- Email: [email protected]
Estimation of backscattering factor for low atomic number elements and their alloys
Surface Science 0 North-Holland
74 (1978) 621-635 Publishing Company
ESTIMATION OF BACKSCATTERING
FACTOR FOR LOW ATOMIC
NUMBER ELEMENTS AND THEIR ALLOYS
Aleksander JABLOfiSKI Department of Catalysis on Metals, Institute of Physical Chemistry, Polish Academy of Sciences, 44152 Kasprzaka Street, 01-224 Warsaw, Poland Received
14 July
1977; manuscript
received
in final form 28 November
1977
The single elastic scattering theory of Everhart was adapted to determine the angular and energy distribution of electrons backscattered from low atomic number solid materials (Z < 40-45). This distribution and the classical ionization cross-section expression of Gryzinski were used in the calculations of the backscattering factor, r, in quantitative AES analysis. The values of r were found to be in reasonable agreement with the results of the Monte Carlo calculations and the existing experimental data. Everhart’s theory was extended for the case of mixtures, and that made possible the determination of the backscattering factor for binary alloys. It was found that neglecting the concentration dependence of r in quantitative AES analysis apparently enriches the surface with the component having the lower atomic number. The experimental data on surface composition of binary alloys measured by AES are discussed in view of the presented theory.
1. Introduction Quantitative measurements by means of Auger electron spectroscopy require taking into account the additional Auger yield due to ionizations by scattered electrons. This effect is accounted for by introducing the so-called backscattering factor into the formula describing the total Auger current [ 11. Estimation of the backscattering factor requires the knowledge of energy and angular distribution of scattered electrons in the surface region. On the other hand, the experimental data on that distribution are rather scarce. In 1960, Everhart presented a simple theory of electron scattering in semi-infinite targets [2]. This theory, also called the large single elastic scattering theory [3], is briefly sketched in the next section. It may be applied to target materials having a low atomic number (below 4045) [3,4]. Everhart’s theory gives reasonable agreement with experimental data on reflection coefficients [2] j and energy distribution of backscattered electrons [4,5]. Recently this theory was extended for the case of double layers and supported thin films [5] giving also reasonable agreement with experimental data. In the present paper, the energy and angular distribution of backscattered electrons is derived from Ever621
622
A. Jabloriski / Backscatteting
factor
hart’s theory, and used in calculations of the backscattering factor. The other extension of the theory presented here is its application to binary alloys.
2. The large single elastic scattering theory Everhart’s theory of electron scattering is based on the following assumptions: (1) Electrons passing through the matter undergo a continuous energy loss, being a function of the distance traversed, as a result of inelastic collisions with electrons of the target. Those collisions do not change the direction of electron velocity. (2) An electron changes its direction as a result of scattering by a Coulomb field of a bare nucleus. (3) Electrons scattered through an angle less than 7r/2 are treated as if they are not scattered at all. Those scattered through an angle greater than 7r/2 are leaving the target. Everhart considered the model of a semi-infinite target with an ideally flat surface. The target was bombarded with a beam of monoenergetic electrons at normal incidence. The Thomson-Whiddington law was accepted to express the continuous energy loss of electrons, i.e. the dependence of electron velocity, u, on the distance x, traversed inside the target: IJ4= u”o- cpx = cp(R - x) )
(1)
where u. is the initial electron velocity, c is a constant, p is the target density, and R is the range ofelectronsinthe target material, i.e. the value of x when u = 0. Largeangle scattering into the solid angle da was described by the Rutherford cross-section: da=-
Z2e4
da
4m2u4 sin4(X/2) ’
where x is the angle of deflection of a scattered electron, Z is the atomic number of the target material, and m and e are the electron mass and charge, respectively. Let us consider the layer parallel to the surface having thickness dx and surface area S. The number of scattering centres cw” in the considered volume is equal to dnF = NopS dx/M ,
(3)
where No is the Avogadro number, and M is the atomic mass of the target material. Let us denote by dnb, 0) the number of electrons scattered at depth y =x/R through an angle x = n - 0 into the solid angle da = 2n sin 0 de. Then we have dn(y, 0) =
[no(uY~ldW da ,
(4)
where no(y) is the number of electrons incident of eqs. (l), (2), and (3) into (4) yields
a0flo(Y)
dn(y, e>=--dy
2 l-y
sin 8 c0s4(e/2)
de 9
on a plane at depthy.
Substitution
(9
623
A. Jabloriski / Backscatteringfactor
where a0 = N,,Z2e%lMm2c. On the basis of assumption the following expression [2,5] dn(y, e) = ;
no(l - v)” ‘- ’ dr co;;;,2)
(3), we obtain eventually
(6)
de .
where no is the number of electrons incident on the target surface. The parameter a0 is a function of atomic number, fundamental constants, and the constant c of the Thomson-Whiddington law. The constant c has been estimated by Terrill [6] to be equal to 5.05 X 1O42 cm6/g sec4. Then the a0 parameter can be expressed as a0 = 0.012 Z since Z/M is approximately constant for all elements. However, Everhart found that the theory is in good agreement with experimental data on the reflection coefficient when a0 = 0.045 Z [2]. The discrepancy was ascribed to the neglect of electron deviations smaller than 7~/2 [3]. The energy and angular distribution of scattered electrons leaving the solid dn&‘, 0), may be derived from eq. (6). An electron reflected at an angle B to the surface normal travels the path I = Ry(1 + set f3) inside the solid. The electron energy, E, when leaving the target is determined from the Thomson-Whiddington law written in the form: I = R[l - (E/E,)2], where E, is the initial beam energy. Hence, we obtain the following relation between the parametery and the energy E: Y=
$$j
Introducing dn~(E,e)=--g-
[l - wEp)21 . eq. (7) into eq. (6) we obtain the distribution 4a’noE
a0-1
sine case (1 tc0sej3
P
function dEdB,
(8)
where w = 1 - (E/Ep)2.
3. The backscattering
factor
Let us consider the solid target being bombarded with the monoenergetic electron beam. Let us assume: (1) Auger electrons are emitted isotropically from an atom. (2) The solid surface is ideally flat. (3) The composition of the solid is homogeneous in the plane parallel to the surface. (4) The primary electron current inside the solid, Zp, is constant in the surface region sampled by Auger electrons. Under those assumptions the Auger electron current from the solid, Z,, for the case of the normal incidence of the primary beam is given by [8,9] 1, = ~nj(Ep) (1 - ~+l) Zp Ai2 T C Niriqi , i=1
(9)
624
A. Jablohki
/ Backscattering factor
where a,@n) is the ionization cross-section of the inner level n,l at a primary electron energy E,, (1 - w,,J is the probability that the Auger process follows the ionization, As2 is the part of the solid angle in which Auger electrons are accepted by the analyzer, T is the transparency of the analyzer grid system. The sum is extended over plane atom layers in the surface region. Ni is the number of atoms per unity surface area, qi is the screening factor and ri is the backscattering factor. The latter takes into account ioizations of a given inner level by scattered electrons. It is defined as a ratio of the total number of ionizations in the i-th atom layer to the number of ionizations caused by primary electrons. Thus, if Z,(E, 0) denotes the energy and angular distribution of the scattered electron current in the i-th layer, the backscattering factor is given by [8-lo]
where E,l is the ionization energy of the inner level. Let us assume further that the energy and angular distribution of scattered electrons is the same in every atom layer in the surface region and is equal to that distribution of electrons leaving the target, i.e. li(E, f3) = I(E, 0) = el dn&,
Q/de dE1 .
