- Email: [email protected]
Calculations of ‘effective’ inelastic mean free paths in solids
applied
surface science EI-SEVIER
Applied Surface Science lOO/lOl
(1996) 47-50
Calculations of ‘effective’ inelastic mean free paths in solids Shigeo Tanuma as*, Shingo Ichimura b, Kazuhiro Yoshihara ’ ’ Japan Energy ARC Co. Ltd.. 3-l 7-35 Nii:o-Minami,
Todu. Saitama 335. Japan ’ Electrotechnical Laboratory, I-I-4 Umezono. Tukuba, Ibaraki 305. Japan ‘ National Reseurch Institute for Metals, 1-2-l Sengen, Tukuba, Ibaraki 305, Japan Received 22 August 1995: accepted 5 November
1995
Abstract We have calculated the effective inelastic mean free paths (EIMFF%) for electrons in some elemental solids; i.e. Li, Na, K, Rb, Al, Si, Cu, Ag, and Au in the range 50-2000 eV using Monte Carlo simulation. We defined the effective inelastic mean free path as the straight distance between two successive inelastic collisions. This quantity is independent on the experimental configuration. In conclusion, the values of EIMFP could be expressed as y X A, where y is a coefficient of the effect of the elastic scattering and h the inelastic mean free path, in the range 100-2000 eV for all calculated elements. The y values obtained from the calculations were 0.85-1.0. Above the 1000 eV region for alkali metals, Al and Si, we could ignore the effect of the elastic scattering because the y values were larger than 0.95. We could not determine the y values for Au and Ag in the range 50- 100 eV because their intensities of the signals calculated by the Monte Carlo method did not vary exponentially with the distance between two successive inelastic collisions.
1. Introduction The electron inelastic mean free paths(IMFPs), attenuation lengths (ALs), and escape depth (ED) are very important physical quantities for the surface analyses by Auger and photoelectron spectroscopy. These terms have different definitions; yet many authors are unfortunately unaware of this fact and use these terms interchangeable [l]. In these quantities, IMFP must be the most basic one. There are still some problems to use IMFPs for the quantitative analysis by XPS and AES directly
* Corresponding author. Tel.: + 8 I-48-4332145; 4332150; e-mail: [email protected]. 0169.4332/96/$15.00 Copyright PII SOl69-4332(96)00254-l
fax: + 81.4%
because it ignores the elastic scattering effect completely. On the other hand, the AL includes the elastic scattering effect because it is determined experimentally using the thin film method. It has, however, been demonstrated that the attenuation of electrons in solids may not be exponentiahy related to path length [2]. The situation is additionally complicated by the fact that the AL is not only a material-dependent parameter but depends on the experimental configuration [2]. Then, we have calculated the ‘effective’ inelastic mean free path (EIMFPs) which was defined as the straight distance between two successive inelastic scatterings for some elemental solids in order to know the effect of the elastic scattering on the IMFP. This quantity is expected to be independent on the experimental configuration.
0 1996 Elsevier Science B.V. All rights reserved
2. Calculation
2.3. Flow of the Monte Carlo program
The elemental solids in which we calculated EIMFPs were lithium, sodium, potassium, rubidium, aluminum, silicon, copper, silver, and gold. The energy range calculated was 50-2000 eV. We need to know the inelastic and elastic mean free paths for electrons in solids in order to calculate the EMFP by the Monte Carlo method. The used values were obtained as described in the following sections.
2.3.1. Determination of the path length S moved between two collisions The mean free paths A, between two collisions can be expressed as 1 1 -_-_.-+&7 &,i
1
The path length obtained from S=
(3)
A,, . S between
two collisions
-&In(r).
(4)
2.1. Electron inelastic mean free paths
Here, r is a random number.
The IMFPs used for these calculations were calculated using the energy loss function obtained from optical measurements of E(W) for each material, which are based on an algorithm developed by Penn [3]. However, we used the IMFP values of rubidium calculated by TPP-2M [4] instead of the above method because we did not have its energy loss data.
