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Depth profiling of elements in surface layers of solids based on angular resolved X-ray photoelectron spectroscopy
1
Journal of Electron Spectroscopy and Related Phenomena, 53 (1990) 1-18 Elsevier Science Publishers B.V., Amsterdam
DEPTH PROFILING OF ELEMENTS IN SURFACE LAYERS SOLIDS BASED ON ANGULAR RESOLVED X-RAY PHOTOELECTRONSPECTROSCOPY
OF
O.A. BASCHENKO, V.I. NEFEDOV N.S. Kurnakov Institute of General and Inorganic Chemistry of the Academy of Sciences of the U.S.S.R., Moscow (U.S.S.R.) (First received 2 August 1989; in final form 24 January 1990)
ABSTRACT A numerical method is proposed for the non-destructive restoration of the depth-concentration profiles for four components, based on the angular distribution of XPS intensities. Calculations performed for a number of model systems demonstrated that ratios of concentrations obtained by using simplified procedures, i.e. from relative XPS intensities for a single escaping angle, led to large errors (up to 200% ) for the upper surface layer. A number of restored profiles are presented using experimental angle-resolved XPS data for samples containing 3-4 components.
INTRODUCTION
Angular resolved X-ray photoelectron spectroscopy (XPS ) allows us to carry out a non-destructive layer-by-layer analysis of the chemical composition - a sort of tomography - of 5-10 nm thick surface layers of solids. The depthconcentration profiles of elements with 1 nm depth resolution, which are obtained from the angular distribution of XPS intensities, present very important, easy-to-interpret and instructive information for studying processes such as oxidation of metals and alloys, segregation, corrosion, adhesion, the effect of ion beams on solids, etc. If needed, depth-concentration profiles can be determined for identical atoms which differ only in their chemical states (e.g. by the oxidation state ) . References l-6 put forward methods of mathematical treatment of angular XPS for finding the depth-concentration profiles. All these methods contribute to removing the major obstacle namely the extremely high sensitivity of restored depth-concentration profiles to the errors of experimental data [ 1,46] because the problem of profile restoration falls into the class of ill-posed problems [ 71. In order to overcome this instability of solutions, it is necessary to use some a priori information about the profile, as is generally done when solving Fredholm integral equations of the first kind [ 7,8]. It was shown in
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Elsevier Science Publishers B.V.
2
ref. 1,that the instability problem can be resolved with the aid of the following a priori information about the solution: the desired function of dependence between concentration and depth should be (a) non-negative, bounded and (b) monotonous. References 4-6 show that the second constraints can be relaxed, i.e. it is sufficient to require that the depth-concentration profile should not have more than one extremum up to the depth of 3A (2 is the mean free path of photoelectrons without inelastic collisions in a solid). References 1 and 4-6 deal with samples containing only two components, where if one of the profiles is monotonous (or has only one extremum) then the other is also monotonous and has only one extremum. When the number of components is more than two the situation is somewhat more complex. For example, for a three-component sample, if two depth-concentration profiles are monotonous the third one may have an extremum. Likewise, if two profiles have an extremum the third profile may have several such extrema. Therefore, we may infer that as the number of components of the sample increases, the problem of depth-concentration profile restoration becomes more difficult. The procedure of numerical solution of such a problem requires considerable modification compared with that for the case of two components. Samples used in most XPS studies have more than three components because in the surface layers oxygen and carbon are adsorbed from the air. In the present work, the numerical method which we proposed in refs. 4-6 for the restoration of depth-concentration profiles of elements in the surface layers of solids with thicknesses up to 100 A, based on the data of angular resolved XPS, is extended to embrace the case of four-component samples. It was shown for a number of model systems that the suggested procedure of accounting the a priori information (i.e. that every depth-concentration profile is a non-negative and bounded function and for three of the four components studied the profiles may have no more than one extremum) enables us to obtain solutions which are resistant to the experimental errors attainable at the present day (about 10% ) . It was found that determination of the ratios of elemental concentrations based on the simplified method, i.e. from the ratios of XPS line intensities at the same take-off angles 19(usually 8=45” ), may lead to considerable errors (up to and over 200% ) for the uppermost layer owing to the neglect of real concentration gradients in the sample. Non-destructive restoration of depth-concentration profiles was conducted on the basis of experimental angular distributions of XPS for several samples with 3-4 components. For a TiOz sample (with a hydrocarbon contamination layer) it was established that the ratio of Ti2p and 01s line intensities at 8= 45’ gives the 1: 3 proportion of atoms, while the obtained profiles indicate the presence of stoichiometric oxide TiOz, but only from depths greater than about 20 A. Above this oxide layer lies a film containing adsorbed oxygen and hydrocarbons.
3 INVESTIGATION METHODS
Angular distributions of XPS intensities for a four-component sample For each of the components (with index i) the intensity of the XPS line is described by the formula m Ii =FiGi
s
?Zi(Z)
t?Xp(
-Z/Ai
COS 0)dZ
(1)
0
where Fi is an instrumental factor (which contains the dependence of the transmissibility of the energy analyser on the kinetic energy Ek of the photoelectron), ai is the cross-section of photoionization of the core level of the element (or component) i, Ai is the mean free path of photoelectrons in a sample without inelastic collisions (all 1 i are assumed to be dependent only on Ek but not on the chemical composition ), ni (z) is the concentration of element i at depth z from the sample’s surface and 19is the take-off angle of photoelectrons measured from the normal to the atomically smooth surface of a sample. We shall consider the ratio of line intensities of the first three components to that of the fourth Fi Ci Ri(e)=jJjjj=
Ii(e)
I
ni(2) exp( -Z/Li
cos
8)dZ
(2)
20 J’4 04
I 0
n,(Z) exp( -Z/n,
co9 8)dZ
We shall henceforth set the ratios Fi: F4 equal to 1.0 for i= 1,2,3, since these quantities depend not on the sample composition but only on the characteristics of spectrometers and energy analysers, and can be determined ‘from independent experiments. If Fi : F4 = 1.0 for the spectrometer used in the experiments then a suitable correction factor may be introduced in the definition of Ri (0). Restoration of the profiles of concentrations ni (z) reduces to the solution of a set of three simultaneous equations
R;(e) =Ry(e)
i= l-3
(3)
where R:(e) are described by eqn. (2) (with Fi:F4=1.0) and R;(e) are the experimentally measured line intensities of the ith component with respect to that of the fourth component. Using the same method as in refs. 1 and 4-6, we replace the integrals in eqns. (1) and (2) by sums which correspond to the partition of the sample in M- 1 identical layers parallel to the surface and of thickness d, so that (M- 1)d = 3A4. Assuming that within every layer the concentrations of elements are constant
4
(i.e. n(z) =n(j) when dfj-l) (M- 1 )d= 3&), instead of eqn. (1) we obtain the following expression
-;li
,:s 8,)1 (4)
Here k= 1,2,...,K, and K is the number of different take-off angles of photoelectrons at which the ratios of XPS intensities have been measured. It should be noted that the choice of the number M of sublayers and the thickness d of every layer also constitute a sort of a priori information which is used for restoration of depth-concentration profiles. In particular, the assumption that it is reasonable to partition into layers the part of the sample lying higher than 3L, in depth is based on the fact that the possibility for photoelectrons to outflow from greater depth (about 5% ) is less than the typical experimental errors. Now, let us introduce new variables Ni(j) z&(j)
fJi
(5)
where j is the layer number and Uiis the atomic (or molecular) volume of the ith element. Proceeding from volume conservation as we go from the jth layer to another, we can write ii1
niti,ui=i~l
Nio’)=l
forj= 1,2,...,M
(6)
