Porous Silicon Formation Models

At first sight the position of the pore tip in the electrode seems to be a badly defined parameter because its position changes while the pore is propagating. However, the pore tip is always farthest away from the bulk of the electrolyte. Diffusion thereby produces a minimal concentration of active ions and a maximum concentration of reaction products at the pore tip. This condition may already be sufficient for a passivation of the pore walls if the passivation mechanism is sensitive to ionic concentrations. 

The above models describe pore formation from a mathematical point of view and the parameters of the models are subsequently assigned to physical values. The models described below are based on the specific chemistry or physics of the semiconducting electrode. 

A basic requirement for electrochemical pore formation is passivation of the pore walls and passivity breakdown at the pore tips. Any model of the pore formation process in silicon electrodes has to explain this difference between pore tip and pore wall conditions. Three different mechanisms have been proposed to explain the remarkable stability of the silicon pore walls against dissolution in HF, as shown in Fig. 6.1. 

Anodic oxide formation suggests itself as a passivating mechanism in aqueous electrolytes, as shown in Fig. 6.1a. However, pore formation in silicon electrodes is only observed in electrolytes that contain HF, which is known to readily dissolve Si02. For current densities in excess of JPS a thin anodic oxide layer covers the Si electrode in aqueous HF, however this oxide is not passivating, but an intermediate of the rapid dissolution reaction that leads to electropolishing, as described in Section 5.6. In addition, pore formation is only observed for current densities below JPS. Anodic oxides can therefore be excluded as a possible cause of pore wall passivation in PS layers. Early models of pore formation proposed a 

Electrochemical Cell Configuration Corresponding Energy-Band Diagrams 

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Porous silicon formation models have been reviewed (Smith and Collins 1992 Allongue 1997 Zhang 2001). A conceptual analysis has been attempted (Zhang 2004). Major theoretical contributions applying to macropore formation are listed in Table 3. 

A. Valance, Theoretical model for early stages of porous silicon formation fromn- andp-Types silicon... 

P. Jaguiro, S. La Monica, S. Lazarouk, and A. Ferrari, Theoretical model of porous silicon formation,... 

In the anode reaction, the silicon consumes the positive carriers and is solubilized through oxidation. Several models for the anodic reactions have been proposed and are outlined in the handbook chapter Porous Silicon Formation by Metal Nanoparticle Assisted Etching. ... 

Theunissen MJJ (1972) Eteh channel formation during anodic dissolution of n-type silicon in aqueous hydrofluoiie acid. J Electrochem Soc 119 351-360 Urata T, Fukami K, Sakka T, Ogata YH (2012) Pore formation in p-type sihcon in solutions containing different types of alcohol. Nanoscale Res Lett 7 329 Valance A (1997) Theoretical model for early stages of porous silicon formation from n- and p-type silicon substrates. Phys Rev B 55 9706-9715... 

Eaton P, West P (2010) Atomic force microscopy. Oxford University Press, Oxford Frohnhoff S, Marso M, Berger MG, Thonissen M, Liith H, Miinder H (1995) An extended quantum model for porous silicon formation. J Electrochem Soc 142(2) 615 Goodhew PJ, Humphreys J, Beanland R (2001) Electron microscopy and analysis, 3rd edn. Taylor Francis, New York... 

An extension of this QC model, including tunneling probabilities between the confined crystallites and the bulk, has been developed [Fr6]. The QC model for microporous silicon formation, however, is still qualitative in character, and a quantitative correlation between anodization parameters and the morphology and properties of the porous structure is at yet beyond the capability of the model. 

For homogeneously doped silicon samples free of metals the identification of cathodic and anodic sites is difficult. In the frame of the quantum size formation model for micro PS, as discussed in Section 7.1, it can be speculated that hole injection by an oxidizing species, according to Eq. (2.2), predominantly occurs into the bulk silicon, because a quantum-confined feature shows an increased VB energy. As a result, hole injection is expected to occur predominantly at the bulk-porous interface and into the bulk Si. The divalent dissolution reaction according to Eq. (4.4) then consumes these holes under formation of micro PS. In this model the limited thickness of stain films can be explained by a reduced rate of hole injection caused by a diffusional limitation for the oxidizing species with increasing film thickness. 