2.3.2. Determination of the type scattering The type of the scattering, either tic, can be decided by the following a uniform random number r, (0 I
de
E)
= If( e>l’+
Is( ev.
Here, 8 denotes the scattering angle, and the scattering amplitudes f(0) and g(0) are obtained by solving the Dirac equation using the partial wave expansion method [6]. We adopted the analytical expression for the Thomas-Fermi-Dirac (TFD) potential, which is taking into account electron screening effects, for the calculation. We have calculated the following function R(E, 9) [51 from Eq. (1) in order to estimate the elastic scattering angle necessary for the Monte Carlo calculation. That is, for a given random number r (0 _
R(E,8)=
/
do”(
E, f3’) de’
de’ 0 Td&‘( E. 0’)
I0
de’
elastic or inelasinequality using r, < 1):
’
T(E)
q E) = (Tinel(E)+8’(E).
The differential cross-section, do “( E)/d 0, for elastic scattering used in the simulation, is given by the following equation [5]: d8’(
of the electron
cr Ine’( E) r1 <
2.2. Electron elastic scattering
can be
de’
.
(2)
(5) where, ue’(E), o.‘?E) represents the total cross section for electron elastic scattering and inelastic scattering cross section, respectively. If an electron having energy E satisfies Eq. (5) at a scattering point, it is judged that an inelastic scattering occurs. Then, we determine the straight-distance between the original point (electron emerged) and the inelastic scattering point.
3. Results and discussion 3.1. Alkaline metals The calculated results of the normalized signal intensity of electrons along the distance (Z(z)/Z) for Li, Na, K and Rb at 50 and 1500 eV are shown in Figs. 1 and 2, respectively. In these figures, the circles show the signal intensity of electrons corresponding to the distance from the emerging point of the electrons. The signal intensity F(z) represented by the dot lines are calculated from the IMFP X by the following equation: F(z)=+exp
i
--:
1
S. Tanuma et al/Applied
.g
2
loo
P.
.z E
10’
s
E 102
.B 3
L
0
100 10'
102
B 103
10-3
Elo-’ z 10-s
Jj
if b
10
20
30
40
50
z
10-4 10-s 0
10.’
x .=
.Z
2 10-2
2 E
.g Tii
40
60
60
g 102
2
10-s
10"
10-4
.N a
10-4
g 105
i?
0
200
400
600
Z 10-e 0
200
400
600
600
distance(a)
distance(A)
Fig. 1. Calculated results of effective inelastic mean free paths for lithium and rubidium at 50 and 1500 eV. Open circles: the normalized intensity calculated by the Monte Carlo method. Solid lines: intensity calculated with Eq. (6) using IMFP value A. Dot lines: curve fit result of the open circles data with Eq. (7).
These figures show that the calculated signal intensity decreases exponentially along the distance. Then, we have fitted the calculated signal intensity to the following equation, which takes into account the effect of elastic scattering as a factor y:
The resulting y factors for alkaline metals are shown in Fig. 2. This figure shows that y values are larger than 0.94 in the range 1000-2000 eV for all ele-
0
ments. In this range, we could, therefore, neglect the elastic scattering effect in a practical analysis of XPS and AES. Since the y for Na and K are 0.92-0.93 under the 100 eV region, we should consider the correction of the elastic scattering effect to the simple quantitative formalism for AES and XPS analysis in this range. 3.2. Copper, silver and gold
10
S ‘D
E lo-$ 10-e
z
20
49
distance@)
distance(A) 21
Surface Science lOO/ 101 (1996) 47-50
500
1000 1500 2000
Energy(eV) Fig. 2. The y values for alkaline metals. These values were obtained from the curve fitting to the normalized intensities calculated by the Monte Carlo method with Eq. (7).
