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The electron mean free path (applicable to quantitative electron spectroscopy)
Surface Science 149 (1985) 349-365 North-Holland, Amsterdam
349
THE ELECTRON MEAN FREE PATH (APPLICABLE QUANTITATIVE ELECTRON SPECTROSCOPY) H. TOKUTAKA, Department Received
K. NISHIMORI
TO
and H. HAYASHI
of Electronics, Faculty of Engineering, Tottori Universrty, Koyama, Tottori, Japan 8 March
1984; accepted
for publcation
27 August
1984
There are two well known methods accepted generally to establish the value of the electron mean free path. One is Seah’s method where he compiled many published experimental data. The other is Penn’s method which is extensively theoretical. Besides, Tarng and Wehner showed that there is a significant difference between the observed electron mean free paths, according to whether the electron pass through MO or W. Here, we propose a general method to calculate the electron mean free path for material of any atomic number, using Tarng and Wehner’s experimental results (MO and W) and our experimental data for Cr. Then, we compare and review these three methods of Seah, Penn and ourselves to learn which method is the most accurate, using published AES and XPS experimental data. Among these three methods, our method shows values closest to the experimental ones. Finally, we must add the following sentence: When the AES experimental data are compared with the theoretical values, the attenuation length of the primary electron beam should be considered.
1. Introduction It is well known that AES (Auger electron spectroscopy) is commonly used to investigate the composition of very thin layers on the solid surface. In this method, it is necessary to know the value of the mean free path of an Auger electron in order to examine the sampling depth along the direction of the surface normal. For this purpose, there are two well known methods to determine the electron mean free paths. One is Seah’s method [l] where he referred to many published works and expressed the famous universal curve of the electron mean free path as a function of the electron energy. The other is Penn’s method [2] which is extensively theoretical. Here, we propose a new method [5] to calcultae the true electron mean free path f,(E) for material of any atomic number (2) by interpolation or extrapolation, using the experimental results from materails with different atomic number: Cr (24) [3], Mo (42) [4] and W (74) [4]. Then, we compare and investigate these three methods of Seah, Penn and ourselves to learn which method is the most accurate. The electron mean free paths are usually measured by an AES and/or XPS (X-ray photoelectron spectroscopy) experiment. Electrons and photons are 0039-6028/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
350
H. Tokutaka
et al. / Electron meun Jree path
used as the primary beam in respectively an AES and XPS experiment. In AES, there is an attenuation of the primary electron beam in the material. This effect is taken into account for the calculation in our case. On the other hand, Seah expressed a universal curve which explains the experimental data themselves. The attenuation of the primary electron beam is not considered in his method. However, Penn’s method is extensively theoretical. In order to compare the above three methods, at first we used the experimental data which Seah referred to. When AES experimental data are examined by our method, it is necessary to know the primary electron beam energy. Therefore, we used the AES data of published papers in which the experimental conditions which we need are fully described. Also, these data should be worthwhile to be referred to and reliable as well. Seah’s paper was published in 1979. We also searched for new data of published papers up to 1983 after Seah by a commercial data-base. Then, we compared the above three methods with each other using these experimental data. The procedure of to compare these three methods is as follows: The difference between the experimental value and the theoretical value was calculated as the error. The averages of the absolute values of the errors were compared for these three methods. The deviation of the error was also calculated in order to learn the distribution of the error. As a conclusion, our method shows values closest to the experimental ones among these three methods to obtain the mean free path. The second is Seah’s and the third is Penn’s method. In section 2, the methods to obtain the electron mean free path as suggested by us, by Seah and by Penn are briefly shown. In section 3, the practical method to compare the experimental value with the theoretical value is discussed. In section 4, the results and the discussion of the evaluations for these three methods are described. Section 5 provides conclusions.
2. Theory
(to describe the true mean free path)
The trajectories of electrons in a solid depend on their energy and the properties of the materials which they pass through. This point was first observed by Tarng and Wehner [4]. They showed that there is a significant difference between the observed mean free paths according to wether the electrons pass through MO (42) or W (74). We also observed a similar behavior in a Cr (24) experiment [3]. Using these experimental results, the theory for obtaining the true mean free path of electrons in the material was fully described in a previous paper [5]. Here, we summarize the results. When the electron energy E 2 350 eV, the true mean free path I,(E) of electrons in material of atomic number Z can be described by the following three equations. For an atomic number Z of less than 24 for Cr or larger than
H. Tokutaka et al. / Electron mean free path
351
74 for w, ln
,
Z
( EJ = ldQ(W3.321
(1.6551-0.2890lnE)+(-3.2563f0.9395lnE).
ln(7.74/3.32)
For an atomic
number
Z of between
In / ( EJ = 1n[Q(%‘3.321 Z
24 for Cr and 42 for MO,
(0.6847 - 0.1169 In E) + (-3.2563
ln(4.50/3.32)
+ 0.9395 In E).
(2) For an atomic number ,n
f
(E)
=
Z
Z of between
ln[Q(Z)/4.501 ln(7.74/4.50)
42 for MO and 74 for W,
(0.9704 - 0.1721 In E) + ( -2.5716
+ 0.8226 In E). (3)
When E < 350 eV, for any atomic
ln / cEJ = ln[Q(Z)/4.501 Z
number
Z,
(0.0107 - 0.0083 In E) + (0.7271 + 0.2595 In E).
ln(7.74/4.50)
(4 In the above equations, the relation Q(Z) = Zp,/M, holds, where pz is the bulk density (g/cm3) and M, is the atomic weight (g/mol). The above four equations are described by four figures of merit. This is the reason why the value of the true mean free path f,(E) can be calculated by two figures of merit after the calculation. Let us explain Seah’s formula [l]. He referred to many published works and expressed the famous universal curve of the true mean free path as a function of the electron energy. However, he also considers the material dependence on the true mean free paths A, and A,, where A, is the path through a single element or an inorganic compound and A, is the path through an organic compound. These values of A, and A, are described as follows: For a single element, A, = 538/E2
+ 0.41(&Y)“*
for an inorganic
material,
A, =2170/E’+
0.72(~E)“~
and for an organic A,=
42/E2
(monolayers),
(5)
(monolayers),
(6)
material,
+ O.llE”’
where a is the monolayer
(mg/m2), thickness;
(7) a is calculated
from the relation
pNna3 =
H. Tokutaka et al. / Electron mean free path
352
1024A, where p is the bulk density (kg/mj), N is Avogadro’s number, n is the number of atoms in the molecule and A is the atomic or molecular weight. Next, let us explain Penn’s result [2]. He approximated the electron trajectories in the solid by the free electron model. The true mean free path I(E) of the electron can be expressed by the following equation: I(E)
= E/P(ln
P and
E + Q).
Q are values
(8)
depending
on the material
which
the electrons
pass
.-(cl mo-
.;
Seall
h = 10 .
a(-%
0.41
/-x
)
EZ 4
r
2
5o
..W --cc
a
,: hi.3
:
--Be
c :
lo-
i : ;
Energy
Electron
lb)
5-
(eV)
I
Penn
I
50
SW
100 Eiectra”
energy
v2Qo
2000 3000
lev I
SO
100
500
1000
Electron energy
2000 3000 (eV1
Fig. 1. The true electron mean free path as a function of the electron energy and the material by (a) our, (b) Penn’s [2] and (c) Seah’s method [l]. (d) The universal curve for mean free path proposed by Seah [l].
H. Tokutaka et al. / Elecrron mean free path
354
through.