Substitution of eqns. (4) and (5) into the set of eqns. (3)) with allowance for eqn. (6)) gives a set of non-linear equations in unknowns Ni(j) (i= l-3, j= 1-M). As in the case of two-component samples (see refs. 4-6), this set of equations may be reduced to the linear form by multiplying both sides of eqn. (3) by the factor 14( ok). Finally, for the unknowns Ni(j) we come to the following set of equations
(7)
5
where ri= ui/u4. Thus, the determination of profiles for a four-component sample through angular distributions of XPS intensities is reduced to the solution of eqn. (7) of linear equations in 3M unknowns Ni (j) (i = l-3, j = 1-M). The numerical method of restoration of non-monotonous profiles with allowance for the a priori information Let us now describe the method of solution of (ill-conditioned when K=M [ 1,4-61) the set of eqns. (7) with due account of the a priori information. This
method is resistant to errors in the input data equal to 5-10%. By a priori information, basically we mean that the solutions are sought in the class of functions satisfying the following restrictions NAj) 20
i= l-4
(3)
N,(j)<1
i= l-4
(9)
and, furthermore, that each of the functions Ni (j) (i = l-3 ) may have no more than one extremum at depths up to 3A4. We shall follow the approach which we have previously developed in refs. 4-6 for solving the problem of profile restoration for two-component samples. The set of eqns. (7) may be recast in the matrix form Q(&> =N=R(e,)
(10)
where N is a 3Mdimensional vector, components of which are the values N1, NZ and N3 of concentration in layers with numbers j= 1,2, .... M; I? is a vector whose components are the experimental values of relative intensities RT( f&), Rg (19,)) R; (0,) with (3,take-cff angles of photoelectrons. The explicit form of the matrix elements &ii is manifest from eqns. (7 ) . Realization of the a priori information consists, firstly, in the change to a new basis [8] p&IV
(11)
where R has components fii(j) = Ni(j) -Ni(j+ 1) when j # M, 2M, 3M and fii(j)=Ni(j) when j=M, 2M, 3M. For l
&ekQ7=Rcek)
(12)
&e,) = @f’, using the method of conjugate gradients with projection on a set of vectors with non-negative components [ 81 (it should be noted that here and previously [ 4-61 the method of conjugate adients is applied to step-by-step minimization of the function F= Ck [Q (0,) 8 -I?( 19,)] “), one obtains solutions which are monotonously non-increasing and non-negative functions (IGiG) > 0 for all i and j, and, owing to the selected form of the transition matrix p, ine-
qualities Ni (j) > Ni (j+ 1) are fulfilled for allj). Now we have to take into consideration one more additional constraint which ensures the fulfilment of condition (9). For this purpose, the numerical minimization procedure contains the projection of the gradient onto the subspace defined by (13) where a vector g’ has the following components: gf = 1 for i=M, 2M, 3M and gi = 0 for all other values of i. To make sure that the set of solutions of eqns. (12) includes functions with a maximum, an approach is used which was advanced in ref. 8 and used by us in developing the numerical algorithm for restoration of two-component depthconcentration profiles [ 4-61. Before solving the set of eqns. ( 12 ), the signs of (QT) matrix columns with numbers j from 1 to q- 1 CM, from M+ 1 to qz - 1~ 2M and from 2M+ 1 to q3 - 1-c 3M, must be reversed, ql, q2 and q3 being the numbers of layers for which the depth-concentration profiles of the first, second and third component, respectively, are supposed to have maxima. For convenience, these sign reversions are described as a vectorp with components Pj =
-1 for l
After finding the solutions of the set of eqns. (12) but before applying the transition inversed to eqn. ( 11)) the signs of the components of vector N should also be reversed as for the columns of matrix (Q p’). In order to fulfil conditions (8) and (9) in the case of non-monotonous profiles with maxima, it is necessary to introduce some more constraint vectors g” (m=2,3, .... M + 3 ). Firstly, the non-negativeness of concentration values N,, N2 and N3 in the uppermost layer must be controlled. Hence, we come to three restriction vectors g2, g3 andg* (the sign prescribed by vector p has been taken into account) Pig?= Pi&=
l,lM 1,2M+l
which define the subspaces: Wgm>,0
m=2-4
(14)
Finally, condition (10) leads to the necessity of introduction of M- 1 more constraints (M is the number of sublayers in the sample partition) g” (m = 5, 6, . ... M+ 3 ). Each one of these constraints serves to control the fulfilment of the condition Ni + N2 + N3 < 1 in all layers, from 1 to M - 1, i.e.
7
iQ%
1
m=5,6,
wherep&=l,
. .. M+3
(15)
l
l,l
.
l,Z
.
pig?+3 =
,
‘*”
1, i=M- 1, M, ZM- 1, ZM, 3M- 1,3M O,l
Thus, the restoration of depth-concentration profiles with one maximum on the basis of angular distribution of XPS intensities for a four-component sample is reduced to the solution of the set of simultaneous eqns. (12) by the method of conjugate gradients in the space of vectors with non-negative components under additional constraints (13)- (15). Selection of the most suitable of the thus-obtained solutions is performed by means of simple exhaustion of the numbers of layers ql, qz and q3, for which the respective profiles N1 (z), N2 (z) and N3 (z) are supposed to have a maximum. The total number of such exhaustions is equal to (M - 1) 3. One more subtlety should be underlined in the process of numerical solution of the set of eqns. (12): the constraint vectorsgm, whichzre also to be projected on the space of positive values of components of vector N, may become linearly dependent. Consequently, in the numerical method of solution of eqns. (12) we must provide for a procedure which allows us to exclude from consideration those vectors g” that have become interdependent. It should be noted that the very problem of verification of linear independence of the constraint vectors g’” also falls into the class of ill-posed problems. Now let us briefly consider the solution of the set of eqns. (12) in the class of functions with one minimum. In this case, before solving eqns. (12), it is necessary to reverse the signs of (@ $) matrix columns with the following numbers: from q1 to M - 1, from q2 to ZM- 1 and from q3 to 3M- 1, where ql, q2 and q3 are the numbers of layers for which the depth-concentration profiles of the first, second and third components, respectively, are supposed to have a minimum. Vector p in this case looks like Pj =
I
-1 for l
ZM+l,
After having solved the set of eqns. (12) but prior to applying the transition inversed to eqn. (11 ), the signs of the components of vector fi should also be reversed in accordance with vectorp. In the class of solutions with a minimum, additional constraints ( 13 )- ( 15 ) must be included. Vector g’ here remains the
same as in the case of solutions with a maximum, and vectors g” for m= 2,3, 4 are given by the expressions Pi& =
Pig4 =
1, q1< iM 1,2M+q,
1, M+q,2M
’
0, i<2M+q,
From all the conditions (15) only the one for m= 5 with the same vector g5 must be taken into account. Lastly, in the numerical method of solution of the set of eqns. (12), control over the linear independence of vectors g” (m = l-5) must be ensured after each stez of projection on the space of nonnegative values of components of vector N. The above-described numerical method was actualized as a program in Fortran 77. Computation of one variant with total exhaustion of extremum locations requires time, which strongly depends on the dimension of the set of eqns. (12): e.g. for M=5, K=4 the IBM PC/XT performs it in about 40 min, and for M= 7, K= 6 it takes about 120 min. RESULTS AND DISCUSSION
Model profiles To begin with, let us consider the possibility of restoration of four-component depth-concentration profiles for a number of model systems using our method. The stability level of the solutions of the set (12) can be ascertained in a similar way to that used in refs. 4-6. For this purpose, the given model depth-concentration profiles are used in eqn. (4) to calculate the intensity ratios RF (0,) (i= l-3), which is afterwards “smeared” by using a randomnumber generator: E(&)
=R(&)
(l+rly)
(16)
where q is a parameter which gives the value of “smearing” (100% at q= 1) and y is a pseudo-random number within the interval [ - 1,l]produced by the generator. After this the values of RP(13,) calculated for 20 different sets of pseudo-random numbers y are entered as the input data into the program for restoration of depth-concentration profiles. The stability of solutions will, therefore, be characterized by the scatter (standard deviation) values in restored concentrations in layers and the difference between mean restored values and original ones. The results of such investigations of correctness of profile restoration by angular resolved XPS data are shown in Figs. l-4. For each of the model systems considered, parameters oi/od, ni/n, and Vi/U4(see eqns.
