Research Article  Open Access
A. Sakly, N. Safta, A. Mejri, H. Mejri, A. Ben Lamine, "The Excited Electronic States Calculated for Quantum Dots Grown by the SolGel Technique", Journal of Nanomaterials, vol. 2010, Article ID 746520, 4 pages, 2010. https://doi.org/10.1155/2010/746520
The Excited Electronic States Calculated for Quantum Dots Grown by the SolGel Technique
Abstract
The present paper is aimed to investigate theoretically the quantum confinement in related quantum dots with x the atomic fraction of Zn. For both electrons and holes, we have calculated the excited bound states with use of the spherical geometry model and assuming a finite potential at the boundary. For electrons, calculations were made by using Bessel function as an orthonormal basis. However, for holes, the confined subbands have been calculated based on squared quantum well envelope wave functions. The subband energies were evaluated for both electrons and holes versus zinc composition as well.
The quantum dots (QDs) are attracting considerable interest in both fundamental and applied research because of their specific properties. In fact, the QDs show a number of striking effects as the quantum confinement, zerodimensional electronic states, and the nonlinear optical behaviour [1ā6]. Among the nanocrystalline IIāVI semiconductors, we mention the S, which has a high potentiality as window layer for solar cells [1, 7ā16]. Concerning the epitaxy ofS QDs, there are many growth methods such as the inverted micelles [17], the selective airgrowth method technique [18], the single source molecular precursors [19], the colloidal method [20, 21], and the solgel technique [1]. In depict of the great interest of QDs based on S, the existing theoretical works seem to be insufficient to explain all the physical aspects [1]. Most approaches have considered that electrons and holes are confined in a spherical QD of radius R and used an infinite potential barrier model [1, 22ā25]. In a recent study, we have investigated the electronic properties of S QDs capped inside a dielectric matrix using the solgel technique. We, indeed, assumed that both electrons and holes can undergo a quantum confinement in a nanosphere of radius R but the boundary of the QD is of a finite barrier height [5]. By restricting the study to the ground state for both electrons and holes, we have calculated the shape of the confinement potentials, the quantized energies, their related envelope wave functions and the QDs sizes. It has been established from this study that the confinement energy shows a significant increase with the ZnS molar fraction for both electrons and holes. Moreover, the radius of S QDs has been found to be decreasing with increased Zn composition. This means that the confinement of carriers increases with composition x. The aim of the present paper is to investigate the electronic structure of S QDs by using the spherical geometry model. A peculiar attention is paid to the computation of electron and hole excited states. Calculations will be made versus the ZnS molar fraction. The paper is organized as follows: after a brief introduction, we present the theoretical formulation, in then, we report numerical results and discussion. Concluding remarks are summarized in the last part.
The system to model consists of an electronhole pair confined in a spherical QD of radius R. The nanocrystallite is assumed to be embedded in a dielectric host matrix. It is considered at a temperature T as well, and no external electric field is applied. Electrons and holes have a potential barrier of finite height. On the other hand, the coupling between electron and hole confinements is disregarded. Also, the Coulomb potential associated with the electronhole interaction has not been taken into account. For the S nanocrystal to model, the electronic bound states can be computed using the following equation: where is the Plankās constant, is the effective mass of the carriers, represents the azimutal momentum operator, is the height of the confinement potential, is the step function, is the confinement energy, and is the eigen wave function. The subscripts e and h refer to the electron and hole particles, respectively. Equation (1) is derived in the framework of the effective mass theory and in the assumption of the band parabolicity. Taking the eigenfunctions under the form where are the spherical harmonics, (1) may be rewritten as: Here, l denotes the quantum number associated with the operator . The function is subjected to be equal to zero at the center of the nanosphere. For l = 0, that is, the ns states, (2) can be solved using the secular equation of a squared quantum well in a direction. For the eigenstates with , Bessel functions can be used as an orthonormal basis to solve such an equation. We distinguish two regions: and . For the first case, (2) can be rewritten as with Here, denote the spherical Bessel functions. For , using Bessel basis functions, (2) is transformed into with As an adequate approximation to calculate the energy levels of a spherical QD, one can take the asymptotic forms to represent the eigen wave functions: In such a case, the secular equation reads As reported in [5], we have calculated the quantized energies for S QDs using the spherical model. Calculations were carried out using the squared quantum approach and have been restricted to the ground states (1s) for electrons and heavy holes. In the present work, we attempt to extend these calculations to the excited energy levels, that is, the 2s state for heavy holes, 1s state for light holes, and 1p state for electrons. For holes, we adopt as a basis, the squared quantum well wave functions to express . Thus, the confinement energies as well as the wave functions for the eigenstates ns in nanocrystals are given by
with and A is a normalised factor.
