Size-dependent energy band gap and dielectric constant within the generalized Penn model applied to a semiconductor nanocrystallite

Abstract

We calculated size-dependent single particle energy eigenvalues, wave functions, energy band gap, and dielectric constant for a semiconductor nanocrystallite, within the framework of the two band Penn model. The finite size effects are presented as size-dependent correction terms to the energy eigenvalues and the energy band gap of the bulk semiconductor. Ours is a self-consistent calculation within the Penn model [Phys. Rev. 128, 2093 (1962)], providing a much deeper understanding of the size dependences of energy band gap and the dielectric constant, as compared to those performed in the past using the so called generalized Penn model. Computed results on energy band gap and dielectric constant from our investigations exhibit very good agreement with those obtained from experimental measurements and the detailed numerical calculations performed using techniques such as density functional approach. Unlike our calculation, prior reported calculations of dielectric constant with the use of the generalized Penn model completely ignore or do not properly take into account the size dependences of oscillator strength and the energy band gap. It is shown that the size dependences of oscillator strength and the energy band gap play an important role and they lead to a fractional exponent (between 1 and 2) of size (radius) that appears in the formula for the dielectric constant of a nanocrystallite.

Dielectric function of colloidal lead chalcogenide quantum dots obtained by a Kramers-Krönig analysis of the absorbance spectrum

“…Experimental data can then be compared to theory, which is mostly focused on understanding the size depen-dence of the electronic dielectric constant ⑀ 0 . ⑀ 0 is often described within the generalized Penn Model, 12 which states that ⑀ 0 strongly decreases with decreasing size, due to an opening of the band gap for smaller Qdots ͑quantum confine-ment͒. However, recent reports have cast doubt on the validity of the Penn model, proposing that the size dependence arises from a reduction in ⑀ 0 near the Qdot surface while it remains bulklike in the inner Qdot volume.…”

The effects of inorganic surface treatments on photogenerated carrier mobility and lifetime in PbSe quantum dot thin films

“…The sensitivity factor for the microwave cell used in this experiment was (2) where was the permittivity of the PbSe QD thin film at the resonance frequency, was the width (0.9 in) and was the height (0.4 in) of the WR90 waveguide, and was the length of the resonant cell (9.136 cm). The used in this work was calculated to be 24 by the generalized Penn approximation [10,58,59] and the effective medium approximation [37,60].…”

Dielectric engineering of Ge nanocrystal/SiO2 nanocomposite thin films with Ge ion implantation: Modeling and measurement

“…For a semiconductor nanocrystal, the decrease of its dielectric constant with size reduction has been reported [25][26][27][28][29]. Such phenomenon is often attributed to the bandgap increase inside the nanocrystal.…”

“…However, when f i > 0 (i.e., nc-Ge distributed region), e i is always larger than e SiO 2 , but smaller than e ncÀGe . It should be noted that e ncÀGe is different from the dielectric constant of the bulk crystalline Ge, because the dielectric constant of a semiconductor nanocrystal decreases with reducing particle dimensions [25][26][27][28][29]. The details of e ncÀGe used in the calculation will be discussed later.…”