Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
C511
0013-4651/2002/149共10兲/C511/10/$7.00 © The Electrochemical Society, Inc.
Quantitative Analysis of the Morphology of Macropores on
Low-Doped p-Si
Minimum Resistivity
J.-N. Chazalviel,a,*,z F. Ozanam,a N. Gabouze,b S. Fellah,b,d and R. B. Wehrspohnc,*
a
Laboratoire de Physique de la Matière Condensée, CNRS-École Polytechnique, 91128 Palaiseau, France
Unité de Dévelopment des Technologies du Silicum, Algiers, Algeria
c
Max-Planck-Institute of Microstructure Physics, D-06120 Halle, Germany
b
The formation of macropores by anodization of low-doped p-Si in HF electrolyte has been investigated quantitatively. As anodization proceeds, structures of increasing characteristic size are formed, then a steady state is reached, where macropores grow
parallel. The intermediate regime is well understood on the basis of a linear stability approach, incorporating the known physics
and chemistry of the Si/electrolyte interface: semiconductor space charge and interface reaction velocity. The characteristic size of
the macropores and their dependence on Si doping and electrolyte resistivity and composition are quantitatively accounted for
after realizing that parallel growth is strongly favored by the channeling of the current in the macropores. Below a critical
resistivity, no macropores are observed. It is shown, through a numerical simulation, that this change of behavior results from a
loss of the insulating character of the walls, due to effects of disorder in a depletion layer when doping increases.
© 2002 The Electrochemical Society. 关DOI: 10.1149/1.1507594兴 All rights reserved.
Manuscript submitted January 28, 2002; revised manuscript received April 23, 2002. Available electronically September 12, 2002.
The mechanism of electrochemical formation of porous silicon
has long been debated. Especially, the formation of microporous
silicon from p-Si has given birth to many different models.1 A few
years ago, we have observed a transition from microporous to
macroporous morphology in highly resistive amorphous silicon and
later in low-doped crystalline p-Si.2,3 Similar observations have
been reported by Lehmann and Rönnebeck,4 and other authors have
encountered related macroporous morphologies in nonaqueous
electrolytes.5-9 These observations have led us to suggest that microporous and macroporous morphologies may both result from
simple Laplacian instabilities of a low-resistive medium 共the electrolyte兲 growing under electrical control into a medium of higher
resistivity, which may be either the space-charge region or the bulk
semiconductor itself.
In a recent work, we have developed an improved version of the
above linear-stability approach in order to address these phenomena
more quantitatively.1,10 Especially, the effects associated with hole
transfer through the space-charge layer in the semiconductor were
addressed in detail. The building elements of this model and its
predictions are briefly summarized in the Summary of previous results section. In order to resolve some apparent discrepancies between these predictions and experiment, we reconsider the model by
giving more attention to the electrolyte side of the interface. The
new predictions are compared to quantitative experimental measurements.
In the following, we report on our systematic experimental investigation of the macropore sizes for a wide range of silicon substrates and electrolytes. Then, after recalling our previous model, the
new theoretical developments are presented and compared to the
experimental results.
Experimental
We have made porous silicon from p-Si, using a broad range for
Si resistivity, electrolyte resistivity, and current density. Silicon resistivity ranged from 10 to 104 ⍀ cm. Samples from 10 to 1000 ⍀
cm were commercially available from ITME, Poland. Samples up to
a few 104 ⍀ cm were obtained from Wacker, Germany, and Topsil,
Denmark. The electrolytes were ternary mixtures HF/H2 O/ethanol
or HF/H2 O/ethylene glycol. The latter kind of mixtures was used for
rising electrolyte resistivity to higher values. Current density was
* Electrochemical Society Active Member.
d
z
Present address: Laboratoire PMC, École Polytechnique, 91128 Palaiseau, France.
E-mail: [email protected]
varied from 1 to 100 mA/cm2 . The porous layers were characterized
with optical microscopy and scanning electron microscopy 共SEM兲.
They were observed in front view, side view after cleavage, and
on bevels at an angle of one-eighth radian. Bevel views are especially useful as they allow one to investigate the progressive variation of morphology as a function of depth in the layer. The bevels
were realized by polishing after impregnation of the porous layer by
wax. In order to minimize stresses on the porous structure, impregnation was realized without exposure to air, by the substitution electrolyte→ethanol→trichloroethylene→saturated solution of
sticky wax in trichloroethylene, and final heating of the sample with
an excess of wax, up to the wax melting temperature 共40°C兲. Typical
SEM pictures 共front view, side view, bevel兲 are shown in Fig. 1-3.
They show the progressive transition from a rather uniform microporous structure to a cellular structure consisting of macropores
filled with microporous silicon 共see Fig. 4a兲. Only in rather dilute
HF electrolyte 共⬍20% HF兲 and at high current densities are empty
pores observed.4
A distinct feature in high current density and low-HFconcentration experiments is the appearance of crystallographic effects, such as preferential pore growth in the 共100兲 directions and
共111兲 faceting of the pore tip, as can be seen in Fig. 4b, for a high
current density 共100 mA/cm2 ) in dilute 共15% ethanolic兲 HF. These
crystallographic effects, similar to those observed on n-Si,11-14 occur
共plausibly兲 when the pore tips are under electropolishing conditions.
They are useful for obtaining large aspect ratio regular pore arrays.15
However, we do not discuss this case here.
In the following, we limit the discussion to the high HF concentration 共low J 0 ) case. In this case, the pores are filled with microporous silicon, the pore tips are rounded, and the parallel
macropore array grows perpendicular to the sample surface, irrespective of the crystallographic orientation of the sample. Extension
of the present work to the case of crystallography-driven pores may
be found elsewhere.16
The characteristic wall thickness and macropore diameter have
been extracted by numerical analysis of the images. Figure 3a shows
a typical bevel view. A digitized column from the image, shown as
Fig. 3b, can be regarded as a function f 共i兲 exhibiting a succession of
bright and dark intervals, of unequal width. Analysis of such data
was carried out as follows.