The calculated results for copper, silver and gold at 50 and 1500 eV are shown in Fig. 3. We have determined the y factors for these elements except at 50 eV because the intensity of signal electrons at 50 eV for silver and gold did not vary exponentially with the distance. Fig. 4 shows that the ratios of elastic and inelastic scattering mean free paths versus energy. In the figure, the values of IMFP for Au and Ag at 50 eV are larger by 300% than those of EMFP. Since the contribution of the elastic scattering effect is very large, the signal intensity may not vary exponentially with the distance. At 1500 eV, we could determined the y factors for those elements. However, all lines obtained by the curve fit with Eq. (7) are larger than those of the calculated signal intensity by the MC method. Furthermore, the difference between the lines obtained by Eq. (7) and those by IMFPs using Eq. (6) are larger compared to the case of alkaline metals. The resulting y vales are shown in Fig. 5. This figure shows that the factor y is increasing according to the increase in electron energy for copper and
o0
1000 Energy(eV)
2000
Fig. 3. The ratios of inelastic mean free paths and elastic mean free paths versus energy for aluminum, silicon, copper, silver. and gold.
S. Tanuma et al. /Applied Sur$ace Science 100 / 101 (1996147-50
50 x
100
2. 100
C * = B
lo-'
-5 g
10-l
G
10-z
E ;
10-z
.g
10-3
.g
10-3
3
3
k z
i z
lo' 10-s 0
10
20
30
40
1o-4 10-s 0
5
distance(a) 2.
100
-5 5
10-i
lo-'
E ;
.z = d
15 20
25
30
a. 100
10~2
5
10'2
.zj 10-3
.;
103
3 E
Tz
B z
10
distance(a)
10' 10-s 0
k z
loA 10-s 0
150
d:anc$)
30
60
90
120 150
distance(A) Fig. 4. Calculated results of the effective inelastic mean free paths for silver and gold at 50 and 1500 eV. See Fig. 1.
silver. Over the 1000 eV region, the differences between IMFP and EIMFP are less than 10%. It might be difficult to determine the y factor if the
1
b 0.9
is
-s ?-0.8 0.7
I.... 0
I
500
0..
1000
8..
1500
I
I
2000
Energy(eV) Fig. 5. The y values for copper, silver, and gold. See Fig. 2.
differences between them are large (over 20’33’0) such as with silver and gold at 50 eV. The reason is that the signal intensity could not vary exponentially with distance due to the elastic scattering effect. We could not. therefore, apply Eq. (7) to the analysis of the calculated signal intensity by the MC method. It is said that the detection angle of electrons is the most important factor for determining AL (or EAL) and DDF. However. our series of calculations indicate that the effective inelastic mean free path, which is considered to be a bulk quantity, largely depends on the elastic scattering effect. This quantity is also independent on the experimental configuration. Especially for Au and Ag at low energies, the attenuation of the signal electron is not exponential with the distance from the starting point of the electron due to the mainly elastic scattering effect (unless the effect of detection angle is included). References [II C.J. Powell, J. Electron Spectrosc. Rel. Phen. 47 (1988) 197. 01 A. Jablonski. Surf. Sci. 188 (1987) 164; A. Jablonski and H. Ebel. Surf. Interf. Anal. 11 (1988) 627: A. Jablonski. Surf. Interf. Anal. 14 (1989) 659; W.H. Gries and W. Werner, Surf. Interf. Anal. 16 (1990) 149: A. Jablonski and S. Tougaard, J. Vat. Sci. Technol. A 8 (1990) 106: A. Jablonski. Surf. Interf. Anal. 15 11990) 559; Z.-J. Ding, Ph.D. Thesis, Osaka University (1990): W.S.M. Werner, W.H. Cries and H. Stori, J. Vat. Sci. Technol. A 9 (1991) 21; S.M. Werner, W.H. Gries and H. Stori, Surf. Interf. Anal. 18 (1992) 217; I.S. Tilinin and W.S.M. Werner. Phys. Rev. B 46 (1993) 13739. 131D.R. Penn, Phys. Rev. B 35 (1987) 182. [41 S. Tanuma, C.J. Powell and D.R. Penn, Surf. Interf. Anal. 21 (1994) 165. 151S. Ichimura. Surf. Sci. Jpn. I I (1990) 604. [61N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions (Oxford University Press, London. 1965).
