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COMPOSITIONAL DEPTH PROFILING
problems. Consequently, angle resolved
XPS is one of the few
ways
of
probing near-surface compositional gradients. There
are
many
ap-
proaches which
can be
taken
to
model such
profiles
but
work
from
the
National Physical Laboratory
in the UK has
produced validated
software
in the
form
of a
spreadsheet which incorporates several ways
of
handling
data,
the
choice depends
on the
type
of
information which
is
desired from
the
ARXPS data set. Named ARCtick,
the
routine
is
fully
described
at
http://www.npl.co.uk/npl/cmmt/sis/arctick.html,
and an
example
of the
output
from
ARCtick
is
provided
in
Chapter
5.
Computational methods
No
unique transformation from angle-dependent intensities
to
depth-
dependent concentration
exists.
This implies that
a
least squares
fit of
trial profiles
to
experimental data
is not
sufficient
to
determine
the
accurate concentration profiles.
The
concept
of
maximum entropy
has
therefore been introduced
to
produce
a
smooth profile, avoiding
the
'over
fitting'
which
a
method based
on
least squares
fitting
would
produce.
In
outline
the
procedure
is as
follows.
A
random depth
profile
is
generated
and the
ARXPS intensities expected
from
such
a
profile
are
calculated.
The
profile
consists
of a
number
of finite
layers
in
which
the
concentrations
of the
various elements
are
calculated.
It is
conveni-
ent
to
restrict
the
number
of
layers
to ~
10–20,
this
value
is a
compro-
mise
between depth resolution
and
computational time.
The
calculated
ARXPS
data
are
compared with
the
experimental data
and the
error
is
calculated:
(
/calc
jobs
\ -
~>
\ *
V'i
'L I
X
=
£—J
where
a is the
standard deviation.
The
entropy term
(S) is
then calcu-
lated
from
the
trial
profile: