Size dependence of the optical gap of ‘‘small’’ silicon quantum dots: Ab initio and empirical correlation schemes Shanawer Niaz b , Emmanuel N. Koukaras b , Nikolaos P. Katsougrakis a , Theodoros G. Kourelis a , Dimitrios K. Kougias b , Aristides D. Zdetsis b,⇑ a Division of Medical Physics, School of Medicine, University of Patras, Patras GR-26500, Greece b Molecular Engineering Laboratory, Department of Physics, University of Patras, Patras GR-26500, Greece article info Article history: Available online 13 July 2013 Keywords: Quantum Dots Silicon Density Functional Theory Quantum Confinement bond-order-length-strength abstract We present a comparative study of the energy-gap dependence on diameter d of ‘‘small’’ (d < 20 Å) hydro- gen-terminated Si quantum dots, using density functional theory (DFT) with the hybrid functional of Becke, Lee, Parr and Yang (B3LYP). These accurate real space ab initio calculations [see [1], Garoufalis et al., Phys. Rev. Lett. 87 (2001) 276402] are used to compare the size dependence of the band gap accord- ing to quantum confinement theory in relation to the empirical bond-order-length-strength (BOLS) cor- relation mechanism, usually applied to larger nanocrystals. Our results for the gap variation, in the range of diameters considered here, are in very good agreement with quantum confinement theory and they reproduce by extrapolation the experimental band gap of bulk silicon with high accuracy (error smaller than 1%). On the contrary, extrapolation of fitted band gaps by BOLS scheme, grossly overestimates the band gap of bulk (by almost 80%); whereas, forcing the agreement of bulk band gap results in largely underestimated dot band gaps, in the range of diameters considered here. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction The visible photoluminescence (PL) of silicon quantum dots or silicon nanocomposed structures, such as of porous Silicon (p-Si) has attracted a lot of attention in recent years, both experimentally and theoretically [1–6]. Numerous applications of silicon quantum dots (QDs) for optical devices such as light emitting devices [7], photovoltaic cell [8] and biocompatible luminescent silicon quan- tum dots [9] have been developed. A critical quantity for most applications is the band gap varia- tion with size for various regions of sizes. Several models have been proposed in the early literature for the band gap variation with size, such as quantum confinement (QC) [10–14], free-exciton collision [15], impurity luminescent centre [16] mechanism and the empirical bond-order-length-strength (BOLS) correlation mechanism [17–19]. Most of the work in this field has been devoted to understand- ing the visible photoluminescence of the QDs and correlating the spectrum with the diameter of the quantum dots. It is widely ac- cepted now [4–6] that the luminescence in the visible of oxygen- free small Si quantum dots (of well defined diameter), is mainly due to quantum confinement of the corresponding quantum dots [1]. However, several times, other alternative mechanisms, such as the ones mentioned above have been also considered for the description of the detailed variation of the gap with the size (diam- eter) of the dots. Knowledge of the detailed size variation of the gap, especially in a large range of sizes (hopefully approaching macroscopic sizes) is essential for ‘‘band gap engineering’’ and for the design of tunable photoluminescence. In our present work we examine in detail the functional depen- dence of the energy gap on the basis of very accurate ab initio DFT calculations based on the B3LYP functional [1] in the range of about 10–20 Å, which is expected to be dominated by quantum confinement. We have paid special attention not only in the basic approximations but also in the technical details (quality of the ba- sis set, mesh of integration, etc.) as is described in the technical de- tails (Section 2). To extrapolate these results to larger diameter scales, dominated usually by empirical schemes, we have fitted our DFT gap-versus-size results according to the predictions of two representative but ‘‘extremely opposite’’ (both in philosophy and in principles) methods: The ‘‘ab initio based’’ Quantum con- finement theory, and the empirical scheme known as the bond-or- der-length-strength (BOLS) correlation mechanism [17–19]. As will be shown below, QC not only fits perfectly the DFT/B3LYP results in the region of 10–20 Å, but also predicts with unexpected accuracy the extrapolated band-gap of the infinite size crystal. On the con- trary, BOLS scheme can be fitted or fixed to describe only one region of diameters, but not both, and certainly not all. This could 0167-9317/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.mee.2013.07.005 ⇑ Corresponding author. Tel./fax: +30 2610 997458. E-mail address: zdetsis@upatras.gr (A.D. Zdetsis). Microelectronic Engineering 112 (2013) 231–234 Contents lists available at ScienceDirect Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee