Account of photoelectron elastic determination of overlayer thickness, in-depth profiling, escape depth, attenuation coefficients and intensities in surface systems

Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Account of photoelectron elastic determination of overlayer thickness, in-depth profiling, escape depth, attenuation coefficients and intensities in surface systems V.I. Nefedov*, I.S. Fedorova Institute of General and Inorganic Chemistry, RAS, Leninski pr. 31, Moscow, Russia Received 29 October 1996; revised 10 February 1997; accepted 10 February 1997

Abstract An analytical connection is found between parameters of Monte Carlo and transport theory methods to account for elastic electron scattering in solids. A simple analytical method is proposed for the intensity calculations in X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES) with account of elastic scattering. The method is applied for solution of many problems in XPS including determination of the overlayer thickness, correction factors for the in-depth profiling, escape depth, attenuation lengths, intensities in layered systems. q 1997 Elsevier Science B.V. Keywords: Elastic scattering; In-depth profiling; Overlayer thickness determination; Attenuation lengths of photoelectrons in solids

1. Introduction The influence of elastic scattering on the photoelectron angular distribution in solids has been generally recognized since the beginning of the 1980s. In the first papers as well as in papers published up to the beginning of the 1990s the main approach to account for the elastic scattering was the Monte Carlo (MC) method [1–12]. Impressive progress was achieved during recent years by application of the transport theory (TT) [13–25]. However the analytical connection between these two methods has not been available so far, although the comparison is usually made. One of the main aims of this paper is to establish such a link. Another aim of this paper is connected with the * Corresponding author.

following observation. The MC method is comparatively simple but entails experience and a lot of computation. TT method is comparatively highly sophisticated although the actual computational side is not time consuming. As a result neither the MC nor TT method is widely used by experimentalists to account for the elastic scattering in their experimental data. That is why this paper attempts to give a comparatively simple approximate analytical method that is simple both in its physical essence and in the computation and gives reasonable accuracy. According to these two aims the paper has the following sections.In Section 2 the different representational forms for the angular distribution of photoelectrons in solids are considered. In Section 3 the connection between the MC and TT methods is discussed. In Section 4 the so-called partial intensity contributions (P) based on the escape probability or

0368-2048/97/$17.00 q 1997 Elsevier Science B.V. All rights reserved PII S 0 36 8- 2 04 8 (9 7 )0 0 03 0 -3

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on the depth distribution function are introduced and an analytical presentation is found. In the following sections the applications of the partial intensity contributions P to in-depth profiling, overlayer thickness determination, attenuation coefficients, escape depth of the XPS analysis and intensity relations in surface systems are demonstrated. In the conclusion the main results are summarized and presented in a form that aids their general application. The notation for angles is given in Fig. 1. For cos v 0 the following equation is valid, where Q 0, a 0, J 0 are the initial angles of the photoelectrons before any scattering occurs: cos v0 = cos g cos a0 + sin g sin a0 cos(J − J0 )

(1)

Fig. 1. The angle notation.

Note the analyzer is in the XZ plane.

2. Representational forms for photoelectron angular distributions The intensity of photoelectrons with a given energy in a semi-infinite homogeneous sample can be represented in the form [26] when elastic scattering is neglected I` = JEQA DS0 ln

dj dQ

(2)

where J is the intensity of the X-ray flux, E is the detector efficiency, Q A is the solid acceptance angle of the analyzer, D is the efficiency of the analyzer, n is the sample density, S 0 is the effective area of the sample from which the photoelectrons reach the analyzer; when cos a = 1, l is the inelastic mean free path of the photoelectron, dj/dQ is the differential photo-ionization cross-section. For free atoms or molecules dj/dQ is given by [27]: dj=dQ = (j=4p)[1 − (b=4)(3 cos2 v − 1)]

(3)

where r is the total photo-ionization cross-section for the level studied, b is the asymmetry parameter, v is the angle between the direction of the photoelectrons and X-ray flux (Fig. 1). We represent Eq. (2) in the form I`

= I0 P0 (g, v → a0 )

P0

= [1 − (b=4)(3 cos v − 1)] 2

(4)

where P 0 is the photo-ionization probability in the direction a 0, J 0 (for angles a 0 and J 0 see Fig. 1). When elastic scattering is neglected (the so called straight line approximation, SLA), the initial direction of the photoelectrons is not changed and a = a 0. The effect of elastic and inelastic scattering will change both the initial direction and intensity and can be described as some transformation T of initial P 0 values to P f values. This transformation has two components, T(i) and T(a), corresponding to isotropic and angular parts of P: T = T(i) + T(a)

(5)

We will designate the transformation applied to a semi-infinite sample as T` (P0 ) = T` (i)(1 + b=4) − T` (a)(3=4)b cos2 v = Pf (g, J → af )

… 6†

The intensity I ` which takes into account both elastic and inelastic scattering is given by I` = I0 T`

(7)

Further on, as in Fig. 1, the notation ‘f’ (final) for the detection angle a will be omitted. In paper [1] (see Eq. (34) in this paper and consider the change of notation) for T(i) and T(a) the following is obtained with the help of the MC method: Pf (MC) = T` (i)(MC)(1 + b=4) − (3=4)bT` (a)(MC) (8) where T(i)(MC) = X , T` (a)(MC) = (Y0 cos g + Y 1 sin 2

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2g cos J/2 + Y2 sin2 g cos2 J + Y3 sin 2g sin 2J)X and Y i are certain complex integrals, which can be determined by the MC method. See Ref. [1] and Appendix A, where calculations of Y i values are described in some detail. Note that the representation Eq. (8) is very convenient for the MC method because a single Monte Carlo calculation is enough to describe the angular dependence, i.e. to calculate X, Y i (see Ref. [1] and Appendix A for details). There is an important analogy with and difference from SLA in the expression for T obtained by the MC method. In the SLA the effective sample area S seen by the analyzer is given by S = S0 =m m = cos a The inelastic scattering attenuation is given by the factor exp( − z/lm)dz and the intensity I for a semi-infinite sample does not depend on m, if v = constant. In the MC method the same is also partly true. All the integrals involved with X and Y i are proportional to m and as in the SLA there is no direct dependence on m connected with the effective sample area. However the indirect dependence on m exists for X even when v = constant. In papers [7,9,10,12] the following representation is proposed based on MC calculations (in our notation): Pfm (g, v → a) = Q[1 − (beff =4)(3 cos2 v − 1)]

(9)

where Q and b eff are parameters depending on m, but the Q × b eff value is nearly constant in the range up to a = 75–808 (Figs. 5–8 in Ref. [23]). Eq. (9) has an evident resemblance to Eq. (4). Eq. (9) is of a remarkable character and not due just to the presence of the ‘magic’ angle cos 2v m = 1/3. Indeed, two isotropic parts present in the initial P expression (Eq. (4)) 1 and b/4 seem to be transformed in a different way by the Monte Carlo process, contrary to Eq. (8) and general considerations. We have made an independent test of the accuracy of Eq. (9) and have found it numerically correct with reasonable accuracy. We shall show below that the reason for the unexpected feature of the representation Eq. (9) is due to the so far theoretically unexpected cancellation of some parts belonging to the isotropic and angular distributions in P 0. We write the corresponding transformation in the form: T` (i)(m) = Q

T` (ia)(m) = Qbeff

In the TT method the following P f expression is obtained [23] in the same form as Eq. (9) rf (TT

ÿ m) = Q[1 − (beff

=

4)(3 cos2 v − 1)]

(10)

where Q = a(D1 + D2 )

a = ltr (l + ltr ) − 1

D1 = (1 − q) − 1 2 H(m, q) q = 1 − a =

beff = ab=Q

p

a1 2 = c = 1 − q =

D 2 is an integral depending on b, q − and angles a, g; l tr is the transport mean free path, q is the single scattering albedo (see Ref. [23] for details); H(m,q) is the Chandrasekhar function, which can be calculated with the accuracy of a few percent by the formula [20] H(m, q) = [1 + m(h − 1)]=[1 + m(h − 1)(1 − q)1 2 ] =

(11)

where h = H(1,1) = 2.908. The integral D 2 is shown to be small in comparison with D 1 for Al 2s, Ag 3s, Au 4f (and some other XPS lines [23]). However even a small dependence of the transformed isotropic part Q on the X-ray incidence angle g is not expected from the general point of view. Now it is time to investigate why the simple and indeed numerically correct forms Eqs. (9) and (10) should be considered as a good approximation both by the MC and TT methods. Using the general TT approach [23,22] one can obtain Pf (TT) = D1 a(1 + b=4) − (3=4)bak

(12)

where 

k = cos2 g(cos2 a − D3 q=2 + qD1 =3) + 

sin 2g sin 2a 2

+ sin2 g(sin2 a − qD4 =4 + qD1 =3) where D3 = H(m, q) D4 = H(m, q)

…1 0

x3 H(m, q) dx x+m

0

xH(m, q) (1 − x2 )dx x+m

…1

D3 = qD1 =3 − D3 q=2

D4 = qD1 =3 − D4 q=4

The corresponding transformation can be written in

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Table 1 The calculated a values for Al2s and Ag3d on the base of Y 0, Y 1 and Y 1 a

Al2s

Ag3d Y1

Y0 a

3 9 15 21 27 33 39 45 51 57 63 69 75 81 87 a

0.829 0.822 0.815 0.813 0.810 0.814 0.815 0.815 0.797 a 0.790 a 0.808 0.805 0.798 0.780 0.750 a a av = 0.82

Y2

0.922 0.818 0.821 0.819 0.814 0.813 0.818 0.817 0.814 0.824 0.821 0.826 0.848 0.910 1.382 a

0.832 0.822 0.819 0.813 0.814 0.825 a 0.815 0.804 0.809 0.808 0.805 0.801 0.794 0.775 a 0.741 a

Y0 a

0.681 0.681 0.675 0.678 0.674 0.675 0.674 0.670 0.634 a 0.759 a 0.695 0.692 0.690 0.662 0.655 a a av = 0.69

Y1

Y2

0.731 0.677 0.687 0.684 0.681 0.681 0.684 0.686 0.683 0.687 0.687 0.701 0.714 0.761 1.160 a

0.683 0.681 0.675 0.687 0.676 0.664 a 0.716 0.692 0.680 0.683 0.676 0.678 0.676 0.650 a 0.633 a

For statistical or some other reasons these values are not reliable.

the form T(i)` (TT) = D1 a

T(a)` (TT) = ak

(13)

Eq. (12) is equivalent to Eq. (10), but while deriving Eq. (12) no cancellation of contributions from isotropic and angular parts was allowed. For example for the integral of the type I 1 one should write 1 I1 = 4p

…



b P0 dQ = 1 + 4 4p



 

− i

b 4

(14) a

and not I1 = 1

We have checked numerically the contribution d in a complete range of q, a and g values. The contribution d to P f (TT) in most cases is under 1%. Only in a few cases does it amount to 2–3% (usually for negative g values, which are not important for practical applications). There is no important X-ray incidence angle dependence of P f (TT). This result has been previously proved for several XPS lines [10,23]. However, when q → 1 and d → 0, P f (TT) → 0 and the d contribution to P f can be large.