A ‘‘local median’’ was defined by taking the average between
maximum and minimum value, on a sliding window of width n 共in
pixels兲. The resulting function was filtered with a Gaussian filter of
root mean square 共rms兲 value n/2, giving a new function g共i兲. The
sign of the function f (i) ⫺ g(i) was then considered. The number
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Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
Figure 1. Typical front view of a porous layer. Preparation conditions: p-Si,
100 ⍀ cm, 35% ethanolic HF, 20 mA/cm2 , 50 min. 共a兲 As made; 共b兲 after
dissolution of the upper microporous layer by 10 s etching in 1 M KOH,
revealing the underlying macroporous structure. For more dilute HF electrolytes and/or prolonged anodization times, the upper microporous layer is
partially dissolved during the anodization, and the macropore structure becomes apparent without KOH stripping.
of pixels with positive 共respectively, negative兲 value, divided by half
the number of sign changes along the column, gives the average
length l d 共respectively, l b) of a dark 共respectively, bright兲 interval.
The value of n was chosen equal to 2 ⫻ (l d ⫹ l b), which was
Figure 3. Same sample as Fig. 1. 共a兲 Top view after one-eighth-radian beveling 共increasing depth from left to right兲. Notice the progressive increase of
pore diameter until a constant value is reached, corresponding to the parallel
array of Fig. 2. 共b兲 Analysis of the picture in 共a兲 along a column. 共c兲 Characteristic wall thickness and pore diameter as a function of depth in the layer,
as deduced from a numerical analysis of 共a兲.
obtained by repeating the procedure until l d and l b come out stable.
Due to geometric considerations for a random cut, the actual average pore diameter was taken as the average length of a dark interval,
multiplied by 4/␲, and the average wall thickness was taken as the
average length of a bright interval, multiplied by 2/␲. It was verified
that this fast automatic procedure gives values in fair agreement
with hand measurements made by a human experimentalist after
averaging over a dozen pores.
Summary of Previous Results
Figure 2. Same sample as Fig. 1. Typical SEM side view. Notice the parallel
macropore structure.
The essence of our previous model is sketched in Fig. 5. The
model assumes that dissolution of a silicon atom is limited by the
electrochemical transfer of a first hole. The interface being under
weak depletion conditions, this transfer can be modeled by a
Schottky-type law, incorporating the finite reaction kinetics at the
interface through a reaction velocity ␷ R . Besides the resistivities ␳ s
共semiconductor兲 and ␳ e 共electrolyte兲 which rule the potential map
Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
Figure 4. SEM view of the pore tips, after cleavage, for samples prepared in
two different conditions. p-Si 关400 ⍀ cm, 共100兲-oriented兴, 100 mA/cm2 , 6
min. 共a兲 25% and 共b兲 15% ethanolic HF. Notice the striking difference in the
shape of the pore tips. In 共a兲, the pores are filled with microporous silicon
共strongly damaged due to the cleavage兲. In 共b兲, the pores are empty. Electropolishing takes place at the pore tips, which exhibit 共111兲 faceting 关notice
the ⬇70° angle at the tip, as expected from two 共111兲 planes兴.
far from the interface, essential parameters are the interface band
bending ⌽ SC , the characteristic diffusion velocity in the space
charge ␷ D , and the reaction velocity ␷ R . This transport model has
been worked out in a linear-stability approach,17 in the same spirit as
several papers in the literature18-21 共though with different ingredients, especially for Ref. 18-20, which deal with the case of n-Si兲.
Namely, an interface was considered with a small sine wave deformation ␦y cos(qx).17 The interface curvature leads to local changes
in the three key parameters ⌽ SC , ␷ D , and ␷ R : near a protrusion
of the electrolyte 共bottom of the sinewave, mimicking a nascent
pore兲, there is 共i兲 a decrease in barrier height, due to a local increase
of the Helmholtz potential drop, 共ii兲 an increase in ␷ D , due to a
local thinning of the space-charge layer, and 共iii兲 a decrease
of ␷ R , due to decreased reaction kinetics induced by steric hindrance effects on the atomic scale. The first two mechanisms tend
to destabilize the flat interface. The strength of this destabilization effect can be lumped into a single dimensionless parameter S
⫽ ␧␧ 0 ⌽ SC /(␭k BTC H) ⫹ ␷ R /(␷ R ⫹ ␷ D), where ␭ is the characteristic space-charge thickness of silicon, C H is Helmholtz capacitance,
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Figure 5. Scheme of the interface in our model. 共a兲 Silicon dissolution rate
is determined by transfer of a hole through the space-charge layer 共velocity
␷ D) and its reaction at the interface 共reaction velocity ␷ R). 共b兲 Typical profile
of the potential in the presence of a small sine wave perturbation of the
interface 共here for q ⬃ 1/␭). Notice the steepening of the space-charge profile in the vicinity of electrolyte protrusions 共increase of ␷ D), together with
the increased Helmholtz-potential drop at the same places 共shown exaggerated for clarity兲.
and ␧, ␧ 0 , k B , and T have their usual meanings. The latter mechanism is stabilizing, but it becomes dominant only at small scales. At
very large scales, the resistivities play a major role, yielding a stabilizing effect if ␳ e ⬎ ␳ s , and a destabilizing effect otherwise. The
rate of variation of the sine wave deformation can be characterized
by the function ␣(q), defined by ␦y ⬀ exp关␣(q)y兴. Positive ␣(q)
means that the interface is unstable in the presence of a perturbation
of wave vector q. A typical representation of ␣(q) is shown in
Fig. 6a.1
From Fig. 6a, several semiquantitative predictions can be made.