9
(4) and (7)) are set equal to 1.0 for i=l-3. Concentrations of elements in every layer are given in units of atomic percent, ci = ni 102/Cf= 1 ni, which are easy to convert to by making use of eqn. (6) and the values of Ni obtained in the process.of solving the set of eqns. ( 12 ). In Figs. l-4 the distance z from the sample surface in units of d4 (with a factor l/4, l/3 or l/2 in parentheses) is plotted as the abscissa. q= 0.1 corresponds to 10% scatter of XPS intensities. It is a matter of great interest to compare the restored depth-concentration profiles with the estimate for the ratio of elemental concentration ci: c4 by the formula -
ci
‘4a4 Rf (&45O)
ei=cQ=x
(17)
widely applied in practice. Values of c”i(i= l-3) are given in the captions to the figures. Figure 1 presents the results of restoration of a four-component profile by angular resolved XPS obtained from eqn. (4) and three values of take-off angle 0, = 0”, 30” and 60” when the sample is partitioned in four layers, the first three of which are A,/3 thick each. Hereafter, the maximum take-off angle of photoelectrons with respect to the normal to the sample surface is limited by 60’ -70’) for at greater angles angular distributions of intensities of XPS lines are strongly affected by the effects of elastic scattering of photoelectrons in
Fig. 1. Restoration of depth-concentration profiles ci(z) (i= l-4) for a four-component sample. Hereafter, 1 - the original profile; 2 - the points and error bars represent mean values and rootmean-square (standard) deviations of restored concentrations obtained using 20 different sets of smeared data. The average relative concentrations obtained from eqn. (17) are El =4.8, &= 7.5 and E,=2.8.
10
? LYlL__. 80
2
60
60
2
60
1
*d
;; -” 40 ”
240
20
20
0
0
0
1
2 ax,/
3 33)
0
3
1
*:X,/T
Fig. 2. Restoration of depth-concentration profiles with a maximum for the fourth component: i;, =0.64; E,= 0.59; E3= 0.2.
100
80
2 ;;’
60
-u v
40 20 i
0
12 Z(h,
0
34 13)
1
2 ZL
100 7 ”
80
;; 260 40 20 0 i,l_ 0
1
2 3 Z(A3/3)
4
Fig. 3. Restoration of depth-concentration profiles for a three-component sample with a maximum for the third component: C1= 1.0; E,=0.64.
the samples [g-13]. The spread of restored values of concentration of elements i= 1-4 in different layers was estimated for 20 cases of the “smearing” of line intensity angular distributions by using eqn. (16). Figure 1 reveals that for
11
OO-
1
2
3
4
Z(k4/3)
Fig. 4. Restoration of depth-concentration of elements: El = Ez= E3= 0.5.
Oo4
1
2 Z(A,
3 13)
profiles for a sample with depth-constant
distribution
model profiles the numerical method described in the previous section yields good results. Let us now consider averaged values of 15~ defined by eqn. (17) for the profiles presented in Fig. 1. For the first component c’i= 4.8 while the actual ratio of concentrations of the first and fourth components in the uppermost layer is equal to 7. The ratio of elemental concentrations close to Eioccurs for a sample layer with thickness dyff z 0.7 A,; for the second component this quantity is gff=&, and for the third component it is Gffw 1.1 A,. The question of the value of the effective depth of analysis by the XPS method, averaged over different shapes of depth-concentration profiles, was discussed in ref. 14. There, it was estimated that d”” = 1.8 A cos 8 and for 19=45’ deffz 1.3 A. Nonetheless, even in analysis as deep as this, the ratios of concentration differ from those predicted by eqn. (17) by more than 40%: for the first and fourth components this ratio is equal to 4 (instead of 4.8); for the second and fourth components it is 10.5 (instead of 7.5), and for the third and fourth components it equals 4 (instead of 2.8). Figure 2 displays the results of restoration for a sample for which the depthconcentration profile of the fourth component has a low maximum. These results show a rather great scatter compared with that given in Fig. 1. The reason for this is, firstly, that the change of relative intensities with a range from 0” to 60’ for depth-concentration profiles in Fig. 2 is significantly less than that for monotonous profiles in Fig. 1. The use of eqn. (17), as in the preceding case, causes considerable errors in estimation of the ratios of concentrations, particularly in the region of the uppermost surface layer. The developed method may also be applied to restoration of depth-concen-
12
tration profiles in samples with less than four components. An example of such restoration is given in Fig. 3. The restored profiles (the third component profile has a maximum at z=O.7 A,) turned out to be less sensitive to errors of input data in comparison with the profiles in Fig. 2. This result serves as one more illustration of the fact, stated earlier by us in refs. 5 and 6, that the sensitivity of the depth-concentration profile restoration to the errors in angular distribution of XPS intensities depends critically on the shape of profiles. Figure 4 presents the results of profile restoration for a sample with depthconstant distribution of elements. They show unexpectedly large root-meansquare deviation of concentration in layers sometimes reaching 100%. Let us look at the data in Table 1 in order to analyse these seemingly anomalous results. The ratios of line intensities for depth-constant profiles also do not depend on the take-off angle 8 of photoelectrons. However the converse turns out to be absolutely untrue. Table 1 presents a set of four-component nonmonotonous profiles that differ greatly from the depth-constant profiles (see c,(z) and c,(z) ) for which, nevertheless, the ratios of line intensities in the range of angles from 0” to 60” appear to be constant within about 2%. It is precisely the data of Table 1 that elucidate the anomalous scatter of the restoration results given in Fig. 2. Results of Fig. 4 and Table 1 corroborate the above-stated qualitative rule TABLE 1 Angular distributions of XPS line intensities (Zi) for depth-concentration profiles c!(z)
c,(z)
0.2
0.2
0.2
0.2
i=l-3 c4 0 lilzb i=l-3
0.4 0” 0.5
0.4 30” 0.5
0.4 60” 0.5
0.4
0.0 0.22 0.31 0.47 0” 0.492 0.484 0.498
0.51 0.22 0.01 0.26 30” 0.511 0.496 0.491
0.51 0.22 0.0 0.27 60” 0.507 0.523 0.487
0.5 0.16 0.3 0.04
c,(z) G(Z) es(z) G(Z)
e
Z,lL
ML L/L
“A - depth constant where z is the depth. bB - non-monotonous.
13
that for low anisotropy of angular distributions of photoelectrons the precision of restoration of depth-concentration profiles is also low. For low-anisotropy angular distribution of XPS, several methods may be proposed for lessening the restoration error. Firstly, the range of angles 0, for which the ratios of XPS line intensities are obtained, may be extended. However, for the case considered in Fig. 4 we have found that the scatter of restoration results (upon 10% “smearing” of angular distributions) does not reduce substantially when the upper bound of 8 is raised to 70’. Usage of angles 0 higher than this boundary value in treating the real experimental data is required by the necessity of due accounting for the effects of photoelectron elastic scattering, the finiteness of entrance angular aperture of the energy analyser, and, probably, the roughness of the sample surface. The most effective approach was found to be the one based on the use of additional a priori information about profiles for enhancing the precision of restoration. For instance, under the additional assumption that all the profiles ci (z) in Fig. 4 are monotonous, the restoration error falls off to a very low level, about l%-2%, in all layers. Experimental profiles For characterization of the quality of restoration we use the same procedure as in the preceding section for model profiles. The experimental intensity ratios were “smeared” by using 20 sets of pseudo-random numbers and we then calculated means and standard deviations of restored concentration values. We assumed a 10% error level in the experimental data. In all the figures in this section (as well as in the preceding section) the distance z from the sample surface is measured in units of ;1for photoelectrons ejected from the core level of the 4th component (Lo and lZTifor Figs. 5-8, respectively). This choice of depth unit (in relative values z/L instead of nanometers or Angstroms) has additional advantages: firstly, all ratios d/ni in eqns. (7) can be replaced by equivalent forms d/&*&/L, and for the numerical profile restoration only ratios li: I, are needed where the errors may be less than those of pi absolute values; secondly, if there are some dependences of 1 on the chemical composition or density it would lead only to local compression or expansion of restored profile along the z/ii axis. The present-day literature lacks experimental data for the angular distribution of XPS intensities for samples with more than two components. The only exception known to us is ref. 3. Using the data from this work (see Table 1 in ref. 3), for a sample which is a C!,F, film plasmochemically applied to quartz, we restored the depth-concentration profiles of elements F, C, Si and 0 (see Fig. 5). The depth-distribution of the elements proved to be close to that expected: at the very top there is a film with approximate proportion C : F z 1: 2.6, under which there lies a substrate of Si : 0 z 1: 2. The film is about
Fig. 5. Distribution of elements F, C, Si and 0 based on the experimentally obtained angular XPS dependences for a C3Fs film applied to quartz [ 31. 4
I
60-I 2
80-i ‘60_
80-
‘I ;
Lrrrrl
c 0
I 40-
f
z40
f
I
20 0
I I
I
1
2
20I
I
lIIilll_IZIW
0
3456789
0
1
2
3
4
$1 0
Z(ho/4.5)
5
6
7
8
1
.