H. Tokutaka
et al. / Elecrron mean free path
These values of P and Q are tabulated
as a function
of the material
[21. The true mean free path of the electrons is calculated by the above three methods of ourselves, of Seah and of Penn. The results are shown in figs. 1 and 2. In fig. 1, the true mean free path is shown as a function of the energy and the material. In fig. 2, the electron energy is fixed at 1000 eV. Then. the true mean free paths are calculated as a function of the atomic number following (a) our method, (b) that of Seah and (c) that of Penn. These three results show that the changes coincide with the periodic table. In our case, the series of the maximum and the minimum values (F. G, H, K, L and A, B, C, D. E in fig. 2a. respectively) decrease as the atomic number increases. However, in Seah’s case. the series of the maximum and the minimum values (F’, G’, H’, K’, L’ and A’. B’, C’, D’, D’, E’ in fig. 2b, respectively) increase as the atomic number increases. In Penn’s case, the series of the maximum and the minimum values (F”, G”, H”, K”, L” and A”, B”, C”, D”, E” in fig. 2c, respectively) increase slowly as the atomic number increases.
3. Analytical method (for the comparison between the experimental the theoretical value of the mean free path)
value and
Mainly, the electron mean free paths are measured by AES and/or XPS. However, in AES, the observed values of the mean free path are affected by the attenuation length c of the primary electron beam [5]. The electrons which have an energy E and are produced at a depth T will attenuate along the
Fig. 3. Production of an Auger electron at normal incidence.
along the detection
angle 0 by the primary
electron
beam
H. Tokutaka
et al. / Electron mean free path
355
detection angle 8 (the angle B of CMA optics is 42”18’) as shown in fig. 3, when they reach the solid surface. Then, the number NA(T) of Auger electrons at the surface can be expressed by the following equation as NA(T)=NA(0)exp(-T/c)exp[-T/I,(E)cos0] =N,(O)exp{-[/,(E)(cos~)+c]
T/f,(E)(cos~)c},
(9)
where NA(0) is the number of Auger electrons in the limit T + 0, I,(E) is the true mean free path by which an electron of energy E passes through the material of atomic number Z, and c is the attenuation length of the primary electron beam. Thus, between the experimental value and eq. (9), the following relation exists: x(E)=X(E)cose=
lz( E) (c0s e) c I,(E)
(cOse)+c'
where x(E) is the experimental value of the mean free path along the direction of the surface normal, X(E) is the experimental value of the mean free path at the detection angle 6 of the optics. In the published papers, the experimental data of the mean free path are expressed by x(E) and/or X(E). Then, the experimental results of x(E) and/or X(E) are compared with the above discussed theories in section 2. 3.1. Our method In AES, at first we calculate the attenuation length c of the primary beam and the true mean free path I,(E) of the Auger electrons using eqs. (l)-(4). Using these values of c and /=(E), the theoretical mean free path which would be compared with the experimental value can be calculated using eq. (10). In XPS, the electron mean free path I,(E) can be calculated using eqs. (l)-(4), when the XPS electron energy E is known. Then, the theoretical value f,(E) can be compared directly with the experimental value using the following equation: x(E)=X(E)/COSo=/,(E).
(11)
3.2. Seah’s method As already discussed in section 2, Seah [l] constructed eqs. (5)-(7) for the electron mean free path. He tried to fit these equations to the experimental data. Therefore, when the electron energy E is known in an AES or XPS experiment, we can select a suitable equation from eqs. (5)-(7) for which the electrons pass through a single element, an inorganic and an organic material, respectively. The obtained theoretical values of I,(E) can be compared with the experimental data by using eq. (11).
H. Tokutaka
356
et al. / Electron mean free path
3.3. Penn’s method Penn [2] says that the mean free path can be approximated by eq. (8). In an AES experiment, we consider the attenuation of the primary electron beam. The attenuation length c and the true mean free path I,(E) of the Auger electron are calculated by using eq. (8). Then, the theoretical value can be compared with the experimental value by using eq. (10). However, when the value lz( E) is compared directly with the experimental value by using eq. (1 l), the error and the deviation which are discussed in section 4 are less than in the case of eq. (10) which considers the attenuation of the primary electron beam. Other researchers also use Penn’s equation without considering the attenuation of the primary electron beam. Thus, in this case also, the theoretical value I,(E) can be compared directly with the experimental value by using eq. (11) like in Seah’s case. In an XPS method, the theoretical value I,(E) is compared with the experimental value by using eq. (11).
4. Results and discussions (for the evaluation of the three theoretical methods using the published experimental data) 4.1. The evaluations
using the AES
experimental
data
4.1.1. The referred data are limited to the data which Seah [I] referred to The error y between the experimental value and the theoretical value calculated using the following equation as
:* *’. H--!I+
y = (the experimental value) - (the theoretical I (the theoretical value)
value)
200
(al
ms t mrs
AES :
[b)
Seah
_.”
100
-
0
-_ 500
1000
Electron ~nergr teU1
5m
1000
Electron energy leV1
(12)
x 100 (%I).
200
1
loo
:
is
0
1. Cc) Es:
Penn
.
.
*
500
IO00
Electron energr (eV1
Fig. 4. The error Y, between the experimental value and the theoretical value using the AES experimental data which are limited to the data that Seah [l] referred to. The data of Tarng and Wehner [4] are excluded [AES(l)].
H. Tokutaka
et al. / Electron mean free path
357
The calculated errors are shown in figs. 4a-4c as a function of the electron energy E for our, Seah’s and Penn’s method, respectively. The average m of the absolute values IT] of the errors is calculated using the following equation as N
vi=c Iyl/N,
(13)
where N is the number of calculated errors y. The calculated values are shown in fig. 5a. The deviation u of the error y is calculated using eq. (14) to learn the distribution of the errors:
u=
[+):/N]“*.
(14)
The calculated values of u are shown in fig. 5b. As discussed in section 2, our method to obtain the mean free path is introduced using the experimental values of Tarng and Wehner [4]. These values are already included in the data which Seah referred to. If these values are included for the evaluation, the average m of the absolute values of the errors and the deviation u of the errors decrease naturally in our case. Therefore, such an evaluation is not fair. Thus, the case which is compared without these values is defined as AES(1). The case which is compared with these values included is defined as AES(2). The numbers of data for AES(l)
r
(a)
r
AES(2)
(b)
AES(l)
AES(Z)
Fig. 5. The evaluations of the mean free path using the AES experimental data. (a) Average IYI of the absolute value [Y,1of the errors (fig. 4) and (b) deviation a of the error Y, (fig. 4). The case which is compared without the experimental data of Tarng and Wehner [4] is defined as AES(1) and the case which is compared including these values is defined as AES(2).
358
H. Tokutaka
et al. / Electron mean free path
and AES(2) are 15 and 21, respectively. In fig. 5, both the value of the average error m and the deviation o are the smallest in our case, the second smallest comes from Seah’s case and the third is for Penn’s case. In every case, AES(2) shows smaller values of F] and u than AES(1). From the result, it is concluded that the experimental values of Tarng and Wehner [4] and of us [3] are accurate and reliable. 4.1.2. The published data in refs. [12]-[15] up to 1983 after Seah [I] are added to section 4.1.1 as the referred data The number of new data is 8. Thus, the numbers of data for AES(1) and AES(2) become 23 and 29, respectively. The errors y between the experimental value and the theoretical value are shown in figs. 6a-6c, as a function of the
200
200
la1
RES :
m
200
1
l-5
(b)
ES
I seah
-l--L z
;
.R ; 100
1000
-Ii1
leV)
Electron
100
L
: &
. : .
0
e
:
I:
-
:
.
.
500
Electron
energy
ICI
FE5
I Penn
I
. ..
.
o.L
:. 1
..
.
.
-
jO0
energ
1,
leV1
Electron
energy
IeVI
Fig. 6. The error Y, between the experimental value and the theoretical value using the AES experimental data where the published data up to 1983 after Seah [l] are added to fig. 4. The data of Tarng and Wehner [4] are excluded (AES(l)].
100
r
(a)
AEStl)
r
AESC-2)
(b)
AESl)
AESc2)
Fig. 7. The evaluations of the mean free path using the AES experimental fig. 6 and (b) the deviation o of fig. 6.
data. (a) Average
1Y / of
H. Tokutaka et al. / Electron mean free path
359
electron energy E. The values of m and a are also shown in figs. 7a and 7b, respectively. The results in fig. 7 are almost the same as compared with the results in fig. 5.