The value of 1/q so that ␣ is maximum gives the scale of the first
structures to be expected. This value is found to be on the order of
a/S, where a is an atomic length and S ⬃ 0.1,1 which is in agreement with the typical size of the smallest structures found in
microporous silicon22 共Ref. 21 failed to get this agreement, due to a
neglect of the barrier-lowering effect and an inappropriate hypothesis on the mechanism responsible for the low-wavelength cutoff兲.10
As Si dissolution proceeds, structures of sizes corresponding to
smaller and smaller ␣ values may be expected. Given the behavior
␣ ⬃ Sq in the mesosize range, the model predicts that structures of
increasing size must appear. Typically, if one considers that microporous silicon is just a homogeneous medium with the resistivity
of the electrolyte, stability of the silicon/microporous silicon interface is ruled by the same ␣(q) function. Hence, for a porous layer of
thickness ⌬, structures of characteristic size S/␣ ⬃ S⌬ are pre-
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Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
that very different morphologies may result from changes in the
electrolyte composition.5-9 However, two observations are clearly
not accounted for by this simple model1,23: The first one is the
observation that when macropores are formed they always tend to
reach a steady state of parallel growth, the number of macropores
and their diameter becoming constant. In contrast to this observation, it is expected from the model that parallel pore growth should
occur only when ␣ becomes negative at low enough q 共large-scale
stability of the pore front兲, which is the case only for ␳ e ⬎ ␳ s . In
this case, the model gives a prediction for the characteristic
macropore size. In the opposite case, a hierarchical arrangement of
macropores would be expected, that is, as growth proceeds, larger
and larger macropores should appear, and the number of active
pores should decrease accordingly. This discrepancy, for the case of
highly resistive silicon, has been addressed here. The second unexplained observation is that no macropores are ever observed when ␳ s
lies below a critical value ␳ c , on the order of a few ohmcentimeters.
In the following, we extend our theory and show that it may be
reconciled with our experimental data. Finally, we address the question on the disappearance of such macrostructures above a critical
doping.
Theory: Linear Stability Analysis of the Pore Front
Figure 6. 共a兲 ␣(q) function as deduced from a linear stability analysis 共after
Ref. 1兲. Notice the unstable behavior 共positive ␣兲 for a wide range of the
wave vector q, extending down to zero for ␳ s ⬎ ␳ e 共here ␳ s ⫽ 0.1 and 100
⍀ cm, ␳ e ⫽ 10 ⍀ cm兲. 共b兲 Characteristic size of the expected structures as a
function of depth in the porous layer, deduced from 共a兲 by a simple change in
the way of plotting the data. The expected unlimited increase in size at larger
depth is not observed experimentally.
dicted to be present near the bottom of the porous layer 共see Fig. 6b,
obtained from Fig. 6a by interchanging the axes兲.
This model correctly predicts the progressive transition from a
homogeneous microporous layer to a macrostructure. It allows one
to understand that the observed macropores may be filled with microporous material, and the predicted orders of magnitude appear
essentially correct. Interestingly, the chemistry is present in the
model, under the form of the reaction velocity ␷ R , which explains
The observation of parallel macropore arrays, which stands in
contrast to the prediction that pores of increasing width should develop, leads us to reconsidering the assumptions of the linearstability model. Our major assumption, when applying a linearstability approach to the growth of thick porous layers, is that
microporous silicon is just a homogeneous medium with the resistivity of the electrolyte. However, this assumption calls for two restrictions: First, it can hold only at scales smaller than the spacecharge layer thickness ␭. In a naive approach, prolonged growth of
structures of a characteristic size 1/q larger than ␭ 共e.g., q␭ Ⰶ 1)
would lead to the formation of highly porous parts separated by bulk
walls, with characteristic dimensions on the order of 1/q. Such thick
walls would no more be depleted, and lateral growth of the porous
parts could go on. This sets ␭ 共actually 2␭11兲 as the upper value of
bulk walls separating macropores. Second, since the porous medium
consists of liquid channels separated by depleted walls, it appears
electrically very anisotropic. This anisotropy should be taken care of
in the linear stability analysis of the growth front.
The growth of pores separated by fully depleted walls has been
examined theoretically in two successive approximations. In a first
approximation, the porous layer was just considered as an anisotropic medium, with a resistivity ␳ e in the direction perpendicular to
the layer plane, and infinite resistivity in the directions parallel to the
plane. In an improved approximation, we took care of the change in
pore diameter induced by a perturbation of the shape of the growing
front. This led us to an ␣(q) function appropriate for macroscopic
scales. Finally, we combined the results of this analysis with the
known contributions to ␣(q) on the microscopic scale and deduced
a prediction for the steady-state macropore size. In the following, we
successively recall the linear stability analysis of a front between
two isotropic resistive media, then present the two successive approximations, and finally build the bridge with the microscopic
scale.
