Effects of Confinement Energies on Lead Sulphide and Indium Phosphide Quantum Dots Within Brus Equation Model

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1 AASCIT Journal of Physics 2017; 3(4): ISSN: (Print); ISSN: (Online) Effects of Confinement Energies on Lead Sulphide and Indium Phosphide Quantum Dots Within Brus Equation Model Uduakobong Sunday Okorie Department of Physics, Akwa Ibom State University, Ikot Akpaden, Nigeria address Keywords Confinement Energies, Quantum Dots (QD), Brus Equation Received: October 19, 2017 Accepted: November 1, 2017 Published: November 25, 2017 Citation Uduakobong Sunday Okorie. Effects of Confinement Energies on Lead Sulphide and Indium Phosphide Quantum Dots Within Brus Equation Model. AASCIT Journal of Physics. Vol. 3, No. 4, 2017, pp Abstract Quantum confinement effect in semiconductor quantum dots (QD s) of Indium Phosphate and Lead Sulphide has been studied within the framework of Brus Equation, using the particle-in-a-box model. The two nanocrystals used exhibit a size dependence phenomenon as predicted by the model used. The results indicate that ground state confinement energy is inversely proportional to the dot size. As such, when the radius of the dot increases, its confinement energy decreases without getting to zero. i.e., the lowest possible energy for the quantum dot sample is never zero. This phenomenon has made the nano-particles considered more relevant even in today s world of technology. 1. Introduction With discovery of physical properties of semiconductor nanostructures, much research has been carried out to make use of this reduced dimensional structure for noble applications. The study of low-dimensional semiconductor heterostructure quantum dots (QDs) is one of the main subjects in condensed matter Physics owing to their application to optoelectronic devices like light emitting diodes [1] and lasers and solar cells [2]. Quantum dots are semiconductor nanoparticle whose excitons are confined in all three spatial dimensions. It is essentially a tiny zero-dimensional semiconductor crystal with size in the order of nanometers, hence, the name dot or island. It is often called artificial atom because of its quantum properties and interactions similar to bulk semiconductor materials. One of the most important optical and electrical properties of Quantum Dots is the ability to adjust their bandgap and therefore control their light absorbance and emission frequencies according to their desired purpose. This is only possible through the quantization of their energy levels. The size of the dots greatly affects the optical properties of these nanocrystals. It goes a long way to change the colour emitted or absorbed by the crystals, as a result of the energy levels within the crystals. The dot size has an inverse relationship with the energy level of its band gap; this phenomenon has effect on the colour and frequency of light emitted. Smaller dots emit higher energy light that is bluer in colour, whereas larger dots emit lower energy light which are redder in colour. The width of the quantum dot band gap depends on its size and chemical composition, making it easy to tune absorption and emission spectra, which is impossible for atoms, but desirable for optical properties [3].

