Abstract
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 antibonding lowestlying 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.
Keywords:
Quantum dots; Adiabatic approximation; Trion; 78.67.n; 71.35.Pq; 73.21.LaBackground
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 selfassembled 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 selfassembled quantum dots (SAQDs) which are formed via the StranskiKrastanow growth mode by depositing a material on a substrate with a different lattice parameter [15]. 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 [1], they are, in general, thin layers and have, for the most part, a small heighttobase aspect ratio whose typical values vary between 0.029 and 0.067 [3]. The motion of the carriers in such structures is quasitwodimensional 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 electronhole pair. It is established that the quantum confinement of few particles systems, e.g., neutral and negatively charged donors (D^{0} 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 [6], diagonalization [79], and finite elements [10]. 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 twodimensional, and they left to aside the effects related to the disk height. Recently, in order to calculate ground state energies of D^{0}, D^{−}, and X in the semiconductor heterostructures, it has been proposed as a simple variational procedure [1113] related to fractal dimension scheme [14]. 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 antibonding lowestlying 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.
Methods
We assume that the adiabatic approximation can be used as a starting point for a qualitative analysis of the ground state energy of X^{+}, taking into account that the effective hole mass in the In_{0.55}Al_{0.45}As material is essentially greater than the effective mass of the electron, and this circumstance is more relevant in conditions of strong confinement. In this approximation the Hamiltonian of the X^{+} can be expressed in terms of the Hamiltonian of the D_{2}^{+}. To analyze the spectrum of the X^{+} within framework of the adiabatic approximation, we first consider this complex under condition when the holes position vectors r_{h1} and r_{h2} are fixed, and this structure can be considered as a donor complex D_{2}^{+}. Being r_{e} as the position vector of the electron and r_{h1}, r_{h2} as the position vectors of the ions, the Hamiltonian for the donor D_{2}^{+} ground state in a structure with the confinement potential V_{e}(r_{e})due the structural nonhomogeneity and in the presence of the external magnetic field B applied along the symmetry axis can be written as:
The effective Bohr radius as the unit of length, the effective Rydberg as the energy unit, and as the unit of the magnetic field strength have been used in Hamiltonian (Equation 1), and the electron and the hole effective masses and as the reduced mass. The following notations have been used: and the cylindrical coordinates r_{k}=(ψ_{k}, ϑ_{k}, z_{k}) labeled by k = h1, h2, e corresponding to the first hole, the second hole, and the electron, respectively. For the sake of simplicity, we consider a model of the QD with infinitebarrier confinement potential, V_{e}(ρ_{e}, z_{e}) = 0, inside the QD and to infinity otherwise, which restricts the electron only inside the QD, and for the holes located away from QDs, V_{h}(ρ_{h}, z_{h}) = 0, otside the QD and to infinity inside them. The eigenfunctions φ_{e}(r_{e}) and eigenvalues of the Hamiltonian (Equation 1) solutions of the wave equation
can be found by using the method described in the previous paper [15]. The positively charged exciton X^{+} Hamiltonian can be written in terms of the D_{2}^{+} Hamiltonian (Equation 1) as follows:
Following the adiabatic approximation standard procedure, we seek the eigenfunction of the Hamiltonian (Equation 3) as , where ψ(r_{h1}, r_{h2}) is the part of the wave function describing a slow holes' motion which is a solution of the following wave equation:
Here, , the energy of the fast electron motion as a function of the holes positions, presents potential curves for the holes' slow motion.
Once the energies for the different electron levels as functions of the holes positions are found, then the wave in Equation 4 can be solved approximately by using the method of symmetry structures [16] in which the problem in Equation 3 is reduced to the following oneparticle wave equation:
Schematically, the structure corresponding to Equation 5 is presented in Figure 1. The different sublevels of the X^{+} corresponding to the different energies of the hole motion in Z direction can be analyzed by solving the onedimensional wave equation (Equation 5). In our calculations we use to aim the trigonometric sweep method [17].