(al
loou
XPS : Olrs
3000
2ooo
4000
Electron energy
(b)
(eU1
XPS I Seuh
. .
. . -. .
0
2; .
-100
’
. -
;. : .-
1000
. .
2000
4000
3000
Electron energy
[e!Jl
500 -. 400
(cl
-
XPS : Penn
5 3@-J F k
203
. . t -
pO-
0 -
-; . :’ .’ : . . .
. .
-‘Y I - I IOW
zoo0
1 I 4GuO
3000
Electron energy
Fig. 8. The error y between the experimental experimental data from Seah’s references [l].
(&I
value and
the theoretical
value using
the XPS
360
H. Tokutaka
4.2. The evaluations
et al. / Electron mean
using the XPS experimental
free path
datu
It is possible to arrange the XPS experimental data into three energy ranges, which are E -C 1000 eV, E < 2000 eV and a region covering all energies. In contrast to the AES experiment, it is not necessary to consider the attenuation of the primary electron beam in a usual XPS experiment. The experimental value can be compared directly with the theoretical one which is derived from eq. (11). 4.2.1. The referred data are limited to the data which Seah [I] referred to Here, the number of data is 39. The error Y between the experimental value and the theoretical value is shown in figs. 8a-8c as a function of the electron energy E for our, Seah’s and Penn’s method, respectively. The average 1Y1 of the absolute value lY1 of the error is shown in fig. 9a. The deviation u of the error is shown in fig. 9b. In fig. 9a, the average m by our method is smallest, the smallest but one is from Seah’s method and the largest is from Penn’s method. In fig. 9b, the deviation CJby our method also shows the smallest value and one but smallest is from Seah’s method. However, Penn’s method shows the largest cr. When the evaluation is carried out using the XPS data, our method shows the best accuracy among these three methods. 4.2.2. The published data in refs. [25]-(281 up to 1983 after Seah [IJ are added to section 4.2.1 as the referred data The number of new data is now 28. Then, the number of all data becomes 67. The error Y is shown in figs. lOa-10c as a function of the electron energy E. The average m is shown in fig. lla. The deviation u of the error is shown in fig. lib. It can be seen that m and (I in fig. 11 show a behavior similar to each
c
E
(b)
(a)
EUOOOeV
all energy
Fig. 9. The evaluations of the mean free path using the XPS experimental data. (a) Average IYI of fig. 8 and (b) the deviation D of fig. 8.
H. Tokuiaka Ed al. / Electron mean free path
500
(al
400 s
361
XPS : krs
300
C 200 k ; 100 0
-. . : . .. . i *-. . .*- : . .*. .. .z* .. . ,* .. * * *_i.
-.
*
:
.
-100
I 000
.
4000
3000
2000 Electron
energy
. . .
let’1
500 XPS : Seah
400 5
3QJ
: 200 &e 100 -
0 -100 1000
2000 Electron
I .
.
3000 enerw
XPS
G -
4m
(et4
: Penn
300
+ 200 b jj 100 0 -100
Fig. 10. The error y between the experimental value and the theoretical value using the XPS experimental data where the published data up to 1983 after Seah [l] are added to fig. 8.
H. Tokutakn et al. / Eleclron mean free purh
362
other. and u shows largest section
When the energy range is extended to cover all energies, the values of fl decrease in our case as well as in that of Seah and of Penn. Our method the smallest values, the smallest but one are from Seah’s method and the are from Penn’s method. The result has a trend similar to the case of 4.2.1. However, each value in fig. 11 is larger than the corresponding
(a)
r
~OOeV E<2000eV
“i
all energy
E<2000eV
all
energy
Fig. 11. The evaluations of the mean free path using the XPS experimental fig. IO and (b) the deviation CJof fig. 10. Table 1 Details of the experimental data and theoretical for AES and (b) for XPS data
values which are used to draw figs. 4 to 11: (a) is
T
ours fl ~iEi
&
5.2 i:: 14.0 6.1 LT.0
0 4.7 -4.9 0 -1.6 0
5.3 1.8 5.3 7.7 9.3
7.5 -12.8 5.7 0 43.0
~
q.’ 5.3 11.6
,%I
(%I 6.7 22.9 3.1 44.7 85.2 64.2 15.1 -35.4 -20.1
-55.a -1.8 -6.4 36.7 21.1 3.8
‘Ei -37:1
T
seall
error
‘2rKor
6.0 3.0 9.7 4.7 5.4 5.3 5.3 8.Z L5.9 5.2 5.5 T.8 ?.5 13.3 5.2
f
data. (a) Average v\ of
3.z 4.0 6.9 3.6 4.‘ 4.3 3.8 8.1, 13.6 4.5 4.8 10.1 !0.2 15.6 4.8
,,:I ,:I
100.0 115.0 44.9 88.9 127.3 102. : 60.5 -36.9 -6.6 -48.9 12.5 -27.1 5.9 3.2 12.5
2.9 4.0 8.5 2.6 2.9 3.0 2.9 5.7 10.9 2.1 2.8 6.2 6.3 11.5 2.B
26.5 11.5 -19.8 22.1 3.4 -*0.9
-i-T 3.2 6.3 21.0 3.4 22.0
_:I”:I 21.7 -23.8 -2.9 44.0 6.7 -56.0
2.6 5.7 2.7 6.2 8.0 8.4 2.4 10.2 -
-
e, IiE)
error C%l
20.3 20.3 24.3 25.7 35.1 17.6 31.1 31.1 52.7 15.1 64.9 77.0 12.2 14.9 11.6 12.0 16.2 16.2 23.c 29.1 29.7 31.1 35.1 33.8 39.2 ,9.2 17.6 33.8 29.1 18.2 17.6
25.7 35.1 29.7
48.1 5O.C 4.5 12.2 12.2 11.9
24.1 40.5 33.8 33.4 32.1 42.6 40.9 64.4 65.2 26.6 20.3 49.8 29.7 63.6 65.6 21.c 23.c I 52.7 16.6 20.4 25.c 29.7 27.2 33.3 36. I 36.9 / - 32.4
‘r b, ” d, ef
: g:::
6.5 8.4 8.5 9.9 9.6 !S.5 26.6 :::;
:::; ‘1.7 41.9 44.8 L5.I 98.1 62.7 23.0 17.3 22.5 22.5 34.2 x8.5 19.9 26.8 27.1 37.5 31.6 46.5 46.5 :::; 20.0 21.6 :::'; 9.2 29.1 33.5 36.1 38.6 43.4 1.1 20.1 ZO.1. L2.8 16.2 26.1 30.3 31.1 44.2 65.7 47.5 24.5 -
A2.2 -46.0 -49.4 -58.3 -60.3 81.4 72.8 -13.3 10.0 -40.5 -10.1 -61.4 -44.5 L29.2 ,8.1 41.2 63.6 68.8 48.4 II.7 -9.2 -5.5 5.1 -O., -6.0 -6.4 -60.1 -26.3 -69.7 -?I.0 -23.5 48.6 56.0 32.0 42.4 29.9 -52.3 -54.5 -55.0 11.7 -23.1 -12.9 -27.3 147.4 129.3 113.0 88.9 LO5.I 99.4 L89.1 -,0.2 48.7 -17.7 64.8 51.2 195.8 14.4 162.2 29.7 25.9 -4.2 -2.0 -12.5 -24.7 -19.7 -22.3 32.2 -
11.7 12.0
13.2 20.5 23.6 10.6
13.7 19.1 19.9 21.6 23.3 34.1 34.7 ?: a:8 11.5 11.6 14.5 18.7 20.5 20.6 20.8 20.9 23.0 23.1 24.7 2L.9 35.5 SO.0 19.7 16.4 20.0 ZO.0 27.6 30.2 19.7 23.9 -24.1 12.0 9.6 19.1 19.7 10.3 $9' 13.5 16.7 17.2 10.1 19.5 20.8 21.5 22.2 "::;
la.5 18.5 13.0 15.5
22.5 25.2
::.z 34:s 35.5 23.8 -
73.5 69.2 84.1 25.4 48.7 69.2 127.0 51.9 164.8 62.5 1?8.5 121.9 223.3 170.9 34.9 ,6.4 40.9 39.7 58.6 58.8 44.2 51.0 68.8 61.7 70.4 69.7 -28.7 ,5.1 -16.3 -63.6 -10.7 56.7 75.5 48.5 76.4 65.6 -51.8 -49.0 -49.4 249.2 153.1 LOS.6 71.6 224.3 205.1 230.2 202.2 285.6 279.1 16,.‘ 4.1 139.4 18.1 186.5 179.i 255.9 24.3 184.9 27.7 31.6 11.1 x7.9 6.3 -0.9 6.4 3.9 36.1