For electrons, we consider Bessel functions under their asymptotic forms as shown in (7). The electron band parameters used in the present work are listed in Table 1. As reported in [5], these parameters were treated as fitting parameters. As an experimental support, we have used absorption data obtained on S nanocrystals grown by solgel technique [1]. The fitting consisted of imposing the optical band gap to coincide with the effective band gap calculated for S QDs. Values of and are 0.7 and 0.23, respectively, in free electron mass [26]. For the S QDs, the effective masses were calculated using Vegardās law. Figure 1 shows the subband energies of the 1s and 2s for heavy holes and 1s for light holes calculated as a function of Zn composition. As shown, the subbands shift up in energy as Zn composition increases going from CdS to ZnS. We have fitted the xdependence of the hole subbands by using polynomial laws. Analytical expressions obtained are summarized in Table 2. As can be noticed from obtained results (i) the subbands, hh1s, hh2s, and lh1s shift up in energy as Zn composition increases, (ii) opposite to the ground state hh1s, the excited subbands hh2s and lh1s show a more rapid increasing with the increase of the ZnS molar fraction. The change in the confinement property of excited holes is mainly due to the fact that the barrier heights are significantly higher for Zn composition near ZnS compared to that at low zinc contents. In S QD structures with higher Zn composition, these states, if occupied by photocreated and/or thermal excited holes, can favour the multilevel emissions from the confined subbands. On the other hand, the excited hole states can interact with acceptor impurities intentionally incorporated, governing the transport properties of S QDs. For electrons, the relevant values of the 1p state are illustrated versus the Zn molar fraction in Figure 2. It has been found that (i) the excited electron state e1p is not detected as a localized state in the quantum well up to the composition , (ii) this state is, however, not resonant in the Zn composition range and increases significantly with x. In Figure 2, we have also plotted the confinement energy of the e1s electron state taken from [5]. Similarly for holes, we have fitted the xdependent subband energies of the electrons by polynomial laws. The results are reported in Table 2. It is worth noting that energy difference between e1p and e1s is particularly higher for Zn composition ranging from 0.8 to 1.0. Such a trend results in the strong increasing of the electron barrier height for higher Zn composition, as reported in [5]. As for holes, the excited e1p state can influence notably the electrical and optical behaviour of QDs based on the S ternary alloy.


In summary, we have attempted to calculate the discrete energy spectrum of S QDs. As a geometry model, we have considered spherical QDs with a finite potential barrier at the boundary. For electrons, we have computed the subband e1p using Bessel functions. For heavy and light holes, however, we have calculated the subbands hh2s and lh1s, respectively. In the latter case, we have adopted the onedimensional quantum well eigenfunctions. Calculations were extended in all the composition range from CdS to ZnS. In fundamental viewpoint, computation of excited bound states for electrons and holes is of great interest. Indeed, it can provide useful information on many physical properties such as multilevel emission processes, interband transitions, and transport of charges. In applied physics, this study can open a possible way to design a new class of nanostructure devices. In the near future, we intend to investigate oscillator strengths for both inter and intrasubband transitions in order to describe the electrooptical behaviour of Srelated QDs.