Stability of a front between two isotropic media (‘‘zeroapproximation’’).—The function ␣(q) for a growth front between a
semiconductor of resistivity ␳ s and an electrolyte of resistivity ␳ e
has been derived in Ref. 2. It is obtained by writing the potential
␾(x,y) in a form satisfying Laplace equation in each medium
Porous layer and electrolyte 共 y ⬍ y B兲
␾ 共 x,y 兲 ⫽ ␳eJ 0 y ⫹ ␦␾ ee qy cos qx
关1a兴
Bulk semiconductor 共 y ⬎ y B兲
␾ 共 x,y 兲 ⫽ ␳sJ 0 y ⫹ ␦␾ se ⫺qy cos qx
关1b兴
Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
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bulk-Si/porous-layer boundary at y ⫽ y B and the electrolyte/
porous-layer boundary at y ⫽ ⫺⌬, where ⌬ is the porous layer
thickness 共see Fig. 7b兲. One can now write the potential as
y ⬍ ⫺⌬
␾ 共 x,y 兲 ⫽ ␳eJ 0 y ⫹ ␦␾ ee q(y⫹⌬) cos qx
⫺⌬ ⬍ y ⬍ y B
关4a兴
␾ 共 x,y 兲
⫽ ␳eJ 0 y ⫹ 关 ␦␾ e ⫹ ␳e␦J 共 y ⫹ ⌬ 兲兴 cos qx
关4b兴
y ⬎ yB
␾ 共 x,y 兲 ⫽ ␳sJ 0 y ⫹ ␦␾ se ⫺qycos qx
关4c兴
where matching of the potential at y ⫽ ⫺⌬ has been ensured. The
matching conditions at y ⫽ y B now become
␳eJ 0 ␦y ⫹ ␦␾ e ⫹ 共 ␳e⌬ 兲 ␦J ⫹ R i␦J ⫽ ␳sJ 0 ␦y ⫹ ␦␾ s
关5a兴
␦J ⫽ q␦␾ e/␳e ⫽ ⫺q␦␾ s/␳s
关5b兴
The new expression for ␣(q) is now
␣共 q 兲 ⫽
Figure 7. Scheme of the calculation. 共a兲 Single boundary. 共b兲 Porous layer
with anisotropic resistivity. 共c兲 Including curvature of the pores near the tips.
where J 0 is average current density and ␦⌽ e and ␦⌽ s are two parameters. The solutions 共potential and current density兲 are matched
at the boundary, defined as y B ⫽ ␦y cos(qx) 共see Fig. 7a兲. To first
order in ␦y, this gives two conditions
␳eJ 0 ␦y ⫹ ␦␾ e ⫽ ␳sJ 0 ␦y ⫹ ␦␾ s
关2a兴
␦J ⫽ q␦␾ e/␳e ⫽ ⫺q␦␾ s/␳s
关2b兴
The current density at the interface is of the form J 0
⫹ ␦J cos(qx), where ␦J is proportional to ␦y 共here J 0 and ␦J are
counted positive ‘‘downward’’, i.e., anodic兲. The stability exponent
␣(q) is derived as ␦J/(J 0 ␦y). The electrochemical kinetics can be
incorporated into this treatment through an interface resistance R i
关in ⍀ cm2 , leading to an extra interface-potential term R i␦J in Eq.
2a兴. This approach 共adopted in Ref. 2兲 is oversimplified, in that it
neglects the detail of the semiconductor space charge, and can hold
at large scale only. However, it represents a good starting point for
considering the physical effects associated with the electrolyte side
of the interface. The final result for ␣(q) is1,10
␣共 q 兲 ⫽
␳s ⫺ ␳e
q
␳s ⫹ ␳e ⫹ qR i
关3兴
The sign of ␣ is governed by the resistivities ␳ e and ␳ s . In the
framework of this approximation, the interface appears stable only if
the electrolyte is more resistive than the Si bulk.
First approximation: porous layer as a simple anisotropic
medium.—A first step for taking into account the anisotropic character of the porous layer is modeling it as an electrically anisotropic
medium, with a resistivity ␳ e in the direction perpendicular to the
surface, and an infinite resistivity in the directions parallel to the
surface. This now requires taking into account two boundaries: the
␳s ⫺ ␳e
q
␳s ⫹ ␳e共 1 ⫹ q⌬ 兲 ⫹ qR i
关6兴
which differs from Eq. 3 by an extra term ␳ eq⌬ in the denominator.
The sign of ␣ is still governed by the difference ␳ s ⫺ ␳ e , but its
magnitude is reduced as compared to Eq. 3: In the porous layer, the
current lines are constrained to stay parallel to the y direction. This
amounts to adding a constant resistance ␳ e⌬ in series with R i ,
which reduces the evolution rate of the sine wave perturbation.
Improved approximation: taking into account variation of pore
size .—The preceding approach is interesting in that it demonstrates
that anisotropy of the porous layer does play an important role for
the stability of the growth front. However, taking a fixed anisotropic
resistivity amounts to assuming that the pores have a constant
diameter and are perfectly rectilinear, and especially that they are
not affected by the sinewave perturbation of the growth front. In the
presence of a perturbation of the growth front, it is observed that
the pores bend in such a way that their growth direction always
remains normal to the front. The pores near a protrusion of the
front then appear wider than those near a depression. Note that a
similar problem is encountered in the growth of ramified metal
electrodeposits.24-26 However, though our treatment might be relevant to the electrodeposition case, it appears essentially different
from that given by Grier et al.26 In a different context, the orthogonality of the pore front with the growing structures has been pointed
out as an important factor for eutectic solidification.27
We now assume that the pore walls always remain perpendicular
to the growth front. In the porous layer, this constrains the current
lines to follow a path perpendicular to the family of curves given
by the front line at preceding times 共see Fig. 7c兲. We must then
take into account the shape of the front at the preceding times, i.e.,
from y ⫽ ⫺⌬ to 0. If the sine wave perturbation of the growth
front when located at y is given by ␦y cos(qx) exp(␣y) 共assuming that ␣ is the same from y ⫽ ⫺⌬ to 0兲, this implies that the
pore widths deviate from their average value by a relative amount
(q 2 ␦y/␣) cos(qx) exp(␣y). Since each pore carries a constant current and its size changes with y, this leads to a redistribution of the
current over the surface. In the porous layer, the first-order current
perturbation ␦J is now a function of y instead of a constant
␦J 共 y 兲 ⫽ ␦J 共 0 兲 ⫹ J 0
q 2 ␦y
共 1 ⫺ e ␣y 兲
␣
关7兴
The first-order contribution to the potential drop in the porous layer
␾ PS is now
Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
C516
冋
␦␾ PS ⫽ ␳e J 0␦y 共 1 ⫺ e ⫺␣⌬ 兲 ⫹
再
⫽ ␳e ␦J 共 0 兲 ⌬ ⫹ J 0␦y
冋
冕
0
⫺⌬
␦J 共 y 兲 dy
q 2⌬
⫹
␣
冉
1⫺
册
q2
␣2
冊
共 1 ⫺ e ⫺␣⌬ 兲
册冎
关8兴
From Eq. 7-8 and the modified boundary conditions
␳eJ 0 ␦ye ⫺␣⌬ ⫹ ␦␾ e ⫹ ␦␾ PS ⫹ R i␦J ⫽ ␳sJ 0 ␦y ⫹ ␦␾ s
关9a兴
␦J 共 ⫺⌬ 兲 ⫽ q␦␾ e /␳e
关9b兴
␦J 共 0 兲 ⫽ ⫺q␦␾ s /␳s
关9c兴
we deduce an equation for ␣
冋
冉
␣⫽
冊
册
q
q 2⌬
q
⫹
1⫺
共 1 ⫺ e ⫺␣⌬ 兲
␣
␣
␣
q 关10兴
␳s ⫹ ␳e共 1 ⫹ q⌬ 兲 ⫹ qR i
␳s ⫺ ␳e 1 ⫹
The negative contribution of ␳ e in the numerator is now enhanced as
compared to Eq. 3 and 6. This may lead ␣ to turn negative 共stable
front兲 for a condition less stringent than ␳ e ⬎ ␳ s . Intuitively, channeling of the current by the macropore walls makes electrolyte resistivity play a more important role than would be the case if the
porous layer were electrically isotropic. Especially, widening of the
pores near a protrusion of the growth front makes such pores feed a
larger interface area than their neighbors, hence, they are affected by
a larger ohmic drop, and the stability of a parallel array turns out to
be much easier than naively expected.