I
I
2
3
4
I
I
I
I
1)
56709 Z( h0/4.51
9
zcxo145)
Fig. 6. Depth-concentration profiles of Si, C and 0 based on the experimental angular distributions of XPS intensities for a 0.6 pm thick pyrolytically prepared SiOp film.
& (about 20 A) thick, II, being the mean free path of photoelectrons knocked out from the 01s core level. It should be noted that the results are substantially different from those in ref. 3. At &32” the ratio of Si2p and 01s line intensities is 1.14 (with allowance for the ratio of corresponding core level photoionization cross-sections), and in ref. 3 the restored profiles for elements Si and 0 have a concentration ratio approximately equal to 1.0. The profiles obtained by us (see Fig. 5) are somewhat different - with ratio Si :0 = 1: 2. A possible
15
b
1 I
60-
I
I
=
I
I
60-
2
2
J 40_ .9 0
c409 v
ao-
o0
’ 1
1
I
I
2
3
4
N 5
o0
6
’ 1
I 2
3 zcx,,
Z(bqi/3)
0
1
2
3
4
Z(hTi
/3)
4
5
5
I
I,
6
/3)
6
Fig. 7. Restored depth-concentration profiles of Ti, C and 0 based on the angular distribution of XPS line intensities for a “TiOz” sample.
;f,~,‘,_,;,.:“~,~ ,=,=, I,r,,,C
~~=,~~~~~~~~~~~~~‘-:-r_: 01234567
0 Z(h,,
/3)
1
2
3
4 Z(XTi
5
6
7
13)
Fig. 8. Restored depth-concentration profiles of Ti, N, C and 0 based on XPS line intensities of a TiN sample held for 500 h in air.
explanation of the latter fact is that for the mean free paths of photoelectrons knocked out from Si2p and 01s levels, Asi:Ao= 1.3: 1, and therefore, the concentration ratio Si : 0 is lower than the ratio of line intensities Si2p : 01s. In addition, it should be mentioned that the concentration of 0 rises sharply from about 0% to about 60% at somewhat greater depths (by about 1,/3) than those depths where the growth of Si is observed. Needless to say, judging only by one set of experimental data, it is hardly possible to make definite conclusions, but
16
apparently there are oxygen-free compounds of Si with F at the film/substrate interface. Figures 6-8 give the results of restoration based on experimental data for “SiOZ”, “ Ti02” and TiN samples after holding them for 500 h in air. The angular distribution of XPS lines were obtained in our laboratory by T. Reich on an ADES-400 spectrometer for SiOZ and for other samples by M.V. Kuznetsov on an ESCALAB VG spectrometer in Professor V.V. Gubanov’s laboratory in the Institute of Inorganic Chemistry of the Ural branch of the Academy of Sciences of the U.S.S.R. The experimental data presented in Tables 2-4 serve mainly to illustrate the restoration method, and therefore we will dwell neither on measuring aspects nor on detailed discussion of the character of obtained profiles. Parameters necessary for calculations are as follows: (1) photoionization cross-sections taken from ref. 15, and the ratio of the mean free paths estimated from eqn. [ 161 hi/Ak= (Ei/Ek)0.7, where Ei and Ek are the kinetic energies of photoelectrons. The angle between the directions of incident X-radiation and of photoelectron emission was equal to 48”. The ratio of atomic volumes Vi: V, was estimated from the corresponding values of covalent radii (here, it must be stressed that this ratio does not affect the results of restoration to a great extent). Figure 6 shows the profiles of elements Si, 0 and C for a thick SiO, oxide TABLE 2 Relative angular distributions of Si2p, Cls and 01s line intensities for a SO, film Take-off angle 0 ( ’
)
0 5 30 45 60
lSiZpllOls
LlIOl.
0.208 0.205 0.201 0.200 0.205
0.147 0.164 0.206 0.265 0.364
TABLE 3 Relative angular distributions of Ols, Cls and Ti2p line intensities for a “TiOz” sample Take-off angle 19(O )
101s11Ti2p
klsIJr12p
15 25 35 45 55
1.092 1.139 1.159 1.187 1.249
0.085 0.097 0.100 0.110 0.123
17 TABLE 4
Relative angular distributions of XPS line intensities for a TIN sample after holding it for 500 h in air Take-off angle 0 15 25 35 45 55
( a )
IOlSlITiZP
zNlslzTi2p
ICdKTi2p
0.483 0.511
0.176
0.097
0.167 0.166
0.106 0.115 0.140
0.560 0.604 0.677
0.163 0.128
0.183
film introduced into the spectrometer from air without preliminary cleaning. As expected, there is a layer of hydrocarbons on the surface under which the sample consists of 0 and Si with a proportion of 2.12 0.2. It should be noted that for the ratio of Si2p and 01s line intensities for 8= 45 o (see Table 2)) eqn. ( 17) gives an 0 : Si proportion equal to about 3. Thus, as shown in the preceding section for model profiles, the use of expression (17) for determining the composition of samples by XPS data may lead to considerable errors because of the neglect of real element concentration gradients in a sample. This conclusion also remains valid for Auger electron spectroscopy, practical application of which for chemical analysis also uses an expression similar to eqn. (17) (with factors of element sensitivity instead of photoionization cross-sections o and photoelectron mean free paths 2). The concentration of C atoms in Fig. 6 does not attain 100 at.% even in the &/4 thick uppermost layer. Surface coating is observed only for such increase of the number of sublayers when the thickness of each sublayer does not exceed about 0.1 2,. A not very sharp decrease of concentration of element C with depth is apparently due to the roughness of the sample surface which leads to depth-inhomogeneity of the hydrocarbon contamination layer. Figure 7 presents the distribution of elements 0, C and Ti for a TiOz sample according to the experimental data given in Table 3. For depths more than about 1.5 Lo the ratio Ti: 0 x 1: 2, while nearer to the surface the concentration of C atoms increases and that of Ti atoms (in oxidated form) decreases. It should be noted that when 0= 45 ’ and the ratio of photoionization cross-sections ools : aTizp = 0.36 [ 151, eqn. (17) gives the ratio of elemental concentration O:Ti=3.3: 1. Finally, Fig. 8 shows the restored profiles of elements 0, C, N and Ti for a TiN sample which has been held for 500 h in air. The strongly nonlmonotonous distribution of element Ti with depth deserves particular attention; this is probably due to the formation of titanium oxide through the extraction of Ti atoms from the nitride layer.
18
REFERENCES
8
9
10 11 12 13 14 15 16
M. Pijolat and G. Hollinger, Surf. Sci., 105 (1981) 114. T.D. Bussing and P.H. Holloway, J. Vat. Sci. Technol. A, 3 (1985) 1973. R.S. Yih and B.D. Ratner, J. Electron Spectrosc. Relat. Phenom., 43 (1987) 61. O.A. Baschenko and V.I. Nefedov, Poverkhn., 7 (1987) 75 (in Russian). O.A. Baschenko and V.I. Nefedov, Poverkhn., 10 (1987) 99 (in Russian). V.I. Nefedov and O.A. Baschenko, J. Electron Spectrosc. Relat. Phenom., 47 (1988) 1. A.N. Tikhonov and V.Ya. Arsenin, Solutions to Ill-Posed Problems, V.H. Winston, Washington DC, 1977. A.N. Tikhonov, A.V. Goncharskii, V.V. Stepanov and A.G. Yagola, Regulyariziruyushchie Algoritmy i Apriornaya Informatsiya (Regularizing Algorithms and A Priori Information), Nauka, Moscow, 1983 (in Russian). O.A. Baschenko and V.I. Nefedov, J. Electron Spectrosc. Relat. Phenom., 17 (1979) 405. O.A. Baschenko and V.I. Nefedov, J. Electron Spectrosc. Relat. Phenom., 21 (1980) 153. O.A. Baschenko and V.I. Nefedov, Metallofizika, 4 (1982) 61 (in Russian). O.A. Baschenko and V.I. Nefedov, Poverkhn., 2 (1982) 87 (in Russian). O.A. Baschenko, G.V. Machavariani and V.I. Nefedov, J. Electron Spectrosc. Relat. Phenom., 34 (1984) 304. V.I. Nefedov, Surf. Interface Anal., 3 (1981) 72. I.M. Band, Yu.1. Kharitonov and M.B. Trzhaskovskaya, Atomic Data Nucl. Data Tables, 23 (1979) 443. D.R. Penn, J. Electron Spectrosc. Relat. Phenom., 9 (1976) 29.