(15)

although it is equivalent from a mathematical point of view. The representation Eq. (12) can be obtained also by using Eq. (15), however the use of Eq. (14) makes the deriving of Eq. (12) more direct. The representations in Eqs. (12) and (13) have a correct form (Eq. (6)) and are in direct correspondence with Eq. (8), derived by the MC method. This connection is discussed in the next section. Now we transform Eq. (12) to the form of Eq. (9): Pf (TT) = aD1 −

1 4ab(3

2

cos v − 1) 2

− 34ab D3 cos g + D4 sin g − (D1 − 1)=3 2

2

= aD1 − 14ab(3 cos2 v − 1) − d(a, i)

3. Connection between the MC and TT methods In the previous section the P f value was obtained in two completely independent ways using the MC method (Eq. (8)) and the TT method (Eq. (12)). It is possible to find the connection between these two methods. Comparing Eqs. (8) and (12) we obtain the following relations: X = aD1 = Hc

Y0 = a(cos2 a − D3 ) Y1 = a sin 2a

p

…17†

3

Y2 = a(sin2 a − D4 ) c = 1 − q

…16†

To get a more detailed insight into the interconnection between the MC and TT methods one can

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

use the approximation Eq. (9). In this case one sets d(ia) = 0 and, using Eq. (16), Eq. (17) becomes Y0 = a(cos2 a − 1=3) + x=3

X = aD1

Y2 = a(sin2 a − 1=3) + x=3

Y1 = a sin 2a

Y3 = 1=3(x − a)

…18†

The elements of Eq. (18) are a simple and convenient way to determine the a value from MC calculations. In the case of the ‘magic’ angle cos 2v = 1/3 one gets X = 3Y0

3Y2 = a + X

(19)

If we introduce Eq. (18) into Eq. (8) we obtain P f (TT-m) with d(ai) = 0 (Eq. (16)). Eqs. (18) and (19) have been checked in a wide range of values using some previous MC calculations [1,2] as well as new ones (see Appendix A). These relations are found to be of a good accuracy. Some results of this check are presented in Table 1. Note that Eqs. (18) and (19) do not depend on the specific scattering cross-section that was used for MC or TT calculations.

225

The intensity according to Ref. [1] is given by I` = JEQA DS0 nl

G 4pl

(24)

The analytical expression for G as well as the means for its calculation by the MC method is given in Ref. [1]. Note the P 0 used in this paper (Eq. (3)) differs by a factor 1/4p from the phn (Q → Q es) in paper [1]. It is completely unimportant for the final results but spares the necessity of writing 1/4p repeatedly if the definition of the present paper is accepted. So in the following the factor 1/4p will be incorporated in the I 0 value as stated by Eqs. (3) and (4). A similar observation is valid also for the definition of G in [1]. It is more convenient to consider G9 = G/l. So it is set that Gs 9 =

W 9 = 4pW4

1 ml

…` 0

Ws 9(z) dz = Gs =4pl

G9 = ∑ Gs 9

…25†

I(`) = JEQA DS0 nlG9 = I0 G9

(26)

s

4. The partial intensity contributions

Now Eq. (26) coincides with Eq. (7) and it is valid that

In paper [1] the probability W z for the photoelectron to escape from depth Z with the escape angle a was introduced. In the SLA

G9 = T`

ÿ

Wz = exp( − z=lm)phn Qin → Qes

1

phn =

1 P 4p 0

(20)

In paper [1] the G function was also introduced as an integral contribution of the photoelectrons coming from all z values to the intensity G=

4p m

…` 0

Wz dz

4p Gs = m

Pzi =

…` 0

Ws dz

(22)

s

1 lm

…z

i

0

W 9(z) dz

(28)

The P values allow the XPS intensity of different systems to be represented in a simple way: the semi-infinite sample I` = I0 P `

(29)

the overlayer of thickness d I(d) = I0 Pd

The integral contribution G to the intensity is G = ∑ Gs

Both G9 and T ` mean the integral change of the intensity of the semi-infinite sample due to the influence of elastic scattering. Now the partial intensity contribution P is introduced. P zi is defined as a contribution of the overlayer from z = 0 to z i to the intensity of the semiinfinite sample for the escape angle a (see Eqs. (7), (11), (25)–(27)).

(21)

The probabilities W s, which describe the probability of the photoelectrons coming from the depth z to reach the surface after s elastic scattering, are considered taking into account the inelastic and elastic scattering. One can write [1]

(27)

(30)

the substrate under overlayer of thickness d (23)

Is (d =s) = I0 (P` − Pd )

(31)

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the layer of thickness t under overlayer of thickness d dIt (d =t=s) = I0 (Pd + t − Pd )

(32)

If t → dz one obtains for W9(z) according to Eqs. (30) and (32) dI 1 W 9(z) =I 0 = dz lm

(33)

There is a direct connection between the ‘escape probability’ W(z), W9(z) introduced in Refs. [1,2] and the depth distribution function F(z) as defined in Refs. [10,11,28]: F(z) =

dI 1 4p W 9(z) = W (z) =I = dz lm lm

(34)

the thickness d. The attenuation factor is determined mainly by the properties of the overlayer but the photoelectron paths in the substrate have also some influence on k values. Nevertheless to a good approximation [3,34,36] the k value for the overlayer of thickness d obtained for the homogeneous substance A can be used for calculations of the intensities in the system, where there is an overlayer of substance A on substrate B. In this case one obtains for overlayer A of thickness d on the B substrate Id (A) = I0 (A)Pd (A)

(38)

for the substrate B under overlayer A of thickness d Is (dA=SB) = I0 (B)[P` (A) − Pd (A)]=P` (A)

(39)

In the SLA the function F(z) has the form of Eq. (35) and normalisation of Eq. (36): F(z) =

1 exp( − z=lm) lm

(35)

The usual normalization is …`

F(z) dz = 1

(36)

0

Note that W9(z) in the SLA corresponds to the attenuation factor exp( − z/lm), which gives the escape probability from the depth z for the escape angle a. Indeed according to Eq. (33) the value (1/lm)W9(z) gives the intensity contribution of the layer dz under overlayer of the thickness z to the intensity of the photoelectrons without energy loss with the exit angle a. The W9(z) values are usually smaller than corresponding exp( − z/lm) values in the SLA for the same z. There are two important exceptions: for z = 0 the back scattering effect which is absent in the SLA makes W9(z) . 1 and W9(z) . exp( − z/lm) for small m (see below, Section 7). The normalization of W9(z) is discussed in Section 5. So far we have considered the part of the same semi-infinite sample. Let us turn to the system, where the overlayer and the substrate consist of different substances. According to Eqs. (31) and (33) the quantity k = (P` − Pd )=P`

(37)

can be regarded as the attenuation factor for the substrate signal due to the presence of the overlayer of

5. Determination of partial intensity contribution In the following the prime of W z9 will be omitted. To use simple Eqs. (29)–(32), (38) and (39) it is necessary to determine W z as a function of z, m, l and some parameters which describe the elastic scattering. In Refs. [14–18] the depth distribution function is determined within the TT approach. In Ref. [8] an analytical expression is derived using MC results. Our approach is based on the results in Refs. [8,14– 18]. The general idea can be described in the following way. It is desirable to obtain the W z function as a simple analytical expression which would model in some way the results of the TT derivations. Some modifications will then be applied to bring the final expression in accordance with the MC results. It was found that a comparatively simple analytical expression for W z does not have the desired accuracy when the whole range of z, m values is considered. To keep the W z expression reasonably simple, the range of z was limited to 3–4l and the exit angle a was limited to 70–758, because only these ranges for z and a are of practical interest. As a matter of fact the range of a could be also limited to 60–708, because the roughness of the majority of sample surfaces prevents reliable measurements for a . 60–708. No special attention is given to the so-called surface effect [8], because the situation near the surface is too complicated and the MC results obtained for z → 0 are

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not expected to be correct for the following reasons. The same is also true for the TT approach. Let us consider an atomic monolayer on the very surface. First, the normal statistical distribution of mean free paths is not valid, especially for m = 1. In the MC or TT approach both the elastic and inelastic scattering take place within this monolayer, i.e. within a single emitter even for m = 1 which is really not the case. Second, the usual MC approach uses the plane wave (PW) as a final photoelectron function and not the real radial and angular part, which must be different for 2s and 2p photoelectrons with the same kinetic energy due to the dipole selections rules DL = 1. It is important when the small distances between the emitter and the scatter have a large statistical weight, which is the case on the very surface. The calculations as well as empirical adjustment of the photoelectron diffraction results [29,30] have shown that the forward scattering amplitude even for large kinetic energies in the PW approximation can be overestimated by a factor of 2 or even 2.5 in comparison to scattered (spherical) wave calculations ˚. if R > 2 A Similar considerations are also partly valid for the second atomic monolayer. Although the adjustment we have made for our W(z) expressions indirectly takes into account the ‘surface’ effect, one can hardly expect either the W(z) expressions or other considered approaches to be realistic for the very surface. That is why it is recommended to use W(z) values reported below the calculations of overlayer intensities for z $ 2d, where d is the thickness of the monolayer, and for z $ d for the substrate intensities. As a matter of fact for such thin overlayers a special theory is needed which takes the discrete character of the surface completely into account, and also accounts for the spherical nature and the selections rules of the photoelectron final wave functions. Now we consider the determination of W z. The W z function consists of isotropic and angular parts Wz = Wz (i) − Wz (a)

(40)

