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The conduction band and electron-donor impurity states in elliptic-shaped GaAs quantum dots under the effect of an externally applied electric field are calculated within the effective mass and adiabatic approximations using two different numerical approaches: a spectral scheme and the finite element method. The resulting energies and wave functions become the basic information needed to evaluate the interstate optical absorption in the system, which is reported as a function of the geometry, the electric field strength, and the temperature.

Semiconductor elliptical quantum dots (QDs) have been the subject of investigation for a number of years due to their prospective applications in optoelectronics. Recent works on electronics and optical properties in this kind of nanosystems can be referred to in [

The calculation of charge carrier states in 3D confined systems heavily depends on the geometry of the structure. In the particular case of elliptic-like QDs the analytical solution of effective mass equations is not possible in general. Furthermore, the inclusion of the effect of an externally applied electric field makes this possibility be unreachable. Therefore, numerical ways of solution for conduction and valence band states are required. In this sense, different approaches appear reported: finite- and boundary-element calculations [

It is well known that even in the case of high quality and high purity samples, semiconductor compounds contain atoms of external elements acting as impurities, not to mention the intentional doping aimed at obtaining a desired carrier concentration in the material. In consequence, the investigation of the influence of impurity atoms on the spectrum of carrier states in semiconductor-based systems is always of significance. On the other hand, the optical response associated with quantum confined carrier states is an important element for both the understanding of the very energy spectrum of the systems and the design of their practical applications. In the case of elliptic-shaped systems a numerical calculation of optical transitions in InAs/GaAs QDs is reported, for example, in [

In this article we are going to present the results of a study on the electron and electron-impurity states in elliptic-shaped GaAs-based QDs, including the influence of external static electric fields. Two kinds of structures are considered, for which Figure

(a, b) Pictorial view of the two kinds of quantum dots considered in the present work: with constant height (a) and with variable height (b).

Here we shall consider the motion of conduction band electrons in the 3D elliptically shaped QD under the effect of in-plane applied electric field

According to the schematic view for the shape of the 3D elliptically shaped QD depicted in Figure

The 3D problem in (

The energy

The eigenfunctions and eigenvalues of (

In this study we have used a basis of sine functions in a region

With the aim of solving the eigenvalues problem given by (

The knowledge of the wave functions

The input parameters for a prototypical GaAs QD are as follows:

Figure

(a, b) Energy of the first ten confined-electron states in an elliptical GaAs quantum dot as a function of the horizontal semilength (

In Figure

In Figure

(a, b) Energy of the first ten confined-electron states in an elliptical GaAs quantum dot as a function of the impurity position along the

Including the influence of the static electric field modifies the above-mentioned picture. This time the electron density of probability becomes displaced inside the QD region. As a consequence, depending on the localization of the donor impurity, the energy values will be shifted upwards or downwards. This is a consequence of the rise or the fall of the expected electron-impurity distance, respectively, implying the weakening or the strengthening of the attractive Coulombic interaction. The field effect is more pronounced when it is applied along the horizontal direction of the elliptical cross section because in that case the field-induced electron delocalization is stronger. It can be seen that the electric field causes the appearance of level anticrossings in cases not appearing when

These features can be confirmed by observing the evolution of the lowest electron confined energy levels as functions of the electric field strength shown in Figure

(a, b) Energy of the first ten confined-electron states in an elliptical GaAs quantum dot as a function of the applied electric field along the

The results for the variation of the energy levels as functions of

(a, b) The results are as in Figure

The reduction in the energy values as a consequence of the increase in the horizontal dot size, as discussed above in the constant height case, is observed as well. It is interesting to note that the inclusion of a nonzero-height strip at the base of the QD that amounts

In Section

(a, b) The results are as in Figure

In first place one may readily noticed that, as it should be expected, the values of the obtained energies are significantly smaller compared with the previously discussed infinite potential barrier cases. However, the behavior of the energies as functions of the horizontal semilength are qualitatively similar, with the same arguments related with the electron spatial localization and the expected electron-impurity distance as the key elements for their explanation. It should be kept in mind that the confining configuration (ii) implies a larger size of the width of the quantum well along the vertical direction. As a consequence, in spite of the presence of a top infinite barrier, the obtained energies are lower in both the uncoupled and Coulombic-coupled situations.