(11)
Then, the backscattering factor is constant in the surface region. Almost all theoretical and semiempirical expressions, determining cross-section of internal electron level, have the following form [ 1 l] o,@)
= (ne4/E$)
Z,, Au)
,
the ionization
(12)
where Z,,, is the number of electrons in the level n, 1, and U = E/Enr is the socalled reduced energy. The function g(U) describes the shape of ionization cross-section dependence on the reduced energy. Gallon [7] introduced an experimental technique allowing the determination of the function g(U). He has found that in the case of silicon and silver the shape of the function g(U) is close to that given by the classical expression of Gryzinski [ 121
g(U)= i(U&)3’2
11 +f (I - $)ln[2.7
+ (U-
1)‘12]) .
This expression was employed in calculations in the present work. Substitution of eqs. (1 l), (12) and (8) into eq. (lo), remembering and introduction of a new variable, U, gives eventually
uP
s UdO r=l ‘L$g(Up,, 4a0
@(a’, w>du,
that I, = nee,
(13)
A. Jablohski / Backscattering factor
625
where Up = BP/E,,,, w = 1 - (E/E&’ = 1 - (U/UrJ2 , and
@(a’,w) =
j’2(1- G
w)=‘-l +“‘c”,fe), (1
d0
0 =
w(fz0
+
1) - 1 - (1 - w/2)“O[w(a0/2 t 1) - l] w2&r”
+ 1)
(14)
Thus, the backscattering factor is a function of the reduced energy of the incident beam, Up, and atomic number 2 (through the parameter a’). The same result was also found by Bishop and Riviere [ 11. The integrand in eq. (13) has a rather complicated form, so that the calculation of the backscattering factor from this formula involves a numerical approach. The values of r reported in the next sections were calculated by the Gaussian quadrature method. The energy distribution of scattered electrons following from Everhart’s theory does not agree with the distribution determined experimentally for electron energies close to the incident beam energy [S]. This deviation was ascribed to the fact that the continuous energy loss law is not valid in the surface region. Also, Everhart’s theory does not account for the presence of the elastic peak. Gerlach and DuCharme [lo] found that the elastic peak makes a small contribution to the backscattered durrent when the difference E, -E,,] is considerably greater than the width of the elastic peak. This condition is usually met in the case of Auger electron spectroscopy. For the same reason, we may expect that the observed discrepancy in the high electron energy region also has an insignificant effect on the results of calculations.
4. Results of calculations Bishop and Riviere [1,13] calculated the reduced energy dependence of the backscattering factor for carbon, aluminu, titanium, and copper using the Monte Carlo method. They found that the value of r increases with the atomic number of the target material and also with incident beam energy. The same conclusion results from the present work, and also from experimental data of Smith and Gallon [ 141. In fig. 1 the reduced energy dependence of r obtained by Bishop and Riviere [l] is compared with that calculated from eq. (13). The curves have the same shape and are in good agreement. The deviations between them do not exceed 15%, and usually are below 10%. This agreement is remarkable in view of the scattering model applied in the present work, which is much simpler than that used by Bishop [ 131. Figs. 2, 3 and 4 compare the backscattering factor calculated for carbon, silicon, and selenium (solid line) with experimental data of Smith and Gallon [14]. The dashed line in fig. 2 corresponds to energy dependence of r as calculated by Bishop and Riviere [l] for carbon. An excellent agreement between theory and experiment
A. Ja~lo~~k~ / Backscatte~ng
626
factor
Atomic number
2.2 L d $j 28 F ‘C ai = * 24 P 5 $ 1.0
0
0.2
0.4
0.6 I/
0.6
1.0
Up
Fig. 1. Reduced energy dependence of the backscattering factor for carbon, aluminum, titanium, and copper. Solid line: present calculations; dashed line: Monte Carlo calculations of Bishop and Riviire [ 1] .
can be seen in the case of carbon and silicon. The triangles in fig. 3 correspond to experimental values of products qr for sihcon obtained by Meyer and Vrakking [8], where q is the screening factor for the outermost atom layer. These values are considerably smaller than the data of Smith and Gallon [ 141 and the theory prediction. This result is to be expected since the screening factor, i.e. the probab~ity that an Auger electron moving towards the analyzer will not be attenuated, is always smaller than unity. In the case of selenium (fig. 4), the expe~ment~ values of the
4 Reduced
6 energy,
8
10
Up
Fig. 2. Comparison of experimental and theoretical values of the backscattering factor for carbon. Solid line: present calculations; dashed line: Monte Carlo calculations of Bishop and Riviere [ 11. Experimental values: Smith and Gallon [ 141.
621
A. Jablohki / Backscatteringfactor
I.8 I
L
1.6 -
b’ s c .f
-
1.4
z s 5 1.2 $
8
2
0
Redtced
en&
U,
10
Fig. 3. Comparison of experimental and theoretical values of the backscattering factor for silicon. Solid line: present calculations. Experimental values: (o) Smith and Gallon [ 141; (*I qr product from Meyer and Vrakking [ 8).
factor are much smaller than the values resulting from theory; the difference reaches 30%. This discrepancy may be due to the relatively large atomic number of selenium (Zse = 34). The number is in the transition range between the applicability of Everhart’s theory and the diffusion theory of electron scattering
backscattering
[3j.
2.4 2.2 ;_ 2.0 0 j 1.8 .f 1.6 0) 5 1.4
0 0 0 0
B fiJ 22 I I.01 0
0
f
’
’
2
’
’
’
4 Reduced
6 energy,
j
1
J
8
IO
Up
Fig. 4. Comparison of experimental and theoretical values of the backscattering factor for germanium. Solid line: present calculations. Experimental values: Smith and Gallon [14].
628
A. Jabloliski / Backscattering factor
5. The large single elastic scattering theory for binary alloys The extension of Everhart’s theory for mixtures is based on the assumption that the Thomson-Whiddington law is also valid for alloys; in that case p denotes the alloy density. Consideration will be limited to the case of binary alloys only. Let us denote components by A and B. As in section 3, we determine the numbers of scattering centres, WA and MB, contained in the layer having surface area S and thickness dx, xANOPs
dNl=
MAXA
h
*s
+Mgn
=
xBNOpS
dX
(15)
MAXA + MBxB ’
B
’
where MA, MB are the atomic weights, and xA, xB are the atom fractions of components A and B, respectively. For the case of an alloy the distribution function dn(y, f3) is defined as
where duA and duB denote the Rutherford scattering cross-section for scattering centres A and B respectively. Substitution of the Rutherford scattering cross-sections, and eqs. (1) and (1.5) into eq. (16) gives sin 0 __ de , cos4 e/2
a no(Y) dn(y, 19)= 6 -dy 2 1 -y
(17)
where ZAnNoe4
MAXAZA ‘=MTA
MB%~B
‘MAxA+MBXB
tMBxB=
~Blh$,e4
M,rn’c
.
(18)
Eq. (17) is identical with eq. (5) except for the definition of parameter a. Repeating the derivation from sections 2 and 3 we would obtain the same expressions for functions dn(y, 8), da&y, 0) and @(a, w) as in the case of pure elements, i.e. we would obtain the formulas (6), (8), and (14) having a0 replaced by a. Let us, as Everhart suggested [2], instead of the constant ZAnNoe4 M,V12C
s-e
ZBrNoe4 M,m2c
= 0.012
)
introduce the constant 0.045 in eq. (18). This procedure, as it was mentioned previously, leads to agreement with the experimental values of the reflection coefficient. Hence, a = 0.045
MALAYA ~ MA~A
+ MB~B~B + MB~B
1.
(19)
Thus, the backscattering factor for alloys may be calculated from eq. (13), in which u” is substituted by eq. (19).
.4. Jablohki / Backscatteringfactor
629
Al
5#
1.4 -
Cu
u=1.5
ii/ m” 1.21.0 0
0.2
0.4
0.6
0.0
1.0 0
0.2
Alloy composition,
0.4
0.6
0.8
1.0
xA
Fig. 5. The dependence of the backscattering factor on alloy composition and reduced energy for NiPd and AlCu ahoy systems. xA denotes the atom fraction of component having larger atomic number.
Figs. 5 and 6 show the dependence of the backscattering factor for NiPd, AlCu, CrFe, and CuNi alloys on composition and reduced energy. It can be seen that the values of r monotonically increase from the element having the smaller atomic number to the element with the larger one. In fact, such a behaviour of r was assumed for the AuPd alloy in discussion of its surface composition [ 1.51. In the case of CrFe and CuNi alloys, the variation of the backscattering factor with composition is negligible due to a small difference in atomic numbers.
0
0.2
0.4
0.6
0.8 20
0
Alloy composition,
0.2
0.4
0.6
0.0
1.0
xA
Fig. 6. The dependence of the backscattering factor on alloy composition and reduced energy for CrFe and CuNi alloy systems. xA denotes the atom fraction of component having larger atomic number.
630
A. Jablohski / Backscattering factor
6. The effect of concentration dependence of the hackscattering factor on results of quantitative AES analysis The most frequently used experimental technique of quantitative AES analysis is the external calibration method. This method involves the comparison of the Auger signal intensity from an alloy with that from the pure metal. As a results of measurements, we obtain the ratio t =1,/e, where the superscript zero refers to the pure component. Let us denote by xi the average concentration of component A in the surface region sampled by Auger electrons. Let this region extend over m atom layers. Then eq. (9) may be written in the form: 1, = km:, where
k = ~nl(Ep) (1 - w,,) Ip Aa TN C qi y i=1
and N is the total number of atoms in the area unity of atom layer. Thus, k is a constant depending on properties of the solid and instrumental parameters. It is the usual procedure to assume that the parameters k and I have the same value for an alloy as for the pure component. Then the ratio of Auger peak heights, &, is approximately equal to the surface concentration of component A. In the present paper, we assume that only the parameter k is independent of the alloy composition, and is equal to parameter k” of the pure metal. This assumption seems to be reasonable since the quantities u,~, (1 - onI), N, and qi have a weak dependence on the alloy composition [16]. Then the ratio of Auger electron intensities, t;A, is equal to (r/r’).-&. Let us suppose that component B has a lower atomic number than component A. Since the backscattering factor is a monotonic function of xA we have always rA/ri
< 1
and rBlrE > 1 ,
so that L4
and
ln>xb.
Thus, neglecting the difference in backscattering factors leads to an apparent enrichment of the surface region with the component having lower atomic number no matter which Auger transition is taken into account. Now we will discuss in detail binary alloys of elements having relatively low atomic numbers, which were the subject of AES quantitative analysis in the past. Those systems are listed in table 1. 6.1. The AlCu alloy system The system forms a one-phase substitutional solid solution in the concentration range O-20% Al. Let us assume that the alloy has the same surface and bulk compositions, i.e. xsAl =xAl. Then, the ratio gAl = (rAI/&)xsAI can be calculated using
[ 17 ]
Stoddart
[ 181 [ 191
[ 221
et al. [24]
[ 231
et al.
Mathieu, Landolt
Laygraf
Helms [ 201 Helms, Yu [21]
Nakayama et al. Takasu, Shimizu
Ferrante
Authors
Table 1 Binary alloy systems
NiPd
NiPd
CrFe
CuNi
CuNi
AlCu
Alloy
of low atomic
number
studied
M2,3M4,sM4,5 L3M4,sMw N2,3N4,sN4,s MsNwN4,s
L3M4,sM4,s MsN4,sNw
LJM,,~MzJ L3Mz~M4.s L3M2,3M4,s L3M4,sMw
MrM4,sM4,s MrM4,sM4,s
L3MwM4,s L3MwMw
L3MwM2,3 MwMwM4,s
Auger transition
elements
Ni Ni Pd Pd
Ni Pd
Cr Cr Fe Fe
cu Ni
cu Ni
Al cu
61 848 43 3261330
848 3261330
489 529 651 703
105 102
920 716
68 58160
W)
(eV)
708.1
68.1 854.7 51.1 334.7
334.7
854.7
2500
5000
1000
2500
111.8 574.5 574.5 708.1
119.8
O-3000
2200 2000
(eV)
Energy of primary beam
931.1 854.7
73.6
72.72
Ionization energy [26]
AES
Auger electron energy [ 25 ]
by quantitative
B
R
G % j* z z :i?.
g 3. z?
3
P
A. Jabloliski / Backscattering factor
632
1.0
-7
0.8 /’ O6
0
/’
4 04- C
0
0
I’
0.2
I
/
/’
/‘///
1’
/’
0.4
0.6
Bulk composition,
0.8
I 1.0
x Al
Fig. 7. The dependence of the ratio [Al on bulk composition calculated for AlCu alloy system assuming no surface segregation. Calculations were carried out for L~M~,JM~,s Al Auger transition, and Ep = 3000 eV. Experimental results of Ferrante [17] for 700°C.
eq. (13) with parameter a0 = 0.045 . 13 = 0.585, and with parameter a given by eq. (19). Results of calculations for the Al Auger transition and E, = 3000 eV are shown in fig. 7. Neglecting the difference in the backscattering factors in the case of the external calibration method, i.e. assuming that the ratio .& is equal to the surface concentration of Al, would clearly indicate the Al surface enrichment. Ferrante“ [ 171 studied the AlCu single crystals having composition 1% Al, 5% Al, and 10% Al. Employing the low energy Auger electron transitions LsMa,&Ia,s Al (68 eV) and Ma,sM4,5M~,s Cu (58/60 eV) he has found significant surface enrichment with Al. He applied an experimental technique resembling the internal calibration method, using an extensively sputtered surface as a standard. The procedure should provide a true surface composition since the internal calibration method removes the effect of the backscattering factor on the results of analysis [16]. Those results are also shown in fig. 7. However, the simple model of binary alloy sputtering introduced by Shimizu et al. [27], together with sputtering yields reported by Laegreid and Wehner [28], predicts that the standard surface should be enriched with Al. It is possible then that Ferrante underestimated the extent of the Al surface segregation. 6.2. The CuNi and CrFe alloy systems The backscattering factors of pure copper and nickel have similar values (fig. 6) due to the small difference in atomic numbers (ZcU = 29,Z,i = 28). The ratio r/r0 deviates from unity by not more than 1% in the case of high energy transitions Ni; E, = 2000 eV) and by not more than 1.6% in (LsM,,sM+s Cu and L&2,3”2,3 the case of low energy transitions (M1M4,sM4,s Cu and Ni; E, = 2500 ev). Thus the ratio of backscattering factors may be neglected in the determination of surface composition.
A. Jablotiski / Backscatteringfactor
633
A similar situation is found in the case of CrFe alloy (2 Cr = 24, Zr+ = 26), where the value of I-/T’ does not differ from unity by more than 2%. 6.3. The NiPd alloy system Mathieu and Landolt [23] have found that extensively sputtered NiPd ahoy surfaces are enriched with Ni. This result is also predicted by the model of Shimizu et al. [27]. These authors reported the dependence of the peak height in the derivative spectra, cw(E3/dE versus E, on the alloy composition. Those results after normalization with respect to the corresponding pure metals are presented in fig. 8. They are compared with the concentration dependence of the ratios ENi and ,& calculated for the case with no surface segregation (solid lines). Calculations were performed for Auger transitions employed by Mathieu and Landolt, and for E, = 5000 eV. An apparent surface enrichment with nickel results from the assumptions ENi = Xhi and &r = xSpd. It follows from fig. 8 that a large part of nickel enrichment observed by Mathieu and Landolt may be due to the neglect of the concentration dependence of the backscattering factor. Theoretical considerations predict that the NiPd ahoy surface, being in equilibrium with the bulk, is enriched with palladium [29,30]. The palladium segregation was found experimentally by Stoddart et al. [24] for thin NiPd films using the external calibration method. A‘s previously, let us assume that XNi = Xhi+ The concentration dependence of the ratio [Ni for M,,3M4,5M4,S (61 eV) and LJM4,sM4,5 (848 eV) nickel Auger transitions and for E, = 2500 eV is shown in fig. 9. The experimental points of Stoddart et al., derived from low energy and high energy
ie0.6 Y
d 0.4
Y
0.2 0 0
Q2 Bulk
Q4
Q6
composition,
0.8
20
xNi
Fig. 8. The dependence of the ratios ENi and tpd on bulk composition calculated for NiPd alloy system, assuming no surface segregation. Calculations were carried out for L3M4,5M4,5 Ni Auger transition, M~N~,~NG,s Pd Auger transition, and Ep = 5000 eV. Experimental results of Mathieu and Landolt [23] after normalization with respect to pure metals (A) derived from Ni Auger transition, (0) derived from Pd Auger transition.
634
A. Jabloriski / Backscattering factor
0.6
0
0.2
0.4
0.6
0.8
1.0
Bulk composition, xNi
Fig. 9. The dependence of the ratio [Ni on bulk composition calculated for NiPd alloy system assuming no surface segregation. Calculations were carried out for LsM4,+M4,5 Ni Auger transition (l), M2,3M4,5M4,5 Ni Auger transition (2), and Ep = 2500 eV. Experimental values of Stoddart et al. [24] (0) derived from high energy Auger transition, (a) derived from low energy Auger transition.
nickel Auger transitions, are also shown in that figure. It is seen that taking into account the concentration dependence of the backscattering factor would noticeably increase_ the surface segregation of palladium.
7. Conclusions The angular and energy distribution of electrons backscattered from a solid may be determined by the single elastic scattering theory of Everhart [2]. The knowledge of this distribution together with the classical ionization cross-section expression of Gryzmski [ 121 makes possible the estimation of the backscattering factor for target materials having an atomic number below 4045. Values of r for elements determined in this way are in good agreement with values calculated by Bishop and Riviere using the Monte Carlo method and with experimental data on r available at the moment. The backscattering factor calculated for binary alloys was found to be a monotonic function of composition increasing from the element with the lower atomic number to the element with the larger one. For this reason, neglecting the concentration dependence of the backscattering factor in the external calibration method of quantitative AES apparently enriches the surface with an element having the lower atomic number. This apparent surface enrichment is stronger when the difference in atomic numbers of elements forming an alloy is larger. For CuNi and CrFe alloys, the backscattering effects were found to be negligible; the ratio r/r0 was equal to unity within 2% across the whole concentration range. In the case of AlCu
A. Jablohski / Backscatteringfactor
and NiPd alloys, taking into account the concentration r/r0 markedly affected the surface composition.
635
dependence
of the ratio
The work was carried out within the Research Project 03.10.
References [l] [2] [3] [4] [S] [6] [7] [8] [9]
H.E. Bishop and J.C. Riviere, J. Appl. Phys. 40 (1969) 1740. T.E. Everhart, J. Appl. Phys. 31 (1960) 1483. G.D. Archard, J. Appl. Phys. 32 (1961) 1505. W.S. McAfee, J. Appl. Phys. 47 (1976) 1179. G.J. Iafrate, W.S. McAfee and A. Ballato, J. Vacuum Sci. Technol. 13 (1976) 843. H.M. Terrill, Phys. Rev. 22(1923) 101. T.E. Gallon, J. Phys. D (Appl. Phys.) 5 (1972) 822. F. Meyer and J.J. Vrakking, Surface Sci. 33 (1972) 271. F. Meyer and J.J. Vrakking, Surface Sci. 45 (1974) 409. [lo] R.L. Gerlach, A.R. DuCharme, Surface Sci. 32 (1972) 329. [ll] C.J. Powell, Rev. Mod. Phys. 48 (1976) 33. [ 121 M. Gryzinski, Phys. Rev. Al38 (1965) 336. [13] H.E. Bishop, Brit. J. Appl. Phys. 18 (1967) 703. [14] D.M. Smith and T.E. Gallon, J. Phys. D (Appl. Phys.) 7 (1974) 151. [ 151 A. Jabfonski, S.H. Overbury and G.A. Somorjai, Surface Sci. 65 (19.77) 578. [ 161 R. Bouwman, J.B. van Mechelen and A.A. Holscher, Surface Sci. 57 (1976) 441. [17] J. Ferrante, Acta Met. 19 (1971) 743. [ 181 K. Nakayama, M. Ono and H. Shimizu, J. Vacuum Sci. Technol. 9 (1972) 749. [ 191 Y. Takasu and H. Shimizu, J. Catalysis 29 (1973) 479. [20] CR. Helms, J. Catalysis 36 (1975) 114. [21] C.R. Helms and K.Y. Yu, J. Vacuum Sci. Technol. 12 (1975) 276. [22] C. Leygraf, G. Hultquist, S. Ekelund and J.C. Eriksson, Surface Sci. 46 (1974) 157. [23] H.J. Mathieu and D. Landolt, Surface Sci. 53 (1975) 228. [24] C.T.H. Stoddart, R.L. Moss and D. Pope, Surface Sci. 53 (1975) 241. [25] P.W. Palmberg, G.E. Riach, R.E. Weber and N.C. McDonald, Handbook of Auger Electron Spectroscopy (Physical Electronics Industries, Edina, Minnesota, 1972). [ 261 K.D. Sevier, Low Energy Electron Spectrometry (Wiley-Interscience, New York, 1972) p. 356. [ 271 H. Shimizu, M. Ono and K. Nakayama, Surface Sci. 36 (1973) 817. [28] N. Laegreid, G.K. Wehner, J. Appl. Phys. 32 (1961) 365. 1291 F.L. Williams and D. Nason, Surface Sci. 45 (1974) 377. [30] A. Jabfofiski, Advan. Colloid Interface Sci. 8 (1977) 213.