References
 B. Bhattacharjee, S. K. Mandal, K. Chakrabarti, D. Ganguli, and S. Chaudhuri, āOptical properties of ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$ nanocrystallites in solgel silica matrix,ā Journal of Physics D, vol. 35, no. 20, pp. 2636ā2642, 2002. View at: Publisher Site  Google Scholar
 K. K. Nanda, S. N. Sarangi, S. Mohanty, and S. N. Sahu, āOptical properties of CdS nanocrystalline films prepared by a precipitation technique,ā Thin Solid Films, vol. 322, no. 12, pp. 21ā27, 1998. View at: Google Scholar
 H. YĆ¼kselici, P. D. Persans, and T. M. Hayes, āOptical studies of the growth of ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$ nanocrystals in borosilicate glass,ā Physical Review B, vol. 52, no. 16, pp. 11763ā11772, 1995. View at: Publisher Site  Google Scholar
 Y. Kayanuma, āQuantumsize effects of interacting electrons and holes in semiconductor microcrystals with spherical shape,ā Physical Review B, vol. 38, no. 14, pp. 9797ā9805, 1988. View at: Publisher Site  Google Scholar
 N. Safta, A. Sakly, H. Mejri, and Y. Bouazra, āElectronic and optical properties of ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$ nanocrystals,ā European Physical Journal B, vol. 51, no. 1, pp. 75ā78, 2006. View at: Publisher Site  Google Scholar
 N. Safta, A. Sakly, H. Mejri, and M. A. ZaĆÆdi, āElectronic properties of multiquantum dot structures in ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$ alloy semiconductors,ā European Physical Journal B, vol. 53, no. 1, pp. 35ā38, 2006. View at: Publisher Site  Google Scholar
 V. Alberts, R. Herberholz, T. Walter, and H. W. Schock, āDevice characteristics of Inrich CuInSe2based solar cells,ā Journal of Physics D, vol. 30, no. 15, pp. 2156ā2162, 1997. View at: Google Scholar
 N. Kohara, T. Negami, M. Nishitani, and T. Wada, āPreparation of devicequality Cu(In, Ga)S${\text{e}}_{2}$ thin films deposited by coevaporation with composition monitor,ā Japanese Journal of Applied Physics, vol. 34, no. 9, pp. L1141āL1144, 1995. View at: Google Scholar
 H. L. Kwok, āA study of ultrathin C${\text{u}}_{\text{x}}$SC${\text{d}}_{\text{y}}$Z${\text{n}}_{1\text{y}}$S polycrystalline solar cells,ā Journal of Physics D, vol. 16, no. 12, pp. 2367ā2377, 1983. View at: Google Scholar
 H. S. Kim, H. B. Im, and J. T. Moon, āEffects of cell width on the photovoltaic properties of sintered ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$/CdTe solar cells,ā Thin Solid Films, vol. 214, no. 2, pp. 207ā212, 1992. View at: Google Scholar
 G. Gordillo, āPhotoluminescence and photoconductivity studies on ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$ thin films,ā Solar Energy Materials and Solar Cells, vol. 25, no. 12, pp. 41ā49, 1992. View at: Google Scholar
 J. W. Bowron, S. D. Damaskinos, and A. E. Dixon, āCharacterization of the anomalous second junction in Mo/CuInS${\text{e}}_{2}$/(CdZn)S/ITO solar cells,ā Solar Cells, vol. 31, no. 2, pp. 159ā169, 1991. View at: Google Scholar
 O. M. Hussain, P. S. Reddy, B. S. Naidu, S. Uthanna, and P. J. Reddy, āCharacterization of thin film ZnCdS/CdTe solar cells,ā Semiconductor Science and Technology, vol. 6, no. 7, pp. 690ā694, 1991. View at: Publisher Site  Google Scholar