A problem with Eq. 10 is that it is hardly consistent with our
hypothesis ␣ ⫽ const. from y ⫽ ⫺⌬ to 0. From now on, our predictions must be taken as orders of magnitude only. First, we can
solve Eq. 10 to see in which conditions the sign of ␣ would reverse.
Expanding the exponential in Eq. 10 in the vicinity of ␣ ⫽ 0 共i.e.,
␣⌬ Ⰶ 1) leads to the simplified equation
冉
␣⬇
冊
q 2⌬ 2
2
q
␳s ⫹ ␳e共 1 ⫹ q⌬ 兲 ⫹ qR i
␳s ⫺ ␳e 1 ⫹ q⌬ ⫹
Figure 8. Effect of changing J 0 on macropore dimensions. Average wall
thickness 共theory: dashed curve, and experiment: triangles兲 and average pore
diameter 共theory: solid curve, and experiment: lozenges兲. p-Si 100 ⍀ cm,
35% ethanolic HF. Curves calculated with ⌽ SC ⫽ 0.1 eV, ␭ ⫽ 0.47 ␮m,
␷ R ⫽ 105 cm/s, ␳ e ⫽ 4 ⍀ cm.
of a characteristic size 1/q ⬃ S/␣ ⬃ S⌬ are to be expected. Hence,
increasing sizes appear as the porous layer becomes thicker. At
scales larger than ␭, ␣ is still strongly positive, with a different
behavior ␣ ⬃ ␭Sq 2 . The characteristic size of the corresponding
structures now reads 1/q ⬃ (␭S/␣) 1/2 ⬃ (␭S⌬) 1/2. The crossover
from this microscopic behavior to the macroscopic ␣ ruled by the
resistivities 关Eq. 11兴 will generally occur in this range of sizes. If we
assume that the crossover size 1/q c is such that q c⌬ Ⰷ 1 Ⰷ q c␭,
and use the further assumption R i ⫽ k BT/(eJ 0 ) Ⰷ ␳ e⌬, the righthand side of Eq. 11 can be simplified considerably and the corresponding expression is found to take over the space-charge contributions for
␳eq 2c ⌬ 2
关11兴
For given resistivities and even in the case ␳ s ⬎ ␳ e , Eq. 11
predicts that ␣ can become negative and the front becomes stable
above some critical value of q⌬. This is a new result as compared to
the prediction from Eq. 3. It explains that steady-state growth of a
parallel pore array may take place for any electrolyte resistivity.
Stability sets in at a given layer thickness and increases as the layer
gets thicker, an effect which explains that prepatterning can be very
sufficient for growing regular arrays.15 However, the above treatment is only valid on a macroscopic scale, the electrochemical
kinetics being modeled by the single parameter R i . At small
scales 共and in the absence of prepatterning兲, the thickness of the
space-charge layer comes into play, bringing positive contributions
to ␣.1,10 We now have to take these contributions into account in
order to determine the conditions where the change of sign of ␣ will
actually take place.
Taking into account microscopic contributions to ␣.—Though
achievable in principle, taking into account the present contributions
to ␣ at the same time as those from our previous microscopic
approach1,10 would represent a rather formidable task. We then content ourselves with an order-of-magnitude argument. In our former
approach, we showed that at scales smaller than the space-charge
layer thickness ␭ 共that is, q␭ Ⰷ 1), we have ␣ ⬃ Sq, where S is a
dimensionless ‘‘instability’’ factor of the order of 0.1. These contributions, largely dominant at these scales, lead to the formation of
micro- and mesostructures. At a given layer thickness ⌬, structures
k BT/ 共 eJ 0 兲
⬃ ␭Sq 2c
With ⌬ c ⬃ 1/␣ c ⬃ 1/(␭Sq 2c ), this gives
冋
1
k BT
⬃ ␭3S 3
qc
eJ 0 ␳e
册
关12兴
1/4
关13兴
This characteristic length is the characteristic pore size above which
parallel pore growth is predicted to occur. This expression is slightly
different from that given in Ref. 1, 23, due to a more careful treatment of the numerator in our Eq. 10. Although Eq. 13 gives an order
of magnitude only, this is a prediction which can be compared directly with experiment, which is done in the following.