Journal of Electron Spectroscopy and Related Phenomena, 53 (1990) 1-18 Elsevier Science Publishers B.V., Amsterdam
DEPTH PROFILING OF ELEMENTS IN SURFACE LAYERS SOLIDS BASED ON ANGULAR RESOLVED X-RAY PHOTOELECTRONSPECTROSCOPY
OF
O.A. BASCHENKO, V.I. NEFEDOV N.S. Kurnakov Institute of General and Inorganic Chemistry of the Academy of Sciences of the U.S.S.R., Moscow (U.S.S.R.) (First received 2 August 1989; in final form 24 January 1990)
ABSTRACT A numerical method is proposed for the non-destructive restoration of the depth-concentration profiles for four components, based on the angular distribution of XPS intensities. Calculations performed for a number of model systems demonstrated that ratios of concentrations obtained by using simplified procedures, i.e. from relative XPS intensities for a single escaping angle, led to large errors (up to 200% ) for the upper surface layer. A number of restored profiles are presented using experimental angle-resolved XPS data for samples containing 3-4 components.
INTRODUCTION
Angular resolved X-ray photoelectron spectroscopy (XPS ) allows us to carry out a non-destructive layer-by-layer analysis of the chemical composition - a sort of tomography - of 5-10 nm thick surface layers of solids. The depthconcentration profiles of elements with 1 nm depth resolution, which are obtained from the angular distribution of XPS intensities, present very important, easy-to-interpret and instructive information for studying processes such as oxidation of metals and alloys, segregation, corrosion, adhesion, the effect of ion beams on solids, etc. If needed, depth-concentration profiles can be determined for identical atoms which differ only in their chemical states (e.g. by the oxidation state ) . References l-6 put forward methods of mathematical treatment of angular XPS for finding the depth-concentration profiles. All these methods contribute to removing the major obstacle namely the extremely high sensitivity of restored depth-concentration profiles to the errors of experimental data [ 1,46] because the problem of profile restoration falls into the class of ill-posed problems [ 71. In order to overcome this instability of solutions, it is necessary to use some a priori information about the profile, as is generally done when solving Fredholm integral equations of the first kind [ 7,8]. It was shown in
036%2048/90/$03.50
0 1990 -
Elsevier Science Publishers B.V.
2
ref. 1,that the instability problem can be resolved with the aid of the following a priori information about the solution: the desired function of dependence between concentration and depth should be (a) non-negative, bounded and (b) monotonous. References 4-6 show that the second constraints can be relaxed, i.e. it is sufficient to require that the depth-concentration profile should not have more than one extremum up to the depth of 3A (2 is the mean free path of photoelectrons without inelastic collisions in a solid). References 1 and 4-6 deal with samples containing only two components, where if one of the profiles is monotonous (or has only one extremum) then the other is also monotonous and has only one extremum. When the number of components is more than two the situation is somewhat more complex. For example, for a three-component sample, if two depth-concentration profiles are monotonous the third one may have an extremum. Likewise, if two profiles have an extremum the third profile may have several such extrema. Therefore, we may infer that as the number of components of the sample increases, the problem of depth-concentration profile restoration becomes more difficult. The procedure of numerical solution of such a problem requires considerable modification compared with that for the case of two components. Samples used in most XPS studies have more than three components because in the surface layers oxygen and carbon are adsorbed from the air. In the present work, the numerical method which we proposed in refs. 4-6 for the restoration of depth-concentration profiles of elements in the surface layers of solids with thicknesses up to 100 A, based on the data of angular resolved XPS, is extended to embrace the case of four-component samples. It was shown for a number of model systems that the suggested procedure of accounting the a priori information (i.e. that every depth-concentration profile is a non-negative and bounded function and for three of the four components studied the profiles may have no more than one extremum) enables us to obtain solutions which are resistant to the experimental errors attainable at the present day (about 10% ) . It was found that determination of the ratios of elemental concentrations based on the simplified method, i.e. from the ratios of XPS line intensities at the same take-off angles 19(usually 8=45” ), may lead to considerable errors (up to and over 200% ) for the uppermost layer owing to the neglect of real concentration gradients in the sample. Non-destructive restoration of depth-concentration profiles was conducted on the basis of experimental angular distributions of XPS for several samples with 3-4 components. For a TiOz sample (with a hydrocarbon contamination layer) it was established that the ratio of Ti2p and 01s line intensities at 8= 45’ gives the 1: 3 proportion of atoms, while the obtained profiles indicate the presence of stoichiometric oxide TiOz, but only from depths greater than about 20 A. Above this oxide layer lies a film containing adsorbed oxygen and hydrocarbons.
3 INVESTIGATION METHODS
Angular distributions of XPS intensities for a four-component sample For each of the components (with index i) the intensity of the XPS line is described by the formula m Ii =FiGi
s
?Zi(Z)
t?Xp(
-Z/Ai
COS 0)dZ
(1)
0
where Fi is an instrumental factor (which contains the dependence of the transmissibility of the energy analyser on the kinetic energy Ek of the photoelectron), ai is the cross-section of photoionization of the core level of the element (or component) i, Ai is the mean free path of photoelectrons in a sample without inelastic collisions (all 1 i are assumed to be dependent only on Ek but not on the chemical composition ), ni (z) is the concentration of element i at depth z from the sample’s surface and 19is the take-off angle of photoelectrons measured from the normal to the atomically smooth surface of a sample. We shall consider the ratio of line intensities of the first three components to that of the fourth Fi Ci Ri(e)=jJjjj=
Ii(e)
I
ni(2) exp( -Z/Li
cos
8)dZ
(2)
20 J’4 04
I 0
n,(Z) exp( -Z/n,
co9 8)dZ
We shall henceforth set the ratios Fi: F4 equal to 1.0 for i= 1,2,3, since these quantities depend not on the sample composition but only on the characteristics of spectrometers and energy analysers, and can be determined ‘from independent experiments. If Fi : F4 = 1.0 for the spectrometer used in the experiments then a suitable correction factor may be introduced in the definition of Ri (0). Restoration of the profiles of concentrations ni (z) reduces to the solution of a set of three simultaneous equations
R;(e) =Ry(e)
i= l-3
(3)
where R:(e) are described by eqn. (2) (with Fi:F4=1.0) and R;(e) are the experimentally measured line intensities of the ith component with respect to that of the fourth component. Using the same method as in refs. 1 and 4-6, we replace the integrals in eqns. (1) and (2) by sums which correspond to the partition of the sample in M- 1 identical layers parallel to the surface and of thickness d, so that (M- 1)d = 3A4. Assuming that within every layer the concentrations of elements are constant
4
(i.e. n(z) =n(j) when dfj-l)
-;li
,:s 8,)1 (4)
Here k= 1,2,...,K, and K is the number of different take-off angles of photoelectrons at which the ratios of XPS intensities have been measured. It should be noted that the choice of the number M of sublayers and the thickness d of every layer also constitute a sort of a priori information which is used for restoration of depth-concentration profiles. In particular, the assumption that it is reasonable to partition into layers the part of the sample lying higher than 3L, in depth is based on the fact that the possibility for photoelectrons to outflow from greater depth (about 5% ) is less than the typical experimental errors. Now, let us introduce new variables Ni(j) z&(j)
fJi
(5)
where j is the layer number and Uiis the atomic (or molecular) volume of the ith element. Proceeding from volume conservation as we go from the jth layer to another, we can write ii1
niti,ui=i~l
Nio’)=l
forj= 1,2,...,M
(6)
Substitution of eqns. (4) and (5) into the set of eqns. (3)) with allowance for eqn. (6)) gives a set of non-linear equations in unknowns Ni(j) (i= l-3, j= 1-M). As in the case of two-component samples (see refs. 4-6), this set of equations may be reduced to the linear form by multiplying both sides of eqn. (3) by the factor 14( ok). Finally, for the unknowns Ni(j) we come to the following set of equations
(7)
5
where ri= ui/u4. Thus, the determination of profiles for a four-component sample through angular distributions of XPS intensities is reduced to the solution of eqn. (7) of linear equations in 3M unknowns Ni (j) (i = l-3, j = 1-M). The numerical method of restoration of non-monotonous profiles with allowance for the a priori information Let us now describe the method of solution of (ill-conditioned when K=M [ 1,4-61) the set of eqns. (7) with due account of the a priori information. This
method is resistant to errors in the input data equal to 5-10%. By a priori information, basically we mean that the solutions are sought in the class of functions satisfying the following restrictions NAj) 20
