Trion X+ in vertically coupled type II quantum dots in threading magnetic field
© Horta-Piñeres et al.; licensee Springer. 2012
Received: 16 July 2012
Accepted: 12 September 2012
Published: 26 September 2012
We analyze the energy spectrum of a positively charged exciton confined in a semiconductor heterostructure formed by two vertically coupled, axially symmetrical type II quantum dots located close to each other. The electron in the structure is mainly located inside the dots, while the holes generally move in the exterior region close to the symmetry axis. The solutions of the Schrödinger equation are obtained by a variational separation of variables in the adiabatic limit. Numerical results are shown for bonding and anti-bonding lowest-lying of the trion states corresponding to the different quantum dots morphologies, dimensions, separation between them, thicknesses of the wetting layers, and the magnetic field strength.
During the last few years, there has been much interest in the study of quantum dots (QDs), which are structures in which charge carriers are confined in all three dimensions. The quantum dots have opened the possibility to fabricate both artificial atoms and molecules with novel and fascinating optoelectronic properties which are not accessible in bulk semiconductor materials. Especially, the self-assembled QDs are considered to be very promising for possible applications, such as QD lasers, due to their large confinement energy and high optical quality. An attractive route for nanostructuring semiconductor materials is offered by the self-assembled quantum dots (SAQDs) which are formed via the Stranski-Krastanow growth mode by depositing a material on a substrate with a different lattice parameter[1–5]. The analogy with atoms can be strengthened by the capture of carriers with opposite charges. In spite of the fact that SAQDs can possess different morphologies such as disks, cones, rings, and lenses, they are, in general, thin layers and have, for the most part, a small height-to-base aspect ratio whose typical values vary between 0.029 and 0.067. The motion of the carriers in such structures is quasi-two-dimensional but due to the tunneling of the carriers through finite height barrier along the crystal growth direction, they can escape outside of the layer, with a considerable probability. Besides, if the probabilities of the electron and the hole tunneling are different, the opposite charge density in the central part inside the SAQD and the peripheral regions around SAQDs can be induced by a captured electron-hole pair. It is established that the quantum confinement of few particles systems, e.g., neutral and negatively charged donors (D0 and D−, respectively), excitons (X), trions (X−, X+) and biexcitons, enhances the binding energy allowing their observation even at the room temperature. The energy spectra of these systems governed by the interplay between tunneling, confinement, and correlation effects have been studied by using different methods, such as variational, diagonalization[7–9], and finite elements. Although these techniques give consistent results with the experimental data, they entail a tedious computational work. Besides, the great majority of the analyzed models are two-dimensional, and they left to aside the effects related to the disk height. Recently, in order to calculate ground state energies of D0, D−, and X in the semiconductor heterostructures, it has been proposed as a simple variational procedure[11–13] related to fractal dimension scheme. In this paper, we analyze the energy spectrum of a positively charged exciton confined in a semiconductor heterostructure formed by two vertically coupled, axially symmetrical type II quantum dots located close to each other. The electron in the structure is mainly located inside the dots, while the holes generally are placed in the exterior region close to the symmetry axis. The solutions of the Schrödinger equation are obtained by a variational separation of variables in the adiabatic limit. Numerical results are shown for bonding and anti-bonding lowest-lying of the trion states corresponding to different quantum dots morphologies, dimensions, the separation between them, thicknesses of the wetting layers, and the magnetic field strength.
Here,, the energy of the fast electron motion as a function of the holes positions, presents potential curves for the holes' slow motion.
Results and discussion
The corresponding potential curves are indicated by two-electron quantum numbers (n e , l e ) and by symbols (+) and (−) for the bonding and anti-bonding levels, respectively. It is seen that the electron energies corresponding to different levels initially descends as the distance between holes and the layers Z h decreases, while the electrostatic attraction between electron and holes is greater than the repulsion from the layer provided by the structural confinement. As Z h further decreases, the repulsion from the layer provided by the increasing confinement becomes significant and the potential energy becomes to climb up sharply. Also, in Figure2 we show the holes' vibrational sublevels (dashed lines) with gaps significantly smaller than the corresponding values between electronic levels (solid lines). Similarly, the gaps between bonding and anti-bonding levels with the same quantum numbers, (n e , l e ), are larger than the gaps between the two levels with the same parity and different quantum numbers. Additionally, one can see that energies corresponding to the bonding states are generally inferior, and there is only one crossover between the lower anti-bonding potential curve and the upper bonding one. The parameter Z h tends to infinitize the energies of the potential curves approach asymptotically to electronic energy levels, while the dash lines in Figure2 show energy levels (ground and excited) of the positively charged trion.
When comparing the curves of DOS for the trion and electron, one can see that the additional attraction between the electron and holes in trion provides a displacement of all the peaks toward the region with lower energies. On the other hand, an external magnetic field enhances the difference between the curves of DOS for the electron and the trion due to the fact that the splitting between energy levels with positive and negative angular momentum states for the trion is larger than for the electron. Therefore, it is seen in Figure3 a more significant broadening of the peaks for the trion in the presence of the magnetic field (γ = 3). Also in Figure3, one can observe a slight displacement of the peaks provided by the magnetic field toward the region of higher energies due to the diamagnetic term in the Hamiltonian.
However, as the magnetic field is sufficiently strong, the electron becomes more strongly confined within a region about the axis and the energy spectrum becomes analogous to one in lens. In consequence, the DOS in disk-shaped vertically coupled QDs for the magnetic field in Figure5 (γ = 3) becomes similar to the lens-shaped QDs in Figure3.
In short we propose a simple numerical procedure for calculating the energy spectrum of a positively charged exciton confined in a semiconductor heterostructure that are formed by two vertically coupled, axially symmetrical type II quantum dots located close to each other. The electron in the structure is mainly located inside the dots, while the holes generally are placed in the exterior region close to the symmetry axis. Our calculation includes some important characteristic of the heterostructure such as the possibility of the variation of the QD morphology. We found that the holes' vibrational sublevels with gaps significantly smaller than the corresponding values between the electronic levels and the gaps between bonding and anti-bonding levels with the same quantum numbers are larger than the gaps between two levels with the same parity and different quantum numbers. Also we found that the splitting between energy levels with positive and negative angular momentum states for the trion is larger than for the electron, and the external magnetic field enhances the difference between curves of DOS for the electron and the trion.
This work was financed by the Universidad del Magdalena through the Vicerrectoría de Investigaciones (Código 01).
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