

CB

76.1

73.3

78.4

77.0

co

7.8

8.5

4.4

4.3

csi

16.1

18.2

17.2

18.7

c(at%)

Nuclear backscattering

w i t h o u t the substrate, c o m p a r e d w i t h R u t h e r f o r d b a c k - s c a t t e r i n g measu r e m e n t s 1 4. Since the latter m e a s u r e m e n t s r e f e r r e d t o specimens on substrate, the results seem to indicate t h a t the p r o c e d u r e o f r e m o v i n g the substrate changes the film c o m p o s i t i o n , increasing the o x y g e n content.

506 REFERENCES

1.

2.

3. 4. 5. 6. 7.

8. 9. 10. 11. 12.

13.

14.

For a comprehensive review on the Monte Carlo technique for electron trajectory calculation and related applications, one can see: Use o f Monte Carlo Calculations in Electron Probe Microanalysis and Scanning Electron Microscopy, K . F J . Heinrich, D.E. Newbury and H. Yakowitz, ~ds., NBS Special Publication No. 460, 1976. A. DESALVO and R. ROSA -- L 'uso del metodo Monte Carlo nella raicroscopia elettronica a scansione e microanalisi a raggi X, in: Microanalisi e microscopia elettronica a scansione, U. Valdr~ ed., in press. K. MURATA, T. MATSUKAWA and R. SHIMIZU - Japan. J. AppL Phys., I0, 678, 1971; K. MURATA, T. MATSUKAWA and R. SHIMIZU - Proc. 6th In~ Conf. X-ray Optics and Microanalysis, Osaka, 1971, G. Shinoda, K. Kohra and T. Ichinokawa eds. (Tokio: University of Tokio Press, 1972), p. 105. D.F. KYSER and K. M U R A T A -- IBM J. Res. Develop., 18, 352, 1974. R. ROSA -- Proc. Simposium "'Problemi Attuali in Scienza dei Materiali", Bressanone, 1978, P. Mazzoldi ed., to be published. A. ARMIGLIATO, A. DESALVO, R. R I N A L D I and R. ROSA - J. Phys., D, 12, 1299, 1979. H.A. BETHE - - A n n a l e n der Physik, 5, 325, 1930. P. DUNCUMB, P.K. SHIELDS-MASON and C. DA CASA - Proc. 5th Int. Conf. X-ray Optics and Microanalysis, Tuhingen, 1968, G. MSllenstedt and K.H. Gaukler eds., Berlin: Springer Verlag, 1969, p. 146. B.P. NIGAM, M.K. SUNDARESAN and TA-YON WU --Phys. Rev., 115, 491, 1959. L.A. KULCHITSKY and G.D. LATYCHEV -- Phys. Rev., 61, 254, 1942. R. SHIMIZU, T. IKUTA, T.E. E V E R H A R T and W J . DEVORE -- J. Appl. Phys., 46, 1581, 1975. C.R. WORTHINGTON and S.G. TOMLIN - - P r o c . Phys. Soc., A69, 401, 1956. A. ARMIGLIATO, G. CELOTTI, S. GUERRI, G. MARTINELLI, P. OSTOJA and R. ROSA -- Proc. 1979 Photovoltaic Solar Energy Conference, Berlin, 1979, p. 784. A. ARMIGLIATO, G.G. BENTINI, A. DESALVO, R. RLNALDI, R. ROSA and G. R U F F I N I -- Proc. 8th Int. Conf. X-ray Optics and Microanalysis, Boston, 1977, Science Press, Princeton, 1979, to be published. P. MAZZOLDI -- this conference.