74 (1978) 621-635 Publishing Company
ESTIMATION OF BACKSCATTERING
FACTOR FOR LOW ATOMIC
NUMBER ELEMENTS AND THEIR ALLOYS
Aleksander JABLOfiSKI Department of Catalysis on Metals, Institute of Physical Chemistry, Polish Academy of Sciences, 44152 Kasprzaka Street, 01-224 Warsaw, Poland Received
14 July
1977; manuscript
received
in final form 28 November
1977
The single elastic scattering theory of Everhart was adapted to determine the angular and energy distribution of electrons backscattered from low atomic number solid materials (Z < 40-45). This distribution and the classical ionization cross-section expression of Gryzinski were used in the calculations of the backscattering factor, r, in quantitative AES analysis. The values of r were found to be in reasonable agreement with the results of the Monte Carlo calculations and the existing experimental data. Everhart’s theory was extended for the case of mixtures, and that made possible the determination of the backscattering factor for binary alloys. It was found that neglecting the concentration dependence of r in quantitative AES analysis apparently enriches the surface with the component having the lower atomic number. The experimental data on surface composition of binary alloys measured by AES are discussed in view of the presented theory.
1. Introduction Quantitative measurements by means of Auger electron spectroscopy require taking into account the additional Auger yield due to ionizations by scattered electrons. This effect is accounted for by introducing the so-called backscattering factor into the formula describing the total Auger current [ 11. Estimation of the backscattering factor requires the knowledge of energy and angular distribution of scattered electrons in the surface region. On the other hand, the experimental data on that distribution are rather scarce. In 1960, Everhart presented a simple theory of electron scattering in semi-infinite targets [2]. This theory, also called the large single elastic scattering theory [3], is briefly sketched in the next section. It may be applied to target materials having a low atomic number (below 4045) [3,4]. Everhart’s theory gives reasonable agreement with experimental data on reflection coefficients [2] j and energy distribution of backscattered electrons [4,5]. Recently this theory was extended for the case of double layers and supported thin films [5] giving also reasonable agreement with experimental data. In the present paper, the energy and angular distribution of backscattered electrons is derived from Ever621
622
A. Jabloriski / Backscatteting
factor
hart’s theory, and used in calculations of the backscattering factor. The other extension of the theory presented here is its application to binary alloys.
2. The large single elastic scattering theory Everhart’s theory of electron scattering is based on the following assumptions: (1) Electrons passing through the matter undergo a continuous energy loss, being a function of the distance traversed, as a result of inelastic collisions with electrons of the target. Those collisions do not change the direction of electron velocity. (2) An electron changes its direction as a result of scattering by a Coulomb field of a bare nucleus. (3) Electrons scattered through an angle less than 7r/2 are treated as if they are not scattered at all. Those scattered through an angle greater than 7r/2 are leaving the target. Everhart considered the model of a semi-infinite target with an ideally flat surface. The target was bombarded with a beam of monoenergetic electrons at normal incidence. The Thomson-Whiddington law was accepted to express the continuous energy loss of electrons, i.e. the dependence of electron velocity, u, on the distance x, traversed inside the target: IJ4= u”o- cpx = cp(R - x) )
(1)
where u. is the initial electron velocity, c is a constant, p is the target density, and R is the range ofelectronsinthe target material, i.e. the value of x when u = 0. Largeangle scattering into the solid angle da was described by the Rutherford cross-section: da=-
Z2e4
da
4m2u4 sin4(X/2) ’
where x is the angle of deflection of a scattered electron, Z is the atomic number of the target material, and m and e are the electron mass and charge, respectively. Let us consider the layer parallel to the surface having thickness dx and surface area S. The number of scattering centres cw” in the considered volume is equal to dnF = NopS dx/M ,
(3)
where No is the Avogadro number, and M is the atomic mass of the target material. Let us denote by dnb, 0) the number of electrons scattered at depth y =x/R through an angle x = n - 0 into the solid angle da = 2n sin 0 de. Then we have dn(y, 0) =
[no(uY~ldW da ,
(4)
where no(y) is the number of electrons incident of eqs. (l), (2), and (3) into (4) yields
a0flo(Y)
dn(y, e>=--dy
2 l-y
sin 8 c0s4(e/2)
de 9
on a plane at depthy.
Substitution
(9
623
A. Jabloriski / Backscatteringfactor
where a0 = N,,Z2e%lMm2c. On the basis of assumption the following expression [2,5] dn(y, e) = ;
no(l - v)” ‘- ’ dr co;;;,2)
(3), we obtain eventually
(6)
de .
where no is the number of electrons incident on the target surface. The parameter a0 is a function of atomic number, fundamental constants, and the constant c of the Thomson-Whiddington law. The constant c has been estimated by Terrill [6] to be equal to 5.05 X 1O42 cm6/g sec4. Then the a0 parameter can be expressed as a0 = 0.012 Z since Z/M is approximately constant for all elements. However, Everhart found that the theory is in good agreement with experimental data on the reflection coefficient when a0 = 0.045 Z [2]. The discrepancy was ascribed to the neglect of electron deviations smaller than 7~/2 [3]. The energy and angular distribution of scattered electrons leaving the solid dn&‘, 0), may be derived from eq. (6). An electron reflected at an angle B to the surface normal travels the path I = Ry(1 + set f3) inside the solid. The electron energy, E, when leaving the target is determined from the Thomson-Whiddington law written in the form: I = R[l - (E/E,)2], where E, is the initial beam energy. Hence, we obtain the following relation between the parametery and the energy E: Y=
$$j
Introducing dn~(E,e)=--g-
[l - wEp)21 . eq. (7) into eq. (6) we obtain the distribution 4a’noE
a0-1
sine case (1 tc0sej3
P
function dEdB,
(8)
where w = 1 - (E/Ep)2.
3. The backscattering
factor
Let us consider the solid target being bombarded with the monoenergetic electron beam. Let us assume: (1) Auger electrons are emitted isotropically from an atom. (2) The solid surface is ideally flat. (3) The composition of the solid is homogeneous in the plane parallel to the surface. (4) The primary electron current inside the solid, Zp, is constant in the surface region sampled by Auger electrons. Under those assumptions the Auger electron current from the solid, Z,, for the case of the normal incidence of the primary beam is given by [8,9] 1, = ~nj(Ep) (1 - ~+l) Zp Ai2 T C Niriqi , i=1
(9)
624
A. Jablohki
/ Backscattering factor
where a,@n) is the ionization cross-section of the inner level n,l at a primary electron energy E,, (1 - w,,J is the probability that the Auger process follows the ionization, As2 is the part of the solid angle in which Auger electrons are accepted by the analyzer, T is the transparency of the analyzer grid system. The sum is extended over plane atom layers in the surface region. Ni is the number of atoms per unity surface area, qi is the screening factor and ri is the backscattering factor. The latter takes into account ioizations of a given inner level by scattered electrons. It is defined as a ratio of the total number of ionizations in the i-th atom layer to the number of ionizations caused by primary electrons. Thus, if Z,(E, 0) denotes the energy and angular distribution of the scattered electron current in the i-th layer, the backscattering factor is given by [8-lo]
where E,l is the ionization energy of the inner level. Let us assume further that the energy and angular distribution of scattered electrons is the same in every atom layer in the surface region and is equal to that distribution of electrons leaving the target, i.e. li(E, f3) = I(E, 0) = el dn&,
Q/de dE1 .