In order to proceed with the investigation of the optical absorption coefficient in the elliptic-shaped QDs we need to evaluate the off-diagonal electric dipole matrix elements associated with the interstate transitions that will be considered. As indicated in (

Dipole matrix elements for circular polarization of the incident radiation considering combinations between the first four confined states. The results are obtained under the same geometrical, compositional, and interaction configurations leading to Figure

Dipole matrix elements for circular polarization of the incident radiation considering combinations between the first four confined states. The results are obtained under the same geometrical, compositional, and interaction configurations leading to Figure

Dipole matrix elements for circular polarization of the incident radiation considering combinations between the first four confined states. The results are obtained under the same geometrical, compositional, and interaction configurations leading to Figure

Dipole matrix elements for circular polarization of the incident radiation considering combinations between the first four confined states. The results are obtained under the same geometrical, compositional, and interaction configurations leading to Figure

Dipole matrix elements for circular polarization of the incident radiation considering combinations between the first four confined states. The results are obtained under the same geometrical, compositional, and interaction configurations leading to Figure

The different, and sometimes jumbled, behaviors of these quantities are governed by two main elements: the spatial extension and the symmetry of the involved wave functions. They, combined with the polarization of the incident light (here considered to be circular), determine whether the dipole matrix element vanishes or remains finite as well as its functional dependence with respect to the varying quantity in the system. For instance, the effect of the anticrossings, which imply the sudden change in the symmetry of certain wave function and, therefore, the imposition of a particular selection rule for the transition under specific circumstances, can be clearly noticed.

The total light absorption coefficient that includes the contribution of the above transitions commented on is shown in Figure

Total absorption coefficient as a function of the incident photon energy for several values of the electric field applied in the

In our calculation, the effect of the temperature is supposed to fall on the population of the transition states. We are assuming that the influence of

Finally, Figure

Total absorption coefficient as a function of the incident photon energy for several values of the horizontal length varying in steps of

The shifts of the resonant absorption peak positions can be explained as above, that is, by observing the behavior of the confined state energies appearing in Figures

In the present work we have addressed the calculation of the conduction band effective mass states of elliptically shaped quantum dots with finite and infinite confinement potentials, the presence of a donor impurity atom, and the influence of externally applied static electric fields. The solution of the Schrödinger-like effective mass differential equation incorporates the adiabatic approximation to uncouple the electron motion along the

Our results confirm the known effects of the presence of the electron-impurity interaction and the application of static electric fields on the spectrum of carrier confined in quantum nanostructures, indeed with the particularities associated with the specific geometry of the system under study.

The light absorption associated with electron transitions between the allowed quantum states in each case is studied making use of the previously calculated energies and wave functions. It is shown that the changes in the electron state energies and probability densities due to modifications in the type of confinement, the geometry, and the presence or absence of the donor impurity center, as well as the variation of the level population with temperature, are all causes for the shift of the absorption resonant peaks and/or the increment or reduction of their corresponding amplitudes.

Quantum dots with the shape here discussed are, actually, experimentally realized systems, with well-identified current and prospective applications. We hope, with this work, to shed some light on the electronic and optical features of a GaAs-based structure of this type.

The authors declare that they have no conflicts of interest.

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 103.01-2015.93. The authors are grateful to the Colombian Agencies CODI-Universidad de Antioquia (Estrategia de Sostenibilidad de la Universidad de Antioquia) and Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (C. A. Duque and A. L. Morales Exclusive Dedication Projects 2016-2017). M. A. Londoño is grateful to Colciencias-Ecopetrol for financial support through the project

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