Journal of the Physical Society of Japan Vol. 68, No. 7, July, 1999, pp. 2218-2220 Letters Self-Organized Pattern Formation in Porous Silicon Using a Lattice Model with Quantum Confinement Effect Hiroshi Furuta, Tokiyoshi Matsuda, Chihiro Yamanaka and Motoji Ikeya Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043 (Received December 11, 1998) A spatial structure of porous silicon formed by anodization was simulated using a lattice model considering the quantum confinement effect. Calculation starting from a random distribution of null cells at the surface yielded the final granular, columnar or flat surface structures depend- ing on the magnitude of the anodizing bias voltage in concordance with previously reported structures. Our calculations suggest that columnar and dendritic structures can be produced in degenerately doped p-type Si wafers by self-organization of nanowires with quantum confinement effects. KEYWORDS: pattern formation, lattice model, quantum confinement, self-organization, simulation, anodize, etch- ing, columnar, granular, porous silicon §1. Introduction Recent developments in nanosize fabrication technolo- gies have made enabled the construction of a circuit formed with quantum wells, wires and dots whose prop- erties are affected by quantum size effects. Techniques for the manipulation of atoms and sufficient resolution to image them using micro-probes such as a scanning tunneling microscope (STM) and an atomic force mi- croscope (AFM) have greatly contributed to the physics and technology of nanosized circuits. There are a num- ber of works dealing with the construction of small-scale quantum-sized circuits using quantum-size effects which will play an important role in future nanosized circuits. The phenomena of self-organization have attracted at- tention as a breakthrough method to fabricate a large- scale quantum-sized circuit. Meanwhile, porous silicon (PS) which is obtained by anodizing from crystalline silicon in HF-based elec- trolytes has a skeletal structure made of Si nano-wires. A Si-based photonic device might be fabricated since vis- ible photoluminescence (PL) is observed at room temper- ature. 1) The mechanism and the origin of PL from PS have not yet been well understood. The combination of quantum confinement and surface modification of Si nanocrystals must be considered in order to explain the PL mechanism. In this work, we report a computer simulation of self- organized pattern formation of the nano-wires in PS us- ing a two-dimensional (2-D) and a three-dimensional (3- D) lattice model. Understanding the mechanism behind the formation of PS and its property will contribute to- ward the organization of a large-scale nanosized circuit. §2. Calculation Procedure A quantum confinement effect model was proposed by Lehmann and G¨ osele for the formation of nanowires in PS. 2) Hole generation in the valence band at the sili- con surface is a rate-controlling step in the anodization of silicon. Silicon dissolves into electrolytes under an anodizing bias voltage larger than the band-gap energy. The band-gap energy of the silicon wire increases as the wire width decreases. Higher bias is needed for the holes to be produced or to reach the surface of the nanowires with higher quantum confinement energy. Further an- odization is suppressed at the nanowires in PS because narrower nanowires have a lower hole concentration. For- mation of PS is thus affected by the anodizing bias volt- age. In our calculation, a lattice model was applied to sim- ulate the formation of a spatial structure of PS. Each state of a filled and a null cell is defined as “1” and “0”, respectively. Necessary conditions for the dissolution of a cell of interest were assumed as follows. Condition 1: A target cell must have at least one null cell in the surrounding cells; the silicon cluster must have a surface which touches the etchant for dissolution. Condition 2: The target cell must have lower confine- ment energy than the anodizing bias voltage; the an- odizing bias voltage was assumed to be the same at the surface. Quantum confinement energy 3) E QC of a parti- cle of size L and mass m * is given by E QC = (¯ h 2 /2m * ) × (π 2 /L 2 ). In 2-D calculations, quantum con- finement energy at a target cell was defined as E QC =1/L 2 =1/S, (1) where S is the sum of filled cells in the surrounding eight cells and target cell itself. The quantum confinement by the nearest neighbor and next-nearest neighbor cells was considered in a calculation. Similarly for the 3-D lattice, quantum confinement en- ergy was defined as E QC =1/L 2 =1/V 2/3 , (2) where V is sum of filled cells in surrounding cells and the 2218