#### 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, zero-dimensional electronic states, and the nonlinear optical behaviour [1–6]. Among the nanocrystalline II–VI semiconductors, we mention the S, which has a high potentiality as window layer for solar cells [1, 7–16]. Concerning the epitaxy ofS QDs, there are many growth methods such as the inverted micelles [17], the selective air-growth method technique [18], the single source molecular precursors [19], the colloidal method [20, 21], and the sol-gel 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, 22–25]. In a recent study, we have investigated the electronic properties of S QDs capped inside a dielectric matrix using the sol-gel 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 electron-hole 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 electron-hole 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 Plank’s 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 sol-gel 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 Vegard’s 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 *x*-dependence 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 e-1p 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 e-1s electron state taken from [5]. Similarly for holes, we have fitted the *x*-dependent subband energies of the electrons by polynomial laws. The results are reported in Table 2. It is worth noting that energy difference between e-1p and e-1s 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 e-1p 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 e-1p using Bessel functions. For heavy and light holes, however, we have calculated the subbands hh**-**2s and lh-1s**,** respectively. In the latter case, we have adopted the one-dimensional 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 intra-subband transitions in order to describe the electrooptical behaviour of S-related QDs.