(11)
Then, the backscattering factor is constant in the surface region. Almost all theoretical and semiempirical expressions, determining cross-section of internal electron level, have the following form [ 1 l] o,@)
= (ne4/E$)
Z,, Au)
,
the ionization
(12)
where Z,,, is the number of electrons in the level n, 1, and U = E/Enr is the socalled reduced energy. The function g(U) describes the shape of ionization cross-section dependence on the reduced energy. Gallon [7] introduced an experimental technique allowing the determination of the function g(U). He has found that in the case of silicon and silver the shape of the function g(U) is close to that given by the classical expression of Gryzinski [ 121
g(U)= i(U&)3’2
11 +f (I - $)ln[2.7
+ (U-
1)‘12]) .
This expression was employed in calculations in the present work. Substitution of eqs. (1 l), (12) and (8) into eq. (lo), remembering and introduction of a new variable, U, gives eventually
uP
s UdO r=l ‘L$g(Up,, 4a0
@(a’, w>du,
that I, = nee,
(13)
A. Jablohski / Backscattering factor
625
where Up = BP/E,,,, w = 1 - (E/E&’ = 1 - (U/UrJ2 , and
@(a’,w) =
j’2(1- G
w)=‘-l +“‘c”,fe), (1
d0
0 =
w(fz0
+
1) - 1 - (1 - w/2)“O[w(a0/2 t 1) - l] w2&r”
+ 1)
(14)
Thus, the backscattering factor is a function of the reduced energy of the incident beam, Up, and atomic number 2 (through the parameter a’). The same result was also found by Bishop and Riviere [ 11. The integrand in eq. (13) has a rather complicated form, so that the calculation of the backscattering factor from this formula involves a numerical approach. The values of r reported in the next sections were calculated by the Gaussian quadrature method. The energy distribution of scattered electrons following from Everhart’s theory does not agree with the distribution determined experimentally for electron energies close to the incident beam energy [S]. This deviation was ascribed to the fact that the continuous energy loss law is not valid in the surface region. Also, Everhart’s theory does not account for the presence of the elastic peak. Gerlach and DuCharme [lo] found that the elastic peak makes a small contribution to the backscattered durrent when the difference E, -E,,] is considerably greater than the width of the elastic peak. This condition is usually met in the case of Auger electron spectroscopy. For the same reason, we may expect that the observed discrepancy in the high electron energy region also has an insignificant effect on the results of calculations.
4. Results of calculations Bishop and Riviere [1,13] calculated the reduced energy dependence of the backscattering factor for carbon, aluminu, titanium, and copper using the Monte Carlo method. They found that the value of r increases with the atomic number of the target material and also with incident beam energy. The same conclusion results from the present work, and also from experimental data of Smith and Gallon [ 141. In fig. 1 the reduced energy dependence of r obtained by Bishop and Riviere [l] is compared with that calculated from eq. (13). The curves have the same shape and are in good agreement. The deviations between them do not exceed 15%, and usually are below 10%. This agreement is remarkable in view of the scattering model applied in the present work, which is much simpler than that used by Bishop [ 131. Figs. 2, 3 and 4 compare the backscattering factor calculated for carbon, silicon, and selenium (solid line) with experimental data of Smith and Gallon [14]. The dashed line in fig. 2 corresponds to energy dependence of r as calculated by Bishop and Riviere [l] for carbon. An excellent agreement between theory and experiment
A. Ja~lo~~k~ / Backscatte~ng
626
factor
Atomic number
2.2 L d $j 28 F ‘C ai = * 24 P 5 $ 1.0
0
0.2
0.4
0.6 I/
0.6
1.0
Up
Fig. 1. Reduced energy dependence of the backscattering factor for carbon, aluminum, titanium, and copper. Solid line: present calculations; dashed line: Monte Carlo calculations of Bishop and Riviire [ 1] .
can be seen in the case of carbon and silicon. The triangles in fig. 3 correspond to experimental values of products qr for sihcon obtained by Meyer and Vrakking [8], where q is the screening factor for the outermost atom layer. These values are considerably smaller than the data of Smith and Gallon [ 141 and the theory prediction. This result is to be expected since the screening factor, i.e. the probab~ity that an Auger electron moving towards the analyzer will not be attenuated, is always smaller than unity. In the case of selenium (fig. 4), the expe~ment~ values of the
4 Reduced
6 energy,
8
10
Up
Fig. 2. Comparison of experimental and theoretical values of the backscattering factor for carbon. Solid line: present calculations; dashed line: Monte Carlo calculations of Bishop and Riviere [ 11. Experimental values: Smith and Gallon [ 141.
621
A. Jablohki / Backscatteringfactor
I.8 I
L
1.6 -
b’ s c .f
-
1.4
z s 5 1.2 $
8
2
0
Redtced
en&
U,
10
Fig. 3. Comparison of experimental and theoretical values of the backscattering factor for silicon. Solid line: present calculations. Experimental values: (o) Smith and Gallon [ 141; (*I qr product from Meyer and Vrakking [ 8).
factor are much smaller than the values resulting from theory; the difference reaches 30%. This discrepancy may be due to the relatively large atomic number of selenium (Zse = 34). The number is in the transition range between the applicability of Everhart’s theory and the diffusion theory of electron scattering
backscattering
[3j.
2.4 2.2 ;_ 2.0 0 j 1.8 .f 1.6 0) 5 1.4
0 0 0 0
B fiJ 22 I I.01 0
0
f
’
’
2
’
’
’
4 Reduced
6 energy,
j
1
J
8
IO
Up
Fig. 4. Comparison of experimental and theoretical values of the backscattering factor for germanium. Solid line: present calculations. Experimental values: Smith and Gallon [14].
628
A. Jabloliski / Backscattering factor
5. The large single elastic scattering theory for binary alloys The extension of Everhart’s theory for mixtures is based on the assumption that the Thomson-Whiddington law is also valid for alloys; in that case p denotes the alloy density. Consideration will be limited to the case of binary alloys only. Let us denote components by A and B. As in section 3, we determine the numbers of scattering centres, WA and MB, contained in the layer having surface area S and thickness dx, xANOPs
dNl=
MAXA
h
*s
+Mgn
=
xBNOpS
dX
(15)
MAXA + MBxB ’
B
’
where MA, MB are the atomic weights, and xA, xB are the atom fractions of components A and B, respectively. For the case of an alloy the distribution function dn(y, f3) is defined as
where duA and duB denote the Rutherford scattering cross-section for scattering centres A and B respectively. Substitution of the Rutherford scattering cross-sections, and eqs. (1) and (1.5) into eq. (16) gives sin 0 __ de , cos4 e/2
a no(Y) dn(y, 19)= 6 -dy 2 1 -y
(17)
where ZAnNoe4
MAXAZA ‘=MTA
MB%~B
‘MAxA+MBXB
tMBxB=
~Blh$,e4
M,rn’c
.
