Pores growth

Sintering of particles occurs when one heats a system of particles to an elevated temperature. It Is caused by an interaction of particle surfaces whereby the surfaces fuse together and form a solid mass. It Is related to a solid state reaction In that sintering is governed by diffusion processes, but no solid state reaction, or change of composition or state, takes place. The best way to illustrate this phenomenon is to use pore growth as an example. 

Such a behavior of the UA(t) or ja(t) functions is consistent with a fairly well established pore growth mechanism.4 According to this mechanism, the linear potential growth (and current density... 

Many theories on the formation mechanisms of PS emerged since then. Beale et al.12 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. Smith et al.13-15 described the morphology of PS based on the hypothesis that the rate of pore growth is limited by diffusion of holes to the growing pore tip. Unagami16 postulated that the formation of PS is promoted by the deposition of a passive silicic acid on the pore walls resulting in the preferential dissolution at the pore tips. Alternatively, Parkhutik et al.17 suggested that a passive film composed of silicon fluoride and silicon oxide is between PS and silicon substrate and that the formation of PS is similar to that of porous alumina. 

The reaction product SiF4 would be gaseous, but it reacts with two HF to Si I7 and two protons and stays in solution [Mellj. The solubility of Si 17, which is in the order of mol 1 1 is significantly reduced in the presence of alkali metal ions. Especially for Rb, K or Cs, a micrometer thick, insoluble layer of metal hexafluoro-silicate may be formed on the electrode surface [Hal2j. The divalent electrochemical dissolution reaction is dominant during PS formation. The effects of the reaction products SiFg and H2 on pore growth are discussed in Section 9.5. 

Let us assume that the total surface of an electrode is in an active state, which supports dissolution, prior to anodization. The application of a constant anodic current density may now lead to formation of a passive film at certain spots of the surface. This increases the local current density across the remaining unpassivated regions. If a certain value of current density or bias exists at which dissolution occurs continuously without passivation the passivated regions will grow until this value is reached at the unpassivated spots. These remaining spots now become pore tips. This is a hypothetical scenario that illustrates how the initial, homogeneously unpassivated electrode develops pore nucleation sites. Passive film formation is crucial for pore nucleation and pore growth in metal electrodes like aluminum [Wi3, He7], but it is not relevant for the formation of PS. 

The process described above is expected to produce a random distribution of active and passive spots on the electrode interface. But the electrode surface may also be artificially patterned prior to anodization in order to form nucleation centers for pore growth. This may be a lithographically formed pattern in said passive film or a predetermined pattern of depressions in the electrode material itself, which become pore tips upon subsequent anodization. The latter case applies to silicon electrodes and is discussed in detail in Chapter 9, which is devoted to macropore formation in silicon electrodes. 

Another consequence of pore geometry is that for crystalline electrodes, other crystal planes are exposed to the electrolyte at the pore tip than at the pore walls. The dependence of pore growth on crystal orientation of the silicon electrode is discussed in Chapters 8 and 9. 

Pore Size Regimes and Pore Growth Rates... 

Fig. 6.4 Dependence of pore growth rates on crystal orientations of the substrate for different substrate doping densities and applied current densities (in HF H20 ethanol, 2 5 1). After data of [Gu4]).

Fig. 6.5 Pore growth rates in (a) 3% aqueous dicated. Note that high growth rates are ob-...

Having discussed the causes of pore wall passivity, we will now focus on the active state of the pore tip, which is caused by its efficiency in minority carrier collection. Usually the current density at the pore tip is determined by the applied bias. This is true for all highly doped as well as low doped p-type Si electrodes and so the pore growth rate increases with bias in these cases. For low doped, illuminated n-type electrodes, however, bias and current density become decoupled. The anodic bias applied during stable macropore formation in n-type substrates is... 

Fig. 9.9 SEM micrograph of an n-type silicon electrode with an etched macropore array (5 2 cm, (100), 3 V, 350 min, 2.5% HF). Pore growth was induced by a square pattern of pits produced by standard lithography and subsequent alkaline etching (inset upper right). In order to measure the depth dependence of the growth rate, the current density was periodically kept at 5 mA cm 2 for 45 min and then reduced to 3.3 mA crrf2 for 5 min. This results in a periodic decrease in the pore diameter, as indicated by the white labels on the left-hand side. After [Le9].

The steady-state condition (/ap=Jps) at the pore tip determines not only the pore diameter but also the pore growth rate. The rate rp of macropore growth can be calculated if the local current density at the pore tip is divided by the dissolution valence nv (number of charge carriers per dissolved silicon atom), the elementary charge e (1.602 xlO-19 C) and the atomic density of silicon Nsi (5xl022 cm-3) ... 

Fig. 9.18 (a) If the pore tip is the only reaction site, a linear HF concentration gradient and a parabolic pore growth is observed, (b)... 

If HF is consumed at the pore walls, also, the concentration gradient is nonlinear and pore growth becomes retarded, (c) Using depth... 

Arrays with pore diameters d as small as about 0.3 pm have been achieved [Lel7]. The lower limit for the pore diameter is established by breakdown, according to Fig. 8.1b, which leads to light-independent pore growth and spiking. There seems to be no upper limit for the pore diameter, because the formation of 100 pm wide pores has been shown to be feasible [Kl3]. Array porosities may range from 0.01 to close to 1. The porosity, which is controlled by the etching current, determines the ratio between pore diameter and pitch of the pore pattern. 

The pore growth direction is along the (100) direction and toward the source of holes. For the growth of perfect macropores perpendicular to the electrode surface (100), oriented Si substrates are required. Tilted pore arrays can be etched on substrates with a certain misorientation to the (100) plane. Misorientation, however, enhances the tendency to branching and angles of about 20° appear to be an upper limit for unbranched pores. For more details see Section 9.3. 

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