surface science EI-SEVIER
Applied Surface Science lOO/lOl
(1996) 47-50
Calculations of ‘effective’ inelastic mean free paths in solids Shigeo Tanuma as*, Shingo Ichimura b, Kazuhiro Yoshihara ’ ’ Japan Energy ARC Co. Ltd.. 3-l 7-35 Nii:o-Minami,
Todu. Saitama 335. Japan ’ Electrotechnical Laboratory, I-I-4 Umezono. Tukuba, Ibaraki 305. Japan ‘ National Reseurch Institute for Metals, 1-2-l Sengen, Tukuba, Ibaraki 305, Japan Received 22 August 1995: accepted 5 November
1995
Abstract We have calculated the effective inelastic mean free paths (EIMFF%) for electrons in some elemental solids; i.e. Li, Na, K, Rb, Al, Si, Cu, Ag, and Au in the range 50-2000 eV using Monte Carlo simulation. We defined the effective inelastic mean free path as the straight distance between two successive inelastic collisions. This quantity is independent on the experimental configuration. In conclusion, the values of EIMFP could be expressed as y X A, where y is a coefficient of the effect of the elastic scattering and h the inelastic mean free path, in the range 100-2000 eV for all calculated elements. The y values obtained from the calculations were 0.85-1.0. Above the 1000 eV region for alkali metals, Al and Si, we could ignore the effect of the elastic scattering because the y values were larger than 0.95. We could not determine the y values for Au and Ag in the range 50- 100 eV because their intensities of the signals calculated by the Monte Carlo method did not vary exponentially with the distance between two successive inelastic collisions.
1. Introduction The electron inelastic mean free paths(IMFPs), attenuation lengths (ALs), and escape depth (ED) are very important physical quantities for the surface analyses by Auger and photoelectron spectroscopy. These terms have different definitions; yet many authors are unfortunately unaware of this fact and use these terms interchangeable [l]. In these quantities, IMFP must be the most basic one. There are still some problems to use IMFPs for the quantitative analysis by XPS and AES directly
* Corresponding author. Tel.: + 8 I-48-4332145; 4332150; e-mail: [email protected]. 0169.4332/96/$15.00 Copyright PII SOl69-4332(96)00254-l
fax: + 81.4%
because it ignores the elastic scattering effect completely. On the other hand, the AL includes the elastic scattering effect because it is determined experimentally using the thin film method. It has, however, been demonstrated that the attenuation of electrons in solids may not be exponentiahy related to path length [2]. The situation is additionally complicated by the fact that the AL is not only a material-dependent parameter but depends on the experimental configuration [2]. Then, we have calculated the ‘effective’ inelastic mean free path (EIMFPs) which was defined as the straight distance between two successive inelastic scatterings for some elemental solids in order to know the effect of the elastic scattering on the IMFP. This quantity is expected to be independent on the experimental configuration.
0 1996 Elsevier Science B.V. All rights reserved
2. Calculation
2.3. Flow of the Monte Carlo program
The elemental solids in which we calculated EIMFPs were lithium, sodium, potassium, rubidium, aluminum, silicon, copper, silver, and gold. The energy range calculated was 50-2000 eV. We need to know the inelastic and elastic mean free paths for electrons in solids in order to calculate the EMFP by the Monte Carlo method. The used values were obtained as described in the following sections.
2.3.1. Determination of the path length S moved between two collisions The mean free paths A, between two collisions can be expressed as 1 1 -_-_.-+&7 &,i
1
The path length obtained from S=
(3)
A,, . S between
two collisions
-&In(r).
(4)
2.1. Electron inelastic mean free paths
Here, r is a random number.
The IMFPs used for these calculations were calculated using the energy loss function obtained from optical measurements of E(W) for each material, which are based on an algorithm developed by Penn [3]. However, we used the IMFP values of rubidium calculated by TPP-2M [4] instead of the above method because we did not have its energy loss data.
