Chemistry Reference
In-Depth Information
nature of the reactions would be revealed later (new morphological features at more
refined scales are still reported today and will probably be the case for some time in
the future).
Based on an extensive investigation of the anodic
i-V
relations, dissolution
valence, and PS morphology on a large matrix of silicon substrates, Beale
et al
.
35
proposed a rather comprehensive model on the formation of PS. This was the first
model that analyzed the
i-V
characteristics and correlated them with the current
conduction mechanisms associated with silicon substrates of different types and doping
concentrations.
Because the spacing between pores is always less than the width of the depletion
layer and PS has a very high resistivity, Beale
et al.
proposed that the material in the
PS is depleted of carriers and the presence of a depletion layer is responsible for current
localization at pore tips where the field is intensified. This intensification of field is
attributed to the small radius of curvature at the pore tips. For lowly doped
p-
Si the
charge transfer is by thermionic emission and the small radius of curvature reduces the
height of the Schottky barrier and thus increases the current density at the pore tips.
For heavily doped materials the current flow inside the semiconductor is by a tunnel-
ing process and depends on the width of the depletion layer. In this case the small radius
of curvature results in a decrease of the width of the depletion layer and increases the
current density at pore tips. The initiation was considered to be associated with the
surface inhomogeneities, which provide the initial localized high current density at
small surface depressions.
The model of Beale
et al.
provided a deeper level of understanding of the
current localization required for PS formation on different silicon substrates and
pointed out the correlation between the relative dimension of pore size and the width
of the depletion layer. Several concepts proposed in their model would be adopted
and further developed in many of the later models such as those by Fo
l,
69
Zhang,
8
and Lehmann.
12,1084
However, because the model considered only the physical
aspects of the semiconductor and none of the chemical reactions, it provided little
insight for the change of pore size and other morphological features with current and
HF concentration. Also, Beale's model assumed that the Fermi level of the semi-
conductor is pinned on the surface on the midgap which does not agree with the later
experimental data.
Carrier Diffusion Model.
Near the end of the 1980s Smith
et
l
.
59,60,62
proposed
a model with commuter simulation to describe the morphology of PS based on the
hypothesis that the rate of pore growth is limited by diffusion of holes from the bulk
of the silicon to the growing pore tip. The pore structure is determined by the intrinsic
nature of the random walk and the magnitude of the diffusion length. A hole randomly
walking toward the growing pore tips is more likely to contact those pores that are
nearest to it, meaning the outer tips of pores have the highest probability of hole capture
and growth. According to this model, the features of PS morphology are essentially
determined by the hole diffusion length
L
which is a function of potential and dopant
concentration. The interpore spacing is then on the order of two diffusion lengths, that
is,
al
L.
The PS density decreases with decreasing diffusion length along with an increas-
ingly interconnected porous structure. To account for the variation in pore diameter and
the transition from PS formation to electropolishing a sticking factor was introduced.
2
