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Orientation dependence of overlayer attenuation of electrons for the cylindrical mirror analyzer and a retarding field analyzer
Journal of Electron Spectroscopy and Related Phenomena, 3 (1974) 417425 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
ORIENTATION DEPENDENCE OF OVERLAYER ELECTRONS FOR THE CYLINDRICAL MIRROR RETARDING FIELD ANALYZER
ATTENUATION OF ANALYZER AND A
J. C. SHELTON Bell Laboratories, Murray Hi/i, NJ.
07974 (U.S.A.)
(Received 19 February 1974)
ABSTRACT
The effect of a homogeneous flat overlayer in attenuating spectral lines is analyzed as a function of orientation for apertures corresponding to a 42.3 ’ cylindrical mirror analyzer and 48” retarding field analyzer. Exponential attenuation with path length is assumed. Results are presented graphically for both isotropic and cosine distributions of sources. The numerical results are particularly useful when practical considerations require the use of an unsymmetrical sample orientation. The results are pertinent to measurements of inelastic mean free paths and are also related to the response of these spectrometers to different depth distributions of elements. INTRODUCTION
This note gives the orientation and overlayer thickness dependence for exponential attenuation of spectral lines by a uniform flat overlayer. Unequal pathlengths through the layer into different elementary solid angles of the spectrometer aperture result in a superposition which is not exponential with overlayer thickness. We give the result of the superposition for apertures of two widely used electron spectrometers: the 42.3’ cylindrical mirror analyzer (CMA)l and a 48 * spherical grid retarding field analyzer (RFA)2. The attenuation by overlayers is pertinent to electron spectroscopies such as Auger and photoemission spectroscopies when the intensities of electrons of characteristic energies are to be quantitatively used for surface studies. MODEL
The overlayer
attenuation
functions
are given by the model equation
418
Figure 1. Geometric parameters. c = semicone angle of the analyzer aperture. r = rotation angle of the layer normal n from the analyzer axis, a. 62 = unit vector toward an elementary solid angle within the aperture, making angle 0 with respect to VZ.x = overlayer thickness.
This equation describes the overlayer attenuation as a function of overlayer rotation, Y, and overlayer thickness, x, under the following assumptions which, since the pioneering work of Harris3, have been widely used in electron spectroscopy4-s. Geometric parameters are defined in Figure 1. a. For the CMA, the aperture RO covers the range of emission directions S2 along the cone of semicone angle c = 42.3 O. For the RFA considered here, the aperture RO covers the range of emission directions &I within the cone of semicone angle c = 48”. Mechanical obstructions and grid transparencies are ignored. The analyzer collects electrons having straight trajectories from the source into elemental solid angles within Q. Q is that part of the analyzer aperture !& which has &&II > 0. b. The overlayer is homogeneous, uniform and flat. An electron which leaves the source along Q escapes the layer with probability given by emPX Se’’ where p is the (uniform) attenuation probability per unit path length, the reciprocal of the inelastic mean free path. @LX is the layer thickness in units of the inelastic mean free path.) Assuming p to be constant results in great simplification. Tracy’ has recently shown that with certain aluminum deposition techniques on some substrates the substrate Auger peak heights fit the exponential model over several decades. This is consistent with Feibelman’s model calculation1 ‘, which for normal emission from jellium at the density of aluminum shows that the interaction probability for a 300 eV electron is approximately given by a constant p, truncated sharply to zero at the surface. c. Two cases of angular dependencekof emission from the source are considered:
419 The isotropic distribution is a first approximation to the emission from single atoms, and may, for example, approximate the angular dependence of emission of an Auger electron in a solid (ignoring structural effects on the emission process). The cosine distribution describes the angular distribution from a uniform isotropically emitting and absorbing continuum and therefore is a first approximation to emission from a substrate. We assume a(0) is independent of the rotation r, and therefore exclude the effect of variation of the excitation by primaries3’ 5V” ’ ” I2 and secondaries” 5* ‘* ’3, 14
.