i= l-4
(3)
N,(j)<1
i= l-4
(9)
and, furthermore, that each of the functions Ni (j) (i = l-3 ) may have no more than one extremum at depths up to 3A4. We shall follow the approach which we have previously developed in refs. 4-6 for solving the problem of profile restoration for two-component samples. The set of eqns. (7) may be recast in the matrix form Q(&> =N=R(e,)
(10)
where N is a 3Mdimensional vector, components of which are the values N1, NZ and N3 of concentration in layers with numbers j= 1,2, .... M; I? is a vector whose components are the experimental values of relative intensities RT( f&), Rg (19,)) R; (0,) with (3,take-cff angles of photoelectrons. The explicit form of the matrix elements &ii is manifest from eqns. (7 ) . Realization of the a priori information consists, firstly, in the change to a new basis [8] p&IV
(11)
where R has components fii(j) = Ni(j) -Ni(j+ 1) when j # M, 2M, 3M and fii(j)=Ni(j) when j=M, 2M, 3M. For l
&ekQ7=Rcek)
(12)
&e,) = @f’, using the method of conjugate gradients with projection on a set of vectors with non-negative components [ 81 (it should be noted that here and previously [ 4-61 the method of conjugate adients is applied to step-by-step minimization of the function F= Ck [Q (0,) 8 -I?( 19,)] “), one obtains solutions which are monotonously non-increasing and non-negative functions (IGiG) > 0 for all i and j, and, owing to the selected form of the transition matrix p, ine-
qualities Ni (j) > Ni (j+ 1) are fulfilled for allj). Now we have to take into consideration one more additional constraint which ensures the fulfilment of condition (9). For this purpose, the numerical minimization procedure contains the projection of the gradient onto the subspace defined by (13) where a vector g’ has the following components: gf = 1 for i=M, 2M, 3M and gi = 0 for all other values of i. To make sure that the set of solutions of eqns. (12) includes functions with a maximum, an approach is used which was advanced in ref. 8 and used by us in developing the numerical algorithm for restoration of two-component depthconcentration profiles [ 4-61. Before solving the set of eqns. ( 12 ), the signs of (QT) matrix columns with numbers j from 1 to q- 1 CM, from M+ 1 to qz - 1~ 2M and from 2M+ 1 to q3 - 1-c 3M, must be reversed, ql, q2 and q3 being the numbers of layers for which the depth-concentration profiles of the first, second and third component, respectively, are supposed to have maxima. For convenience, these sign reversions are described as a vectorp with components Pj =
-1 for l
After finding the solutions of the set of eqns. (12) but before applying the transition inversed to eqn. ( 11)) the signs of the components of vector N should also be reversed as for the columns of matrix (Q p’). In order to fulfil conditions (8) and (9) in the case of non-monotonous profiles with maxima, it is necessary to introduce some more constraint vectors g” (m=2,3, .... M + 3 ). Firstly, the non-negativeness of concentration values N,, N2 and N3 in the uppermost layer must be controlled. Hence, we come to three restriction vectors g2, g3 andg* (the sign prescribed by vector p has been taken into account) Pig?= Pi&=
l,l
which define the subspaces: Wgm>,0
m=2-4
(14)
Finally, condition (10) leads to the necessity of introduction of M- 1 more constraints (M is the number of sublayers in the sample partition) g” (m = 5, 6, . ... M+ 3 ). Each one of these constraints serves to control the fulfilment of the condition Ni + N2 + N3 < 1 in all layers, from 1 to M - 1, i.e.
7
iQ%
1
m=5,6,
wherep&=l,
. .. M+3
(15)
l
l,l
.
l,Z
.
pig?+3 =
,
‘*”
1, i=M- 1, M, ZM- 1, ZM, 3M- 1,3M O,l
Thus, the restoration of depth-concentration profiles with one maximum on the basis of angular distribution of XPS intensities for a four-component sample is reduced to the solution of the set of simultaneous eqns. (12) by the method of conjugate gradients in the space of vectors with non-negative components under additional constraints (13)- (15). Selection of the most suitable of the thus-obtained solutions is performed by means of simple exhaustion of the numbers of layers ql, qz and q3, for which the respective profiles N1 (z), N2 (z) and N3 (z) are supposed to have a maximum. The total number of such exhaustions is equal to (M - 1) 3. One more subtlety should be underlined in the process of numerical solution of the set of eqns. (12): the constraint vectorsgm, whichzre also to be projected on the space of positive values of components of vector N, may become linearly dependent. Consequently, in the numerical method of solution of eqns. (12) we must provide for a procedure which allows us to exclude from consideration those vectors g” that have become interdependent. It should be noted that the very problem of verification of linear independence of the constraint vectors g’” also falls into the class of ill-posed problems. Now let us briefly consider the solution of the set of eqns. (12) in the class of functions with one minimum. In this case, before solving eqns. (12), it is necessary to reverse the signs of (@ $) matrix columns with the following numbers: from q1 to M - 1, from q2 to ZM- 1 and from q3 to 3M- 1, where ql, q2 and q3 are the numbers of layers for which the depth-concentration profiles of the first, second and third components, respectively, are supposed to have a minimum. Vector p in this case looks like Pj =
I
-1 for l
ZM+l,
After having solved the set of eqns. (12) but prior to applying the transition inversed to eqn. (11 ), the signs of the components of vector fi should also be reversed in accordance with vectorp. In the class of solutions with a minimum, additional constraints ( 13 )- ( 15 ) must be included. Vector g’ here remains the
same as in the case of solutions with a maximum, and vectors g” for m= 2,3, 4 are given by the expressions Pi& =
Pig4 =
1, q1< i
1, M+q,
’
0, i<2M+q,
From all the conditions (15) only the one for m= 5 with the same vector g5 must be taken into account. Lastly, in the numerical method of solution of the set of eqns. (12), control over the linear independence of vectors g” (m = l-5) must be ensured after each stez of projection on the space of nonnegative values of components of vector N. The above-described numerical method was actualized as a program in Fortran 77. Computation of one variant with total exhaustion of extremum locations requires time, which strongly depends on the dimension of the set of eqns. (12): e.g. for M=5, K=4 the IBM PC/XT performs it in about 40 min, and for M= 7, K= 6 it takes about 120 min. RESULTS AND DISCUSSION
Model profiles To begin with, let us consider the possibility of restoration of four-component depth-concentration profiles for a number of model systems using our method. The stability level of the solutions of the set (12) can be ascertained in a similar way to that used in refs. 4-6. For this purpose, the given model depth-concentration profiles are used in eqn. (4) to calculate the intensity ratios RF (0,) (i= l-3), which is afterwards “smeared” by using a randomnumber generator: E(&)
=R(&)
(l+rly)
(16)
where q is a parameter which gives the value of “smearing” (100% at q= 1) and y is a pseudo-random number within the interval [ - 1,l]produced by the generator. After this the values of RP(13,) calculated for 20 different sets of pseudo-random numbers y are entered as the input data into the program for restoration of depth-concentration profiles. The stability of solutions will, therefore, be characterized by the scatter (standard deviation) values in restored concentrations in layers and the difference between mean restored values and original ones. The results of such investigations of correctness of profile restoration by angular resolved XPS data are shown in Figs. l-4. For each of the model systems considered, parameters oi/od, ni/n, and Vi/U4(see eqns.
9
(4) and (7)) are set equal to 1.0 for i=l-3. Concentrations of elements in every layer are given in units of atomic percent, ci = ni 102/Cf= 1 ni, which are easy to convert to by making use of eqn. (6) and the values of Ni obtained in the process.of solving the set of eqns. ( 12 ). In Figs. l-4 the distance z from the sample surface in units of d4 (with a factor l/4, l/3 or l/2 in parentheses) is plotted as the abscissa. q= 0.1 corresponds to 10% scatter of XPS intensities. It is a matter of great interest to compare the restored depth-concentration profiles with the estimate for the ratio of elemental concentration ci: c4 by the formula -
ci
‘4a4 Rf (&45O)
ei=cQ=x
(17)
widely applied in practice. Values of c”i(i= l-3) are given in the captions to the figures. Figure 1 presents the results of restoration of a four-component profile by angular resolved XPS obtained from eqn. (4) and three values of take-off angle 0, = 0”, 30” and 60” when the sample is partitioned in four layers, the first three of which are A,/3 thick each. Hereafter, the maximum take-off angle of photoelectrons with respect to the normal to the sample surface is limited by 60’ -70’) for at greater angles angular distributions of intensities of XPS lines are strongly affected by the effects of elastic scattering of photoelectrons in
Fig. 1. Restoration of depth-concentration profiles ci(z) (i= l-4) for a four-component sample. Hereafter, 1 - the original profile; 2 - the points and error bars represent mean values and rootmean-square (standard) deviations of restored concentrations obtained using 20 different sets of smeared data. The average relative concentrations obtained from eqn. (17) are El =4.8, &= 7.5 and E,=2.8.