Discussion
Comparison with experiment.—Our prediction 关Eq. 13兴 has been
compared with the experimental results when the different parameters J 0 , ␳ s , and ␳ e are varied 共staying in the range of pores insensitive to crystallography, as specified above兲. The corresponding results are shown in Fig. 8-10.
Figure 8 shows the effect of changing current density on the
morphology. Experimental parameters were ␳ e ⫽ 4 ⍀ cm 共35%
ethanolic HF兲, ␳ s ⫽ 100 ⍀ cm and J 0 from 1 to 100 mA/cm2 . The
theoretical values taken were ␭ ⫽ 0.47 ␮m, ⌽ SC ⫽ 0.1 eV, ␷ R
⫽ 105 cm/s, whence ␷ D ⫽ 106 cm/s, S ⫽ 0.1. As expected, the
wall thickness is independent of J 0 and falls close to the predicted
value 2␭. The pore diameter is significantly larger than the wall
thickness and exhibits a weak decrease with increasing current den-
Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
Figure 9. Effect of changing Si resistivity on macropore dimensions. Average wall thickness 共theory: dashed curve, and experiment: triangles兲 and
average pore diameter 共theory: solid curve, and experiment: lozenges兲.
35% ethanolic HF 共open symbols: 25%兲. ␳ e ⫽ 4 ⍀ cm, J 0 ⫽ 10 mA/cm2 ,
⌽ SC ⫽ 0.1 eV, ␭/␳ s1/2 共␮m ⍀ ⫺1/2 cm⫺1/2) ⫽ 0.047, ␷ R ⫽ 105 cm/s.
sity. The theoretical prediction is represented as a solid line in Fig. 8.
We have plotted ␲/q c , which is more appropriate than 1/q c for a
comparison with pore diameter. The absolute magnitude and the
decrease of pore diameter with increasing J 0 are both correctly
reproduced.
Figure 9 shows the effect of changing silicon resistivity. Since
this is realized through a change of the dopant concentration, the
space-charge layer thickness ␭ is also varied. For the theoretical
estimate, we have assumed that ␭ scales exactly as ␳ s1/2, ⌽ SC , and
␷ R being the same as for Fig. 8. This is certainly a good approximation for samples of resistivity up to a few hundred ohm-centimeters.
In this range, the wall thickness is indeed found to match the theoretical prediction. However, samples of higher resistivity are probably compensated, so that the ␳ s1/2 scaling law leads to an overestimate for ␭. This accounts for the deviations between theory and
experiment observed for the wall thickness at and above 1000 ⍀ cm.
The pore diameter exhibits a variation about similar to that of the
wall thickness, while staying significantly larger. Both trends agree
with the theoretical prediction.
Figure 10 shows the effect of changing electrolyte resistivity on
pore diameter. Electrolyte resistivity was changed over a wide range
by changing HF concentration, and using ethylene glycol instead of
Figure 10. Effect of changing electrolyte resistivity on average macropore
diameter 共theory: solid curve, and experiment: lozenges兲. p-Si 1500 ⍀ cm,
J 0 ⫽ 10 mA/cm2 , electrolyte is 50% aqueous HF mixed with ethylene
glycol in variable proportions. The curve has been calculated using
⌽ SC ⫽ 0.1 eV, ␭ ⫽ 1 ␮m and ␷ R 共cm/s兲 ⫽ 3•103 ⫻ HF%.
C517
ethanol as the dilutant. The measured resistivity ␳ e is shown in the
same figure. When HF concentration decreases, resistivity is seen to
increase, and pore diameter is seen to decrease markedly. For the
interpretation, a problem is that changing the HF concentration is
expected to change the reaction velocity, a parameter which enters
the expression of S and which is hardly accessible to experimental
measurement. We have then assumed that ␷ R is simply proportional
to HF concentration, starting from the value of 105 cm/s already
used for 35% HF.
Notice that the electropolishing current density exhibits a nonlinear dependence on HF concentration.11,28,29 However, that current
density is determined by the dissolution rate of the interfacial oxide,
which is not relevant in the conditions of porous silicon formation.
In the latter case, the commonly accepted ideas on the reaction
mechanism make the hypothesis of a linear relationship between ␷ D
and HF concentration more plausible.
The resulting theoretical prediction is seen to give a very good fit
to the experimental data. When the HF concentration is decreased,
note that the decrease of ␷ R and the increase of ␳ e both tend to lower
the pore diameter. However, the effect of ␷ R is stronger than that of
␳ e , which enters Eq. 13 with the rather weak exponent of 1/4.
In physical terms, the process can be regarded in terms of
Laplacian instabilities. The small-scale instability is due to the
nearly insulating character of the space charge: at a scale much
smaller than ␭, the electrolyte represents the low-resistivity medium,
which makes the interface unstable, leading to the formation of microporous and mesoporous silicon. This effect was first pointed out
by Lehmann and Rönnebeck4 in terms of space-charge effects associated with interface curvature. At large scale, the space charge just
acts as a series interface resistance, and the microporous layer is an
anisotropic medium which channels the current lines in its pores. As
the layer grows thicker, this channeling effect makes the series resistance of the electrolyte play an increasingly important role, ultimately stabilizing the pore front and setting a characteristic size for
the largest pores. This is the main new effect quantitatively pushed
forward in the present paper. Equation 13 results from the combination of these two competing effects, together with chemical and
electrochemical effects 共incorporated with space-charge effects in
the instability parameter S兲. Notice that a larger instability parameter
pushes the hierarchical growth to later times and larger sizes 共S in
the numerator兲, whereas a larger electrolyte resistivity sets the parallel growth sooner (␳ e in the denominator兲.