 T. L. Chu, S. S. Chu, J. Britt, C. Ferekides, and C. Q. Wu, āCadmium zinc sulfide films and heterojunctions,ā Journal of Applied Physics, vol. 70, no. 5, pp. 2688ā2693, 1991. View at: Publisher Site  Google Scholar
 H. H. GĆ¼rel, Ć. Akinci, and H. ĆnlĆ¼, āTight binding modeling of CdSe/ZnS and CdZnS/CdS IIVI heterostructures for solar cells: role of dorbitals,ā Thin Solid Films, vol. 516, no. 20, pp. 7098ā7104, 2008. View at: Publisher Site  Google Scholar
 M. Gunasekaran and M. Ichimura, āPhotovoltaic cells based on pulsed electrochemically deposited SnS and photochemically deposited CdS and ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$,ā Solar Energy Materials and Solar Cells, vol. 91, no. 9, pp. 774ā778, 2007. View at: Publisher Site  Google Scholar
 L. Cao, S. Huang, and E. Shulin, āZnS/CdS/ZnS quantum dot quantum well produced in inverted micelles,ā Journal of Colloid and Interface Science, vol. 273, no. 2, pp. 478ā482, 2004. View at: Publisher Site  Google Scholar
 H. Kumano, A. Ueta, and I. Suemune, āModified spontaneous emission properties of CdS quantum dots embedded in novel threedimensional microcavities,ā Physica E, vol. 13, no. 2ā4, pp. 441ā445, 2002. View at: Publisher Site  Google Scholar
 Y. C. Zhang, W. W. Chen, and X. Ya Hu, āIn air synthesis of hexagonal ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$ nanoparticles from singlesource molecular precursors,ā Materials Letters, vol. 61, no. 26, pp. 4847ā4850, 2007. View at: Publisher Site  Google Scholar
 K. Tomihira, D. Kim, and M. Nakayama, āOptical properties of ZnSCdS alloy quantum dots prepared by a colloidal method,ā Journal of Luminescence, vol. 112, no. 1, pp. 131ā135, 2005. View at: Publisher Site  Google Scholar
 J. Ouyang, M. Vincent, D. Kingston et al., āNoninjection, onepot synthesis of photoluminescent colloidal homogeneously alloyed CdSeS quantum dots,ā Journal of Physical Chemistry C, vol. 113, no. 13, pp. 5193ā5200, 2009. View at: Publisher Site  Google Scholar
 K. K. Nanda, S. N. Sarangi, S. Mohanty, and S. N. Sahu, āOptical properties of CdS nanocrystalline films prepared by a precipitation technique,ā Thin Solid Films, vol. 322, no. 12, pp. 21ā27, 1998. View at: Google Scholar
 Q. Pang, B. C. Guo, C. L. Yang et al., ā${\text{Cd}}_{1\text{x}}{\text{Mn}}_{\text{x}}\text{S}$ quantum dots: new synthesis and characterization,ā Journal of Crystal Growth, vol. 269, no. 2ā4, pp. 213ā217, 2004. View at: Publisher Site  Google Scholar
 M. C. Klein, F. Hache, D. Ricard, and C. Flytzanis, āSize dependence of electronphonon coupling in semiconductor nanospheres: the case of CdSe,ā Physical Review B, vol. 42, no. 17, pp. 11123ā11132, 1990. View at: Publisher Site  Google Scholar
 H. YĆ¼kselici, P. D. Persans, and T. M. Hayes, āOptical studies of the growth of ${\text{Cd}}_{1\u0101\x88\x92\text{x}}{\text{Zn}}_{\text{x}}\text{S}$ nanocrystals in borosilicate glass,ā Physical Review B, vol. 52, no. 16, pp. 11763ā11772, 1995. View at: Publisher Site  Google Scholar
 W. Ekardt, K. LĆ¶sch, and D. Bimberg, āDetermination of the analytical and the nonanalytical part of the exchange interaction of InP and GaAs from polariton spectra in intermediate magnetic fields,ā Physical Review B, vol. 20, no. 8, pp. 3303ā3314, 1979. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2010 A. Sakly et al. 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.