Let us begin with the isotropic part W z(i). In accordance with the results in Refs. [8,14–18] the

analytical expression for W z(i) is sought with the following boundary conditions. W z(i) is given as a sum of some exponential functions 1(i) W (i) = ∑ (k)ak exp( − z=lym) 2(i) ∑ (k)ak = H(mq) for z = 0 or ak < ak 9H(m, „ q), ∑ ak 9 = 1  3(i) 0` Wz (i)dz = mlH 1 − q = mlHc 4(i) The preferential y values must be c, c 2 and c 2n(in the last case m = 1), where n is defined in Ref. [16] and for q , 0.5 it is valid that n = 1 + 2 exp( − 2/q). The connection between c and l tr and l is given in Eq. (10). 5(i) The asymptotic value for the term with exp( − z/lnc 2) must be of the type m ak < exp( − z=lnc2 ) n−m It is started with the largest y value, i.e. y = c.

p

In this case due to condition 3(i) one obtains Wz (i) = H exp( − z=lcm) (41) Condition 3(i) prevents any other contribution of the second term a 2 exp( − z/lym). That is why two exponential functions are looked for with y = c 2 and c 2n. In accordance with equation 15 in Ref. [14] one obtains H [(nc − m)exp( − z=lmc2 ) Wz (i) = c(n − m) + m(1 − c)exp( − z=lc2 n)]

…42†

Eqs. (41) and (42) have been previously considered as a good approximation for the depth distribution function [14,15], i.e. Eq. (41) for m , 0.5 and Eq. (42) for 0 , m , 1. This conclusion is based on comparison with the theoretical depth distribution function in the TT approach. The analysis has shown that the best fit to the MC results can be achieved in the following way. The W z(i) defined by Eqs. (41) and (42) should be taken with the same statistical weight: 

Wz (i) = H 0:5 exp( − z=lc) +

0: 5 c(n − m)



× [(nc − m)exp( − z=lmc ) + m(1 − c)exp( − z=lc n)] 2

2

…43†

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As a matter of fact more elaborated weighting gives a better fit to the MC results. However this improvement is not critical and in the paper the most simple expression for W z(i) is investigated. The partial intensity contribution P d value (Eq. (28)) equals Pd =

1 lm

…d



W (z)dz

The P values in the general case are (Eqs. (28), (44) and (45)) 

Pd = Hc 0:5[1 − exp( − dlmc)] +



+ n(1 − c)(1 − exp( − d =lc n))]

0

2

= Hc 0:5[1 − exp( − d =clm)] +

−

0: 5 [ − (nc − m)[1 − exp( − d =c2 lm)] n−m 

+ n(1 − c)[1 − exp( − d =lc2 n)] − ]

…44†



Wz = H 0:5[1 − exp( − z=lmc)] +

0: 5 [(nc − m)exp( − z=c2 ml) c(n − m) 

+ m(1 − c)exp( − z=c2 nl)]

… †

Wz (a) = ∑ bk exp( − z=lym) k

2(a)

F(z) = (1=lm)W (z)

∑ bk = 1 3(a) Wz (a)dz = alm

where a = 1 − q = c 2 is given by Eq. (10). To begin with the one-exponential form was tried, Wz (a) = exp( − z=lmc2 )

…46†

b − [exp( − z=lc2 m)](3 cos2 v − 1) 47 4 The depth distribution function F can be presented by Eq. (34):

1(a)

0

c2 b [1 − exp( − d =lc2 m)](3 cos2 v − 1) 4

The W equals (Eqs. (40), (43) and (45))

Some possible improvements for W z(i) are discussed below. Now we turn to the determination of the angular W z(a) form. Conditions similar to Eqs. (1)–(4) are used.

…`

0: 5 [(nc − m)(1 − exp( − d =c2 ml)) c(n − m)

(45)

The agreement between Eq. (45) and MC calculations was very good and no further improvements were tried. The good correspondence between W z(a) according to Eq. (45) and MC calculations have been already expected due to the good accuracy of Eq. (18). Indeed, when a values are found on the basis of the MC determined Y i values which are responsible for the angular distribution of the photoelectrons in solids, the correspondence between Eq. (45) and the MC calculations depends mainly on the accuracy of the relations in Eq. (18).

(48)

The H and n values are given by Eq. (11) and in condition 4(i). Note the relations of Eq. (18) allow 3n independent determination of a values, where n is the number of Da regions in the MC calculations. In our case Da = 6 and n = 15. Of course, due to the statistical errors in the MC calculations as well to the approximate nature of the ‘magic’ form the a values show some scattering around some average values. There are also some trends. For example, the a values tend to increase with a values if a is calculated on the base of Y 1 values. Nevertheless up to a = 75–808 the a value can be regarded as a constant or slowly changing parameter. In this paper the relations of Eq. (18) are used to determine the a value. Just a single MC calculation as described in Appendix A is enough for the determination of the escape probability W z and of partial intensity contribution P, which allow the detailed description of the photoelectron distribution in solids.

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

229

Fig. 3. The angular part W z(a) of Ag3d-line for different a values (q = 0.31, v = 458, b = 1.22, z in l) A-MC, x-th. Fig. 2. The comparison of W z(i)-values for Mo, E kin = 500 eV, z in ˚ . The solid line is in accordance with [33]. The present calcuA lations with a = 0.65 (Table 1B), K-th1 and A-th3.

The direct use of MC results for many purposes require more effort than the determination of the a value and further use of the W z function. Of course, the correctness of the a value depends on the precision of the MC calculation. For general use of the simple Eqs. (46)–(48)) the a = c 2 value is needed. There are also two theoretical ways for calculation of a values. The most simple one is to use numerous data for l tr values calculated in Ref. [23] and the relation a = ltr =(l + ltr ) The other way is to calculate the so-called transport elastic cross-section as described in Ref. [23] and then l tr. The a values found in this way we shall call theoretical values. The second way is described in Appendix B, where the a values for 45 elements for the photoelectrons with kinetic energies 250, 500, 1000 and 1500 eV are given. Usually the theoretical a values are in good agreement with a values found by MC calculations (Eq. (18)). For example, Eq. (18) gives a values

equal to 0.82, 0.77 and 0.69 for Al 2s, Au 4f and Ag 3d, respectively (using our MC calculations). Using l tr in Ref. [23] for Mg K a radiation one obtains 0.89, 0.66 and 0.67. The last two values are in accordance with MC calculations [23] but for Al 2s the value < 0.80 is obtained by the MC method [23]. A comparatively large difference for the Au 4f line between our a value and the a value in Ref. [23] can be explained by the difference in the elastic scattering cross-section used in the present paper and in Ref. [23]. In Ref. [23] the relativistic cross-section was used and the a value in this paper is expected to be more reliable. Note that the a value found according to Eq. (18) reproduces the corresponding MC calculation when Eqs. (46)–(48) are applied. The elastic cross-section has no or little influence on the correctness of Eq. (18). Some examples will be discussed below. In Fig. 2 the W(i) function calculated according to the analytical form given in Refs. [8,33] for photoelectrons with E kin = 500 eV in Mo is compared with the calculations according to Eqs. (43) and (50) using an a value obtained from the MC calculations in Refs. [8,33] (Appendix B). Both calculations agree with the MC results [33].

230

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Table 2 The comparison of analytical (theoretical) and MC determined (1/lm)Wz (i) function (z in l) for Al2s (q = 0.18) and Au4f (q = 0.23) a

z

0.00 Al MC th1 Au MC th1 0.15 Al MC th1 Au MC th1 0.25 Al MC th1 Au MC th1 1.00 Al MC th1 Au MC th1 2.00 Al MC th1 Au MC th1 3.00 Al MC th1 Au MC th1 4.00 Al MC th1 Au MC th1

3

15

33

51

75

81

1.02 1.07 1.03 1.09

1.06 1.10 1.06 1.12

1.22 1.27 1.24 1.29

1.64 1.68 1.66 1.70

4.09 3.99 4.17 4.03

6.89 6.53 7.00 6.58

0.91 0.90 0.92 0.91

0.92 0.93 0.93 0.94

1.04 1.04 1.07 1.05

1.32 1.28 1.33 1.29

2.12 2.06 2.05 2.02

1.87 2.18 1.75 2.10

0.83 0.81 0.84 0.82

0.85 0.83 0.85 0.84

0.93 0.91 0.93 0.92

1.11 1.07 1.11 1.08

1.29 1.33 1.22 1.29

0.90 1.06 0.80 1.00

0.38 0.35 0.38 0.35

0.37 0.35 0.37 0.35

0.35 0.34 0.33 0.33

0.29 0.29 0.29 0.28

0.076 0.067 0.080 0.064

0.045 0.023 0.050 0.026

0.12 0.12 0.12 0.11

0.12 0.11 0.11 0.11

0.090 0.091 0.087 0.085

0.051 0.052 0.047 0.048

0.008 0.007 0.010 0.008

0.005 0.006 0.006 0.006

0.037 0.038 0.034 0.035

0.034 0.035 0.031 0.032

0.023 0.025 0.021 0.022

0.009 0.010 0.009 0.009

0.0013 0.0019 0.0015 0.0020

0.0008 0.0016 0.0010 0.0017

0.0095 0.0125 0.0042 0.0108

0.0088 0.0112 0.0031 0.0097

0.0047 0.0067 0.0022 0.0058

0.0012 0.0022 0.0010 0.0019

0.0005 0.0006 0.0003 0.0005

0.0001 0.0005 0.0002 0.0005

We have also found an agreement between the calculated W(i) functions (Eq. (43)) and a values calculated from MC results in Ref. [2]. Note that the scattering cross-section used for the MC calculations in Refs. [2,8,33] and in the present paper differ. Nevertheless the a values found from these MC calculations according to Eq. (18) reproduce the corresponding MC results when eqns (46) and (47) are used. This is an additional proof of independence of the correctness of Eqs. (18), (46) and (47) on the scattering cross-sections used in MC

calculation. However it is evident that realistic scattering cross-sections will give a realistic a value and realistic W z functions which can produce realistic MC results. The fact is that Eqs. (18), (46) and (47) can also reproduce unrealistic MC calculation with good accuracy, i.e. Eqs. (18), (46) and (47), as follows from their derivation, do not rely on correct elastic and inelastic scattering cross-sections. One of the main aims of this paper is to show that Eqs. (18), (46) and (47) are of use for reproducing of the MC results.