13.0 13.5 15.5 28.5 36.0 8.4 12.6 22.2 22.6 25.7 29.1 55.8 ‘E 4:s 4.7 6.9 6.9 9.7 14.3 16.6 16.7 16.9 17.1 19.9 20.0 18.4 18.7 :;:t 19.1 12.3 16.1 16.7 27.6 32.0 13.7 18.5 -18.6 13.5 13.5 22.2 22.2 7.3 7.5 10.0 10.7 14.8 15.3 5.7 15.2 :::; 18.8 20.5 3.3 12.a 12.8 8.8 11.4 i9.9 23.8 24.5 38.0 19.6 41.5 16.7 -
56.2 50.4 56.8 -9.8 -2.5 109.5 146.8 40.1 133.2 ,6.6 12Y3.0 38.0 101.1 496.0 157.8 155.3 134.8 134.8 117.1 107.7 73.9 86.2 107.7 97.7 97.0 96.0 -4.3 80.7 -8.9 -64.6 -7.9 108.9 110.2 77.8 76.4 56.3 -30.7 -34.1 -34.4 2 lo. 4 ml.0 82.4 52.3 357.5 328.3 326.0 281.3 335.1 326.1 366.7 33.6 194.7 65.9 238.3 220.0 536.4 79.7 311.7 88.6 78.9 25.6 24.8 11.0 -12.4 -7.3 -11.1 94.0
The material where the efectron penetrates. Auger electron energy. Primary electron beam energy. D(E) are the experimental data. X(E) is the value which is read in the B direction, where il is the angle between the surface normal direction and the optics. When D(E) is read in the 8 direction, the value is defined as X(E). When D(E) is read in the surface normal direction, the value X(E) is defined as D( E),‘cos
8.
is the value which is calculated from eqs. (l)-(4). tsf E) is the vatue which is calculated from eqs. (S)-(7). I,(E) is the value which is calculated from eq. (8). ( ) are the data which are read in the surface normal direction. Photoelectron energy.
‘) I(E)
s) hi ‘) ‘)
35.1 37.6 G.0 61.6 88.5 9.7 18.0 46.6 47.9 59.0 72.2
17.1
364
H. Tokutaka
et al. / Electron mean free path
value in fig. 9. This is probably because of the large number of used data points. The number of data used here is 67 instead of 39 in section 4.2.1. Thus, our method also shows a much better result than that of Seah and that of Penn in the XPS case. The details of the experimental data and the theoretical values which are used to draw figs. 4 to 11 are tabulated in table 1.
5. Conclusions (1) When the AES data as the experimental values are compared with the theoretical values, our method shows results which are closest to the experimental data among the three theoretical methods of ourselves [5], Seah [l] and Penn [2]. This is caused by the fact that the attenuation length of the primary electron beam is considered in our method. (2) Let us take the case that the XPS data as the experimental values are compared with the theoretical values. Our method also shows the best result among these three methods. This is probably due to the fact that our method is based on the experimental values of Tarng and Wehner [4] of MO and W and of Cr of our experiment [3], which may be fairly accurate. (3) In our method, the series of the maximum and the minimum values (F, G, H, K, L and A, B, C, D, E in fig. 2a, respectively) decrease as the atomic number increases. Tarng and Wehner [4] say that the inelastic scattering cross-section increases as the number of outer shell electrons increases. Therefore, the mean free path should decrease in the order of F, G, H, K, L and A. B, C, D, E as shown in fig. 2a. Our result in fig. 2a is obtained by interpolation or extrapolation, using the experimental results of Cr (24) [3], MO (42) [4] and W (74) [4], the atomic numbers of which are very different from each other. Therefore, our results which is obtained by using the experimental result can be more practical and accurate than Seah’s and Penn’s results as discussed in section 4. (4) After Seah compared his result with that of Penn, he concluded that the mean free path obtained by him is better than that obtained by Penn. Our analysis also shows that Seah’s result is better than that of Penn. However, our method shows values closest to the experimental ones among these three methods to obtain the electron mean free path.
Acknowledgement This work was partly supported by a Grant-in-Aid from the Ministry of Education of Japan.
for Scientific
Research
H. Tokutaka
et al. / Electron mean free path
365
References [l] [2] [3] (41 [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
M.P. Seah and W.A. Dench, Surface Interface Anal. 1 (1979) 2. D.R. Penn, J. Electron Spectrosc. 9 (1976) 29. K. Nishimori, H. Tokutaka and K. Takashima, Surface Sci. 100 (1980) 665. M.L. Tarng and G.K. Wehner, J. Appl. Phys. 44 (1973) 1534. H. Tokutaka, K. Nishimori, K. Takashima and T. Ichinokawa, Surface Sci. 133 (1983) 547. C.R. Brundle, J. Vacuum Sci. Technol. 11 (1974) 212. M.P. Seah, Surface Sci. 32 (1972) 703. J.W.T. Ridgway and D. Haneman, Surface Sci. 24 (1971) 451. J.W.T. Ridgway and D. Haneman, Surface Sci. 26 (1971) 683. D.C. Jackson, T.E. Gallon and A. Chambers, Surface Sci. 36 (1973) 381. P.W. Palmberg and T.N. Rhodin, J. Appl. Phys. 39 (1968) 2425. B.C. De Cooman, V.D. Vankar and R.W. Vook, Surface Sci. 128 (1983) 128. P.W. Davies, M.A. Quinlan and G.A. Somorjai, Surface Sci. 121 (1982) 290. M. Matten and H. Liith, Surface Sci. 126 (1983) 502. S. Nakanishi and T. Horiguchi, Surface Sci. 125 (1983) 635. R.G. Steinhardt, J. Hudis and M.L. Perlman. Phys. Rev. B5 (1972) 1016. T.A. Carlson and G.E. McGuire, J. Electron Spectrosc. 1 (1972) 161. M. Klasson, A. Berndtsson, J. Hedman, R. Nilsson, R. Nyholm and C. Nordling, J. Electron Spectrosc. 3 (1974) 427. M. Klasson, J. Hedman, A. Berndtsson. R. Nilsson, C. Nordling and P. Melnik, Phys. Scripta 5 (1972) 93. H. Kanter. Phys. Rev. Bl (1970) 2357. J. Szajman, J.G. Jenkin, J. Liesegang and R.C.G. Leckey, J. Electron Spectrosc. 14 (1978) 41. R. Flitsch and S.I. Raider, J. Vacuum Sci. Technol. 12 (1975) 305. W.A. Fraser, J.V. Florio, W.N. Delgass and W.D. Robertson, Surface Sci. 36 (1973) 661. Y. Baer, P.F. Heden, J. Hedman, M. Klasson and C. Nordling, Solid State Commun. 8 (1970) 1479. P. Cadman and G.M. Gossedge, J. Electron Spectrosc. 18 (1980) 161. SM. Hall, J.D. Andrade, S.M. Ma and R.N. King, J. Electron Spectrosc. 17 (1979) 181. W.F. Egelhoff, Jr., Appl. Surface Sci. 11/12 (1982) 761. D.T. Clark. M.M. Abu-Shbak and W.J. Brennan, J. Electron Spectrosc. 28 (1982) 11.