2 29 Uduakobong Sunday Okorie: Effects of Confinement Energies on Lead Sulphide and Indium Phosphide Quantum Dots Within Brus Equation Model One of the most fascinating effects of nanoparticles occurs within the ambient studies of the physics of electrons, atoms and photons. It s characteristic effects is observed in dot particles of various shapes in the range of little nanometers. Interesting electronic and optical properties have been acquired by these tiny and unseen nanoparticles. Quantum dots combination can be controlled sufficiently to obtain a perfect crystal. This phenomenon is macroscopically impossible. Louis Brus first determined that an electron and hole created when the dot absorbs light are bound together within the confines of a box using perturbation theory [4]. This led him to the equation called Brus Equation. QDs are quite interesting as they enable the study of semiconductors on small length scale. In these materials, photon of energy greater than band gap causes the maximum length of separation between an electron and hole at which they are still linked by Coulombic attraction forces is called the exciton BohrRadius. Its value varies depending on the semiconductor material [5]. As the particle size approaches the exciton Bohr radius, the charge carriers are confined in three dimensions. This phenomenon known as quantum confinement causes the continuous band of the bulk to split into discrete, quantized levels [6]. Confinement in quantum dots can be seen arise from reduction of the dot s dimension and doping of the dot material, in which the resultant effect is the increase of the dot s confinement energy [7]. Most times, Quantum confinementnormally results in the enlargement of the band-gap. This in-turn decreases the size of the quantum dots [8]. This confinement results in properties that are not seen in bulk form of materials. A typical example is silicon which is known to be a poor light emitter in its bulk form due to its indirect band gap. When it is confined as quantum dot, it emits light [9]. Two fundamental factors contribute to the variance observed between quantum dot properties and its bulk counterpart. Firstly, there exist a larger surface to volume ratio in nanoparticles; Secondly, QDs have a tunableband-gap as shown in figure 1 below: Figure 1. Splitting of valence band and conduction band into discrete energy levels as a result of quantum confinement effect [8]. Figure 2. Changes in the photoluminescence colour of colloidal solutions of CdSe QDs [10]. The change in colour of an optically clear solution of Cadmium Selenide quantum dots with variation in particle size is shown in Figure 2 above [10]. Semiconductor QDs areattracting growing interest from the sensor research. One of these lie within the ambient of advanced IR image sensors and THz detectors. This has been viewed recently as a potential solution inaddressing challenges in diagnostics and therapeutics [11]. In the manufacturing processes, the dot size can be rebranded to obtain a nanocrystal suitable for optical imaging [12]. Quantum dot technology has been used recently to manufacture a start up device called Store Dot which is used to revive dead phone batteries back to life within a very short period. These dots are peptides that are altered to possess optical properties and the ability of generating charges for optimum operation of device being used. The Store Dot uses nanocrystal solution in the place of electrolyte, being used in traditional batteries to generate electrons. Quantum Dot application has yielded much interest in structural and functional imaging to study the interactions between cells and between a cell and its environment in diseased tissues [11], in cancer diagnosis [13], in lymph-node mapping during biopsy and surgery [14] and in biomedical applications [15]. Several theoretical methods have been used to investigate this concept. This includes: Tight-Binding Approach (TBA) [16], the K.P. method [17], Effective-Mass Approximation (EMA) [4] and most recently, the Finite-Depth Square-Well Effective-Mass Approximation (FWEMA) model [18], Potential-Morphing Method (PMM) [19] and Single Band Toy Model (SBTM) [20]. Baskoutas et al. [21] calculated the exciton energy of the narrow band gap colloidal PbS, PbSe and InAs QD using the PMM, using an assumption of a single dependent dielectric function. Kumar et al. [22] also used k.p. model to calculate the shape and size dependent electronic properties of GaAs/AlGaAs QD s. This model was adopted due to its accuracy for modeling the band structure near the first

3 AASCIT Journal of Physics 2017; 3(4): Brillouin zone [23]. Ekong and Osiele [24] employed a quantum confinement model to study different shapes of nanocrystalline silicon (nc Si) QD, within the limits of an effective diameter of 3nm. This research seeks to demonstratehow the Brus equation can be used to obtain the confinement energy at various dots radii in other to deduce the confinement nature associated with the individual dot understudy, which are Lead Sulfide (Pbs) and Indium Phosphide (InP). The theoretical framework of this research is presented in section 2, results and discussion in section 3, and finally conclusion in section Theoretical Framework The theoretical framework adopted for this discussion was first proposed by Brus [4]. This framework relies on Effective mass Approximation, where an exciton confined to a spherical volume of the crystallite is put into consideration with the mass of electron and hole being replaced with effective masses ( m e andm h ) to define the wave function: 2 2 h e Eg( qd) = Ebulk + + 8R m m 4πε ε R 2 * * 2 e h o r * * Here, h, e, R, m e, m h, ε o, ε o, are Planck s constant, electron charge, radius of quantum dot, Effective Mass of excited electron, Effective mass of excited hole, Permittivity of vacuum, and Relative permittivity respectively. The first term in the right hand side of Equation (1) represents the band gap energy of bulk materials, which are the characteristics of the material. The second additive term of the equation represents the additional energy due to quantum confinement having a dependence on the band gap energy (also known as ground state confinement energy). The third subtractive term stands for the columbic interaction energy exciton. Neglecting the coulombic interaction energy exciton due to high dielectric constant of the semiconductor material, the overall equation for calculating the emission energy is given as: (1) 2 h 1 1 E ( R) = E + + g( R) (2) 2 * 8R me mh * E = Emission energy E = Band gap energy g 3. Results and Discussion Lead sulfide (Pbs) and Indium Phosphide (InP) quantum dots, in addition to its necessary parameters as shown in the tables below were used for this computation. Table 1. Showing material parameter used for the computation of the confinement energies at various radii which is less than the Bohr radius a B [27]. Quantum Dot InP Pbs * m e m * h bulk 0.08m o 0.6m o 0.11m o 0.9m o E at 300k 1.344eV 0.41eV a (Bohr radius) 15nm 20nm B Table 2. Showing confinement and emission energies obtained at different dot radii for InP semiconductor quantum dot. Dots Radius (nm) Confinement Energy (ev) Emission Energy (ev) Table 3. Showing confinement and emission energies obtained using different dot radii for Pbs semiconductor quantum dot. Dot Radius (nm) Confinement Energy (ev) Emission Energy (ev)

4 31 Uduakobong Sunday Okorie: Effects of Confinement Energies on Lead Sulphide and Indium Phosphide Quantum Dots Within Brus Equation Model Figure 3. A plot of Confinement Energy (ev) against Radius (nm) for InP Quantum Dot. Figure 4. A plot of Emission Energy (ev) against Radius (nm) for InP Quantum Dot.