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248 Table 3 Comparison of (1/lm)W i(z) values for Ag3d (q = 0.31) calculated in different approximations a

z

0 MC th1 th2 th3 0.15 MC th1 th2 th3 0.25 MC th1 th2 th3 0.60 MC th1 th2 th3 1.00 MC th1 th2 th3 2.00 MC th1 th2 th3 3.00 MC th1 th2 th3 4.00 MC th1 th2 th3 5.00 MC th1 th2 th3

3

15

33

51

75

81

1.11 1.13 1.14 1.13

1.13 1.16 1.18 1.16

1.31 1.33 1.35 1.33

1.75 1.75 1.77 1.75

4.31 4.09 4.12 4.09

7.19 6.65 6.68 6.65

0.97 0.94 0.96 0.94

0.99 0.97 0.98 0.97

1.11 1.07 1.08 1.07

1.37 1.31 1.31 1.31

1.94 1.96 1.92 1.95

1.55 1.95 1.87 1.94

0.87 0.84 0.84 0.84

0.89 0.86 0.86 0.86

0.98 0.93 0.93 0.93

1.15 1.08 1.08 1.07

1.10 1.21 1.18 1.19

0.69 0.89 0.85 0.86

0.58 0.55 0.55 0.55

0.57 0.55 0.55 0.55

0.60 0.56 0.56 0.56

0.56 0.55 0.54 0.54

0.23 0.26 0.24 0.24

1.137 0.098 0.097 0.097

0.36 0.34 0.33 0.33

0.34 0.34 0.33 0.33

0.31 0.32 0.31 0.30

0.24 0.26 0.25 0.24

0.067 0.061 0.059 0.059

0.044 0.031 0.033 0.034

0.093 0.100 0.095 0.087

0.084 0.095 0.090 0.083

0.063 0.075 0.071 0.064

0.034 0.042 0.039 0.034

0.0078 0.0083 0.0083 0.0079

0.0057 0.0070 0.0070 0.0067

0.020 0.029 0.026 0.022

0.019 0.026 0.024 0.020

0.012 0.018 0.016 0.013

0.0051 0.0073 0.0066 0.0052

0.0012 0.0019 0.0018 0.0015

0.0008 0.0016 0.0015 0.0012

0.0054 0.0083 0.0073 0.0055

0.0046 0.0073 0.0064 0.0048

0.0026 0.0042 0.0037 0.0027

0.00110 0.00140 0.00121 0.00085

0.00025 0.00045 0.00040 0.00029

0.0002 0.0003 0.0003 0.0002

0.0013 0.0024 0.0020 0.0014

0.0010 0.0020 0.0017 0.0012

0.00054 0.00101 0.00084 0.00055

0.00022 0.00029 0.00023 0.00015

0.000064 0.000106 0.000088 0.000055

0.0000 0.0000 0.0000 0.0000

231

232

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Table 4 Angular part (1/lm)Wz (a) for Al2s (q = 0.18, v = a) z 0 MC th 0.1 MC th 0.2 MC th 0.5 MC th 1 MC th 2 MC th 3 MC th 4 MC th a

3

15

33

51

75

81

1.00 1.00

0.93 0.93

0.66 0.66

0.139 0.149

−1.63 −1.54

−3.16 −2.96

0.88 0.88

0.82 0.82

0.56 0.57

0.091 0.123

−1.04 −0.96

−1.31 −1.36

0.79 0.78

0.72 0.72

0.49 0.50

0.070 0.101

−0.61 −0.60

−0.51 −0.62

0.55 0.54

0.50 0.50

0.31 0.32

0.037 0.056

−0.11 −0.15

−0.051 −0.060

0.30 0.29

0.27 0.26

0.15 0.15

0.027 0.021

−0.005 −0.001

−0.0001 −0.0012

0.087 0.086

0.074 0.075

0.039 0.036

0.010 0.003

0.002 −0.001

0.0014 −0.0000 a

0.024 0.026

0.021 0.021

0.010 0.008

0.0029 0.0004

0.0006 −0.0000

0.0004 +0.0000 a

0.006 0.008

0.005 0.006

0.002 0.002

0.0006 0.0001

0.0001 −0.0000 a

0.0001 +0.0000

The absolute value is less than 5 × 10 −5.

6. The W function and MC calculations

is accepted:

Before using Eqs. (46)–(48) for the calculations of various parameters of photoelectron angular distributions in solids it is reasonable to compare the functions W(i), W(a) and their combinations with the MC results. Only a part of the accomplished comparison is presented in this paper. In Tables 2, 3 and 4 and Figs. 3 and 4 such a comparison is given. The MC calculations used for comparison are described in Appendix A. The W(i) functions (Tables 2 and 3) are in reasonable correspondence with the MC calculations for a values up to 75–808. Note that the contribution from layers with z . 3l and a . 708 to the total intensity of the semi-infinite solid is comparatively small. The dependence of the information depth I d on q values is not strong. In the SLA (q = 0) the values for z = 1, 2, 3, 4 and m = 1 are 63.2, 86.1, 91.0, 98.2%. The modification of W(i) (see below) has also no essential influence on the information depth. For the information depth I d the definition proposed in Ref. [10]

Id =

…d

…`

F(z)dz)= 0

F(z)dz 0

Table 2 demonstrates the importance of q values: although there are large differences both in absolute and relative l and l e values for Al 2s and Au 4f lines ˚ , 14.3 A ˚ , and 14.5 A ˚ , 7.3 A ˚ , respec(they equal 21.6 A tively) the similar q values for these two lines result in a good correspondence between W(i) functions. The analytical W(i) functions follow all the small differences between these two lines as revealed by MC calculation. The calculated W(i) functions are found also to be in good correspondence with graphical presentations in Ref. [16] for Be (E kin = 1000 eV, q = 0.1), Ag (E kin = 1000 eV, q = 0.30) and Co (E kin = 250 eV, q = 0.42). A systematic deviation is found between analytical and MC calculations of W(i) functions, namely: the MC values for z . 3l are smaller than the analytical values. The effect is increasing with increasing q

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

233

This last empirical correction is found to be the most effective and is recommended for the practical determination of the overlayer thickness and the correction factors for the in-depth profiling. Table 3 demonstrate good correspondence of the th3 calculations with MC results even for z = 5l. In Table 4 and Figs. 3 and 4 the angular part W(a) is considered. The correspondence is rather good and no empirical corrections are needed. The same good correspondence is found for different experimental arguments. The comparison was made for the following six arrangements: v = 908, v = 408, v = 08, v = a, v = 908 + a, v = 458 + a. The calculated ratio Al2s/Al2p (Eq. (46), d → `) is in a good agreement with MC and TT calculations, as well as with experiments (Fig. 3 in Ref. [18]).

7. The escape depth Fig. 4. As Fig. 2, but v = 908.

values. This deviation is not critical and can be empirically corrected. Two types of empirical correction are found to improve the correspondence between theoretical calculated W(i) functions and MC results. The calculations without any corrections based on Eqs. (46)–(48) are labelled as th1. The calculations with an increased q value, namely q9 = 1.1q, are labelled as th2. For the th3 calculations the following W(i) function was used: W (i)(th3) = W (i)(th1, q9 = q)e − z /l + W(i)(th1, but q9 = 1.3q)(1 − e − z /l): 8

Pd (th3) = 0:5Hc PId (q) − PId (q9 = 1:3q) + Pd (q9 = 1:3q)

9

…49†

D=

…`

zF(z)dz

However it is important to note that the other definitions give D values very close to the D value defined by Eq. (50). This situation is discussed in some more detail in Section 8 where attenuation lengths are considered. From Eqs. (46)–(48) (th1) using the normalized F(z) function one can get (in SLA units, where D(SLA) = lm)



1 nc − m [1 − exp( − d(1 + mc2 )=lmc2 )] n − m 1 + c2 m 

n(1 − c) [1 − exp( − d(1 + c2 n)=lc2 n)] + 1 + c2 n P d is given in Eq. (46).



D 0:5cH bc3 (3 cos2 v − 1) = m(1 + c) + nc(1 − c) − 4A lm Am (51)

1 [1 − exp( − d(1 + mc)=lmc)] 1 + mc +

(50)

0



where PId (q) =

Various definitions of the escape depth D are considered in Ref. [10]. The decay of the photoelectron intensity in solids is not exactly exponential, therefore in the present paper the definition proposed in Ref. [28] is accepted. According to this definition, D is equal to the mean value of the normalized depth distribution function

where A = H − (bc=4)(3 cos2 v − 1) Due to the presence of the factor (3 cos 2v − 1), the D value oscillates in anti-phase with this factor when b . 0 and in phase when b , 0. The anti-phase

234

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Table 5 Escape depth (in SLA units): (a) for isotropic source; (b) for v = 40; (c) for v = 90 q

(a) 0.1 thJ th1 th2 0.2 thJ th1 th2 0.3 thJ th1 th2 0.4 thJ th1 th2 (b) 0.1 thJ th1 th2 0.2 thJ th1 th2 0.3 thJ th1 th2 0.4 thJ th1 th2 (c) 0.1 thJ th1 th2 0.2 thJ th1 th2 0.3 thJ th1 th2 0.4 thJ th1 th2

