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the molar ratio of Co ions and Si atoms in every sublayer. The stepped curve
represents 30 sublayers with an offset of about 10% for the surface at
z
=
0nm
and with a maximum of 33% in a depth of about 60 nm. The actual dose was
determined by integration to be 1.2
×
10
17
ions per cm
2
, 20% above the nominal
value. This profile was checked by Rutherford backscattering spectrometry
(RBS), which is a well-established and reliable method for such transition
layers. Both profiles match quite well with regard to their maxima and integrals
(implantation dose).
In support of GI-XRF measurements, the specular reflectivity of a flat
sample can simply be observed. A theoretical evaluation of XRR is possible by
assuming a model of the structured sample with values for the layer thickness,
concentration of different elements, density, coefficients
δ
and
β
, and the
experimental parameters
E
,and
Δ
E
of the primary beam, and
ε
of the detector.
R
(
α
) can be calculated according to this model and compared with the
experimental values of the reflection/angle scan, again with a
χ
-square test.
4.6.5IncludingtheSurfaceRoughness
In practice, perfectly flat surfaces and interfaces are nonexistent. A certain
nanometric roughness will be left on bulk samples even after careful polishing.
Even an extremely flat and smooth wafer surface deviates from an ideal plane
at the atomic scale level. For other flat samples, an average roughness of some
10 to a few 100 nm often occurs in practice.
The roughness of a surface or interface is defined by different measures. The
most common quantity is the average roughness
R
a
, which is the average
deviation of
z
values (height above a zero level of the
xy
plane) from a mean
value of the respective surface or layer. Another customary parameter is the
root-mean-squared roughness,
R
RMS
, which is the root of the mean of squared
deviations (RMS) of
z
values and which is often termed
σ
. Generally, a stylus
profilometer or laser interferometer can be used for the determination of
surface roughness ranging from 10 nm to 100
μ
m. A scanning tunneling micro-
scope (STM) and an atomic force microscope (AFM) can measure a roughness
down to 0.2 nm or even 0.1 nm. XRR measurements can also be applied for this
task, and several papers have been published with regard to roughness. Si
wafers can be polished to a roughness of 0.1 nm (1 nm down to 0.1 nm). For
single layers and multilayers, the roughness depends on the production process,
on the substrate and on the type of materials. It increases usually with the
thickness and the number of layers.
Various approaches have sought to cope with such rough surfaces [89,91-93].
Three different models were developed in order to implement roughness in the
calculation of standing waves [47]:
1. The first model supposes that only the reflectivity is reduced by rough
surfaces and interfaces. The beam-path, however, is assumed to be not
disturbed. A“Debye-Waller”factor can be included in the coefficients of
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