Historically, the first reports of porous silicon layers were by Uhlir [59] and Turner [60]. These authors reported on the electropolishing of silicon and noted that under certain conditions a porous layer was formed at the silicon surface. The first models for porous layer formation assumed that the layer was formed on the silicon substrate by a deposition process thought to involve the reduction of divalent silicon to amorphous Si via a disproportionation reaction in solution [61]. Subsequently, Theunissen [62] showed that the porous structure was the result of a selective etching process within the silicon, contradicting the silicon deposition model. 

One of the arguments used in the porous silicon literature to explain the typical dimension of the structures states that the pore formation is governed by the width of the space charge region formed near the semi-conductor/electrolyte junction (see, for example, [17]). The argument seems to be very plausible as it works well for the mesoporous structures (10-100 nm pore size) obtained from n-type Si. The model has also been proposed for porous SiC [18] and elaborated in Shishkin et al. [14],... 

Recently, Wolkin et al. observed an upper limit of the PL emission energy of 2.1 eV in oxidized porous silicon even if the nanocrystal size became smaller than 2 nm. This behavior, which seems to contradict quantum confinement, was explained in terms of the formation of stabilized electronic states on Si=0 bonds at the surface. For nanocrystals with diameters smaller than 2.8 nm, the widening of the band gap due to quantum confinement makes them appear as inner band gap states. Including the results of Wolkin et al. in our model calculations, we obtained nice agreement with the experimental data. " ... 

In certain papers (Aroutiounian and Ghulinyan, 2000 Aroutiounian et al., 2000), a fractal model of a porous layer formation was proposed.The consideration of the time-dependent pore growth process has allowed us to calculate important parameters of the porous matrix, such as the formed surface area, and the surface and volume porosity values. We have theoretically shown that the formed surface area is strongly dependent on the difference between the pore size growth velocities parallel and perpendicular to the surface (i.e. the crystallographic orientation of the silicon surface). The volume and surface porosity values and the formed porous surface area are linear functions of the density of the anodization current. These results are in agreement with other theoretical and experimental data. 

From the following observations that (1) Si dissolves with formation of molecular H2 and protons, (2) only Si forms mi-croporous layers with a single crystalline skeleton, and (3) hydrogen (Deuterium) species penetrate into the Si substrate upon porous Si formation, a model was proposed to explain pore initiation. It was assumed that hydrogen incorporation induces structural defects in Si, which may act as active sites for the localized Si dissolution. The model relates pore initiation to the selective dissolution of the hydrogen-induced structural defects at the surface of bulk silicon [117,118]. 

Balagurov LA, Yarldn DG, Petrova EA (2000) Electronic transport in porous silicon of low porosity made on a p" substrate. Mater Sci Eng B 69-70 127 Balberg I (2000) Transport in porous silicon The pea-pod model. Philos Mag B 80 691 Beale MU, Benjamin JD, Uren MJ, Chew NG, Cullis AG (1985) An experimental and theoretical study of the formation and microstmcture of porous sihcon. J Cryst Growth 73 622 Ben-Chorin M, Moller F, Koch F (1994) Nonlinear electrical transport in porous silicon. Phys Rev B 49 2981... 

The quantum confinement model was extended by Frohnhoff etal to account for the wide distribution of pore diameters of the PS formed on p-Si. Tunneling of holes through silicon crystallites was proposed as a process also involved in the formation of the quantum size porous structure. The tunneling current oscillates with the crystallite size which was considered to be responsible for the uneven pore size distribution and for the stability of very small crystallites in the PS. The quantum confinement model... 

Another model proposed by Kohl and coworkers [77, 78] is based upon the strain - induced preferential etching described earlier. The model accounts for the formation of macropores and highly branched micropores when the silicon is rendered porous in either nonaque-ous or aqueous HF solutions, respectively. 

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