2.3.2. Determination of the type scattering The type of the scattering, either tic, can be decided by the following a uniform random number r, (0 I
de
E)
= If( e>l’+
Is( ev.
Here, 8 denotes the scattering angle, and the scattering amplitudes f(0) and g(0) are obtained by solving the Dirac equation using the partial wave expansion method [6]. We adopted the analytical expression for the Thomas-Fermi-Dirac (TFD) potential, which is taking into account electron screening effects, for the calculation. We have calculated the following function R(E, 9) [51 from Eq. (1) in order to estimate the elastic scattering angle necessary for the Monte Carlo calculation. That is, for a given random number r (0 _
R(E,8)=
/
do”(
E, f3’) de’
de’ 0 Td&‘( E. 0’)
I0
de’
elastic or inelasinequality using r, < 1):
’
T(E)
q E) = (Tinel(E)+8’(E).
The differential cross-section, do “( E)/d 0, for elastic scattering used in the simulation, is given by the following equation [5]: d8’(
of the electron
cr Ine’( E) r1 <
2.2. Electron elastic scattering
can be
de’
.
(2)
(5) where, ue’(E), o.‘?E) represents the total cross section for electron elastic scattering and inelastic scattering cross section, respectively. If an electron having energy E satisfies Eq. (5) at a scattering point, it is judged that an inelastic scattering occurs. Then, we determine the straight-distance between the original point (electron emerged) and the inelastic scattering point.
3. Results and discussion 3.1. Alkaline metals The calculated results of the normalized signal intensity of electrons along the distance (Z(z)/Z) for Li, Na, K and Rb at 50 and 1500 eV are shown in Figs. 1 and 2, respectively. In these figures, the circles show the signal intensity of electrons corresponding to the distance from the emerging point of the electrons. The signal intensity F(z) represented by the dot lines are calculated from the IMFP X by the following equation: F(z)=+exp
i
--:
1
S. Tanuma et al/Applied
.g
2
loo
P.
.z E
10’
s
E 102
.B 3
L
0
100 10'
102
B 103
10-3
Elo-’ z 10-s
Jj
if b
10
20
30
40
50
z
10-4 10-s 0
10.’
x .=
.Z
2 10-2
2 E
.g Tii
40
60
60
g 102
2
10-s
10"
10-4
.N a
10-4
g 105
i?
0
200
400
600
Z 10-e 0
200
400
600
600
distance(a)
distance(A)
Fig. 1. Calculated results of effective inelastic mean free paths for lithium and rubidium at 50 and 1500 eV. Open circles: the normalized intensity calculated by the Monte Carlo method. Solid lines: intensity calculated with Eq. (6) using IMFP value A. Dot lines: curve fit result of the open circles data with Eq. (7).
These figures show that the calculated signal intensity decreases exponentially along the distance. Then, we have fitted the calculated signal intensity to the following equation, which takes into account the effect of elastic scattering as a factor y:
The resulting y factors for alkaline metals are shown in Fig. 2. This figure shows that y values are larger than 0.94 in the range 1000-2000 eV for all ele-
0
ments. In this range, we could, therefore, neglect the elastic scattering effect in a practical analysis of XPS and AES. Since the y for Na and K are 0.92-0.93 under the 100 eV region, we should consider the correction of the elastic scattering effect to the simple quantitative formalism for AES and XPS analysis in this range. 3.2. Copper, silver and gold
10
S ‘D
E lo-$ 10-e
z
20
49
distance@)
distance(A) 21
Surface Science lOO/ 101 (1996) 47-50
500
1000 1500 2000
Energy(eV) Fig. 2. The y values for alkaline metals. These values were obtained from the curve fitting to the normalized intensities calculated by the Monte Carlo method with Eq. (7).