5 AASCIT Journal of Physics 2017; 3(4): Figure 5. A plot of Confinement Energy (ev) against Radius (nm) for Pbs Quantum Dot. Figure 6. A plot of Emission Energy (ev) against Radius (nm) for Pbs Quantum Dot.

6 33 Uduakobong Sunday Okorie: Effects of Confinement Energies on Lead Sulphide and Indium Phosphide Quantum Dots Within Brus Equation Model Figure 7. A Plot of Confinement Energies for InP and Pbs Quantum Dots. Figure 8. A plot of Emission Energies for InP and Pbs Quantum Dots. The graphs of ground state confinement energy against size (radius) for lead sulfide (Pbs) and Indium phosphide (InP) semiconductor quantum dots in Figures 3 and 5 respectively shows the dependence of confinement on the size of quantum dots. The resulthere shows an inverseproportionality ratio between the ground state confinement energy and the dot size (radius). The graphs are asymptotic to the radius (horizontal) axis. Thus, as one increases the radius (size), the confinement energy decreases, gradually approaching zero.

7 AASCIT Journal of Physics 2017; 3(4): The confinement energy is observed in quantum dots through an increase in the energy of the band gap. Confinement begins when radius of the quantum dot sample is comparable to the order of the exciton Bohr radius, a B (15 nm for Indium Phosphate and 20 nm for Lead sulfide). In order words, the size is comparable to 2 a B that is (doubles the exciton Bohr radius). The confinement energy increases as the size of the quantum dot is gradually reduced until the cluster and magic number limit for the particular crystal is reached. At this limit, Brus Equation no longer holds, hence, the crystal losses its stability. We can say here that the energy spectrum is discrete rather than continuous in the confinement regime. As such, only certain energies are allowed for a quantum dot of a given size. The confinement region is subdivided into strong confinement regime and weak confinement regime. It must however be noted that in the weak confinement regime, the energy levels form a near continuum. In Figure 3, sharp increase in confinement energy begins at r = 1.40 nm. Thus, the limit of strong confinement for Indium Phosphide is at size 1.95 nm, which corresponds to confinement energy of about ev. Beyond this limit, the discrete nature of the energy spectrum becomes more apparent until one gets to the cluster and magic number limit. Similarly, Figure 5 shows that a sharp increase in confinement energy for Lead Sulfide begins from size 1 nm up to the cluster and magic number limit. Thus, the limit or threshold for strong confinement is at 1.45 nm which corresponds to energy of about ev. Figures 4 and 6 also show the size dependence of these dots on emission energy. It is observed vividly that quantum dots used also demonstrate an inverse dependence characteristic on the emission energy. For comparison, the plots showing the confinement energy and the emission against the dot radius in Figures 7 and 8 shows that both the confinement and emission energies is higher in InP quantum dots than Pbs quantum dots. In other words, it can be said that the smaller the size or radius of the dot, the higher or more effective the confinement energy, hence, the more efficiency of the electronic device it will be applied to. When comparing with results of similar worksof [25] and [26] using Brus Equation, it was found that the nanocrystals exhibit the size dependence predicted by the particle-in-a-box model and that the confinement energy exhibits an inverse proportionality phenomenon with the dot radius. Hence, the theoretical model considered here are in perfect agreement with the experimental observation of the QD s size dependence on the confinement energy. 4. Conclusion The simple models obtained for the two different semiconductor nanocrystals exhibits the size dependence predicted by the particle-in-a-box model. Also the confinement energy exhibits an inverse proportionality phenomenon with the dot radius. Hence, the theoretical model considered here are in perfect agreement with the experimental observation of the QD s size dependence on the confinement energy. The level of confinement is discovered to be stronger in Indium Phosphide, as compared to Lead Sulfide. The confinement of electrons in semiconductor quantum dots tends to increase as the dot size decreases. It is found that the tunable range of the QD is solely dependent on the size of the exciton Bohr radius. 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