Ion implantation is frequently performed through masking layers consisting
of oxide (
), nitride (
) or photo resist. Thin dielectric
layers (screening oxide) are used to avoid contamination of the substrate or
for scattering of the implanted ions to reduce the channeling effect.
The basic assumptions of analytical approaches based on moments of a distribution function restrain their applicability to half spaces. Extentions to multilayered structures mostly give acceptable results but suffer from lacking underlying physics.
Each layer usually has a different stopping power for the ions and, therefore, the covering layer has a strong influence on the velocity distribution of the ions at the interface to the underneath layer. An approach to model the velocity (energy) distribution analytically has been proposed by Pantic [Pan89]. As mentioned before, the Monte Carlo and the Boltzmann Transport method are applicable to arbitrary target structures.
For simplicity we first assume a two-layer target (see
Figure 2.2-2) composed of a mask (layer 1) and a substrate (layer 2). Analytical models presume that a profile 
in pure mask material and 
in pure substrate material is known.
Dependent on the thickness of the mask 
and the properties
of mask and substrate material, the profiles 
and 
are stretched,
shifted and scaled to get the profile 
in the compound target.
A first analytical model for the calculation of the profile in multilayered structures has been published by Ishiwara et al. [Ish75]. Several modifications and improvements have been proposed later on, e.g. by Ryssel [Rys83a], [Rys87], [Wie89]. Comparisons of analytical models with Monte Carlo results are collected in [Rys87] and [Hob87b].
We use the numerical range scaling (NRS) method proposed by Ryssel
[Rys83a]. The profile in a two-layer structure is given by
(2.2-27). The scaling factor 
has to be chosen to satisfy the
dose matching condition 
and
consequently the normalization condition (2.2-28).
In theories describing the stopping of particles in matter, it is generally
assumed that the range is inversely proportional to the
material density. Therefore, the shift 
is calculated from
(2.2-29). This model is sufficiently correct for very thin
masking layers as well as thick masking layers. The condition of dose
conservation is fulfilled inherently.
For some applications, however, the predicted profiles do not agree
sufficiently well with measurements. Therefore, an approved numerical range
scaling model has been developed [Wie89]. This model uses a modified
standard deviation (2.2-30) for the substrate layer for the
calculation of .
The generalization of the numerical range scaling to a target with 
layers is evident. Within layer 
(
), where 
denotes the position of interface between layer and layer 
(see Figure 2.3-5), the distribution 
is given by
(2.2-31) with 
the distribution function for infinitely
thick material . For the first layer the shift 
vanishes and
the profile has not to be scaled (2.2-32).
Then, the shift 
in layer is calculated from
(2.2-33) using the thickness 
of layer 
and the
projected ranges 
and 
.
The normalization condition 
yields
the relation for the scaling factor 
(2.2-35).
From detailed discussions of multilayer models, e.g. in [Rys87], [Hob88a], it seems obvious that only Ryssels numerical range scaling model is capable of describing profiles in multilayer targets with larger differences in stopping power to a reasonable degree of accuracy.
In Figure 2.2-3, you find a comparison between a Monte Carlo
simulation and an analytical description using the numerical range scaling
algorithm for a boron implantation with 
and a dose of
into amorphous silicon covered by 
nitride. The
result of the NRS algorithm shows reasonable agreement with the Monte Carlo
simulations.
For most common combinations in silicon technology, e.g. 
, 
and 
on silicon, and some 
semiconductor technologies the
profiles obtained by the numerical range scaling model fit best with Monte
Carlo results. However, it leads to totally incorrect profiles if the
coating layer consists of atoms much heavier or much lighter than the target
atoms, e.g. tungsten on top of silicon [Hob88a].