a 0

10

20

30

40

50

60

70

80

0.94 0.95 0.94

0.93 0.95 0.94

0.93 0.95 0.94

0.93 0.95 0.94

0.93 0.95 0.94

0.94 0.96 0.95

0.95 0.97 0.96

0.97 0.99 0.99

1.04 1.06 1.07

0.85 0.89 0.86

0.85 0.89 0.87

0.85 0.89 0.87

0.86 0.90 0.87

0.86 0.90 0.88

0.87 0.91 0.89

0.90 0.93 0.91

0.94 0.97 0.96

1.08 1.09 1.10

0.77 0.83 0.78

0.77 0.83 0.79

0.77 0.83 0.79

0.78 0.83 0.79

0.79 0.84 0.80

0.81 0.86 0.82

0.84 0.88 0.85

0.90 0.94 0.91

1.10 1.10 1.09

0.69 0.76 0.70

0.69 0.76 0.70

0.69 0.76 0.70

0.70 0.77 0.71

0.72 0.78 0.72

0.74 0.79 0.74

0.78 0.82 0.77

0.86 0.89 0.84

1.11 1.08 1.05

0.94 0.97 0.96

0.94 0.97 0.96

0.94 0.98 0.96

0.94 0.98 0.96

0.95 0.98 0.97

0.96 0.99 0.98

0.98 1.01 1.00

1.01 1.04 1.04

1.12 1.14 1.16

0.87 0.93 0.89

0.87 0.93 0.90

0.87 0.93 0.90

0.88 0.94 0.90

0.89 0.95 0.92

0.99 0.97 0.94

0.94 0.99 0.97

1.01 1.05 1.04

1.21 1.23 1.25

0.80 0.86 0.82

0.80 0.88 0.82

0.80 0.88 0.82

0.81 0.89 0.83

0.83 0.90 0.85

0.85 0.92 0.87

0.89 0.96 0.91

0.99 1.04 1.00

1.27 1.27 1.27

0.72 0.81 0.73

0.72 0.81 0.73

0.73 0.81 0.74

0.74 0.82 0.74

0.75 0.84 0.76

0.79 0.86 0.79

0.84 0.90 0.83

0.95 0.99 0.93

1.30 1.27 1.24

0.92 0.93 0.92

0.92 0.94 0.93

0.92 0.93 0.93

0.92 0.93 0.93

0.92 0.94 0.93

0.93 0.94 0.94

0.93 0.95 0.94

0.95 0.96 0.96

1.00 1.01 1.01

0.83 0.86 0.85

0.83 0.86 0.85

0.84 0.87 0.85

0.84 0.87 0.83

0.84 0.87 0.86

0.85 0.88 0.86

0.87 0.89 0.88

0.90 0.92 0.91

0.99 1.01 1.00

0.75 0.79 0.76

0.75 0.79 0.76

0.75 0.79 0.76

0.76 0.80 0.77

0.76 0.80 0.78

0.78 0.81 0.79

0.80 0.83 0.81

0.85 0.87 0.85

0.98 0.98 0.98

0.67 0.72 0.67

0.67 0.72 0.67

0.67 0.72 0.67

0.68 0.72 0.68

0.69 0.73 0.69

0.70 0.74 0.70

0.73 0.77 0.73

0.79 0.81 0.77

0.97 0.95 0.93

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

oscillations have been found previously by MC calculations [11]. Eq. (51) reproduces these oscillations in good accordance with MC results. The physics of these oscillations has been already discussed in Ref. [11]. The escape depths for an isotropic source are given in Table 5, part (a). The D values decrease with the increasing q values. This is because, since q = l/(l + l tr), increasing q corresponds to increasing elastic scattering and the photoelectron path length in solids increases with stronger elastic scattering. There is also an increase in D with a values which follows from our choice of units (lm). As a matter of fact the absolute escape depth decreases with growing a values. It is interesting to note that the value D . 1 for large a values. This situation has already been discussed in Refs. [2,3]. The photoelectrons with large emission angles come from the deep layers in the solid more or less perpendicularly to the surface and the emission angle is changed near the surface due to the elastic scattering. The resulting path in the solid is shorter than in the SLA for a large emission angle and therefore D . 1 (process A). In Table 5, parts (b) and (c), the D values are given for angles v = 408 and 908, (b = 2). For v = 408 the factor (3 cos 2v − 1) is positive, but for v = 908 this factor is negative. That is why D values for v = 90 are larger but for v = 408 they are smaller than for the isotropic source. A similar situation also occurs for the attenuation lengths and the correction factors for the in-depth profiling: the isotropic source has values between those for v = 908 and v = 408. In paper [24] a slightly different Eq. (52) is proposed for ED. The calculated ED are in good agreement with MC results. The equation in Ref. [24] can be easily transformed into Eq. (52): D c[H(c + qx=2m) − c2 B] = lm H − cB

(52)

where bc B = (3 cos2 − 1) x = 4

…1

mH(m, q)dm

235

Using relations n < 1, c < 1 − q/2, Eq. (51) can be presented in similar form:  



q q c H 1+ − c2 B − D 4m 4 = H − cB lm



(53)

The X value for 0 , q , 0.4 is in the range 0.5–0.6 and one obtains for the H coefficient in Eq. (52) q q 1− + 2 4m which coincides with the similar quantity in Eq. (53) up to the quantity q/4. That is why both Eqs. (52) and (53) give similar numerical results, especially for the practically important range of q values (q , 0.4) and v angles (408 , v , 908) (see Table 5). Eq. (52) gives escape depths in good agreement with MC calculations and the thJ results in the table can be considered as MC results. The th1 give slightly larger values in comparison with thJ, however the th2 give already the values (not shown) close to those of thJ. If in the th2 calculations we set q9 = 1.2q the obtained results nearly coincide with those of thJ (Table 7).

8. The attenuation length (AL) There are several definitions of the attenuation lengths. The ASTM E-42 Committee definition indirectly assumes the absence of the elastic collisions and defines the AL as the average distance between successive inelastic collisions, i.e. collisions without the loss of kinetic energy of the electron. In the present paper the published definitions are analyzed and the connection between them is considered. The most important point is that all these definitions coincide when elastic scattering is ignored or the exponential decay law for the electron intensity flux in a solid is valid, which is a less strict assumption. The first AL definition that accounts for elastic scattering is given in Ref. [3] and is based on the overlayer method: f1 = d[m ln I0 =Id s ] − 1 =

(54)

0

The D/lm values according to Eq. (52) are presented in Table 5 as thJ.

where f 1 is the effective AL, d is the thickness of the overlayer, I d/s is the intensity of substrate covered by the film of thickness d and I 0 is the intensity from the

236

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Table 6 The comparison of the different isotropic attenuation lengths (q = 0.31, z and f in l, th1) a z/a 0.1 f1 f2 f3 0.5 f1 f2 f3 1.0 f1 f2 f3 f4 2.0 f1 f2 f3 3.0 f1 f2 f3 4.0 f1 f2 f3 a

0

10

20

30

40

50

60

70

80

0.83 0.85 0.84

0.83 0.85 0.84

0.83 0.84 0.84

0.83 0.84 0.83

0.83 0.82 0.83

0.83 0.81 0.81

0.83 0.80 0.80

0.84 0.79 0.79

0.8 0.7 0.7

0.83 0.84 0.83

0.83 0.84 0.83

0.83 0.84 0.83

0.83 0.83 0.83

0.83 0.83 0.83

0.84 0.82 0.83

0.85 0.81 0.83

0.89 0.81 0.85

1.0 0.8 1.0

0.82 0.84 0.82 0.82

0.82 0.83 0.82 0.82

0.82 0.83 0.83 0.82

0.83 0.83 0.83 0.83

0.84 0.83 0.83 0.84

0.85 0.83 0.85 0.85

0.88 0.83 0.88 0.88

0.98 0.86 1.02 0.93

1.5 1.1 2.8 1.1

0.82 0.83 0.81

0.82 0.83 0.81

0.82 0.83 0.82

0.83 0.83 0.83

0.84 0.83 0.84

0.87 0.85 0.89

0.94 0.89 1.02

1.20 1.03 1.65

2.2 1.7 3.9

0.81 0.82 0.80

0.81 0.82 0.80

0.82 0.82 0.81

0.83 0.83 0.82

0.84 0.84 0.85

0.89 0.87 0.93

1.01 0.95 1.18

1.37 1.20 1.97

2.6 2.1 3.9

0.81 0.81 0.80

0.81 0.82 0.80

0.81 0.82 0.81

0.82 0.83 0.82

0.85 0.84 0.86

0.91 0.88 0.96

1.07 1.01 1.29

1.49 1.34 2.02

2.8 2.3 3.9

Eqs. (46)–(48), (55)–(58) are used.

free substrate. The f 1 and d are given in l units. These units are used for all f i in this section. This definition assumes the dependence of f 1 on the angular arrangements, d and b values. But the complicated character of the f 1 value makes it practically important because of the relation between the reduced film thickness d = t/l and the effective AL = f 1 (see also Section 10). The f 1 value was shown to be nearly a constant within a sufficiently large a interval especially for small d (see Fig. 1 in Ref. [3]). This conclusion is also confirmed by the present consideration (see Table 6 and Table 7). That means that it is possible to find the approximate value d/f 1 via Eq. (54) using experimental dependence of d/f 1 on m. In Ref. [3] the explicit expression for f 1 is given depending on d, b and l for v = 908. Now there is a more simple way using the previous results: f1 = − d[m ln(1 − Pd =P` )] − 1

(55)

The f 1 definition is useful for the layer thickness determination (Section 10). The definition f 2 [8] is useful for the layer between z 1 , z , z 2: f2 = (z2 − z1 )=m ln[F(z1 )=F(z2 )]

(56)

For z 2 → z 1 one obtains 

f3 = −

m] ln F(z1 ) ]z1

−1

(57)

The f 3 refers to the slope of the depth distribution function at the definite depth in semilogarithmic scale. The AL definition of f 4 in Ref. [28] is equivalent to the escape depth D if D is measured in l units (see Eq. (50)): 1 f4 = m

…`

zF(z)dz

(58)

0

The f i values are given in Table 8. For f 2 we set

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

237

Table 7 The calculations of the isotropic f attenuation length in different approximations ( f 1 and z in l, q = 0.31) a

z

0.25 MC th1 th2 th3 0.6 MC th1 th2 th3 1.0 MC th1 th2 th3 2.0 MC th1 th2 th3 3.0 MC th1 th2 th3 4.0 MC th1 th2 th3

3

15

33

51

75

81

0.84 0.84 0.80 0.85

0.83 0.84 0.80 0.82

0.83 0.84 0.81 0.81

0.84 0.84 0.81 0.81

0.91 0.88 0.86 0.86

1.08 0.94 0.93 0.94

0.81 0.84 0.80 0.81

0.80 0.84 0.80 0.80

0.80 0.84 0.81 0.79

0.81 0.85 0.82 0.81

1.00 0.97 0.97 0.96

1.41 1.26 1.29 1.28

0.79 0.84 0.80 0.79

0.79 0.84 0.80 0.78

0.78 0.84 0.81 0.78

0.81 0.86 0.83 0.81

1.16 1.13 1.14 1.12

1.76 1.70 1.73 1.71

0.75 0.83 0.79 0.76

0.75 0.83 0.79 0.76

0.76 0.84 0.80 0.77

0.82 0.88 0.85 0.82

1.45 1.53 1.52 1.48

2.29 2.47 2.45 2.39

0.73 0.83 0.78 0.74

0.73 0.83 0.79 0.75

0.75 0.84 0.80 0.76

0.85 0.91 0.87 0.83

1.63 1.79 1.75 1.69

2.63 2.91 2.84 2.73

0.72 0.82 0.78 0.74

0.72 0.83 0.78 0.74

0.76 0.84 0.80 0.76

0.88 0.93 0.89 0.84

1.80 1.96 1.90 1.81

2.92 3.20 3.09 2.99

z 1 = 0. A remarkable numerical coincidence between various f i values is found for a , 608. The explanation is quite simple. The F(z) function in this angle interval can be represented as an exponential function with good accuracy [8,9,15]: F(z) ~ exp( − z=AL × m)

(59)

In this case the various AL definitions are identical for the same q values. Due to the numerical resemblance of various f i values the f 1 values only are discussed below. The AL values decrease with increasing z values. Usually AL , 1, however for large z and a values AL . 1, because the process A becomes more important (see above, Section 7). The most essential

results are nearly equal values of AL for given d in the large a interval as well as the slow decrease of AL with increasing d. This allows experimental determination of the overlayer thickness on the basis of Eq. (54) if the q value for this overlayer is known (Section 10). The th3 calculations of f 1 values are in close agreement with the MC results (Table 7). The f 1 values for v = 908 are given in Table 8. The situation is similar to that discussed above for the escape depth D. The dependence of AL on E kin is considered in Refs. [31–33]. Note that the attenuation length should not be considered as a physical constant, because the attenuation law for photoelectrons in solids is not exponential. The attenuation length depends on the

238

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Table 8 The f 1 attenuation length (in l, Al2s, v = 90, b = 2) a

z

0.2 MC th3 0.5 MC th3 1 MC th3 2 MC th3 3 MC th3 4 MC th3

3

15

33

51

63

75

81

0.92 0.87

0.90 0.87

0.89 0.87

0.88 0.87

0.87 0.87

0.84 0.88

0.91 0.90

0.90 0.87

0.89 0.86

0.88 0.87

0.87 0.87

0.86 0.88

0.92 0.92

1.17 1.03

0.88 0.86

0.88 0.86

0.87 0.86

0.87 0.87

0.88 0.90

1.08 1.04

1.53 1.49

0.85 0.86

0.85 0.86

0.85 0.86

0.87 0.89

0.94 0.96

1.35 1.42

2.06 2.28

0.82 0.85

0.83 0.85

0.83 0.86

0.85 0.91

0.99 1.05

1.66 1.71

2.70 2.77

0.79 0.85

0.79 0.85

0.80 0.86

0.85 0.93

1.13 1.14

2.11 1.91

3.49 3.11

definition and conditions of the experiment [3,34]; however it is useful for semi-quantitative estimates. The f 3 value can be regarded as a physical constant of the material for large z values [8,35].

overlayer thickness determination with account of elastic scattering. One of the possible methods is described in the following section.

9. The intensity in layered systems

10. The overlayer thickness determination

The intensities of the overlayer and the substrate under the overlayer intensity, as well as the ratio of these two intensities, are important for the overlayer thickness determination and for analytical XPS applications. In this section we demonstrate the usefulness of the obtained W functions for the calculations of the intensity relations in layered systems. Note the nature of the substrate has little influence on the overlayer intensities [3,34,36]. The th3 approach is proved to be more accurate than th1 and th2 calculations for q = 0.31 (Ag 3d) and in Figs. 5–7 the only the th3 results for Al 2s are presented. The SLA has a reasonable accuracy only for the overlayer intensities but completely breaks down when substrate under overlayer intensity or the ratio of the overlayer/substrate-intensity are calculated. The results in Figs. 5–7 are self-explanatory and demonstrate the principal possibility of the

There are two popular ways for the overlayer thickness XPS determination based on relative XPS intensities (see also reviews [38,39]). The first method is based on the intensity ratio of the substrate under overlayer I s(d/s) to the free substrate I `. According to Eqs. (30) and (31) one obtains k1 = Is (d =s)=I` = 1 − Pd =P`

(60)

Using this ratio the overlayer thickness d can be expressed with the help of the attenuation length f 1 (see Section 8, Eq. (54)): d =f1 = − m ln(1 − Pd =P` ) = − m ln k1

(61)

Note no approximation is involved in Eq. (61), because f 1 is not a constant but a function which depends on experimental arrangements and physical

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Fig. 5. The intensity I of the overlayer (d in l, A12s, q = 0.18, v = 908, b = 2.00) A-MC, K-th3, X-SLA.

239

Fig. 7. The ratio Y of the overlayer and substrate intensities. Y = Id (d =s)/Is (d =s), d in l, A12s, q = 0.18, v = 908, A-MC, K-th3, XSLA.

constant. As a matter of fact Eq. (61) is the definition for the attenuation length f 1. The second method is based on the intensity ratio K 2 of the overlayer A to the substrate B under the overlayer (see Eqs. (30) and (39)): K2 = Id (A)=Is (dA=sB) = P` (A)I0 (A)Pd (A)=I0 (B)P` (B)[P` (A) − Pd (A)]

…62†

Similar to arguments in Ref. [3,37] we can represent the intensity I d of the overlayer A and intensity I s(d/s) of the substrate as Id ~ I` (A)[1 − exp( − d =mf5 )]

…63†

Is (d =s) = I` (B) exp( − d =mf5 )

where f 5 is the attenuation length depending on experimental arrangements and physical constants. Eqs. (62) and (63) give d =f = − m ln(1 − Pd =P` ) = m ln(K2 =m + 1)

…† …†

f5 = f1 where m = I` A =I` B Fig. 6. The intensity I s of the substitute under overlayer (d in l, A12s, q = 0.18, v = 908, b = 2.00) A-MC, K-th3, X-SLA.

:

…64†

Note m < I0 (A)/I 0(B), if q(A) < q(B) and P `(A) < P `(B).

240

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Table 9 The determination of the overlayer thickness d (in l), (Ag3d, q = 0.31, th3, isotropic) a

z

d real = 0.25 1st appr. Best fit SLA d real = 0.60 1st appr. Best fit SLA d real = 1.00 1st appr. Best fit SLA d real = 2.00 1st appr. Best fit SLA d real = 3.00 1st appr. Best fit SLA d real = 4.00 1st appr. Best fit SLA

3

15

33

51

63

75

81

0.24 0.25 0.31

0.24 0.25 0.31

0.24 0.25 0.31

0.24 0.25 0.31

0.25 0.26 0.32

0.23 0.26 0.30

0.19 0.23 0.25

0.59 0.60 0.75

0.59 0.60 0.76

0.60 0.61 0.77

0.59 0.62 0.76

0.58 0.62 0.74

0.47 0.58 0.61

0.33 0.55 0.43

1.00 1.00 1.28

1.00 1.00 1.29

1.02 1.02 1.30

0.99 1.02 1.26

0.92 1.01 1.18

0.68 0.98 0.87

0.45 0.98 0.57

2.09 2.01 2.68

2.10 2.04 2.69

2.08 2.05 2.66

1.92 2.02 2.46

1.64 2.00 2.10

1.08 2.06 1.38

0.68 2.09 0.87

3.27 3.10 4.19

3.23 3.10 4.14

3.13 3.05 4.01

2.78 2.96 3.56

2.26 2.96 2.90

1.42 3.08 1.82

0.89 3.12 1.14

4.47 4.24 5.73

4.41 4.19 5.66

4.20 4.09 5.38

3.56 3.88 4.57

2.79 3.87 3.58

1.70 3.98 2.18

1.07 4.04 1.37

Eq. (64) indicates that the P values being good for the first way (Eq. (61)) must be also good for the second way of the overlayer thickness XPS determination. That is why in the present paper only the first method will be considered. The determination of the overlayer thickness with account of elastic scattering is based on the weak dependence of the calculated (th3) f 1 attenuation length on the overlayer thickness d (see Table 8). This dependence decreases with decreasing q value. The procedure of the overlayer thickness determination can be performed in the following steps: 1. The intensity ratio k 1 is measured and the d/f 1 value is calculated according to Eq. (61). 2. The f 1 length for the given q value of the overlayer is calculated using theoretical P d and P ` values (Eq. (61)) and the average ¯f 1 value for the expected d range is found for 0 , a , 30. For example, if q = 0.31 and d is expected to be between 0 and 4l the average value is calculated

to be about 0.76. The exact average is not necessary, the rough estimate will also do. 3. The d/f 1 values found in step 1 are multiplied by ¯f 1. This is the first approximation for d (see Tables 9 and 10). The k 1 values used for calculation of d/f 1 are found by the MC procedure in which the substrate and the overlayer are the same material (Ag in Tables 9 and 10). 4. The first approximation already gives d 1 values with good accuracy (Tables 9 and 10). However, the second iteration is advised. The f 1 lengths are calculated for d 1 values and initial d/f 1 values (step 1) are multiplied by these f11 lengths. According to our experience the third iteration is not necessary. The obtained d 2 values practically coincide with the values (Tables 9 and 10) obtained when ¯f 1 values corresponding to the real (exact) d thickness are used. (Of course, roughness and statistical errors are not taken into account in this paper.) 5. The d value should be taken as an average for the

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Fig. 8. The determination of the overlayer thickness. d in l, A12sisotropic, A-1-st appr., K-best fit, X-SLA.

different a angles (to minimize the roughness effect and errors in f 1 calculations angles a larger than 608 should be avoided). The method seems to be reliable up to d values of about 3–4l. The SLA results for d are also given for comparison in Tables 9 and 10 and Figs. 8 and 9. With increasing q and d values the SLA becomes unreliable.

