Fractal nature of porous silicon nanocrystallites T. Nychyporuk, V. Lysenko, and D. Barbier Materials Physics Laboratory (LPM), CNRS UMR-5511, INSA de Lyon, 7 Avenue Jean Capelle, Bâtiment Blaise Pascal, 69621 Villeurbanne Cedex, France Received 8 May 2004; revised manuscript received 5 January 2005; published 1 March 2005 Experimental study of the hydrogen coverage of a nanoporous silicon-specific surface reveals a fractal nature of the surface of silicon nanocrystallites constituting the porous layer. A fractal model describing the nanocrystallite morphology is elaborated. The evolution of the nanocrystallites’ fractal dimension 2.1–2.4 along with porosity is deduced from a correlation of the model to the experimental measurements of hydrogen concentration. DOI: 10.1103/PhysRevB.71.115402 PACS numbers: 68.65.-k, 61.46.+w, 61.43.Hv, 61.43.Gt I. INTRODUCTION Porous materials are frequently encountered examples of naturally disordered media. Porous silicon PS nanostruc- tures and their physical properties were also objects of nu- merous attempts to be described by fractal models. 1,2 From a comparison of electron microscopy data with computer models, 3 from small-angle x-ray 4,5 and neutron 6 scattering as well as from atomic force microscopy observations, 7,8 it is known that the PS network has a fractal structure. A fractal model of pore formation in a PS layer was recently proposed 9 by Aroutiounian et al. The fractal structure of PS was found to govern its ac conductivity in a low-frequency regime 10 and was recently supported by Raman and photolu- minescence spectroscopies. 11 Being relatively well described at a macroscopic scale i.e., at the level of the whole PS network, the porosity dependence of the fractal nature of the nanocrystallites forming the PS structure was never, to our knowledge, brought to the fore, while the fractal nature of the nanocrystallites constituting PS layers was already quan- titatively treated in one of our recent papers. 12 Now, in this paper we present i experimental results reflecting clearly a porosity-dependent fractal nature of the specific-surface of the silicon nanocrystallites constituting the PS nanostructure, ii a fractal model describing the specific-surface nanostruc- turing, and finally, iii the porosity dependence of the specific-surface fractal dimensions obtained from a compari- son of the experimental results and the elaborated model. II. EXPERIMENTAL METHODS A. Porous silicon formation and structural characterization Our PS samples were produced according to a standard procedure 13 of electrochemical etching of monocrystalline 100-oriented boron-doped 1–10  cm Si wafers at cur- rent densities, 2 – 300 mA cm 2 . The etching solutions were 9:1 and 3:1 by volume mixtures of concentrated aqueous hydrofluoric acid 48% and ethanol. Depending on the current density and on the etching so- lution composition, the porosity of the layers 40–90 % was estimated from the PS refractive index measurements per- formed by means of a Perkin-Elmer GSX-2 Fourier transfor- mation infrared FTIR spectrometer used in reflective back- scattering geometry and from the correlation of the index values to porosity by using Bruggeman’s effective media model. 14 The PS refractive indexes were deduced from the spectral position of the interferential fingers detected in the 3000–5000 cm -1 spectral range for the optically thin PS samples using the simple relation, n PS = 1 2d  1  k - 1  k+1  -1 1 where d denotes the thickness of the porous layer, and  k is the wavelength of the kth fringe. The thickness of all porous films determined from optical microscopy measurements was about 10 m. The diameter of the Si nanocrystallites constituting the porous layer was estimated by Raman mi- crospectroscopy using a method described in details elsewhere. 15 B. Hydrogen concentration measurements Hydrogen concentration in the fresh as-prepared PS samples was measured 12 by means of absorption infrared spectra of Si-H X stretching bonds obtained by an FTIR spec- trometer in attenuated total reflection ATR measurement mode. A germanium single crystal with a refractive index of 4 was used for the infrared wave guiding and the incident beam angle was 45°. The ATR mode was chosen mainly because of the extremely small volume of the studied porous samples interacting with the testing evanescent electromag- netic field of the infrared light, in order to avoid complete loss of the detected signal due to a strong absorption from a huge number of the Si-H x bonds, as it occurs, for example, in the transmission measurement mode. The hydrogen concentration N H mmol g -1  is estimated from the absorption spectra by using the following relation used earlier for an estimation of the hydrogen content in amorphous Si layers: 12 N H = 1  S  Si 1- P  h  h dh = I S  S  Si 1- P , 2 where I S cm -1  is the integrated absorption of the stretching band,  Si is the monocrystalline Si density 2.33 g cm -3 , P is the porosity of the PS layer, cm -1  is the absorption PHYSICAL REVIEW B 71, 115402 2005 1098-0121/2005/7111/1154025/$23.00 ©2005 The American Physical Society 115402-1