Extension of the theory to the low-resistivity limit.—From the
above discussion, it appears that there is an overall good agreement
between theory and experiment. A technical improvement of the
theory might be taking into account fluoride depletion at the growth
front. However, this effect is not expected to play a major role in the
conditions used here. At this point, a much more important experimental fact is still unexplained, namely, the disappearance of macrostructures for resistivities below a critical value. This disappearance was emphasized by Lehmann and Rönnebeck, who reported a
critical value on the order of 3 ⍀ cm for aqueous electrolytes.4
However, macroporous structures have been obtained at somewhat
lower resistivity values in nonaqueous electrolytes.5-9 This disappearance is not accounted for by the above theoretical model, which
must be revised accordingly.
As stated above, a key element leading to parallel macropore
growth is the insulating character of the pore walls. However, when
acceptor concentration N A increases, the wall thickness decreases
proportionally to N A⫺1/2. Meanwhile, the average interimpurity distance decreases as N A⫺1/3. The two quantities cross for a doping on
the order of 1018 cm⫺3 共see Fig. 11兲. However, the crossing is so
smooth (N A⫺1/2 vs. N A⫺1/3) that even for dopings on the order of
1016 cm⫺3 , the two quantities are of the same order of magnitude.
This means that, in the vicinity of a path crossing the wall, only a
small number of impurities 共possibly zero兲 may be found. The classical description of the depleted wall in terms of a parabolic poten-
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Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
Figure 11. When doping increases, the space-charge layer thickness
(2␧␧ 0 ⌽ SC /e 2 N A) 1/2 becomes smaller than the average interimpurity distance (6/␲N A) 1/3 above a critical acceptor concentration.
tial profile then becomes inappropriate, and the discrete nature of the
charges in the wall has to be taken into account. One may infer that
the associated statistical fluctuations lead to a breakdown of the
insulating character of the wall.
In order to test this idea, we performed a numerical simulation of
a depleted wall. A portion of wall, of thickness d, was modeled as a
parallelepiped, where a number of negatively charged impurities
was thrown at random. The resulting potential ␾共r兲 was determined
with periodic boundary conditions on the side borders and ␾ ⫽ 0
on the two opposite faces of the wall. Figure 12 shows the change in
Figure 12. Map of the potential across a depleted wall. Two-dimensional
map in a section of the wall for N Ad 3 ⫽ 100 共a兲 and 5 共c兲. Profile of the
potential averaged over the portion of wall for N Ad 3 ⫽ 100 共b兲 and 5 共d兲.
Notice the large fluctuations in 共c兲 and 共d兲.
Figure 13. Insulating character of a depleted wall in the (N A , d) plane, as
deduced from our numerical simulations. The dotted lines represent the classical critical value given by twice the space-charge-layer thickness. The
points represent the critical value determined by numerical simulations in a
parallelepiped d ⫻ 2d ⫻ 2d. The dispersion arises from the random character of a given impurity set. The solid curves have been deduced from these
simulations by making a statistical analysis of the results and extrapolating
them to the case of a parallelepiped ⬃d ⫻ 5d ⫻ 5d, which is a better
representation of the wall covering the surface of a typical macropore.
shape of this potential when the number of impurities in the wall is
changed. For a large number of impurities, the classical parabolic
shape is recovered. However, for a small number of impurities, even
though the average potential follows the parabolic law, fluctuations
around this average value are very large. The ground state of a hole
in such a potential was then determined, with two different methods
共see Appendix兲. Up to this point of the calculation, the only parameters were the wall thickness and the doping. A third parameter now
appears, namely, the band bending ⌽ SC at the interface. If the extra
binding energy of the hole ground state is larger than ⌽ SC 共extra,
i.e., as compared with the usual binding energy of an acceptor兲, this
ground state is occupied, so that the wall is not fully depleted and is
not insulating. In the opposite case 共extra binding energy of the
ground state smaller than ⌽ SC), the wall is fully depleted and insulating. The result of our calculations is shown in Fig. 13 in the form
of a map in the plane (N A , d). The map gives the insulating character of such a model wall for ⌽ SC ⫽ 0.1 eV and 0.2 eV. At low
dopings, the limiting thickness between insulating and noninsulating
character coincides with the value (8␧␧ 0 ⌽ SC /e 2 N A) 1/2 expected
from classical Poisson-Boltzmann theory. However, systematic deviations from the Poisson-Boltzmann prediction appear when doping increases. At dopings higher than a few 1016 cm⫺3 , it is seen that
the thickness of an insulating wall drops to very small values, where
tunneling may take place. In practice the wall can no more become
insulating. The critical doping above which this behavior occurs
increases with ⌽ SC . For a value of 0.1 eV, the critical doping of
⬃1016 cm⫺3 is in fair agreement with the observed critical resistivity
of a few ⍀ cm 4 共actually, 1016 cm⫺3 corresponds to 1.4 ⍀ cm兲. For
⌽ SC ⫽ 0.2 eV, the critical doping density appears an order of magnitude higher 共see Fig. 13兲. This mechanism of breakdown of the
Journal of The Electrochemical Society, 149 共10兲 C511-C520 共2002兲
insulating character of a depleted wall should apply as well to the
case of n-Si. However, in that case, the barrier height is much larger
than for p-Si. The critical doping is then much larger, which explains
that macropores can be made on n-Si at least down to a resistivity of
0.1 ⍀ cm.
We think that our simulation strongly supports the idea that disappearance of the macrostructures at higher dopings is due to a loss
of the insulating character of the pore walls. It stands as a further
confirmation of our theoretical model. Interestingly, it predicts that
higher values of ⌽ SC would lead to the observation of macrostructures at higher dopings. Such values may probably be obtained in
other 共nonaqueous兲 electrolytes, or in the presence of cationic
surfactants,15 which should shift the flatband potential to more positive values, increasing the value of ⌽ SC . This may be a way to
realize macropores of characteristic sizes smaller than are obtainable
at present.