11. In-depth profiling The in-depth profiling methods [40–45] are usually based on consideration of the individual layer contribution to the total sample intensity. In the SLA the following expression is valid for the intensity of the layer of thickness t under layer of the thickness d: It (m) = kjlP(bf)exp( − d =lm)[1 − exp( − t=lm)]cm (65) where c m is the concentration of the m component. In the general case with elastic scattering the correction coefficient B must be introduced [43]: It 9(m) = It B

(66)

241

Fig. 9. The determination of the overlayer thickness. d in l, A12s, q = 0.18, v = 908, b = 2.00. A-1-st appr., K-best fit, X-SLA.

where B can be expressed using Eq. (32) as Pd + t − Pd B= exp( − d =lm)[1 − exp( − t=lm)]

(67)

In Tables 11, 12 and 13 the B corrections are calculated in three different approximations. Th2 is shown to be the best fit to MC results, though the differences between different calculations are not large. The same q value shown in the table captions (0.18) was taken for all the layers but Eq. (67) allows the calculation of correction factors for q values specific for each overlayer. The calculated B factors in general reproduce MC results for z # 3l and a angles up to 65–708. The calculated results for v = 908 and isotropic sources are in better agreement with MC correction factors in comparison with results for v = 408. The explanation is related to the decrease of the calculated intensity in the last case which makes the errors in the calculated value more important. The account of elastic scattering does not change the calculated concentration profile in general but improves the depth scale by 10–25% [43,44]. The improvement can be estimated on the base of the f 1

242

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Table 10 The determination of the overlayer thickness (q = 0.31, Ag3d, b = 1.22, v = 90) a

z

d real = 0.25 1st appr. Best fit SLA d real = 0.60 1st appr. Best fit SLA d real = 1.00 1st appr. Best fit SLA d real = 2.00 1st appr. Best fit SLA d real = 3.00 1st appr. Best fit SLA d real = 4.00 1st appr. Best fit SLA

3

15

33

51

63

75

81

0.24 0.25 0.32

0.24 0.25 0.32

0.24 0.25 0.32

0.24 0.25 0.32

0.25 0.26 0.33

0.24 0.25 0.31

0.20 0.23 0.26

0.58 0.59 0.77

0.59 0.60 0.78

0.60 0.60 0.79

0.60 0.61 0.79

0.58 0.61 0.77

0.49 0.57 0.64

0.34 0.53 0.45

0.99 0.99 1.31

1.00 1.00 1.32

1.01 1.01 1.34

0.99 1.01 1.31

0.93 1.00 1.23

0.63 0.94 0.90

0.44 0.92 0.58

2.06 2.01 2.73

2.07 2.02 2.74

2.06 2.04 2.73

1.92 2.00 2.55

1.63 1.93 2.16

0.99 1.84 1.32

0.59 1.74 0.78

3.18 3.08 4.21

3.17 3.07 4.19

3.10 3.06 4.10

2.77 2.94 3.66

2.10 2.71 2.79

1.11 2.36 1.47

0.63 2.16 0.83

4.27 4.11 5.66

4.29 4.15 5.68

4.17 4.11 5.52

3.46 3.74 4.58

2.31 3.17 3.06

1.14 2.64 1.51

0.63 2.38 0.84

values (Section 8) and d overlayers calculations (Section 10). However the change in the relative concentrations cannot be neglected either. That is why the application of the B factors is recommended in general. Note the change in the depth scale due to the B factors is not linear (Section 10).

Acknowledgements

12. Conclusion

Appendix A

In the present paper the partial intensity contributions P are defined (Eqs. (28), (46) and (49)) which allow the simple intensity calculations for semiinfinite sample, overlayer and substrate under overlayer (Eqs. (29)–(32), (38) and (39)). The applications are shown for the calculation of the escape depths (Section 7), attenuation length (Section 8), overlayer thickness determination (Section 10) and in-depth profiling (Section 11). The determination of a values which are necessary for the calculations of partial intensity contribution is described in Appendix B.

The MC calculations are similar to those described in Ref. [46]. We have been interested in the W z values with the best possible resolution Dz for a semi-infinite sample. In the algorithm described below the resolution Dz is equal to zero. The Monte Carlo simulation of the substrate photoelectron trajectories in polycrystalline or amorphous overlayer was performed in the following way. The polar and azimuthal angle regions were divided into 15 and 60 subsections, respectively. The small detector acceptance angle 6 × 68

The authors thank the Russian Fund for Fundamental Research for financial support. The support from Deutsche Forschungsgemeinschaft under Sz 58/4-1 is gratefully acknowledged.

243

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248 Table 11 The comparison of analytical and MC determined isotropic B correction factors for in-depth profiling (Al2s, q = 0.18) ta

a 3

0–0.05 MC th1 th2 th3 0.15–0.2 MC th1 th2 th3 0.8–1.0 MC th1 th2 th3 1.0–1.3 MC th1 th2 th3 1.65–2.0 MC th1 th2 th3 2.0–2.5 MC th1 th2 th3 2.5–3.0 MC th1 th2 th3 3.0–3.5 MC th1 th2 th3 a

15

33

63

75

81

1.02 1.06 1.07 1.05

1.03 1.06 1.07 1.06

1.03 1.06 1.06 1.06

1.04 1.05 1.05 1.05

1.05 1.04 1.04 1.04

1.06 1.02 1.02 1.02

1.04 1.00 1.00 1.00

1.04 1.05 1.05 1.04

1.04 1.05 1.05 1.05

1.04 1.04 1.04 1.04

1.05 1.02 1.02 1.02

1.04 1.00 0.99 0.99

1.02 0.97 0.93 0.94

0.93 0.87 0.86 0.87

1.05 0.97 0.97 0.97

1.02 0.97 0.96 0.96

0.98 0.95 0.94 0.94

0.93 0.90 0.89 0.88

0.81 0.85 0.83 0.82

0.85 0.79 0.78 0.77

2.73 1.41 1.45 1.50

1.03 0.95 0.94 0.94

1.00 0.94 0.94 0.93

0.96 0.92 0.91 0.90

0.89 0.87 0.86 0.84

0.78 0.82 0.80 0.78

1.11 0.90 0.89 0.89

6.59 3.63 3.88 4.12

0.93 0.88 0.87 0.85

0.91 0.87 0.86 0.84

0.85 0.85 0.83 0.81

0.78 0.80 0.78 0.74

0.82 0.81 0.80 0.77

3.34 2.55 2.66 2.80

109 103 110 118

0.88 0.84 0.82 0.80

0.86 0.83 0.82 0.78

0.79 0.81 0.79 0.75

0.74 0.78 0.75 0.72

0.89 0.88 0.87 0.84

7.08 6.59 6.91 7.30

580 768 810 862

0.80 0.80 0.77 0.73

0.78 0.79 0.77 0.72

0.73 0.76 0.74 0.69

0.68 0.76 0.73 0.68

1.00 1.06 1.05 1.03

19.7 23.9 24.8 25.8

5618 10201 10612 11059

0.70 0.75 0.72 0.67

0.70 0.74 0.72 0.66

0.64 0.72 0.69 0.64

0.61 0.76 0.73 0.67

1.11 1.42 1.41 1.38

55.0 88.8 91.2 92.3

59392 135495 139051 141287

t shows d i and d f for the layer under investigation: t = d f − d i.

was chosen to compare with usual experimental values about 2–48. The resulting 900 sections defined by polar and azimuthal angles of a i, ai + 1 and Ji , J i+1, respectively, determine the intensity 1

51

l units are used.

distribution before and after randomization. The following escape depths (and corresponding overlayer thicknesses) are introduced: z = 0, 0.05, 0.15, 0.25 or 0.27, 0.5 or 0.6 or 0.8, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0, 6.01 in each angular section. To account for the geometric conditions the

244

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Table 12 The comparison of analytical and MC determined B correction factors for in-depth profiling (Al2s, q = 0.18, v = 90, b = 2.00) ta

a 3

0–0.05 MC th1 th2 th3 0.15–0.2 MC th1 th2 th3 0.8–1.0 MC th1 th2 th3 1.0–1.3 MC th1 th2 th3 1.65–2.0 MC th1 th2 th3 2.0–2.5 MC th1 th2 th3 2.5–3.0 MC th1 th2 th3 3.0–3.5 MC th1 th2 th3 a

15

33

51

63

75

1.01 1.04 1.05 1.03

1.02 1.04 1.04 1.04

1.02 1.04 1.04 1.04

1.02 1.03 1.03 1.03

1.03 1.02 1.02 1.02

1.05 1.01 1.01 1.01

1.02 1.02 1.02 1.02

1.02 1.02 1.02 1.02

1.02 1.01 1.01 1.01

1.02 0.99 0.99 0.99

1.01 0.97 0.97 0.97

0.93 0.91 0.91 0.91

0.98 0.92 0.92 0.92

0.95 0.92 0.91 0.91

0.92 0.90 0.89 0.89

0.86 0.85 0.84 0.83

0.74 0.78 0.77 0.76

0.72 0.69 0.68 0.67

0.96 0.89 0.89 0.89

0.93 0.89 0.88 0.88

0.89 0.86 0.85 0.85

0.81 0.80 0.80 0.79

0.69 0.74 0.73 0.71

0.88 0.73 0.72 0.72

0.85 0.81 0.80 0.79

0.82 0.80 0.79 0.78

0.77 0.77 0.76 0.74

0.68 0.71 0.70 0.68

0.66 0.68 0.67 0.65

2.28 1.77 1.85 1.94

0.79 0.77 0.75 0.73

0.77 0.76 0.75 0.72

0.71 0.73 0.71 0.69

0.62 0.67 0.66 0.63

0.67 0.70 0.70 0.67

4.23 4.45 4.66 4.92

0.72 0.71 0.70 0.67

0.69 0.71 0.69 0.66

0.64 0.67 0.66 0.63

0.55 0.64 0.62 0.59

0.66 0.80 0.79 0.77

10.2 15.9 16.6 17.3

6 7 7

0.62 0.66 0.65 0.61

0.62 0.66 0.64 0.60

0.55 0.65 0.61 0.57

0.47 0.61 0.59 0.56

0.68 1.02 1.01 0.99

24.1 59.2 60.8 62.9

22 90 92 94

6 6 7 7

See footnote to Table 11.

factor f i for the initial intensity which corresponds to the X and Y values (see below) and the factor f f for the calculated intensity were introduced: fi = (cos ai − cos ai + 1 )

(A.1)

ff = fi − 1 =cos(ai 9)

(A.2)

where cos(a i9) is the average value for cos a in the subsection. In Eq. (A.1) the initial a values and in Eq. (A.2) the final a values are considered. The factor f i accounts for the statistical weight of the polar angle and the factor cos(a i9) accounts for the dependence of the sample effective surface on the a angle.