349
THE ELECTRON MEAN FREE PATH (APPLICABLE QUANTITATIVE ELECTRON SPECTROSCOPY) H. TOKUTAKA, Department Received
K. NISHIMORI
TO
and H. HAYASHI
of Electronics, Faculty of Engineering, Tottori Universrty, Koyama, Tottori, Japan 8 March
1984; accepted
for publcation
27 August
1984
There are two well known methods accepted generally to establish the value of the electron mean free path. One is Seah’s method where he compiled many published experimental data. The other is Penn’s method which is extensively theoretical. Besides, Tarng and Wehner showed that there is a significant difference between the observed electron mean free paths, according to whether the electron pass through MO or W. Here, we propose a general method to calculate the electron mean free path for material of any atomic number, using Tarng and Wehner’s experimental results (MO and W) and our experimental data for Cr. Then, we compare and review these three methods of Seah, Penn and ourselves to learn which method is the most accurate, using published AES and XPS experimental data. Among these three methods, our method shows values closest to the experimental ones. Finally, we must add the following sentence: When the AES experimental data are compared with the theoretical values, the attenuation length of the primary electron beam should be considered.
1. Introduction It is well known that AES (Auger electron spectroscopy) is commonly used to investigate the composition of very thin layers on the solid surface. In this method, it is necessary to know the value of the mean free path of an Auger electron in order to examine the sampling depth along the direction of the surface normal. For this purpose, there are two well known methods to determine the electron mean free paths. One is Seah’s method [l] where he referred to many published works and expressed the famous universal curve of the electron mean free path as a function of the electron energy. The other is Penn’s method [2] which is extensively theoretical. Here, we propose a new method [5] to calcultae the true electron mean free path f,(E) for material of any atomic number (2) by interpolation or extrapolation, using the experimental results from materails with different atomic number: Cr (24) [3], Mo (42) [4] and W (74) [4]. Then, we compare and investigate these three methods of Seah, Penn and ourselves to learn which method is the most accurate. The electron mean free paths are usually measured by an AES and/or XPS (X-ray photoelectron spectroscopy) experiment. Electrons and photons are 0039-6028/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
350
H. Tokutaka
et al. / Electron meun Jree path
used as the primary beam in respectively an AES and XPS experiment. In AES, there is an attenuation of the primary electron beam in the material. This effect is taken into account for the calculation in our case. On the other hand, Seah expressed a universal curve which explains the experimental data themselves. The attenuation of the primary electron beam is not considered in his method. However, Penn’s method is extensively theoretical. In order to compare the above three methods, at first we used the experimental data which Seah referred to. When AES experimental data are examined by our method, it is necessary to know the primary electron beam energy. Therefore, we used the AES data of published papers in which the experimental conditions which we need are fully described. Also, these data should be worthwhile to be referred to and reliable as well. Seah’s paper was published in 1979. We also searched for new data of published papers up to 1983 after Seah by a commercial data-base. Then, we compared the above three methods with each other using these experimental data. The procedure of to compare these three methods is as follows: The difference between the experimental value and the theoretical value was calculated as the error. The averages of the absolute values of the errors were compared for these three methods. The deviation of the error was also calculated in order to learn the distribution of the error. As a conclusion, our method shows values closest to the experimental ones among these three methods to obtain the mean free path. The second is Seah’s and the third is Penn’s method. In section 2, the methods to obtain the electron mean free path as suggested by us, by Seah and by Penn are briefly shown. In section 3, the practical method to compare the experimental value with the theoretical value is discussed. In section 4, the results and the discussion of the evaluations for these three methods are described. Section 5 provides conclusions.
2. Theory
(to describe the true mean free path)
The trajectories of electrons in a solid depend on their energy and the properties of the materials which they pass through. This point was first observed by Tarng and Wehner [4]. They showed that there is a significant difference between the observed mean free paths according to wether the electrons pass through MO (42) or W (74). We also observed a similar behavior in a Cr (24) experiment [3]. Using these experimental results, the theory for obtaining the true mean free path of electrons in the material was fully described in a previous paper [5]. Here, we summarize the results. When the electron energy E 2 350 eV, the true mean free path I,(E) of electrons in material of atomic number Z can be described by the following three equations. For an atomic number Z of less than 24 for Cr or larger than
H. Tokutaka et al. / Electron mean free path
351
74 for w, ln
,
Z
( EJ = ldQ(W3.321
(1.6551-0.2890lnE)+(-3.2563f0.9395lnE).
ln(7.74/3.32)
For an atomic
number
Z of between
In / ( EJ = 1n[Q(%‘3.321 Z
24 for Cr and 42 for MO,
(0.6847 - 0.1169 In E) + (-3.2563
ln(4.50/3.32)
+ 0.9395 In E).
(2) For an atomic number ,n
f
(E)
=
Z
Z of between
ln[Q(Z)/4.501 ln(7.74/4.50)
42 for MO and 74 for W,
(0.9704 - 0.1721 In E) + ( -2.5716
+ 0.8226 In E). (3)
When E < 350 eV, for any atomic
ln / cEJ = ln[Q(Z)/4.501 Z
number
Z,
(0.0107 - 0.0083 In E) + (0.7271 + 0.2595 In E).
ln(7.74/4.50)
(4 In the above equations, the relation Q(Z) = Zp,/M, holds, where pz is the bulk density (g/cm3) and M, is the atomic weight (g/mol). The above four equations are described by four figures of merit. This is the reason why the value of the true mean free path f,(E) can be calculated by two figures of merit after the calculation. Let us explain Seah’s formula [l]. He referred to many published works and expressed the famous universal curve of the true mean free path as a function of the electron energy. However, he also considers the material dependence on the true mean free paths A, and A,, where A, is the path through a single element or an inorganic compound and A, is the path through an organic compound. These values of A, and A, are described as follows: For a single element, A, = 538/E2
+ 0.41(&Y)“*
for an inorganic
material,
A, =2170/E’+
0.72(~E)“~
and for an organic A,=
42/E2
(monolayers),
(5)
(monolayers),
(6)
material,
+ O.llE”’
where a is the monolayer
(mg/m2), thickness;
(7) a is calculated
from the relation
pNna3 =
H. Tokutaka et al. / Electron mean free path
352
1024A, where p is the bulk density (kg/mj), N is Avogadro’s number, n is the number of atoms in the molecule and A is the atomic or molecular weight. Next, let us explain Penn’s result [2]. He approximated the electron trajectories in the solid by the free electron model. The true mean free path I(E) of the electron can be expressed by the following equation: I(E)
= E/P(ln
P and
E + Q).
Q are values
(8)
depending
on the material
which
the electrons
pass
.-(cl mo-
.;
Seall
h = 10 .
a(-%
0.41
/-x
)
EZ 4
r
2
5o
..W --cc
a
,: hi.3
:
--Be
c :
lo-
i : ;
Energy
Electron
lb)
5-
(eV)
I
Penn
I
50
SW
100 Eiectra”
energy
v2Qo
2000 3000
lev I
SO
100
500
1000
Electron energy
2000 3000 (eV1
Fig. 1. The true electron mean free path as a function of the electron energy and the material by (a) our, (b) Penn’s [2] and (c) Seah’s method [l]. (d) The universal curve for mean free path proposed by Seah [l].
H. Tokutaka et al. / Elecrron mean free path
354
through.
H. Tokutaka
et al. / Elecrron mean free path
These values of P and Q are tabulated
as a function
of the material
[21. The true mean free path of the electrons is calculated by the above three methods of ourselves, of Seah and of Penn. The results are shown in figs. 1 and 2. In fig. 1, the true mean free path is shown as a function of the energy and the material. In fig. 2, the electron energy is fixed at 1000 eV. Then. the true mean free paths are calculated as a function of the atomic number following (a) our method, (b) that of Seah and (c) that of Penn. These three results show that the changes coincide with the periodic table. In our case, the series of the maximum and the minimum values (F. G, H, K, L and A, B, C, D. E in fig. 2a. respectively) decrease as the atomic number increases. However, in Seah’s case. the series of the maximum and the minimum values (F’, G’, H’, K’, L’ and A’. B’, C’, D’, D’, E’ in fig. 2b, respectively) increase as the atomic number increases. In Penn’s case, the series of the maximum and the minimum values (F”, G”, H”, K”, L” and A”, B”, C”, D”, E” in fig. 2c, respectively) increase slowly as the atomic number increases.