10
? LYlL__. 80
2
60
60
2
60
1
*d
;; -” 40 ”
240
20
20
0
0
0
1
2 ax,/
3 33)
0
3
1
*:X,/T
Fig. 2. Restoration of depth-concentration profiles with a maximum for the fourth component: i;, =0.64; E,= 0.59; E3= 0.2.
100
80
2 ;;’
60
-u v
40 20 i
0
12 Z(h,
0
34 13)
1
2 ZL
100 7 ”
80
;; 260 40 20 0 i,l_ 0
1
2 3 Z(A3/3)
4
Fig. 3. Restoration of depth-concentration profiles for a three-component sample with a maximum for the third component: C1= 1.0; E,=0.64.
the samples [g-13]. The spread of restored values of concentration of elements i= 1-4 in different layers was estimated for 20 cases of the “smearing” of line intensity angular distributions by using eqn. (16). Figure 1 reveals that for
11
OO-
1
2
3
4
Z(k4/3)
Fig. 4. Restoration of depth-concentration of elements: El = Ez= E3= 0.5.
Oo4
1
2 Z(A,
3 13)
profiles for a sample with depth-constant
distribution
model profiles the numerical method described in the previous section yields good results. Let us now consider averaged values of 15~ defined by eqn. (17) for the profiles presented in Fig. 1. For the first component c’i= 4.8 while the actual ratio of concentrations of the first and fourth components in the uppermost layer is equal to 7. The ratio of elemental concentrations close to Eioccurs for a sample layer with thickness dyff z 0.7 A,; for the second component this quantity is gff=&, and for the third component it is Gffw 1.1 A,. The question of the value of the effective depth of analysis by the XPS method, averaged over different shapes of depth-concentration profiles, was discussed in ref. 14. There, it was estimated that d”” = 1.8 A cos 8 and for 19=45’ deffz 1.3 A. Nonetheless, even in analysis as deep as this, the ratios of concentration differ from those predicted by eqn. (17) by more than 40%: for the first and fourth components this ratio is equal to 4 (instead of 4.8); for the second and fourth components it is 10.5 (instead of 7.5), and for the third and fourth components it equals 4 (instead of 2.8). Figure 2 displays the results of restoration for a sample for which the depthconcentration profile of the fourth component has a low maximum. These results show a rather great scatter compared with that given in Fig. 1. The reason for this is, firstly, that the change of relative intensities with a range from 0” to 60’ for depth-concentration profiles in Fig. 2 is significantly less than that for monotonous profiles in Fig. 1. The use of eqn. (17), as in the preceding case, causes considerable errors in estimation of the ratios of concentrations, particularly in the region of the uppermost surface layer. The developed method may also be applied to restoration of depth-concen-
12
tration profiles in samples with less than four components. An example of such restoration is given in Fig. 3. The restored profiles (the third component profile has a maximum at z=O.7 A,) turned out to be less sensitive to errors of input data in comparison with the profiles in Fig. 2. This result serves as one more illustration of the fact, stated earlier by us in refs. 5 and 6, that the sensitivity of the depth-concentration profile restoration to the errors in angular distribution of XPS intensities depends critically on the shape of profiles. Figure 4 presents the results of profile restoration for a sample with depthconstant distribution of elements. They show unexpectedly large root-meansquare deviation of concentration in layers sometimes reaching 100%. Let us look at the data in Table 1 in order to analyse these seemingly anomalous results. The ratios of line intensities for depth-constant profiles also do not depend on the take-off angle 8 of photoelectrons. However the converse turns out to be absolutely untrue. Table 1 presents a set of four-component nonmonotonous profiles that differ greatly from the depth-constant profiles (see c,(z) and c,(z) ) for which, nevertheless, the ratios of line intensities in the range of angles from 0” to 60” appear to be constant within about 2%. It is precisely the data of Table 1 that elucidate the anomalous scatter of the restoration results given in Fig. 2. Results of Fig. 4 and Table 1 corroborate the above-stated qualitative rule TABLE 1 Angular distributions of XPS line intensities (Zi) for depth-concentration profiles c!(z)
c,(z)
0.2
0.2
0.2
0.2
i=l-3 c4 0 lilzb i=l-3
0.4 0” 0.5
0.4 30” 0.5
0.4 60” 0.5
0.4
0.0 0.22 0.31 0.47 0” 0.492 0.484 0.498
0.51 0.22 0.01 0.26 30” 0.511 0.496 0.491
0.51 0.22 0.0 0.27 60” 0.507 0.523 0.487
0.5 0.16 0.3 0.04
c,(z) G(Z) es(z) G(Z)
e
Z,lL
ML L/L
“A - depth constant where z is the depth. bB - non-monotonous.
13
that for low anisotropy of angular distributions of photoelectrons the precision of restoration of depth-concentration profiles is also low. For low-anisotropy angular distribution of XPS, several methods may be proposed for lessening the restoration error. Firstly, the range of angles 0, for which the ratios of XPS line intensities are obtained, may be extended. However, for the case considered in Fig. 4 we have found that the scatter of restoration results (upon 10% “smearing” of angular distributions) does not reduce substantially when the upper bound of 8 is raised to 70’. Usage of angles 0 higher than this boundary value in treating the real experimental data is required by the necessity of due accounting for the effects of photoelectron elastic scattering, the finiteness of entrance angular aperture of the energy analyser, and, probably, the roughness of the sample surface. The most effective approach was found to be the one based on the use of additional a priori information about profiles for enhancing the precision of restoration. For instance, under the additional assumption that all the profiles ci (z) in Fig. 4 are monotonous, the restoration error falls off to a very low level, about l%-2%, in all layers. Experimental profiles For characterization of the quality of restoration we use the same procedure as in the preceding section for model profiles. The experimental intensity ratios were “smeared” by using 20 sets of pseudo-random numbers and we then calculated means and standard deviations of restored concentration values. We assumed a 10% error level in the experimental data. In all the figures in this section (as well as in the preceding section) the distance z from the sample surface is measured in units of ;1for photoelectrons ejected from the core level of the 4th component (Lo and lZTifor Figs. 5-8, respectively). This choice of depth unit (in relative values z/L instead of nanometers or Angstroms) has additional advantages: firstly, all ratios d/ni in eqns. (7) can be replaced by equivalent forms d/&*&/L, and for the numerical profile restoration only ratios li: I, are needed where the errors may be less than those of pi absolute values; secondly, if there are some dependences of 1 on the chemical composition or density it would lead only to local compression or expansion of restored profile along the z/ii axis. The present-day literature lacks experimental data for the angular distribution of XPS intensities for samples with more than two components. The only exception known to us is ref. 3. Using the data from this work (see Table 1 in ref. 3), for a sample which is a C!,F, film plasmochemically applied to quartz, we restored the depth-concentration profiles of elements F, C, Si and 0 (see Fig. 5). The depth-distribution of the elements proved to be close to that expected: at the very top there is a film with approximate proportion C : F z 1: 2.6, under which there lies a substrate of Si : 0 z 1: 2. The film is about
Fig. 5. Distribution of elements F, C, Si and 0 based on the experimentally obtained angular XPS dependences for a C3Fs film applied to quartz [ 31. 4
I
60-I 2
80-i ‘60_
80-
‘I ;
Lrrrrl
c 0
I 40-
f
z40
f
I
20 0
I I
I
1
2
20I
I
lIIilll_IZIW
0
3456789
0
1
2
3
4
$1 0
Z(ho/4.5)
5
6
7
8
1
.