The calculated results for copper, silver and gold at 50 and 1500 eV are shown in Fig. 3. We have determined the y factors for these elements except at 50 eV because the intensity of signal electrons at 50 eV for silver and gold did not vary exponentially with the distance. Fig. 4 shows that the ratios of elastic and inelastic scattering mean free paths versus energy. In the figure, the values of IMFP for Au and Ag at 50 eV are larger by 300% than those of EMFP. Since the contribution of the elastic scattering effect is very large, the signal intensity may not vary exponentially with the distance. At 1500 eV, we could determined the y factors for those elements. However, all lines obtained by the curve fit with Eq. (7) are larger than those of the calculated signal intensity by the MC method. Furthermore, the difference between the lines obtained by Eq. (7) and those by IMFPs using Eq. (6) are larger compared to the case of alkaline metals. The resulting y vales are shown in Fig. 5. This figure shows that the factor y is increasing according to the increase in electron energy for copper and
o0
1000 Energy(eV)
2000
Fig. 3. The ratios of inelastic mean free paths and elastic mean free paths versus energy for aluminum, silicon, copper, silver. and gold.
S. Tanuma et al. /Applied Sur$ace Science 100 / 101 (1996147-50
50 x
100
2. 100
C * = B
lo-'
-5 g
10-l
G
10-z
E ;
10-z
.g
10-3
.g
10-3
3
3
k z
i z
lo' 10-s 0
10
20
30
40
1o-4 10-s 0
5
distance(a) 2.
100
-5 5
10-i
lo-'
E ;
.z = d
15 20
25
30
a. 100
10~2
5
10'2
.zj 10-3
.;
103
3 E
Tz
B z
10
distance(a)
10' 10-s 0
k z
loA 10-s 0
150
d:anc$)
30
60
90
120 150
distance(A) Fig. 4. Calculated results of the effective inelastic mean free paths for silver and gold at 50 and 1500 eV. See Fig. 1.
silver. Over the 1000 eV region, the differences between IMFP and EIMFP are less than 10%. It might be difficult to determine the y factor if the
1
b 0.9
is
-s ?-0.8 0.7
I.... 0
I
500
0..
1000
8..
1500
I
I
2000
Energy(eV) Fig. 5. The y values for copper, silver, and gold. See Fig. 2.
differences between them are large (over 20’33’0) such as with silver and gold at 50 eV. The reason is that the signal intensity could not vary exponentially with distance due to the elastic scattering effect. We could not. therefore, apply Eq. (7) to the analysis of the calculated signal intensity by the MC method. It is said that the detection angle of electrons is the most important factor for determining AL (or EAL) and DDF. However. our series of calculations indicate that the effective inelastic mean free path, which is considered to be a bulk quantity, largely depends on the elastic scattering effect. This quantity is also independent on the experimental configuration. Especially for Au and Ag at low energies, the attenuation of the signal electron is not exponential with the distance from the starting point of the electron due to the mainly elastic scattering effect (unless the effect of detection angle is included). References [II C.J. Powell, J. Electron Spectrosc. Rel. Phen. 47 (1988) 197. 01 A. Jablonski. Surf. Sci. 188 (1987) 164; A. Jablonski and H. Ebel. Surf. Interf. Anal. 11 (1988) 627: A. Jablonski. Surf. Interf. Anal. 14 (1989) 659; W.H. Gries and W. Werner, Surf. Interf. Anal. 16 (1990) 149: A. Jablonski and S. Tougaard, J. Vat. Sci. Technol. A 8 (1990) 106: A. Jablonski. Surf. Interf. Anal. 15 11990) 559; Z.-J. Ding, Ph.D. Thesis, Osaka University (1990): W.S.M. Werner, W.H. Cries and H. Stori, J. Vat. Sci. Technol. A 9 (1991) 21; S.M. Werner, W.H. Gries and H. Stori, Surf. Interf. Anal. 18 (1992) 217; I.S. Tilinin and W.S.M. Werner. Phys. Rev. B 46 (1993) 13739. 131D.R. Penn, Phys. Rev. B 35 (1987) 182. [41 S. Tanuma, C.J. Powell and D.R. Penn, Surf. Interf. Anal. 21 (1994) 165. 151S. Ichimura. Surf. Sci. Jpn. I I (1990) 604. [61N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions (Oxford University Press, London. 1965).