3. Analytical method (for the comparison between the experimental the theoretical value of the mean free path)
value and
Mainly, the electron mean free paths are measured by AES and/or XPS. However, in AES, the observed values of the mean free path are affected by the attenuation length c of the primary electron beam [5]. The electrons which have an energy E and are produced at a depth T will attenuate along the
Fig. 3. Production of an Auger electron at normal incidence.
along the detection
angle 0 by the primary
electron
beam
H. Tokutaka
et al. / Electron mean free path
355
detection angle 8 (the angle B of CMA optics is 42”18’) as shown in fig. 3, when they reach the solid surface. Then, the number NA(T) of Auger electrons at the surface can be expressed by the following equation as NA(T)=NA(0)exp(-T/c)exp[-T/I,(E)cos0] =N,(O)exp{-[/,(E)(cos~)+c]
T/f,(E)(cos~)c},
(9)
where NA(0) is the number of Auger electrons in the limit T + 0, I,(E) is the true mean free path by which an electron of energy E passes through the material of atomic number Z, and c is the attenuation length of the primary electron beam. Thus, between the experimental value and eq. (9), the following relation exists: x(E)=X(E)cose=
lz( E) (c0s e) c I,(E)
(cOse)+c'
where x(E) is the experimental value of the mean free path along the direction of the surface normal, X(E) is the experimental value of the mean free path at the detection angle 6 of the optics. In the published papers, the experimental data of the mean free path are expressed by x(E) and/or X(E). Then, the experimental results of x(E) and/or X(E) are compared with the above discussed theories in section 2. 3.1. Our method In AES, at first we calculate the attenuation length c of the primary beam and the true mean free path I,(E) of the Auger electrons using eqs. (l)-(4). Using these values of c and /=(E), the theoretical mean free path which would be compared with the experimental value can be calculated using eq. (10). In XPS, the electron mean free path I,(E) can be calculated using eqs. (l)-(4), when the XPS electron energy E is known. Then, the theoretical value f,(E) can be compared directly with the experimental value using the following equation: x(E)=X(E)/COSo=/,(E).
(11)
3.2. Seah’s method As already discussed in section 2, Seah [l] constructed eqs. (5)-(7) for the electron mean free path. He tried to fit these equations to the experimental data. Therefore, when the electron energy E is known in an AES or XPS experiment, we can select a suitable equation from eqs. (5)-(7) for which the electrons pass through a single element, an inorganic and an organic material, respectively. The obtained theoretical values of I,(E) can be compared with the experimental data by using eq. (11).
H. Tokutaka
356
et al. / Electron mean free path
3.3. Penn’s method Penn [2] says that the mean free path can be approximated by eq. (8). In an AES experiment, we consider the attenuation of the primary electron beam. The attenuation length c and the true mean free path I,(E) of the Auger electron are calculated by using eq. (8). Then, the theoretical value can be compared with the experimental value by using eq. (10). However, when the value lz( E) is compared directly with the experimental value by using eq. (1 l), the error and the deviation which are discussed in section 4 are less than in the case of eq. (10) which considers the attenuation of the primary electron beam. Other researchers also use Penn’s equation without considering the attenuation of the primary electron beam. Thus, in this case also, the theoretical value I,(E) can be compared directly with the experimental value by using eq. (11) like in Seah’s case. In an XPS method, the theoretical value I,(E) is compared with the experimental value by using eq. (11).
4. Results and discussions (for the evaluation of the three theoretical methods using the published experimental data) 4.1. The evaluations
using the AES
experimental
data
4.1.1. The referred data are limited to the data which Seah [I] referred to The error y between the experimental value and the theoretical value calculated using the following equation as
:* *’. H--!I+
y = (the experimental value) - (the theoretical I (the theoretical value)
value)
200
(al
ms t mrs
AES :
[b)
Seah
_.”
100
-
0
-_ 500
1000
Electron ~nergr teU1
5m
1000
Electron energy leV1
(12)
x 100 (%I).
200
1
loo
:
is
0
1. Cc) Es:
Penn
.
.
*
500
IO00
Electron energr (eV1
Fig. 4. The error Y, between the experimental value and the theoretical value using the AES experimental data which are limited to the data that Seah [l] referred to. The data of Tarng and Wehner [4] are excluded [AES(l)].
H. Tokutaka
et al. / Electron mean free path
357
The calculated errors are shown in figs. 4a-4c as a function of the electron energy E for our, Seah’s and Penn’s method, respectively. The average m of the absolute values IT] of the errors is calculated using the following equation as N
vi=c Iyl/N,
(13)
where N is the number of calculated errors y. The calculated values are shown in fig. 5a. The deviation u of the error y is calculated using eq. (14) to learn the distribution of the errors:
u=
[+):/N]“*.
(14)
The calculated values of u are shown in fig. 5b. As discussed in section 2, our method to obtain the mean free path is introduced using the experimental values of Tarng and Wehner [4]. These values are already included in the data which Seah referred to. If these values are included for the evaluation, the average m of the absolute values of the errors and the deviation u of the errors decrease naturally in our case. Therefore, such an evaluation is not fair. Thus, the case which is compared without these values is defined as AES(1). The case which is compared with these values included is defined as AES(2). The numbers of data for AES(l)
r
(a)
r
AES(2)
(b)
AES(l)
AES(Z)
Fig. 5. The evaluations of the mean free path using the AES experimental data. (a) Average IYI of the absolute value [Y,1of the errors (fig. 4) and (b) deviation a of the error Y, (fig. 4). The case which is compared without the experimental data of Tarng and Wehner [4] is defined as AES(1) and the case which is compared including these values is defined as AES(2).
358
H. Tokutaka
et al. / Electron mean free path
and AES(2) are 15 and 21, respectively. In fig. 5, both the value of the average error m and the deviation o are the smallest in our case, the second smallest comes from Seah’s case and the third is for Penn’s case. In every case, AES(2) shows smaller values of F] and u than AES(1). From the result, it is concluded that the experimental values of Tarng and Wehner [4] and of us [3] are accurate and reliable. 4.1.2. The published data in refs. [12]-[15] up to 1983 after Seah [I] are added to section 4.1.1 as the referred data The number of new data is 8. Thus, the numbers of data for AES(1) and AES(2) become 23 and 29, respectively. The errors y between the experimental value and the theoretical value are shown in figs. 6a-6c, as a function of the
200
200
la1
RES :
m
200
1
l-5
(b)
ES
I seah
-l--L z
;
.R ; 100
1000
-Ii1
leV)
Electron
100
L
: &
. : .
0
e
:
I:
-
:
.
.
500
Electron
energy
ICI
FE5
I Penn
I
. ..
.
o.L
:. 1
..
.
.
-
jO0
energ
1,
leV1
Electron
energy
IeVI
Fig. 6. The error Y, between the experimental value and the theoretical value using the AES experimental data where the published data up to 1983 after Seah [l] are added to fig. 4. The data of Tarng and Wehner [4] are excluded (AES(l)].
100
r
(a)
AEStl)
r
AESC-2)
(b)
AESl)
AESc2)
Fig. 7. The evaluations of the mean free path using the AES experimental fig. 6 and (b) the deviation o of fig. 6.
data. (a) Average
1Y / of
H. Tokutaka et al. / Electron mean free path
359
electron energy E. The values of m and a are also shown in figs. 7a and 7b, respectively. The results in fig. 7 are almost the same as compared with the results in fig. 5.
