Chemistry Reference
In-Depth Information
where is the resistivity of the material and and
r
are the diameters of the inner
sphere and outer sphere, respectively. The potential drop, for a current
where
i
is the current density on the inner surface, flowing through the solid is then
described by
Figure 8.65 shows that the potential drop in the material with a curved surface increases
nonlinearly with increasing distance from the inner sphere and most of the potential
drop occurs within a distance of a few times the diameter of the inner sphere. Also, for
the same thickness of the solid the total potential drop increases with increasing radius
of curvature of the inner sphere, reaching the maximum at an infinitely large radius of
curvature, that is, a flat surface. The value of the potential drop is in the millivolt range
under the conditions shown in Fig. 8.65. It will be larger
if the current density or the resistivity is higher. Also, the results shown in Fig. 8.65
are for a spherical surface. In the case of curved pore bottoms, which are roughly semi-
spherical, the potential drop from individual pore bottom into the solid must be sig-
nificantly larger than that of an isolated sphere because the current flow from one pore
overlaps those from neighboring pores.
The results of the above analysis suggest that the formation of macro PS on lowly
doped materials can be associated with a nonlinear potential distribution in the solid
of a curved surface due to the high resistivity of the solid. The formation of two-layer
PS on
p
for an inner diameter of
indicates that there are two different physical layers in which the poten-
tial-current relations are sensitive to the radius of curvature. The space charge layer of
p
-Si
-Si
is thin under an anodic potential and is associated with the formation of the micro
PS. The nonlinear resistive effect of the substrate is responsible for that of the macro
at a given
current density, the radius of curvature must be smaller for a material of larger resis-
tivity according to Eq. (8.12). This means that the pore diameter decreases with increas-
ing resistivity, which agrees with experimental results.
1027
As a further deduction, the effect of high substrate resistivity should also occur
in the case of lowly doped
n-
Si
.
However, the width of the space charge layer under
an anodic potential, at which macro PS is formed, is on the same order of magnitude
as the dimension of the resistive layer. The effect of the space charge layer and the
resistive layer on the pore diameter are not distinguishable under normal conditions. If
the conditions can be controlled such that the pores formed due to the effect of the
space charge layer and those due to the effect of the resistive layer have a size differ-
ence of at least one order of magnitude, it would be possible to obtain PS with two dis-
tinct distributions of pore diameters on high-resistivity
n-
Si as illustrated in Fig. 8.66(a).
Such a condition might exist at a low potential (small space charge layer thickness) on
a back-illuminated substrate. Also, if this is true it should then be possible to obtain PS
with three distinct distributions of pore diameters on a front-illuminated sample as illus-
trated in Fig. 8.66(b).
Anisotropic Effect.
As discussed in Chapter 7, anisotropic etching occurs when
the dissolution reactions depend on the concentration of the active surface species. In
PS. Also, to have the same change of potential drop in the substrate,