C519
here again, using the associated equation ⳵⌿/⳵t ⫽ K 关 ⌬⌿⫺ 2m * e␾⌿/ប 2 兴 and the
relaxation method. However, this method proved to be time costly and rather inaccurate:
in most practical cases, very large array sizes would be needed for ensuring that the
discretization mesh be smaller than the effective Bohr radius. If such is not the case, the
kinetic-energy term in the Hamiltonian is systematically underestimated, and the resulting ground-state energy is incorrect. These difficulties have led us to use an alternate
approximate procedure for the determination of the ground-state energy. For each
impurity site i, the potential induced by the other sites at ri was determined
␾ iloc共 ri兲 ⫽
兺 4␲␧␧ 兩r ⫺ r 兩 ⫹ f 共 r 兲
⫺e
i
j⫽i
0
i
If the ground state wave function is essentially located around site i, the corresponding
energy is just e␾ iloc(ri) minus the effective Rydberg of the medium. The ground state
energy was then taken as the minimum of this quantity, when i is varied from 1 to n
E 0 ⫽ Infi关 e␾ iloc共 ri兲兴 ⫺
Conclusion
We think that the formation of macrostructures in porous silicon
formed on low-doped p-type substrates is well accounted for by our
linear-stability approach. The existence of such macrostructures is a
simple corollary of the mechanisms giving rise to the formation of
microporous silicon. Microporous silicon and macroporous silicon
in p-type silicon are based on the same physical mechanism: electrostatic instabilities during the dissolution of silicon. We have developed a model based on a linear stability analysis which fits all
experimental data for pore growth in p-type silicon very well, at
least in the high HF concentration 共low current density兲 regime for
which crystallographic effects are not present and the model has
been mathematically worked out completely. The existence of a
steady state of growth of parallel macropores results from the stabilizing effect of ohmic drops in the electrolyte. In contrast to previous
belief, such a stabilization may occur even if the resistivity of the
semiconductor is larger than that of the electrolyte, because the role
of the latter is enhanced by the anisotropic character of the porous
layer. The characteristic pore sizes are quantitatively accounted for,
over a wide range of properties of the substrate and the electrolyte,
in the framework of a model incorporating the microscopic and
macroscopic 共ohmic兲 aspects of the problem. Finally, the disappearance of macrostructures for Si substrates of resistivities below a
critical value can be quantitatively justified by a loss of the insulating properties of a depleted wall for dopings exceeding a critical
level in excellent agreement with the experimental data.
Acknowledgment
We are indebted to Dr. M. Plapp for several useful discussions
and for bringing Ref. 27 to our attention.
The Max Planck Institute of Microstructure Physics assisted in meeting
the publication costs of this article.
Appendix
Computer Simulation Procedure
The potential ␾共r兲 generated in a square portion of a wall by a random distribution
of charged impurity centers located at positions ri (i ⫽ 1, . . . , n) was written under
the form
i⫽n
␾ 共 r兲 ⫽
兺 4␲␧␧ 兩r ⫺ r 兩 ⫹ f 共 r兲
i⫽1
⫺e
0
关A-1兴
i
where the function f (r) must fulfill Laplace equation with the boundary conditions
␾共r兲⫽0 on the surfaces of the wall and periodic boundary conditions of ␾共r兲 on the side
boundaries. The solution for f was determined by solving the associated equation
⳵ f /⳵t⫽ K⌬ f , where K is an arbitrary constant, using a finite difference method and an
explicit recursion scheme 共‘‘relaxation’’ method兲.
In a first approach, the ground state of a hole in the potential e␾(r) was determined
by numerical solution of the eigenenergy equation
⫺ប 2
2m *
⌬⌿ 共 r兲 ⫹ e␾ 共 r兲 ⌿ 共 r兲 ⫽ E⌿ 共 r兲
关A-2兴
关A-3兴
j
e2
8␲␧␧ 0 a 0*
关A-4兴
This procedure amounts to neglecting the extra binding energy of ‘‘two-center’’ hole
states 共similar to the H⫹
2 molecular ion兲. This is a fair approximation for acceptor
concentrations below that leading to formation of an impurity band.
If the whole procedure is carried out for n impurities in a wall of thickness d 0 and
surface area S 0 共corresponding to an acceptor concentration N A0 ⫽ n/S 0 d 0 ), a simple
scaling argument allows one to use the same potential map for a wall of thickness
d ⫽ kd 0 , a surface area S ⫽ k 2 S 0 , i.e., an acceptor concentration N A ⫽ N A0 /k 3 .
The scaled potential is then ␾(r) ⫽ ⌽ 0 (r/k)/k, and the ground-state energy is then
E ⫽ (E 0 ⫹ Ry)/k ⫺ Ry 共where Ry stands for the effective Rydberg e 2 /8␲␧␧ 0 a 0 * ).
Calculation of a single potential map then leads to the value of the groundstate energy along a line (N A0 /k 3 , kd 0 ) in the (N A , d) plane. The limiting point
such that the ground-state energy is ⫺Ry ⫺ ⌽ SC 共conducting wall/insulating wall
transition兲 is deduced from E ⫽ (E 0 ⫹ Ry)/k ⫺ Ry ⫽ ⫺Ry ⫺ ⌽ SC , whence k
⫽ ⫺(E 0 ⫹ Ry)/⌽ SC. Calculation of potential maps for various sets of impurities and
a given value of n leads to a set of points (N A , d). The spreading of these points is due
to the random character of the ground state, which depends on the impurity set. The
larger the portion of wall area considered, the deeper the ground state is. In principle,
the relevant area is the area of a macropore, which is typically between one and two
orders of magnitude larger than d 2 . However, direct numerical simulation of such a
large area is unpractical. We then made ten numerical simulations of a portion of wall of
volume d ⫻ 2d ⫻ 2d, and took the second lowest value of the ground state obtained
from such simulations. This method amounts to calculating the average value of the
ground state for an area on the order of 20d 2 , and provides results slightly less noisy
than a direct 共unpractical兲 simulation of a single such area. The corresponding results
are shown as the solid lines in Fig. 13.
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