245

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248 Table 13 The comparison of analytical and MC determined B correction factors for in-depth profiling (Al2s, q = 0.18, v = 40, b = 2.00) ta

a

0–0.05 MC th1 th2 th3 0.8–1.0 MC th1 th2 th3 0.8–1.0 MC th1 th2 th3 1.0–1.3 MC th1 th2 th3 1.65–2.0 MC th1 th2 th3 2.0–2.5 MC th1 th2 th3 2.5–3.0 MC th1 th2 th3 a

3

15

33

51

63

75

81

1.04 1.11 1.12 1.08

1.05 1.10 1.12 1.10

1.06 1.10 1.11 1.10

1.08 1.08 1.10 1.09

1.11 1.07 1.08 1.07

1.14 1.04 1.05 1.04

1.11 1.02 1.02 1.02

1.11 1.10 1.11 1.09

1.11 1.10 1.11 1.10

1.19 1.09 1.10 1.09

1.15 1.07 1.07 1.07

1.16 1.04 1.04 1.04

1.05 0.99 0.98 0.98

0.77 0.93 0.90 0.92

1.20 1.07 1.06 1.06

1.16 1.06 1.06 1.05

1.12 1.04 1.03 1.03

1.05 1.01 0.99 0.97

0.87 0.97 0.94 0.92

0.77 0.99 0.97 0.95

2.32 2.09 2.17 2.24

1.19 1.05 1.04 1.04

1.14 1.15 1.04 1.02

1.10 1.03 1.01 1.00

0.99 0.99 0.97 0.95

0.80 0.97 0.94 0.91

0.98 1.21 1.20 1.19

5.48 5.73 6.13 6.51

1.10 1.01 0.99 0.96

1.05 1.01 0.98 0.95

0.95 0.99 0.96 0.92

0.80 0.97 0.94 0.89

0.74 1.06 1.03 0.98

2.84 3.98 4.16 4.38

1.05 0.98 0.96 0.91

0.99 0.98 0.95 0.90

0.87 0.96 0.93 0.87

0.71 0.97 0.93 0.87

0.75 1.21 1.18 1.13

5.95 10.50 11.00 11.70

545 1239 1306 1391

0.96 0.95 0.91 0.85

0.90 0.94 0.90 0.84

0.78 0.95 0.89 0.82

0.61 0.99 0.94 0.87

0.77 1.55 1.53 1.49

16.3 38.4 39.9 41.6

5332 16453 17117 17838

98.4 166.0 177.0 190.0

See footnote under Table 11.

The inelastic scattering was taken into account by the factor K = exp( − s/l) for calculated intensity where s is the path of the electron and l is the inelastic mean free path. The maximum s value for the MC procedure was 6.1 for z # 3.5 and s = 2z for z $ 4. If s $ s max, the trajectory was neglected. For each section of the initial intensity distribution N = 10 000 trajectories were calculated. This number was found by an empirical way starting from lower values to ensure a good reproducibility of about several percentage points, especially for the compara-

tively small a values because the results for larger a values are not reliable or absent both from the experimental side and from the statistical consideration for calculation. The total number of trajectories was about 10 8. The trajectories were calculated by the usual MC procedure. For the calculation of the free elastic path l e and the scattering angles v i, J i the following equations are used: le 9 = − le ln g

246

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

Table 14 ˚) The a and l values (energy in eV, l in A Element

Energy 250

500

1000

1500

li

a

li

a

li

a

li

a

Be C Mg Al Si Ca Si Ti V Cr Mn

6.6 7.5 8.6 7.3 7.1 10.7 8.2 7.3 6.8 6.6 6.6

0.83 0.68 0.79 0.75 0.77 0.79 0.73 0.70 0.66 0.61 0.61

11.0 12.2 14.3 12.0 11.6 17.8 13.4 11.8 10.9 10.5 10.4

0.87 0.74 0.81 0.78 0.80 0.81 0.75 0.72 0.69 0.66 0.66

18.8 20.6 24.4 20.4 19.7 30.4 22.6 19.8 18.1 17.4 17.2

0.93 0.81 0.85 0.82 0.84 0.84 0.78 0.75 0.72 0.70 0.70

26.0 28.3 33.8 28.0 27.2 42.1 31.0 27.1 24.8 23.7 23.5

0.95 0.84 0.87 0.85 0.86 0.86 0.81 0.77 0.74 0.72 0.71

Fe Co Ni Cu Zn Ga Ge

6.6 6.5 6.7 6.9 6.9 8.3 7.9

0.59 0.58 0.58 0.57 0.65 0.65 0.68

10.4 10.3 10.5 10.5 11.0 13.2 12.6

0.64 0.44 0.40 0.61 0.67 0.68 0.71

17.2 17.0 17.3 17.4 18.3 22.0 22.1

0.68 0.66 0.68 0.66 0.69 0.71 0.73

23.4 23.2 23.5 23.6 25.0 30.1 28.8

0.69 0.67 0.67 0.69 0.72 0.72 0.75

As Se Sr Y

7.4 7.4 12.7 9.5

0.69 0.73 0.76 0.72

11.8 11.9 20.8 15.4

0.71 0.75 0.76 0.72

19.7 19.9 35.3 25.8

0.73 0.76 0.78 0.74

26.9 27.3 48.7 35.4

0.75 0.78 0.78 0.75

Zr Nb Mo Ru Rh Pd Ag Cd In

8.1 7.2 6.6 5.9 5.8 5.9 6.1 6.4 9.3

0.67 0.64 0.61 0.59 0.60 0.61 0.66 0.69 0.65

12.8 11.3 10.3 9.2 9.1 9.1 9.6 10.1 14.7

0.77 0.66 0.65 0.62 0.62 0.65 0.67 0.69 0.67

21.3 18.6 17.0 15.0 14.8 14.9 15.7 16.7 24.4

0.71 0.68 0.66 0.66 0.66 0.67 0.69 0.71 0.68

29.1 25.3 23.0 20.4 20.0 20.2 21.3 22.7 33.2

0.72 0.69 0.68 0.67 0.68 0.68 0.70 0.73 0.69

Sn Sb Te Ba La Hf Ta

9.0 8.2 8.0 13.9 10.3 8.2 6.9

0.70 0.70 0.73 0.73 0.69 0.66 0.65

14.4 13.0 12.7 22.6 16.5 12.6 10.6

0.72 0.71 0.73 0.73 0.69 0.60 0.60

24.0 21.6 21.2 38.1 27.4 20.5 17.1

0.73 0.72 0.74 0.74 0.70 0.61 0.63

32.8 29.4 28.9 52.3 37.5 27.8 23.1

0.74 0.73 0.75 0.74 0.71 0.63 0.61

W Re Os Ir Pt Au Pb

6.0 5.4 5.0 4.8 4.8 5.0 9.1

0.67 0.67 0.66 0.67 0.64 0.70 0.68

9.2 8.2 7.5 7.2 7.2 7.6 14.2

0.61 0.61 0.61 0.64 0.67 0.67 0.65

14.8 13.2 11.9 11.5 11.6 12.3 23.2

0.64 0.64 0.65 0.65 0.66 0.67 0.67

19.9 17.7 16.1 15.4 15.6 16.5 31.5

0.62 0.65 0.66 0.67 0.67 0.68 0.65

V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

…v

F(v)dv = g

0

Ji = 2pg where g is a random number between 0 and 1, F(v) = (f 2 + g 2)sin(v)(1/j e) is the probability for the scattering angle v [47–49] and j e is the total elastic scattering cross-section. In simulating trajectories the same l e and l values were used both for the overlayer and for the substrate because homogeneous samples are considered. The X and Y i values determine completely the final angular distribution (Ref. [1], Eq. (7)). The final intensity distributions for the calculation of the X, Y i values in accordance with Ref. [1] are multiplied by factors for

X Y0 Y1 Y2

1 cos 2a sin 2a cos J sin 2a cos 2J

Note that the analyzer position is in our case in plane XZ (J = 0). One calculation is enough for the determination of the X, Y i values because the same MC trajectories can be used for them. The final intensities I(ai ,ai + 1 ,z k) for X and Y i divided by the number of the trajectories N = 10 000 lead to the differential X(z) and Y i(z) values and the escape probability W(ai 9,zk ) for the depth z k, when W z is calculated by Eq. (33). The total X and Y i values can be obtained by integration, i.e. X=

…`

X (z)dz

0

For this purpose the values X(z k) for k and k − 1 are used to express X between z k and z k−1 as Xk, k − 1 = ak e − bk z One obtains X= ∑ k

= ∑ k

…z

k

zk − 1

ak exp( − bk z)dz

(zk − zk − 1 )(Xk − Xk − 1 ) ln(Xk =Xk − 1 )

Similar results are valid for Y i values.

247

Appendix B In Ref. [33] the attenuation length values l a are given which can be represented as l tr [16], i.e. a < la =l

(B.1)

In Table 14 the a values (Eq. (B.1)) are given along with the l i values used in the calculation. The a values increase with the kinetic energy of the photoelectron, though the situation seems to be more complicated for the heavy elements. The a values for Co and Ni (E = 500 eV) are unusually small and should not be taken into account. There is a reasonable correspondence between a values in Table 14 that can be regarded as MC values and theoretical values that can be calculated based on the data in Ref. [23]. In the literature there is no proper discussion on the a values in chemical compounds. So far two possibilities can be considered in the first approximation: 1. the additive scheme based on the elemental a values 2. the use of equation 6 in Ref. [33], which for our purpose can be rewritten as a = − 1:095=l + 0:79

(B.2)

for a , 0.79. This problem needs further consideration however: the dependence of the W z function on the a value is not supersensitive (Table 2), the dependence of the a value E kin is also rather weak (Table 14) and the use of such an average a value which equals about 0.70– 0.80 in the usual E kin range can be a reasonable approximation. References [1] O.A. Baschenko, V.I. Nefedov, J. Electron Spectrosc. Related Phenomena 17 (1979) 409. [2] O.A. Baschenko, V.I. Nefedov, J. Electron Spectrosc. Related Phenomena 21 (1980) 153. [3] O.A. Baschenko, V.I. Nefedov, J. Electron Spectrosc. Related Phenomena 27 (1982) 109. [4] O.A. Baschenko, G.V. Machavariani, V.I. Nefedov, J. Electron Spectrosc. Related Phenomena 34 (1984) 305. [5] A. Jablonski, H. Ebel, Surf. Interface Anal. 6 (1984) 21. [6] H. Elel, M.F. Ebel, J. Wern ish, A. Jablonski, Surf. Interface Anal. 6 (1984) 140.

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V.I. Nefedov, I.S. Fedorova/Journal of Electron Spectroscopy and Related Phenomena 85 (1997) 221–248

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