(al
loou
XPS : Olrs
3000
2ooo
4000
Electron energy
(b)
(eU1
XPS I Seuh
. .
. . -. .
0
2; .
-100
’
. -
;. : .-
1000
. .
2000
4000
3000
Electron energy
[e!Jl
500 -. 400
(cl
-
XPS : Penn
5 3@-J F k
203
. . t -
pO-
0 -
-; . :’ .’ : . . .
. .
-‘Y I - I IOW
zoo0
1 I 4GuO
3000
Electron energy
Fig. 8. The error y between the experimental experimental data from Seah’s references [l].
(&I
value and
the theoretical
value using
the XPS
360
H. Tokutaka
4.2. The evaluations
et al. / Electron mean
using the XPS experimental
free path
datu
It is possible to arrange the XPS experimental data into three energy ranges, which are E -C 1000 eV, E < 2000 eV and a region covering all energies. In contrast to the AES experiment, it is not necessary to consider the attenuation of the primary electron beam in a usual XPS experiment. The experimental value can be compared directly with the theoretical one which is derived from eq. (11). 4.2.1. The referred data are limited to the data which Seah [I] referred to Here, the number of data is 39. The error Y between the experimental value and the theoretical value is shown in figs. 8a-8c as a function of the electron energy E for our, Seah’s and Penn’s method, respectively. The average 1Y1 of the absolute value lY1 of the error is shown in fig. 9a. The deviation u of the error is shown in fig. 9b. In fig. 9a, the average m by our method is smallest, the smallest but one is from Seah’s method and the largest is from Penn’s method. In fig. 9b, the deviation CJby our method also shows the smallest value and one but smallest is from Seah’s method. However, Penn’s method shows the largest cr. When the evaluation is carried out using the XPS data, our method shows the best accuracy among these three methods. 4.2.2. The published data in refs. [25]-(281 up to 1983 after Seah [IJ are added to section 4.2.1 as the referred data The number of new data is now 28. Then, the number of all data becomes 67. The error Y is shown in figs. lOa-10c as a function of the electron energy E. The average m is shown in fig. lla. The deviation u of the error is shown in fig. lib. It can be seen that m and (I in fig. 11 show a behavior similar to each
c
E
(b)
(a)
EUOOOeV
all energy
Fig. 9. The evaluations of the mean free path using the XPS experimental data. (a) Average IYI of fig. 8 and (b) the deviation D of fig. 8.
H. Tokuiaka Ed al. / Electron mean free path
500
(al
400 s
361
XPS : krs
300
C 200 k ; 100 0
-. . : . .. . i *-. . .*- : . .*. .. .z* .. . ,* .. * * *_i.
-.
*
:
.
-100
I 000
.
4000
3000
2000 Electron
energy
. . .
let’1
500 XPS : Seah
400 5
3QJ
: 200 &e 100 -
0 -100 1000
2000 Electron
I .
.
3000 enerw
XPS
G -
4m
(et4
: Penn
300
+ 200 b jj 100 0 -100
Fig. 10. The error y between the experimental value and the theoretical value using the XPS experimental data where the published data up to 1983 after Seah [l] are added to fig. 8.
H. Tokutakn et al. / Eleclron mean free purh
362
other. and u shows largest section
When the energy range is extended to cover all energies, the values of fl decrease in our case as well as in that of Seah and of Penn. Our method the smallest values, the smallest but one are from Seah’s method and the are from Penn’s method. The result has a trend similar to the case of 4.2.1. However, each value in fig. 11 is larger than the corresponding
(a)
r
~OOeV E<2000eV
“i
all energy
E<2000eV
all
energy
Fig. 11. The evaluations of the mean free path using the XPS experimental fig. IO and (b) the deviation CJof fig. 10. Table 1 Details of the experimental data and theoretical for AES and (b) for XPS data
values which are used to draw figs. 4 to 11: (a) is
T
ours fl ~iEi
&
5.2 i:: 14.0 6.1 LT.0
0 4.7 -4.9 0 -1.6 0
5.3 1.8 5.3 7.7 9.3
7.5 -12.8 5.7 0 43.0
~
q.’ 5.3 11.6
,%I
(%I 6.7 22.9 3.1 44.7 85.2 64.2 15.1 -35.4 -20.1
-55.a -1.8 -6.4 36.7 21.1 3.8
‘Ei -37:1
T
seall
error
‘2rKor
6.0 3.0 9.7 4.7 5.4 5.3 5.3 8.Z L5.9 5.2 5.5 T.8 ?.5 13.3 5.2
f
data. (a) Average v\ of
3.z 4.0 6.9 3.6 4.‘ 4.3 3.8 8.1, 13.6 4.5 4.8 10.1 !0.2 15.6 4.8
,,:I ,:I
100.0 115.0 44.9 88.9 127.3 102. : 60.5 -36.9 -6.6 -48.9 12.5 -27.1 5.9 3.2 12.5
2.9 4.0 8.5 2.6 2.9 3.0 2.9 5.7 10.9 2.1 2.8 6.2 6.3 11.5 2.B
26.5 11.5 -19.8 22.1 3.4 -*0.9
-i-T 3.2 6.3 21.0 3.4 22.0
_:I”:I 21.7 -23.8 -2.9 44.0 6.7 -56.0
2.6 5.7 2.7 6.2 8.0 8.4 2.4 10.2 -
-
e, IiE)
error C%l
20.3 20.3 24.3 25.7 35.1 17.6 31.1 31.1 52.7 15.1 64.9 77.0 12.2 14.9 11.6 12.0 16.2 16.2 23.c 29.1 29.7 31.1 35.1 33.8 39.2 ,9.2 17.6 33.8 29.1 18.2 17.6
25.7 35.1 29.7
48.1 5O.C 4.5 12.2 12.2 11.9
24.1 40.5 33.8 33.4 32.1 42.6 40.9 64.4 65.2 26.6 20.3 49.8 29.7 63.6 65.6 21.c 23.c I 52.7 16.6 20.4 25.c 29.7 27.2 33.3 36. I 36.9 / - 32.4
‘r b, ” d, ef
: g:::
6.5 8.4 8.5 9.9 9.6 !S.5 26.6 :::;
:::; ‘1.7 41.9 44.8 L5.I 98.1 62.7 23.0 17.3 22.5 22.5 34.2 x8.5 19.9 26.8 27.1 37.5 31.6 46.5 46.5 :::; 20.0 21.6 :::'; 9.2 29.1 33.5 36.1 38.6 43.4 1.1 20.1 ZO.1. L2.8 16.2 26.1 30.3 31.1 44.2 65.7 47.5 24.5 -
A2.2 -46.0 -49.4 -58.3 -60.3 81.4 72.8 -13.3 10.0 -40.5 -10.1 -61.4 -44.5 L29.2 ,8.1 41.2 63.6 68.8 48.4 II.7 -9.2 -5.5 5.1 -O., -6.0 -6.4 -60.1 -26.3 -69.7 -?I.0 -23.5 48.6 56.0 32.0 42.4 29.9 -52.3 -54.5 -55.0 11.7 -23.1 -12.9 -27.3 147.4 129.3 113.0 88.9 LO5.I 99.4 L89.1 -,0.2 48.7 -17.7 64.8 51.2 195.8 14.4 162.2 29.7 25.9 -4.2 -2.0 -12.5 -24.7 -19.7 -22.3 32.2 -
11.7 12.0
13.2 20.5 23.6 10.6
13.7 19.1 19.9 21.6 23.3 34.1 34.7 ?: a:8 11.5 11.6 14.5 18.7 20.5 20.6 20.8 20.9 23.0 23.1 24.7 2L.9 35.5 SO.0 19.7 16.4 20.0 ZO.0 27.6 30.2 19.7 23.9 -24.1 12.0 9.6 19.1 19.7 10.3 $9' 13.5 16.7 17.2 10.1 19.5 20.8 21.5 22.2 "::;
la.5 18.5 13.0 15.5
22.5 25.2
::.z 34:s 35.5 23.8 -
73.5 69.2 84.1 25.4 48.7 69.2 127.0 51.9 164.8 62.5 1?8.5 121.9 223.3 170.9 34.9 ,6.4 40.9 39.7 58.6 58.8 44.2 51.0 68.8 61.7 70.4 69.7 -28.7 ,5.1 -16.3 -63.6 -10.7 56.7 75.5 48.5 76.4 65.6 -51.8 -49.0 -49.4 249.2 153.1 LOS.6 71.6 224.3 205.1 230.2 202.2 285.6 279.1 16,.‘ 4.1 139.4 18.1 186.5 179.i 255.9 24.3 184.9 27.7 31.6 11.1 x7.9 6.3 -0.9 6.4 3.9 36.1