(18)
Eq. (17) is identical with eq. (5) except for the definition of parameter a. Repeating the derivation from sections 2 and 3 we would obtain the same expressions for functions dn(y, 8), da&y, 0) and @(a, w) as in the case of pure elements, i.e. we would obtain the formulas (6), (8), and (14) having a0 replaced by a. Let us, as Everhart suggested [2], instead of the constant ZAnNoe4 M,V12C
s-e
ZBrNoe4 M,m2c
= 0.012
)
introduce the constant 0.045 in eq. (18). This procedure, as it was mentioned previously, leads to agreement with the experimental values of the reflection coefficient. Hence, a = 0.045
MALAYA ~ MA~A
+ MB~B~B + MB~B
1.
(19)
Thus, the backscattering factor for alloys may be calculated from eq. (13), in which u” is substituted by eq. (19).
.4. Jablohki / Backscatteringfactor
629
Al
5#
1.4 -
Cu
u=1.5
ii/ m” 1.21.0 0
0.2
0.4
0.6
0.0
1.0 0
0.2
Alloy composition,
0.4
0.6
0.8
1.0
xA
Fig. 5. The dependence of the backscattering factor on alloy composition and reduced energy for NiPd and AlCu ahoy systems. xA denotes the atom fraction of component having larger atomic number.
Figs. 5 and 6 show the dependence of the backscattering factor for NiPd, AlCu, CrFe, and CuNi alloys on composition and reduced energy. It can be seen that the values of r monotonically increase from the element having the smaller atomic number to the element with the larger one. In fact, such a behaviour of r was assumed for the AuPd alloy in discussion of its surface composition [ 1.51. In the case of CrFe and CuNi alloys, the variation of the backscattering factor with composition is negligible due to a small difference in atomic numbers.
0
0.2
0.4
0.6
0.8 20
0
Alloy composition,
0.2
0.4
0.6
0.0
1.0
xA
Fig. 6. The dependence of the backscattering factor on alloy composition and reduced energy for CrFe and CuNi alloy systems. xA denotes the atom fraction of component having larger atomic number.
630
A. Jablohski / Backscattering factor
6. The effect of concentration dependence of the hackscattering factor on results of quantitative AES analysis The most frequently used experimental technique of quantitative AES analysis is the external calibration method. This method involves the comparison of the Auger signal intensity from an alloy with that from the pure metal. As a results of measurements, we obtain the ratio t =1,/e, where the superscript zero refers to the pure component. Let us denote by xi the average concentration of component A in the surface region sampled by Auger electrons. Let this region extend over m atom layers. Then eq. (9) may be written in the form: 1, = km:, where
k = ~nl(Ep) (1 - w,,) Ip Aa TN C qi y i=1
and N is the total number of atoms in the area unity of atom layer. Thus, k is a constant depending on properties of the solid and instrumental parameters. It is the usual procedure to assume that the parameters k and I have the same value for an alloy as for the pure component. Then the ratio of Auger peak heights, &, is approximately equal to the surface concentration of component A. In the present paper, we assume that only the parameter k is independent of the alloy composition, and is equal to parameter k” of the pure metal. This assumption seems to be reasonable since the quantities u,~, (1 - onI), N, and qi have a weak dependence on the alloy composition [16]. Then the ratio of Auger electron intensities, t;A, is equal to (r/r’).-&. Let us suppose that component B has a lower atomic number than component A. Since the backscattering factor is a monotonic function of xA we have always rA/ri
< 1
and rBlrE > 1 ,
so that L4
and
ln>xb.
Thus, neglecting the difference in backscattering factors leads to an apparent enrichment of the surface region with the component having lower atomic number no matter which Auger transition is taken into account. Now we will discuss in detail binary alloys of elements having relatively low atomic numbers, which were the subject of AES quantitative analysis in the past. Those systems are listed in table 1. 6.1. The AlCu alloy system The system forms a one-phase substitutional solid solution in the concentration range O-20% Al. Let us assume that the alloy has the same surface and bulk compositions, i.e. xsAl =xAl. Then, the ratio gAl = (rAI/&)xsAI can be calculated using
[ 17 ]
Stoddart
[ 181 [ 191
[ 221
et al. [24]
[ 231
et al.
Mathieu, Landolt
Laygraf
Helms [ 201 Helms, Yu [21]
Nakayama et al. Takasu, Shimizu
Ferrante
Authors
Table 1 Binary alloy systems
NiPd
NiPd
CrFe
CuNi
CuNi
AlCu
Alloy
of low atomic
number
studied
M2,3M4,sM4,5 L3M4,sMw N2,3N4,sN4,s MsNwN4,s
L3M4,sM4,s MsN4,sNw
LJM,,~MzJ L3Mz~M4.s L3M2,3M4,s L3M4,sMw
MrM4,sM4,s MrM4,sM4,s
L3MwM4,s L3MwMw
L3MwM2,3 MwMwM4,s
Auger transition
elements
Ni Ni Pd Pd
Ni Pd
Cr Cr Fe Fe
cu Ni
cu Ni
Al cu
61 848 43 3261330
848 3261330
489 529 651 703
105 102
920 716
68 58160
W)
(eV)
708.1
68.1 854.7 51.1 334.7
334.7
854.7
2500
5000
1000
2500
111.8 574.5 574.5 708.1
119.8
O-3000
2200 2000
(eV)
Energy of primary beam
931.1 854.7
73.6
72.72
Ionization energy [26]
AES
Auger electron energy [ 25 ]
by quantitative
B
R
G % j* z z :i?.
g 3. z?
3
P
A. Jabloliski / Backscattering factor
632
1.0
-7
0.8 /’ O6
0
/’
4 04- C
0
0
I’
0.2
I
/
/’
/‘///
1’
/’
0.4
0.6
Bulk composition,
0.8
I 1.0
x Al
Fig. 7. The dependence of the ratio [Al on bulk composition calculated for AlCu alloy system assuming no surface segregation. Calculations were carried out for L~M~,JM~,s Al Auger transition, and Ep = 3000 eV. Experimental results of Ferrante [17] for 700°C.
eq. (13) with parameter a0 = 0.045 . 13 = 0.585, and with parameter a given by eq. (19). Results of calculations for the Al Auger transition and E, = 3000 eV are shown in fig. 7. Neglecting the difference in the backscattering factors in the case of the external calibration method, i.e. assuming that the ratio .& is equal to the surface concentration of Al, would clearly indicate the Al surface enrichment. Ferrante“ [ 171 studied the AlCu single crystals having composition 1% Al, 5% Al, and 10% Al. Employing the low energy Auger electron transitions LsMa,&Ia,s Al (68 eV) and Ma,sM4,5M~,s Cu (58/60 eV) he has found significant surface enrichment with Al. He applied an experimental technique resembling the internal calibration method, using an extensively sputtered surface as a standard. The procedure should provide a true surface composition since the internal calibration method removes the effect of the backscattering factor on the results of analysis [16]. Those results are also shown in fig. 7. However, the simple model of binary alloy sputtering introduced by Shimizu et al. [27], together with sputtering yields reported by Laegreid and Wehner [28], predicts that the standard surface should be enriched with Al. It is possible then that Ferrante underestimated the extent of the Al surface segregation. 6.2. The CuNi and CrFe alloy systems The backscattering factors of pure copper and nickel have similar values (fig. 6) due to the small difference in atomic numbers (ZcU = 29,Z,i = 28). The ratio r/r0 deviates from unity by not more than 1% in the case of high energy transitions Ni; E, = 2000 eV) and by not more than 1.6% in (LsM,,sM+s Cu and L&2,3”2,3 the case of low energy transitions (M1M4,sM4,s Cu and Ni; E, = 2500 ev). Thus the ratio of backscattering factors may be neglected in the determination of surface composition.