d. The functions are normalized the values of r = 0 O, px = 0. The working equations are: for the CMA:
for each analyzer
and s(0) distribution
to
x
ACMA (r, px) =
&- se-""
se= ’~(0)
dp
(2)
0
rpm
e(F) = a-n =
arccos 51 n = arccos (cos c cos r 0 [ arccos (cotc cotr):
<
sin c sin r cos 50)
90” -
c
r > 90” -
c
r
and for the RFA: (r, px) =
p*
L
-$
, rf
urn
Ic - rl
e- Irx ‘==’~(0)
I
1
(3)
0 Ic +
A2 =
(A, + Ad
q,(e)
= arccos
r>c
d6’
cos c - cos 8 cos r
(
sin 6 sin r
)
The four normalizing constants A, are chosen such that A(O,O) = 1 for each spectrometer-angular distribution combination. RESULTS
AND Figure
DISCUSSION
2 gives A::*
(I-, px),
the attenuation
function computed
from eqn. (2)
420 for the CMA for an isotropically distributed source, and Figure 3 gives AztA(r, PX) for a cosine source. At large PX the emission through the overlayer is strongly peaked along the outward normal, resulting in a relative maximum in A@, FX) as this peak is rotated toward the annular aperture. The format for these figures was chosen for economy of presentation. (Besides the attenuation function varying with r, so does the rate of excitation, not included in the figures. The excitation rate may contribute comparable r-dependence of the measured intensity of characteristic electrons.) Figure 4 gives thickness variation for r = 45 a taken from Figures 2 and 3, each renormalized to zero thickness and 45”. r = 45” was chosen because it is a typical arrangement for many analyzers lacking coaxial excitation sources, for carrousei holders and for ion miiiing studies. For comparison, e - ’.’ ’ px recommended
-‘t=:--r=q& y-*+\
-+-
+I+
-+\
PX
,o.o
+ +\$\
+.+?>\i$+\+~~---~:~5
‘+.
+-0.2
+4+ +\
+\
0.02
+-
0.5
+‘+.
++\ +--+-. +-+ 1 +u\
Id
-I
1.o
“+-
-+-+-+_ +-%+
‘+1+_
‘+\+-
2.0
2.0
1
IO-* t
a/
I+-+‘+
“\
/ I
0
20
I
40
/so *
60
I
80
i 3
16’;
+/+-+-+L+
\ I
20
I
r,
40
----5.0 GO
rotation (degrees)
80
!! 0
r, rotation (degrees) Figure 2 (left). AisocMA(r, ,ux) for the CMA detecting an isotropically emitting source through a flat ovedayer of thickness x. ,u is the interaction probability per unit pathlength in the overlayer. In the figures, smooth curves join the computed points, marked by “t” or “0”. Figure 3 (right). AcosCMA(r, ,ux) as for Figure 2 with a cosine distribution of the emitting source,
421
0.1 -
COSINE
DISTRIBUTED
ISOTROPlCALLY DISTRIBUTED EMISSION
0.01 -
i 0.001’
I I
I 2
I 3
I 4
I\ 5
I 6
LLX
Figure
4. A~~~CMA(~~~, ,uux)/A~~~C~*(~~~, 0) and AcosCMA(450, ,ux)IA~~~~~*(45~, 0) compared to ,-I.332 . by Seah6 for A,,, CMA(45O,px) is seen to be a reasonable approximation, being 7 oA high at pcx = 0.5 and 50 % low at PX = 5. Similarly, Figures 5 and 6 give A::*@, PX) and AtE*(r, PX). Figure 7 gives thickness dependence for r = 60 O, each renormalized to ARFA(60 ‘, 0) and for comparison, e - ’.3 3 Ps which for A~~*(60°, PX) is in error by 25 ‘A at px = 1. Y = 60” was chosen as a convenient rotation for good lateral resolution from an electron gun normal to the spectrometer axis. Although in Figures 4 and 7 the attenuation of the cosine distribution is slower than of the isotropic, this result is not general. The fraction of the analyzer aperture solid angle having 82. n > 0 is given for the CMA by AkF*(r, 0) f rom Figure 2 and for the RFA by Ax*(r, 0) from Figure 5. Within the modeI, the functions A(r, px) are directly applicable to the attenuation by a uniform overlayer. They are especially useful when the excitation of the source of characteristic electrons is negligibly affected by the overlayer. For example, in Auger spectroscopy, it is sufficient that the overlayer be thin compared to the
,j 20
40
60
80 r, rotation
+-yy_, 0
100 (degrees)
20
, 40
,
100 60 80 r, rotation (degrees)
Figure 5 (left). AL~~RF*(~,,ux), as for Figure 2 for a 48” semicone angle RFA with an isotropically emitting source. Figure 6 (right). AeosRFA(~, ,Ux ) as for Figure 5 with a cosine distribution of the emitting source.
