Journal of Nanomaterials

Volume 2015, Article ID 481087, 9 pages

http://dx.doi.org/10.1155/2015/481087

## Electronic and Optical Properties of Small Hydrogenated Silicon Quantum Dots Using Time-Dependent Density Functional Theory

School of Applied Physics, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 29 April 2015; Revised 31 August 2015; Accepted 31 August 2015

Academic Editor: Bin Dong

Copyright © 2015 Muhammad Mus-’ab Anas and Geri Gopir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a systematic study of the absorption spectrum of various sizes of small hydrogenated silicon quantum dots of quasi-spherical symmetry using the time-dependent density functional theory (TDDFT). In this study, real-time and real-space implementation of TDDFT involving full propagation of the time-dependent Kohn-Sham equations were used. The experimental results for SiH_{4} and Si_{5}H_{12} showed good agreement with other earlier calculations and experimental data. Then these calculations were extended to study larger hydrogenated silicon quantum dots with diameter up to 1.6 nm. It was found that, for small quantum dots, the absorption spectrum is atomic-like while, for relatively larger (1.6 nm) structure, it shows bulk-like behavior with continuous plateau with noticeable peak. This paper also studied the absorption coefficient of silicon quantum dots as a function of their size. Precisely, the dependence of dot size on the absorption threshold is elucidated. It was found that the silicon quantum dots exhibit direct transition of electron from HOMO to LUMO states; hence this theoretical contribution can be very valuable in discerning the microscopic processes for the future realization of optoelectronic devices.

#### 1. Introduction

Current nanostructures, including clusters, biomolecules, and molecular nanodevices, have become the core of many fundamental and technological research projects. The study of static and dynamic electron-electron correlation makes it possible to characterize the electronic, structural, and bonding properties in nanostructure regime. Additional to that, its necessity is also related to the optical, electronic, and time-resolved spectroscopies. Since the structural and electronic properties of nanoparticle, in particular silicon quantum dots, are absolutely sensitive to the changes of atomic configuration, impurities, and doping effect [1], these challenges urged researchers to understand the phenomena involved specifically from quantum mechanics perspective. One can obtain the electronic structure information from the optical analysis; in particular, the optical response of the quantum dots will provide informative view of its dependency on their size and geometrical structure. This is an important feature, since the determination of the structure is, in general, a very sensitive task, either for prototype construction or for sophisticated relaxation of total energy minimization. Furthermore, the knowledge of the geometrical structure for modeling is highly required from solid state physics as a basis for understanding many of the properties of the nanostructure material.

Relatively large quantum dots consisting of thousands of atoms need very high computational cost in order to solve their many-body problem wave functions. Here, the importance of parallel processing combined with perfect computational strategy urged researchers to seek more efficient alternatives to overcome these challenges. Real-space grids are a powerful alternative tool for the simulation of electronic systems. One of the main advantages of this approach is the flexibility and simplicity of working directly in real space where the different fields are discretized on a grid, combined with competitive numerical performance and great potential for parallelization. In this study, this paper is divided into two main parts. In the first section, the electronic properties of ground state hydrogenated silicon quantum dots using density functional theory by three different methods will be briefly reviewed. This comparison step is taken to act as a benchmark for further study of the optical spectra using the time-dependent density functional theory (TDDFT) [2, 3]. Secondly, the basic framework of TDDFT is used for the calculation of the quantum dot optical spectra. This framework is applied in real space by the first principles code of OCTOPUS [4] that allows the study of electron-ion dynamics of many-electron systems under the presence of arbitrary external perturbation.

In particular, this paper will investigate the effect of quantum confinement on the energy gap of hydrogen passivated silicon quantum dots with various sizes. In this aspect, this study is focused to compute the energy gap and optical absorption spectra as a function of dot size, in order to examine the behavior of optical gap under quantum confinement effect. This study is also focused to get brief insight at the microscopic level of the electron transition and bonding characteristic of the dots, identified from the distribution of the highest occupied molecular orbitals (HOMO) and the lowest unoccupied molecular orbitals (LUMO) isosurface.

Since quite a number of computational studies on silicon quantum dots were reported earlier, calculation results were compared with previous studies as a benchmark of this extended study conducted. Serious selection of computational method and parameters is very important in order to achieve ideal counter and balance between efficiency and computational cost. Before the calculation starts, important related background study for comparison purpose was conducted. These results were compared with Zdetsis et al. [8], where the same orbital basis-set representation was used, but it was different in parameters. Models for various prototype structures were done, especially for the structure below 1.8 nm. Those models also covered the missing structure reported from Hirao and Uda [6] DFT calculation with the same exchange-correlation functional. Those possible prototypes were discovered during structural construction while preserving quasi-spherically structured quantum dots.

Computational works done by Onida and Andreoni [12], Xue Jiang et al. [13], and Vasiliev [14] were reported by many; these reports covered small quantum dots (below 0.8 nm) which also considered as small clusters were used as early benchmark for the calculation conducted in this study. Since larger quantum dots (~1.6 nm) fell into quantum confinement regime of electron movement, it was decided to extend this study and analyze computationally atomistic* ab initio* method to understand quantum behavior of electron under confinement region, where all electron and orbital were taken into account. Using the computational facilities available, combined with good calculation approach, this study is capable of analyzing time-dependent optical properties of silicon quantum dots under 1.6 nm in diameter corresponding to 160 atoms (Si_{87}H_{76}).