13.0 13.5 15.5 28.5 36.0 8.4 12.6 22.2 22.6 25.7 29.1 55.8 ‘E 4:s 4.7 6.9 6.9 9.7 14.3 16.6 16.7 16.9 17.1 19.9 20.0 18.4 18.7 :;:t 19.1 12.3 16.1 16.7 27.6 32.0 13.7 18.5 -18.6 13.5 13.5 22.2 22.2 7.3 7.5 10.0 10.7 14.8 15.3 5.7 15.2 :::; 18.8 20.5 3.3 12.a 12.8 8.8 11.4 i9.9 23.8 24.5 38.0 19.6 41.5 16.7 -
56.2 50.4 56.8 -9.8 -2.5 109.5 146.8 40.1 133.2 ,6.6 12Y3.0 38.0 101.1 496.0 157.8 155.3 134.8 134.8 117.1 107.7 73.9 86.2 107.7 97.7 97.0 96.0 -4.3 80.7 -8.9 -64.6 -7.9 108.9 110.2 77.8 76.4 56.3 -30.7 -34.1 -34.4 2 lo. 4 ml.0 82.4 52.3 357.5 328.3 326.0 281.3 335.1 326.1 366.7 33.6 194.7 65.9 238.3 220.0 536.4 79.7 311.7 88.6 78.9 25.6 24.8 11.0 -12.4 -7.3 -11.1 94.0
The material where the efectron penetrates. Auger electron energy. Primary electron beam energy. D(E) are the experimental data. X(E) is the value which is read in the B direction, where il is the angle between the surface normal direction and the optics. When D(E) is read in the 8 direction, the value is defined as X(E). When D(E) is read in the surface normal direction, the value X(E) is defined as D( E),‘cos
8.
is the value which is calculated from eqs. (l)-(4). tsf E) is the vatue which is calculated from eqs. (S)-(7). I,(E) is the value which is calculated from eq. (8). ( ) are the data which are read in the surface normal direction. Photoelectron energy.
‘) I(E)
s) hi ‘) ‘)
35.1 37.6 G.0 61.6 88.5 9.7 18.0 46.6 47.9 59.0 72.2
17.1
364
H. Tokutaka
et al. / Electron mean free path
value in fig. 9. This is probably because of the large number of used data points. The number of data used here is 67 instead of 39 in section 4.2.1. Thus, our method also shows a much better result than that of Seah and that of Penn in the XPS case. The details of the experimental data and the theoretical values which are used to draw figs. 4 to 11 are tabulated in table 1.
5. Conclusions (1) When the AES data as the experimental values are compared with the theoretical values, our method shows results which are closest to the experimental data among the three theoretical methods of ourselves [5], Seah [l] and Penn [2]. This is caused by the fact that the attenuation length of the primary electron beam is considered in our method. (2) Let us take the case that the XPS data as the experimental values are compared with the theoretical values. Our method also shows the best result among these three methods. This is probably due to the fact that our method is based on the experimental values of Tarng and Wehner [4] of MO and W and of Cr of our experiment [3], which may be fairly accurate. (3) In our method, the series of the maximum and the minimum values (F, G, H, K, L and A, B, C, D, E in fig. 2a, respectively) decrease as the atomic number increases. Tarng and Wehner [4] say that the inelastic scattering cross-section increases as the number of outer shell electrons increases. Therefore, the mean free path should decrease in the order of F, G, H, K, L and A. B, C, D, E as shown in fig. 2a. Our result in fig. 2a is obtained by interpolation or extrapolation, using the experimental results of Cr (24) [3], MO (42) [4] and W (74) [4], the atomic numbers of which are very different from each other. Therefore, our results which is obtained by using the experimental result can be more practical and accurate than Seah’s and Penn’s results as discussed in section 4. (4) After Seah compared his result with that of Penn, he concluded that the mean free path obtained by him is better than that obtained by Penn. Our analysis also shows that Seah’s result is better than that of Penn. However, our method shows values closest to the experimental ones among these three methods to obtain the electron mean free path.
Acknowledgement This work was partly supported by a Grant-in-Aid from the Ministry of Education of Japan.
for Scientific
Research
H. Tokutaka
et al. / Electron mean free path
365
References [l] [2] [3] (41 [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
M.P. Seah and W.A. Dench, Surface Interface Anal. 1 (1979) 2. D.R. Penn, J. Electron Spectrosc. 9 (1976) 29. K. Nishimori, H. Tokutaka and K. Takashima, Surface Sci. 100 (1980) 665. M.L. Tarng and G.K. Wehner, J. Appl. Phys. 44 (1973) 1534. H. Tokutaka, K. Nishimori, K. Takashima and T. Ichinokawa, Surface Sci. 133 (1983) 547. C.R. Brundle, J. Vacuum Sci. Technol. 11 (1974) 212. M.P. Seah, Surface Sci. 32 (1972) 703. J.W.T. Ridgway and D. Haneman, Surface Sci. 24 (1971) 451. J.W.T. Ridgway and D. Haneman, Surface Sci. 26 (1971) 683. D.C. Jackson, T.E. Gallon and A. Chambers, Surface Sci. 36 (1973) 381. P.W. Palmberg and T.N. Rhodin, J. Appl. Phys. 39 (1968) 2425. B.C. De Cooman, V.D. Vankar and R.W. Vook, Surface Sci. 128 (1983) 128. P.W. Davies, M.A. Quinlan and G.A. Somorjai, Surface Sci. 121 (1982) 290. M. Matten and H. Liith, Surface Sci. 126 (1983) 502. S. Nakanishi and T. Horiguchi, Surface Sci. 125 (1983) 635. R.G. Steinhardt, J. Hudis and M.L. Perlman. Phys. Rev. B5 (1972) 1016. T.A. Carlson and G.E. McGuire, J. Electron Spectrosc. 1 (1972) 161. M. Klasson, A. Berndtsson, J. Hedman, R. Nilsson, R. Nyholm and C. Nordling, J. Electron Spectrosc. 3 (1974) 427. M. Klasson, J. Hedman, A. Berndtsson. R. Nilsson, C. Nordling and P. Melnik, Phys. Scripta 5 (1972) 93. H. Kanter. Phys. Rev. Bl (1970) 2357. J. Szajman, J.G. Jenkin, J. Liesegang and R.C.G. Leckey, J. Electron Spectrosc. 14 (1978) 41. R. Flitsch and S.I. Raider, J. Vacuum Sci. Technol. 12 (1975) 305. W.A. Fraser, J.V. Florio, W.N. Delgass and W.D. Robertson, Surface Sci. 36 (1973) 661. Y. Baer, P.F. Heden, J. Hedman, M. Klasson and C. Nordling, Solid State Commun. 8 (1970) 1479. P. Cadman and G.M. Gossedge, J. Electron Spectrosc. 18 (1980) 161. SM. Hall, J.D. Andrade, S.M. Ma and R.N. King, J. Electron Spectrosc. 17 (1979) 181. W.F. Egelhoff, Jr., Appl. Surface Sci. 11/12 (1982) 761. D.T. Clark. M.M. Abu-Shbak and W.J. Brennan, J. Electron Spectrosc. 28 (1982) 11.