projection onto the surface normal of the inelastic mean free path of the primary electrons. The results are ylso useful as a kernel for estimating the thickness dependence of the signal strength from any depth distribution of sources E(X) (e.g., with isotropic angular distribution) via either a continuum model,
EMU>] =
7 Aiso (r, /CC’) E(x’) dx’
(4)
0
or a discrete model, which might be written as r[&(?U?)] = n$ where 8 describes
4so
G-,
Wa)
(5)
E (na)
the depth’ distribution
of the emission,
and p is constant.
This
423 I
0.1
0.0 I-
CL00 t-
Figure 7. &,,RF*(60°, e--1.33Lkcm
,ux)/A-~~~~*(~O~, 0) and AcosRFA(600,
,ux)/A~o~~~~(60~,
0) ~~~p~ed
to
,
generalization of the model of Gallon4 which does not account for the analyzer geometry is similar to that of Seah 6. For example, thickness dependence of signal generated by a uniformly emitting overlayer would be found in the continuum model by 3c s
Aiso (I, /CC’)dx’.
0
More generally, if the excitation rate per atom emitting the characteristic electrons and the p describing the attenuation are constant, then moments of the depth distribution of the emitting element can be obtained from the intensity of each of its spectral lines relative to the intensity in pure material of the element. This relative intensity is given by
424 TABLE
1
EFFECTIVE ATTENUATION THE CMA AND THE RFA
DEPTH AS MULTIPLE FOR ,ux RANGING FROM
Cylindrical mirror analyzer
OF OVERLAYER 0.05 TO 5
THICKNESS
IN
Retarding field analyzer
D/x at r =
cos
IS0
DJx at r =
COS
IS0
0” 45”
1.35 1.80 to 1.17
1.35 3.96 to 1.29
10” 20” 60”
1.22 to 1.16 1.27 to 1.16 1.88 to 1.31
1.24 to 1.18 1.34 to 1.19 3.50 to 1.41
where pP and eP are uniform values with depth in the pure material and =*, r=o
0.740 for the CMA and 0.835 for the RFA.
Since an independent result is obtained from each different line (because of the dependence of p on electron energy1 5), this approach may be useful in analysis of depth distributions of elements with several characteristic lines. Approximating the attenuation functions with an exponential A(r, q)/ A(r, 0) N eePD where D is the effective attenuation depth is exact for the CMA at r = 0”; there, D = x set 42.3 o = 1.35 x. Otherwise, this approximation is not exact since exponentials with different multipliers, set 8, are superposed, and the contributions with low 0 dominate with increasing PX. I.e., D/X is not constant but decreases with increasing px as may be seen in Figures 4 and 7. The value of D/x varies significantly, as typified by Table 1. Conversely, erroneous values for the inelastic mean free path may be obtained if the attenuation of line strength with overlayer thickness is taken as exponential and a constant geometric correction factor is used. On the other hand, the departure from exponential (especially for large I) may be used to advantage to test the model assumptions or to provide an independent measure of overlayer thickness by comparison of experimental line strengths as a function of overlayer thickness, to both the shape and magnitude of the appropriate model (e.g. Figure 4). The test may be especially valuable if several characteristic lines from a source are expected to follow the model assumptions, for they should have the same dependence on overlayer thickness, scaled by the inelastic mean free paths at the energies of the spectral lines. CONCLUSIONS
We have given model results for attenuation uniformly attenuating flat layer for two commonly
of lines in a spectrum by a used electron spectrometers.
425 The relevant parameters are the orientation of the layer, the over-layer thickness in units of the inelastic mean free path, and the angular distribution of the emission from the source. Results have been given graphically for the simplest angular distributions: isotropic (approximating an isolated atom or thin layer) and cosine (approximating a uniformly emitting source which is thick compared to the inelastic mean free path for electrons in the spectral line). These results may be used to rapidly estimate the effects of nonuniform depth distributions, by superposition. Deviation of the model attenuation from an exponential with overlayer thickness is large if the overlayer normal is not nearly along the axis of the analyzer. The quantitative results are pertinent to assessment of the inelastic mean free path model, to measurement of inelastic mean free paths, and to application of the model in practical situations. ACKNOWLEDGEMENTS
Useful conversations with C. C. Chang and H. R. Patil and the support of the Materials Science Center at Cornell University during initial stages of this work are gratefully acknowledged. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
P. W. Palmberg, G. K. Bohn, J. C. Tracy, Appl. Phys. Lett., 15 (1969) 254. eg. Varian LEED optics 981-24. L. A. Harris, (a) J. Appf. Phys., 39 (1968) 1419, (b) Surf. Sci., 15 (1969) 77. T. E. GaIlon, Surf Ski., 17 (1969) 486. K. Jacobi, J. Holzl, Surf: Sci., 26 (1971) 54. M. P. Seah, Surf; Sci., 32 (1972) 703. M. L. Tarng, G. K. Wehner, J. Appl. Phys., 44 (1973) 1534. M. P. Seah, J. Phys. F, 3 (1973) 1538. J. C. Tracy, J. Vat. Sci. Technol., 14 (1974) 280. Peter J. Feibelman, Surf. Sci., 36 (1973) 558. J. J. Vrakking, F. Meyer, Surf: Sci., 35 (1973) 34. T. W. Busch, J. P. Bertino, W. P. Ellis, Appl. P&s. Left., 23 (1973) 359. T. E. Gallon, J. Phys. D, 5 (1972) 822. J. H. Neave, C. T. Foxon, B. A. Joyce, Surf. Sci., 29 (1972) 411. For dependenceof p on electronenergy in jellium, see J. C. Shelton, Surf. Sci., in press. Experimental results for ,u (uncorrectedfor the geometrical effects described herein) are summarized by J. C. Rivi&re,Contemp. Phys., 14 (1973) 513.
ORIENTATION DEPENDENCE OF OVERLAYER ELECTRONS FOR THE CYLINDRICAL MIRROR RETARDING FIELD ANALYZER
ATTENUATION OF ANALYZER AND A
J. C. SHELTON Bell Laboratories, Murray Hi/i, NJ.
07974 (U.S.A.)
(Received 19 February 1974)
ABSTRACT
The effect of a homogeneous flat overlayer in attenuating spectral lines is analyzed as a function of orientation for apertures corresponding to a 42.3 ’ cylindrical mirror analyzer and 48” retarding field analyzer. Exponential attenuation with path length is assumed. Results are presented graphically for both isotropic and cosine distributions of sources. The numerical results are particularly useful when practical considerations require the use of an unsymmetrical sample orientation. The results are pertinent to measurements of inelastic mean free paths and are also related to the response of these spectrometers to different depth distributions of elements. INTRODUCTION
This note gives the orientation and overlayer thickness dependence for exponential attenuation of spectral lines by a uniform flat overlayer. Unequal pathlengths through the layer into different elementary solid angles of the spectrometer aperture result in a superposition which is not exponential with overlayer thickness. We give the result of the superposition for apertures of two widely used electron spectrometers: the 42.3’ cylindrical mirror analyzer (CMA)l and a 48 * spherical grid retarding field analyzer (RFA)2. The attenuation by overlayers is pertinent to electron spectroscopies such as Auger and photoemission spectroscopies when the intensities of electrons of characteristic energies are to be quantitatively used for surface studies. MODEL
The overlayer
attenuation
functions
are given by the model equation
418
Figure 1. Geometric parameters. c = semicone angle of the analyzer aperture. r = rotation angle of the layer normal n from the analyzer axis, a. 62 = unit vector toward an elementary solid angle within the aperture, making angle 0 with respect to VZ.x = overlayer thickness.
This equation describes the overlayer attenuation as a function of overlayer rotation, Y, and overlayer thickness, x, under the following assumptions which, since the pioneering work of Harris3, have been widely used in electron spectroscopy4-s. Geometric parameters are defined in Figure 1. a. For the CMA, the aperture RO covers the range of emission directions S2 along the cone of semicone angle c = 42.3 O. For the RFA considered here, the aperture RO covers the range of emission directions &I within the cone of semicone angle c = 48”. Mechanical obstructions and grid transparencies are ignored. The analyzer collects electrons having straight trajectories from the source into elemental solid angles within Q. Q is that part of the analyzer aperture !& which has &&II > 0. b. The overlayer is homogeneous, uniform and flat. An electron which leaves the source along Q escapes the layer with probability given by emPX Se’’ where p is the (uniform) attenuation probability per unit path length, the reciprocal of the inelastic mean free path. @LX is the layer thickness in units of the inelastic mean free path.) Assuming p to be constant results in great simplification. Tracy’ has recently shown that with certain aluminum deposition techniques on some substrates the substrate Auger peak heights fit the exponential model over several decades. This is consistent with Feibelman’s model calculation1 ‘, which for normal emission from jellium at the density of aluminum shows that the interaction probability for a 300 eV electron is approximately given by a constant p, truncated sharply to zero at the surface. c. Two cases of angular dependencekof emission from the source are considered:
419 The isotropic distribution is a first approximation to the emission from single atoms, and may, for example, approximate the angular dependence of emission of an Auger electron in a solid (ignoring structural effects on the emission process). The cosine distribution describes the angular distribution from a uniform isotropically emitting and absorbing continuum and therefore is a first approximation to emission from a substrate. We assume a(0) is independent of the rotation r, and therefore exclude the effect of variation of the excitation by primaries3’ 5V” ’ ” I2 and secondaries” 5* ‘* ’3, 14
.
d. The functions are normalized the values of r = 0 O, px = 0. The working equations are: for the CMA:
for each analyzer
and s(0) distribution
to
x
ACMA (r, px) =
&- se-""
se= ’~(0)
dp
(2)
0
rpm
e(F) = a-n =
arccos 51 n = arccos (cos c cos r 0 [ arccos (cotc cotr):
<
sin c sin r cos 50)
90” -
c
r > 90” -
c
r
and for the RFA: (r, px) =
p*
L
-$
, rf
urn
Ic - rl
e- Irx ‘==’~(0)
I
1
(3)
0 Ic +
A2 =
(A, + Ad
q,(e)
= arccos
r>c
d6’
cos c - cos 8 cos r
(
sin 6 sin r
)
The four normalizing constants A, are chosen such that A(O,O) = 1 for each spectrometer-angular distribution combination. RESULTS
AND Figure
DISCUSSION
2 gives A::*
(I-, px),
the attenuation
function computed
from eqn. (2)
420 for the CMA for an isotropically distributed source, and Figure 3 gives AztA(r, PX) for a cosine source. At large PX the emission through the overlayer is strongly peaked along the outward normal, resulting in a relative maximum in A@, FX) as this peak is rotated toward the annular aperture. The format for these figures was chosen for economy of presentation. (Besides the attenuation function varying with r, so does the rate of excitation, not included in the figures. The excitation rate may contribute comparable r-dependence of the measured intensity of characteristic electrons.) Figure 4 gives thickness variation for r = 45 a taken from Figures 2 and 3, each renormalized to zero thickness and 45”. r = 45” was chosen because it is a typical arrangement for many analyzers lacking coaxial excitation sources, for carrousei holders and for ion miiiing studies. For comparison, e - ’.’ ’ px recommended
-‘t=:--r=q& y-*+\
-+-
+I+
-+\
PX
,o.o
+ +\$\
+.+?>\i$+\+~~---~:~5
‘+.
+-0.2
+4+ +\
+\
0.02
+-
0.5
+‘+.
