Quantitative model of electron energy loss in XPS
A. Cohen Simonsen
Department of Physics, University of Odense, DK-5230 Odense M, Denmark
F. Yubero
Instituto de Ciencia de Materiales de Sevilla, University of Sevilla, 41092 Sevilla, Spain
S. Tougaard
Department of Physics, University of Odense, DK-5230 Odense M, Denmark
Received 16 December 1996
A model for the inelastic-scattering cross section of electrons in XPS experiments is presented. The calcu-
lation is analogous to a previous model for reflection-electron energy-loss spectroscopy by Yubero et al. Phys.
Rev. B 53, 9719 1996. The model treats the general case of photoelectron creation at arbitrary depth and
with arbitrary exit angle and electron energy. The effect of the core hole as well as surface effects are included
in the model. We study systematically the behavior of the model when the energy, exit angle, and depth of
origin are varied. Furthermore, the effect of the core hole on the energy loss is investigated in detail. The
results are compared with experimental XPS spectra from aluminum metal. S0163-18299701127-2
I. INTRODUCTION
Surface electron spectroscopies such as XPS x-ray pho-
toelectron spectroscopy, AES Auger electron spectros-
copy, and REELS reflection-electron energy-loss spectros-
copy are highly influenced by inelastic-scattering events
experienced by electrons. Thus, for the purpose of quantifi-
cation a thorough understanding of the energy loss is impor-
tant.
The usual quantity employed to describe the energy loss
is the inelastic cross section, which gives the probability den-
sity per unit path length of losing the energy . It can be
expressed in terms of the complex dielectric function
( k, ) of the particular medium, and for electrons traveling
in an infinite medium it is given by
1,2
K E
0
, =
1
E
0
a
0
k
-
k
+ dk
k
Im
1
k ,
, 1
where the following quantities are introduced: E
0
is the ini-
tial energy of the electron, a
0
is the Bohr radius, and k is the
wave vector transferred from the electron. k
=(2 m /
2
)
1/2
E
0
1/2
( E
0
- )
1/2
are the limits on the k
vector imposed by energy and momentum conservation dur-
ing the inelastic scattering. This model, however, does not
reproduce the surface loss features observed in REELS and
XPS. Attempts have been made to model the inelastic cross
section as a linear combination of Im1/ and Im1/(1
+ )].
3
Although a reasonable fit to experiment can be made,
the values of the fitting parameters seem unphysical and in-
consistent.
A more realistic model for REELS has been developed to
describe the electron energy loss for normal incidence and
exit geometry.
4
It treats the total transport process for an
electron elastically backscattered a certain depth below the
surface. The model was recently extended to include general
incidence and exit angles
5
and was compared to experimen-
tal inelastic cross sections
6
deconvoluted from experimental
REELS spectra. The quantitative agreement regarding the
dependence on incidence and exit angle as well as on initial
energy was good and all the main loss features of the experi-
ment were reproduced.
The purpose of the present paper is to take a similar ap-
proach to the case of XPS. The system was previously stud-
ied by Gervasoni and Arista
7
who determined the energy-
loss rate, but not the energy-loss distribution, i.e., the cross
section. Seymour et al.
8
modeled the plasmon loss intensity
in photoemission on the basis of a hydrodynamical equation.
However, the model produced spurious structures in the
losses that was not reproduced experimentally. Recently
Chen and Chen proposed a model that incorporates both sur-
face and bulk losses into the Landau formula for the energy
distribution.
9
This procedure however, has three main weak-
nesses: First, they integrate the surface loss function over all
depths; i.e., they assume that the contribution from surface
losses is the same for an electron excited at the surface and
one excited deep in the bulk. Second, they incorporate this
path-integrated surface loss function directly into the Landau
formula. This implies that they allow for multiple surface
losses to infinite order. Third, they ignore the effect of the
core hole on the energy loss of the photoelectron. The first
two procedures are not properly justified and the last ap-
proximation is shown in the present work not to hold.
We treat the general case of photoelectron creation at ar-
bitrary depth and with arbitrary exit angle of the electron.
The procedure is based on the ‘‘specular reflection
model,’’
7,10
which allows one to solve the electrodynamic
problem with the proper boundary conditions. The result is
expressed in terms of the effective inelastic cross section
giving the energy-loss distribution for the total photoemis-
sion process.
II. THEORY
We consider the following situation as a model of the
experimental XPS process, Fig. 1a: a semi-infinite medium
PHYSICAL REVIEW B 15 JULY 1997-I VOLUME 56, NUMBER 3
56 0163-1829/97/563/16128/$10.00 1612 © 1997 The American Physical Society