Quantum Confinement Effects on the Dielectric Constant of Porous Silicon

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R. TSU**a, L. IORIATTI***, J. F. HARVEY*b, H. SHEN*, AND R. A. LUX* *US Army Res. Lab., Ft. Monmouth, NJ 07703 **Univ. of North Carolina, Charlotte, NC 28223 ***Inst. Fisica e Quimica, USP, Sao Carlos, Brazil ABSTRACT The reduction of the dielectric constant due to quantum confinement is studied both experimentally and theoretically. Angle resolved ellipsometry measurements with Ar- and HeNe-lasers give values for the index of refraction far below what can be accounted for from porosity alone. A modified Penn model to include quantum size effects has been used to calculate the reduction in the static dielectric constant (E)with extreme confinement. Since the binding energy of shallow impurities depends inversely on E2 , the drastic decrease in the carrier concentration as a result of the decrease in f leads to a self-limiting process for the electrochemical etching of porous silicon. INTRODUCTION The observation of photoluminescence in electrochemically etched porous silicon [1] is consistent with quantum confinement [2]. As the physical size approaches several nanometers, the reduction in the static dielectric constant results in an increase in the binding energy of shallow impurities. This increase reduces carrier concentration which in turn renders a decrease in electrochemical reaction as the particle size decreases in etching. The drastic decrease in the index of refraction was first reported by Harvey et al [3]. Theoretical treatment of the dielectric constant in quantum confined systems [4],[5], shows that a significant reduction takes place when the width of the quantum well is reduced to 2.5nm or less. For a three-dimensionally quantum confined system, due to the large separation in the discrete energy states, the use of k • P energy bands to calculate the dielectric constant may not be fruitful, therefore in this work, a modified Penn model [6] is used which takes into account the finite particle size. The calculated static dielectric constant of a silicon sphere of 3nm is only half the value of the bulk silicon. As we know that the binding energy Eb is increased due to quantum confinement even without taking into account the reduction of Eas in the work of Bastard [7] and loriatti and Tsu [8]. However, the reduced E results in a reduced Bohr radius such that the ground state hydrogenic wavefunction is already significantly reduced at the boundary of a particle. Therefore, we may use the reduction in Ein the expression for the binding energy of a hydrogenic shallow impurity, to estimate the increase in the binding energy. A factor of two decrease in Ethus, results in a four fold increase in Eb, and since the carrier concentration n - exp (-F_./kT), there is a drastic reduction in the carrier density. DIELECTRIC CONSTANT OF A Si-SPHERE Dielectric constant is a measure of virtual optical transitions. Quantum confinement increases the energy denominator, resulting in a reduction of the dielectric constant. In the case of moderate quant