++\ +--+-. +-+ 1 +u\
Id
-I
1.o
“+-
-+-+-+_ +-%+
‘+1+_
‘+\+-
2.0
2.0
1
IO-* t
a/
I+-+‘+
“\
/ I
0
20
I
40
/so *
60
I
80
i 3
16’;
+/+-+-+L+
\ I
20
I
r,
40
----5.0 GO
rotation (degrees)
80
!! 0
r, rotation (degrees) Figure 2 (left). AisocMA(r, ,ux) for the CMA detecting an isotropically emitting source through a flat ovedayer of thickness x. ,u is the interaction probability per unit pathlength in the overlayer. In the figures, smooth curves join the computed points, marked by “t” or “0”. Figure 3 (right). AcosCMA(r, ,ux) as for Figure 2 with a cosine distribution of the emitting source,
421
0.1 -
COSINE
DISTRIBUTED
ISOTROPlCALLY DISTRIBUTED EMISSION
0.01 -
i 0.001’
I I
I 2
I 3
I 4
I\ 5
I 6
LLX
Figure
4. A~~~CMA(~~~, ,uux)/A~~~C~*(~~~, 0) and AcosCMA(450, ,ux)IA~~~~~*(45~, 0) compared to ,-I.332 . by Seah6 for A,,, CMA(45O,px) is seen to be a reasonable approximation, being 7 oA high at pcx = 0.5 and 50 % low at PX = 5. Similarly, Figures 5 and 6 give A::*@, PX) and AtE*(r, PX). Figure 7 gives thickness dependence for r = 60 O, each renormalized to ARFA(60 ‘, 0) and for comparison, e - ’.3 3 Ps which for A~~*(60°, PX) is in error by 25 ‘A at px = 1. Y = 60” was chosen as a convenient rotation for good lateral resolution from an electron gun normal to the spectrometer axis. Although in Figures 4 and 7 the attenuation of the cosine distribution is slower than of the isotropic, this result is not general. The fraction of the analyzer aperture solid angle having 82. n > 0 is given for the CMA by AkF*(r, 0) f rom Figure 2 and for the RFA by Ax*(r, 0) from Figure 5. Within the modeI, the functions A(r, px) are directly applicable to the attenuation by a uniform overlayer. They are especially useful when the excitation of the source of characteristic electrons is negligibly affected by the overlayer. For example, in Auger spectroscopy, it is sufficient that the overlayer be thin compared to the
,j 20
40
60
80 r, rotation
+-yy_, 0
100 (degrees)
20
, 40
,
100 60 80 r, rotation (degrees)
Figure 5 (left). AL~~RF*(~,,ux), as for Figure 2 for a 48” semicone angle RFA with an isotropically emitting source. Figure 6 (right). AeosRFA(~, ,Ux ) as for Figure 5 with a cosine distribution of the emitting source.
projection onto the surface normal of the inelastic mean free path of the primary electrons. The results are ylso useful as a kernel for estimating the thickness dependence of the signal strength from any depth distribution of sources E(X) (e.g., with isotropic angular distribution) via either a continuum model,
EMU>] =
7 Aiso (r, /CC’) E(x’) dx’
(4)
0
or a discrete model, which might be written as r[&(?U?)] = n$ where 8 describes
4so
G-,
Wa)
(5)
E (na)
the depth’ distribution
of the emission,
and p is constant.
This
423 I
0.1
0.0 I-
CL00 t-
Figure 7. &,,RF*(60°, e--1.33Lkcm
,ux)/A-~~~~*(~O~, 0) and AcosRFA(600,
,ux)/A~o~~~~(60~,
0) ~~~p~ed
to
,
generalization of the model of Gallon4 which does not account for the analyzer geometry is similar to that of Seah 6. For example, thickness dependence of signal generated by a uniformly emitting overlayer would be found in the continuum model by 3c s
Aiso (I, /CC’)dx’.
0
More generally, if the excitation rate per atom emitting the characteristic electrons and the p describing the attenuation are constant, then moments of the depth distribution of the emitting element can be obtained from the intensity of each of its spectral lines relative to the intensity in pure material of the element. This relative intensity is given by
424 TABLE
1
EFFECTIVE ATTENUATION THE CMA AND THE RFA
DEPTH AS MULTIPLE FOR ,ux RANGING FROM
Cylindrical mirror analyzer
OF OVERLAYER 0.05 TO 5
THICKNESS
IN
Retarding field analyzer
D/x at r =
cos
IS0
DJx at r =
COS
IS0
0” 45”
1.35 1.80 to 1.17
1.35 3.96 to 1.29
10” 20” 60”
1.22 to 1.16 1.27 to 1.16 1.88 to 1.31
1.24 to 1.18 1.34 to 1.19 3.50 to 1.41
where pP and eP are uniform values with depth in the pure material and =
0.740 for the CMA and 0.835 for the RFA.
Since an independent result is obtained from each different line (because of the dependence of p on electron energy1 5), this approach may be useful in analysis of depth distributions of elements with several characteristic lines. Approximating the attenuation functions with an exponential A(r, q)/ A(r, 0) N eePD where D is the effective attenuation depth is exact for the CMA at r = 0”; there, D = x set 42.3 o = 1.35 x. Otherwise, this approximation is not exact since exponentials with different multipliers, set 8, are superposed, and the contributions with low 0 dominate with increasing PX. I.e., D/X is not constant but decreases with increasing px as may be seen in Figures 4 and 7. The value of D/x varies significantly, as typified by Table 1. Conversely, erroneous values for the inelastic mean free path may be obtained if the attenuation of line strength with overlayer thickness is taken as exponential and a constant geometric correction factor is used. On the other hand, the departure from exponential (especially for large I) may be used to advantage to test the model assumptions or to provide an independent measure of overlayer thickness by comparison of experimental line strengths as a function of overlayer thickness, to both the shape and magnitude of the appropriate model (e.g. Figure 4). The test may be especially valuable if several characteristic lines from a source are expected to follow the model assumptions, for they should have the same dependence on overlayer thickness, scaled by the inelastic mean free paths at the energies of the spectral lines. CONCLUSIONS
We have given model results for attenuation uniformly attenuating flat layer for two commonly
of lines in a spectrum by a used electron spectrometers.
425 The relevant parameters are the orientation of the layer, the over-layer thickness in units of the inelastic mean free path, and the angular distribution of the emission from the source. Results have been given graphically for the simplest angular distributions: isotropic (approximating an isolated atom or thin layer) and cosine (approximating a uniformly emitting source which is thick compared to the inelastic mean free path for electrons in the spectral line). These results may be used to rapidly estimate the effects of nonuniform depth distributions, by superposition. Deviation of the model attenuation from an exponential with overlayer thickness is large if the overlayer normal is not nearly along the axis of the analyzer. The quantitative results are pertinent to assessment of the inelastic mean free path model, to measurement of inelastic mean free paths, and to application of the model in practical situations. ACKNOWLEDGEMENTS
Useful conversations with C. C. Chang and H. R. Patil and the support of the Materials Science Center at Cornell University during initial stages of this work are gratefully acknowledged. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
P. W. Palmberg, G. K. Bohn, J. C. Tracy, Appl. Phys. Lett., 15 (1969) 254. eg. Varian LEED optics 981-24. L. A. Harris, (a) J. Appf. Phys., 39 (1968) 1419, (b) Surf. Sci., 15 (1969) 77. T. E. GaIlon, Surf Ski., 17 (1969) 486. K. Jacobi, J. Holzl, Surf: Sci., 26 (1971) 54. M. P. Seah, Surf; Sci., 32 (1972) 703. M. L. Tarng, G. K. Wehner, J. Appl. Phys., 44 (1973) 1534. M. P. Seah, J. Phys. F, 3 (1973) 1538. J. C. Tracy, J. Vat. Sci. Technol., 14 (1974) 280. Peter J. Feibelman, Surf. Sci., 36 (1973) 558. J. J. Vrakking, F. Meyer, Surf: Sci., 35 (1973) 34. T. W. Busch, J. P. Bertino, W. P. Ellis, Appl. P&s. Left., 23 (1973) 359. T. E. Gallon, J. Phys. D, 5 (1972) 822. J. H. Neave, C. T. Foxon, B. A. Joyce, Surf. Sci., 29 (1972) 411. For dependenceof p on electronenergy in jellium, see J. C. Shelton, Surf. Sci., in press. Experimental results for ,u (uncorrectedfor the geometrical effects described herein) are summarized by J. C. Rivi&re,Contemp. Phys., 14 (1973) 513.