A. Jablotiski / Backscatteringfactor
633
A similar situation is found in the case of CrFe alloy (2 Cr = 24, Zr+ = 26), where the value of I-/T’ does not differ from unity by more than 2%. 6.3. The NiPd alloy system Mathieu and Landolt [23] have found that extensively sputtered NiPd ahoy surfaces are enriched with Ni. This result is also predicted by the model of Shimizu et al. [27]. These authors reported the dependence of the peak height in the derivative spectra, cw(E3/dE versus E, on the alloy composition. Those results after normalization with respect to the corresponding pure metals are presented in fig. 8. They are compared with the concentration dependence of the ratios ENi and ,& calculated for the case with no surface segregation (solid lines). Calculations were performed for Auger transitions employed by Mathieu and Landolt, and for E, = 5000 eV. An apparent surface enrichment with nickel results from the assumptions ENi = Xhi and &r = xSpd. It follows from fig. 8 that a large part of nickel enrichment observed by Mathieu and Landolt may be due to the neglect of the concentration dependence of the backscattering factor. Theoretical considerations predict that the NiPd ahoy surface, being in equilibrium with the bulk, is enriched with palladium [29,30]. The palladium segregation was found experimentally by Stoddart et al. [24] for thin NiPd films using the external calibration method. A‘s previously, let us assume that XNi = Xhi+ The concentration dependence of the ratio [Ni for M,,3M4,5M4,S (61 eV) and LJM4,sM4,5 (848 eV) nickel Auger transitions and for E, = 2500 eV is shown in fig. 9. The experimental points of Stoddart et al., derived from low energy and high energy
ie0.6 Y
d 0.4
Y
0.2 0 0
Q2 Bulk
Q4
Q6
composition,
0.8
20
xNi
Fig. 8. The dependence of the ratios ENi and tpd on bulk composition calculated for NiPd alloy system, assuming no surface segregation. Calculations were carried out for L3M4,5M4,5 Ni Auger transition, M~N~,~NG,s Pd Auger transition, and Ep = 5000 eV. Experimental results of Mathieu and Landolt [23] after normalization with respect to pure metals (A) derived from Ni Auger transition, (0) derived from Pd Auger transition.
634
A. Jabloriski / Backscattering factor
0.6
0
0.2
0.4
0.6
0.8
1.0
Bulk composition, xNi
Fig. 9. The dependence of the ratio [Ni on bulk composition calculated for NiPd alloy system assuming no surface segregation. Calculations were carried out for LsM4,+M4,5 Ni Auger transition (l), M2,3M4,5M4,5 Ni Auger transition (2), and Ep = 2500 eV. Experimental values of Stoddart et al. [24] (0) derived from high energy Auger transition, (a) derived from low energy Auger transition.
nickel Auger transitions, are also shown in that figure. It is seen that taking into account the concentration dependence of the backscattering factor would noticeably increase_ the surface segregation of palladium.
7. Conclusions The angular and energy distribution of electrons backscattered from a solid may be determined by the single elastic scattering theory of Everhart [2]. The knowledge of this distribution together with the classical ionization cross-section expression of Gryzmski [ 121 makes possible the estimation of the backscattering factor for target materials having an atomic number below 4045. Values of r for elements determined in this way are in good agreement with values calculated by Bishop and Riviere using the Monte Carlo method and with experimental data on r available at the moment. The backscattering factor calculated for binary alloys was found to be a monotonic function of composition increasing from the element with the lower atomic number to the element with the larger one. For this reason, neglecting the concentration dependence of the backscattering factor in the external calibration method of quantitative AES apparently enriches the surface with an element having the lower atomic number. This apparent surface enrichment is stronger when the difference in atomic numbers of elements forming an alloy is larger. For CuNi and CrFe alloys, the backscattering effects were found to be negligible; the ratio r/r0 was equal to unity within 2% across the whole concentration range. In the case of AlCu
A. Jablohski / Backscatteringfactor
and NiPd alloys, taking into account the concentration r/r0 markedly affected the surface composition.
635
dependence
of the ratio
The work was carried out within the Research Project 03.10.
References [l] [2] [3] [4] [S] [6] [7] [8] [9]
H.E. Bishop and J.C. Riviere, J. Appl. Phys. 40 (1969) 1740. T.E. Everhart, J. Appl. Phys. 31 (1960) 1483. G.D. Archard, J. Appl. Phys. 32 (1961) 1505. W.S. McAfee, J. Appl. Phys. 47 (1976) 1179. G.J. Iafrate, W.S. McAfee and A. Ballato, J. Vacuum Sci. Technol. 13 (1976) 843. H.M. Terrill, Phys. Rev. 22(1923) 101. T.E. Gallon, J. Phys. D (Appl. Phys.) 5 (1972) 822. F. Meyer and J.J. Vrakking, Surface Sci. 33 (1972) 271. F. Meyer and J.J. Vrakking, Surface Sci. 45 (1974) 409. [lo] R.L. Gerlach, A.R. DuCharme, Surface Sci. 32 (1972) 329. [ll] C.J. Powell, Rev. Mod. Phys. 48 (1976) 33. [ 121 M. Gryzinski, Phys. Rev. Al38 (1965) 336. [13] H.E. Bishop, Brit. J. Appl. Phys. 18 (1967) 703. [14] D.M. Smith and T.E. Gallon, J. Phys. D (Appl. Phys.) 7 (1974) 151. [ 151 A. Jabfonski, S.H. Overbury and G.A. Somorjai, Surface Sci. 65 (19.77) 578. [ 161 R. Bouwman, J.B. van Mechelen and A.A. Holscher, Surface Sci. 57 (1976) 441. [17] J. Ferrante, Acta Met. 19 (1971) 743. [ 181 K. Nakayama, M. Ono and H. Shimizu, J. Vacuum Sci. Technol. 9 (1972) 749. [ 191 Y. Takasu and H. Shimizu, J. Catalysis 29 (1973) 479. [20] CR. Helms, J. Catalysis 36 (1975) 114. [21] C.R. Helms and K.Y. Yu, J. Vacuum Sci. Technol. 12 (1975) 276. [22] C. Leygraf, G. Hultquist, S. Ekelund and J.C. Eriksson, Surface Sci. 46 (1974) 157. [23] H.J. Mathieu and D. Landolt, Surface Sci. 53 (1975) 228. [24] C.T.H. Stoddart, R.L. Moss and D. Pope, Surface Sci. 53 (1975) 241. [25] P.W. Palmberg, G.E. Riach, R.E. Weber and N.C. McDonald, Handbook of Auger Electron Spectroscopy (Physical Electronics Industries, Edina, Minnesota, 1972). [ 261 K.D. Sevier, Low Energy Electron Spectrometry (Wiley-Interscience, New York, 1972) p. 356. [ 271 H. Shimizu, M. Ono and K. Nakayama, Surface Sci. 36 (1973) 817. [28] N. Laegreid, G.K. Wehner, J. Appl. Phys. 32 (1961) 365. 1291 F.L. Williams and D. Nason, Surface Sci. 45 (1974) 377. [30] A. Jabfofiski, Advan. Colloid Interface Sci. 8 (1977) 213.