I
I
2
3
4
I
I
I
I
1)
56709 Z( h0/4.51
9
zcxo145)
Fig. 6. Depth-concentration profiles of Si, C and 0 based on the experimental angular distributions of XPS intensities for a 0.6 pm thick pyrolytically prepared SiOp film.
& (about 20 A) thick, II, being the mean free path of photoelectrons knocked out from the 01s core level. It should be noted that the results are substantially different from those in ref. 3. At &32” the ratio of Si2p and 01s line intensities is 1.14 (with allowance for the ratio of corresponding core level photoionization cross-sections), and in ref. 3 the restored profiles for elements Si and 0 have a concentration ratio approximately equal to 1.0. The profiles obtained by us (see Fig. 5) are somewhat different - with ratio Si :0 = 1: 2. A possible
15
b
1 I
60-
I
I
=
I
I
60-
2
2
J 40_ .9 0
c409 v
ao-
o0
’ 1
1
I
I
2
3
4
N 5
o0
6
’ 1
I 2
3 zcx,,
Z(bqi/3)
0
1
2
3
4
Z(hTi
/3)
4
5
5
I
I,
6
/3)
6
Fig. 7. Restored depth-concentration profiles of Ti, C and 0 based on the angular distribution of XPS line intensities for a “TiOz” sample.
;f,~,‘,_,;,.:“~,~ ,=,=, I,r,,,C
~~=,~~~~~~~~~~~~~‘-:-r_: 01234567
0 Z(h,,
/3)
1
2
3
4 Z(XTi
5
6
7
13)
Fig. 8. Restored depth-concentration profiles of Ti, N, C and 0 based on XPS line intensities of a TiN sample held for 500 h in air.
explanation of the latter fact is that for the mean free paths of photoelectrons knocked out from Si2p and 01s levels, Asi:Ao= 1.3: 1, and therefore, the concentration ratio Si : 0 is lower than the ratio of line intensities Si2p : 01s. In addition, it should be mentioned that the concentration of 0 rises sharply from about 0% to about 60% at somewhat greater depths (by about 1,/3) than those depths where the growth of Si is observed. Needless to say, judging only by one set of experimental data, it is hardly possible to make definite conclusions, but
16
apparently there are oxygen-free compounds of Si with F at the film/substrate interface. Figures 6-8 give the results of restoration based on experimental data for “SiOZ”, “ Ti02” and TiN samples after holding them for 500 h in air. The angular distribution of XPS lines were obtained in our laboratory by T. Reich on an ADES-400 spectrometer for SiOZ and for other samples by M.V. Kuznetsov on an ESCALAB VG spectrometer in Professor V.V. Gubanov’s laboratory in the Institute of Inorganic Chemistry of the Ural branch of the Academy of Sciences of the U.S.S.R. The experimental data presented in Tables 2-4 serve mainly to illustrate the restoration method, and therefore we will dwell neither on measuring aspects nor on detailed discussion of the character of obtained profiles. Parameters necessary for calculations are as follows: (1) photoionization cross-sections taken from ref. 15, and the ratio of the mean free paths estimated from eqn. [ 161 hi/Ak= (Ei/Ek)0.7, where Ei and Ek are the kinetic energies of photoelectrons. The angle between the directions of incident X-radiation and of photoelectron emission was equal to 48”. The ratio of atomic volumes Vi: V, was estimated from the corresponding values of covalent radii (here, it must be stressed that this ratio does not affect the results of restoration to a great extent). Figure 6 shows the profiles of elements Si, 0 and C for a thick SiO, oxide TABLE 2 Relative angular distributions of Si2p, Cls and 01s line intensities for a SO, film Take-off angle 0 ( ’
)
0 5 30 45 60
lSiZpllOls
LlIOl.
0.208 0.205 0.201 0.200 0.205
0.147 0.164 0.206 0.265 0.364
TABLE 3 Relative angular distributions of Ols, Cls and Ti2p line intensities for a “TiOz” sample Take-off angle 19(O )
101s11Ti2p
klsIJr12p
15 25 35 45 55
1.092 1.139 1.159 1.187 1.249
0.085 0.097 0.100 0.110 0.123
17 TABLE 4
Relative angular distributions of XPS line intensities for a TIN sample after holding it for 500 h in air Take-off angle 0 15 25 35 45 55
( a )
IOlSlITiZP
zNlslzTi2p
ICdKTi2p
0.483 0.511
0.176
0.097
0.167 0.166
0.106 0.115 0.140
0.560 0.604 0.677
0.163 0.128
0.183
film introduced into the spectrometer from air without preliminary cleaning. As expected, there is a layer of hydrocarbons on the surface under which the sample consists of 0 and Si with a proportion of 2.12 0.2. It should be noted that for the ratio of Si2p and 01s line intensities for 8= 45 o (see Table 2)) eqn. ( 17) gives an 0 : Si proportion equal to about 3. Thus, as shown in the preceding section for model profiles, the use of expression (17) for determining the composition of samples by XPS data may lead to considerable errors because of the neglect of real element concentration gradients in a sample. This conclusion also remains valid for Auger electron spectroscopy, practical application of which for chemical analysis also uses an expression similar to eqn. (17) (with factors of element sensitivity instead of photoionization cross-sections o and photoelectron mean free paths 2). The concentration of C atoms in Fig. 6 does not attain 100 at.% even in the &/4 thick uppermost layer. Surface coating is observed only for such increase of the number of sublayers when the thickness of each sublayer does not exceed about 0.1 2,. A not very sharp decrease of concentration of element C with depth is apparently due to the roughness of the sample surface which leads to depth-inhomogeneity of the hydrocarbon contamination layer. Figure 7 presents the distribution of elements 0, C and Ti for a TiOz sample according to the experimental data given in Table 3. For depths more than about 1.5 Lo the ratio Ti: 0 x 1: 2, while nearer to the surface the concentration of C atoms increases and that of Ti atoms (in oxidated form) decreases. It should be noted that when 0= 45 ’ and the ratio of photoionization cross-sections ools : aTizp = 0.36 [ 151, eqn. (17) gives the ratio of elemental concentration O:Ti=3.3: 1. Finally, Fig. 8 shows the restored profiles of elements 0, C, N and Ti for a TiN sample which has been held for 500 h in air. The strongly nonlmonotonous distribution of element Ti with depth deserves particular attention; this is probably due to the formation of titanium oxide through the extraction of Ti atoms from the nitride layer.
18
REFERENCES
8
9
10 11 12 13 14 15 16
M. Pijolat and G. Hollinger, Surf. Sci., 105 (1981) 114. T.D. Bussing and P.H. Holloway, J. Vat. Sci. Technol. A, 3 (1985) 1973. R.S. Yih and B.D. Ratner, J. Electron Spectrosc. Relat. Phenom., 43 (1987) 61. O.A. Baschenko and V.I. Nefedov, Poverkhn., 7 (1987) 75 (in Russian). O.A. Baschenko and V.I. Nefedov, Poverkhn., 10 (1987) 99 (in Russian). V.I. Nefedov and O.A. Baschenko, J. Electron Spectrosc. Relat. Phenom., 47 (1988) 1. A.N. Tikhonov and V.Ya. Arsenin, Solutions to Ill-Posed Problems, V.H. Winston, Washington DC, 1977. A.N. Tikhonov, A.V. Goncharskii, V.V. Stepanov and A.G. Yagola, Regulyariziruyushchie Algoritmy i Apriornaya Informatsiya (Regularizing Algorithms and A Priori Information), Nauka, Moscow, 1983 (in Russian). O.A. Baschenko and V.I. Nefedov, J. Electron Spectrosc. Relat. Phenom., 17 (1979) 405. O.A. Baschenko and V.I. Nefedov, J. Electron Spectrosc. Relat. Phenom., 21 (1980) 153. O.A. Baschenko and V.I. Nefedov, Metallofizika, 4 (1982) 61 (in Russian). O.A. Baschenko and V.I. Nefedov, Poverkhn., 2 (1982) 87 (in Russian). O.A. Baschenko, G.V. Machavariani and V.I. Nefedov, J. Electron Spectrosc. Relat. Phenom., 34 (1984) 304. V.I. Nefedov, Surf. Interface Anal., 3 (1981) 72. I.M. Band, Yu.1. Kharitonov and M.B. Trzhaskovskaya, Atomic Data Nucl. Data Tables, 23 (1979) 443. D.R. Penn, J. Electron Spectrosc. Relat. Phenom., 9 (1976) 29.