#### 2. Theoretical Background

Time-dependent density functional theory (TDDFT) [2, 3] can be used to obtain the optical spectra from relaxed geometries. TDDFT is an exact reformulation of time-dependent quantum mechanics, in which the fundamental variable is no longer the many-body wave function but the time-dependent density. TDDFT is an extension of DFT with the time-dependent domain to describe what happens when a time-dependent perturbation is applied. For the sake of completeness, the essentials of this method can be summarized explicitly. In TDDFT, the basic variable is the one electron density , which is obtained with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The interacting system can be represented with the time-dependent Kohn-Sham orbitals , using the time-dependent Kohn-Sham equation,The Kohn-Sham potential is defined aswhere is the external potential, is Hartree potential, and is the exchange and correlation potential. In this study, the adiabatic local density approximation (ALDA) for the whole simulation was used, where it is mapped from homogenous electron gas (HEG),

Since the exchange and correlation functional is the heart of density functional theory calculation, it is very important that this crucial part be reviewed before proceeding into the further calculation. There are numbers of publications that explore the reliability of various exchange-correlation functionals used to study electronic properties of silicon quantum dots. As reported by Zdetsis et al. [8], result of calculated energy gap using semiempirical hybrid exchange-correlation functional (B3LYP) overestimated the experimental value especially for the dots with diameter below 1.8 nm, while it is accurate for larger structure around 1.8–2.0 nm. Even though this hybrid functional was optimized by chemist to overcome the weaknesses of coulomb potential in local density approximation (LDA) for molecular structure study, here the reliability of LDA functional was demonstrated to explore the electronic and optical properties of small sized silicon quantum dots as presented in this paper. This is also confirmed by earlier computational research reported by Hirao and Uda [6], where the LDA functional is also used to study hydrogenated silicon quantum dots. For the sake of reducing computational cost while maintaining its accuracy, it was found that LDA results were acceptable to study quantum dot sizes silicon semiconductor as benchmarked with reported experimental results.

Next, the absorption spectrum was calculated using the explicit propagation of the time-dependent Kohn-Sham equations. Throughout this approach, the system was first excited from its ground state by applying a delta electric field, . The unit vector determines the polarization direction of the electric field and is its magnitude, which must be small enough if one is interested in linear response. The reaction of the noninteracting Kohn-Sham system caused by the perturbation can be readily computed. The mechanism described that each ground state Kohn-Sham orbital instantaneously phase-shifted, where . The Kohn-Sham equations are then propagated forward in real time, and by then the time-dependent density can be computed. The induced dipole moment variation is an explicit functional of the density:The super index indicates that the perturbation has been applied along the th Cartesian direction. Then, the component of the dynamical dipole polarizability tensor is directly related to the Fourier transform of the induced dipole moment function:The spatially averaged absorption cross section is trivially obtained from the imaginary part of the dynamical polarizability:where as the function of is the spatial average, or trace, of the tensorIt is well known that the simpler approach of taking the differences of eigenvalues between Kohn-Sham orbitals gives peaks at lower frequencies in disagreement with the experimental spectra [15]. On the other hand, TDDFT within the ALDA exchange-correlation functional typically produces the optical linear response of the molecular compound that is in good agreement with experimental results with accuracy below 0.2 eV.

#### 3. Computational Detail

Hydrogen passivated silicon quantum dots (Si-QDs) with selected sizes up to 1.6 nm were constructed by repeating tetrahedral silicon geometry from its crystal geometrical structure (bulk structural properties) until the desired repetition was achieved. The prototype quantum dots are constructed in the order of quasi-spherical shape. Next, the quantum dots structures were passivated with hydrogen atoms on the surface. All the ground state relaxed structures were obtained after performing geometry optimization using the quasi-Newton method to get an ideal relaxation structure*.* The quasi-Newton method used is the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [16–19], all in Cartesian coordinates. This optimized structure is then used to calculate the electronic and optical properties, in particular the energy gap and absorption spectrum linear response. All the computational approaches were successfully done in real space and implemented using OCTOPUS [4]. The core electrons with strong coulombic effect were treated using the Troullier and Martins pseudo potentials [20] throughout this simulation. For the ground state calculation, the computational parameters used in our previous work were used exactly [5] to solve for the Kohn-Sham eigenvalues. Then the time-dependent calculations of the perturbed system were conducted. Subsequently, the time reversal symmetry propagator in the algorithms was used to approximate the evolution operator.

All linear response calculations were done using OCTOPUS [4] implementing the Perdew-Zunger [21] parameterization of the local density approximation (LDA) for the exchange-correlation potential. The wave functions representation in real space was mapped onto a uniform grid with a spacing of 0.175 Å and spheres of radius 4 Å around every atom. The system was perturbed by short electrical pulses 0.01 Å^{−1}. A time step of 0.0017 femtoseconds ensured the stability of the time propagation, and a total propagation time of 10 femtoseconds allowed a resolution of about 0.01 eV in the resulting energy spectrum with almost 6000 steps. The results obtained are summarized in Table 2 and also Figure 2 where the absorption spectrum was plotted for the whole perturbation.

#### 4. Results and Discussion

##### 4.1. Ground State Properties

At the beginning of this simulation, the accuracy of ground states calculation was examined by comparing the Kohn-Sham energy gaps obtained using different types of wave function representation and exchange-correlation function. This comparison acts as a benchmark for the simulation reliability of the selected parameter used. Table 1 shows the comparison of Kohn-Sham energy gaps (, in eV) obtained from the ground state calculations of geometry optimized hydrogenated silicon quantum dots. Three different wave function representations were used to solve the Kohn-Sham equations, namely, the numerical atomic orbital (NAO), plane-waves (PWs), and real-space (RS) basis set. All the reported data were analyzed using the same computational approach (DFT) but in different wave function representation (PWs, NAO, and RS basis set) and exchange-correlation potential. Then, the approximated values were checked. This calculation result using real-space method for the ground state eigenvalues is found to be in good agreement with others’ theoretical report.