Figure 1. Scheme of a trion X^{+ }confined in a type II vertically coupled quantum dots.
Results and discussion
In Figure 2 we present the energies of some lower electron levels as functions of the distance z_{h} between the hole and the adjacent semiconductor layer in the framework of the symmetrical structure model for the case when the height of the lensshaped QD over the wetting layer of h_{w} = 0.2a_{0}* thickness is also equal to h = 0.2a_{0}*, while the separation between layers is d = 0.6a_{0}*.
Figure 2. Potential curvesfor the electron confined in vertically coupled lensshaped QDs. Solid lines represent energy levels of the Trion X^{+} and dashed lines, Trion X^{+}. With separation, d, between layers and distance, Z_{h}, between the hole and the layer for zeromagnetic field case (γ = 0).
The corresponding potential curves are indicated by twoelectron quantum numbers (n_{e}, l_{e}) and by symbols (+) and (−) for the bonding and antibonding 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 Figure 2 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 antibonding 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 antibonding 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 Figure 2 show energy levels (ground and excited) of the positively charged trion.
In Figure 3 we present the density of states (DOS) corresponding to the electron (dash lines) and to the positively charged trion (solid lines) for cases of the zero magnetic field (γ = 0) and (γ = 3). In both cases one can observe the presence of three separated peaks which we associate to the bonding, mixed, and antibonding states, respectively. The first peak is a superposition of the lines corresponding to the lower bonding states, the intermediate peak is related to a superposition of the upper bonding and the lower antibonding states, and the last peak is due to a superposition of the upper antibonding states.
Figure 3. Density of states for trion and electron in vertically coupled lensshaped QDs. Density of states for trion (solid lines) and electron (dashed lines) in vertically coupled lensshaped QDs for (left image), (γ = 0) and (right image), (γ = 3).
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 Figure 3 a more significant broadening of the peaks for the trion in the presence of the magnetic field (γ = 3). Also in Figure 3, 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.
In Figures 4 and 5 we display the similar results for vertically coupled diskshaped QDs. It is seen that the potential curves for the disks are analogous to those for the lens in Figure 2 with a single but very important difference: the electron energies and gaps between them in the lens and in Figure 4 are lower than the correspondent values in the lens in Figure 2. It is due to the fact that the inplane confinement in lens is stronger than one in disk resulting in a weak resolution of the three peaks in the zeromagnetic field case (γ = 0) as it is seen in Figure 5, where the third peak is transformed in a rightside shoulder at DOS curve.
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 diskshaped vertically coupled QDs for the magnetic field in Figure 5 (γ = 3) becomes similar to the lensshaped QDs in Figure 3.
Conclusions
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 antibonding 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. JSO created the analytic model with contributions from IM, SHP and GES performed numerical calculations and wrote the manuscript. All authors discussed the results and implications and commented on the manuscript at all stages. All authors read and approved the final manuscript.
Acknowledgments
This work was financed by the Universidad del Magdalena through the Vicerrectoría de Investigaciones (Código 01).
References

Jacak L, Hawrylak P, Wójs A: Quantum Dots. Springer, Berlin; 1997.

Leonard D, Pond K, Petroff PM: Critical layer thickness for selfassembled InAs islands on GaAs.
Phys Rev B 1994, 50:1168711692. Publisher Full Text

Lorke A, Luyken RJ, Govorov AO, Kotthaus JP: Spectroscopy of nanoscopic semiconductor rings.
Phys Rev Lett 2000, 84:22232226. PubMed Abstract  Publisher Full Text

Granados D, García JM: In(Ga)As selfassembled quantum ring formation by molecular beam epitaxy.
Appl Phys Lett 2003, 82:2401. Publisher Full Text

Raz T, Ritter D, Bahir G: Formation of InAs selfassembled quantum rings on InP.
Appl Phys Lett 2003, 82:1706. Publisher Full Text

Szafran B, Stébé B, Adamowski J, Bednarek S: Excitonic trions in single and double quantum dots.

Xie W, Chen C: Excitonic trion X− in GaAs quantum dots.
Physica E 2000, 8:77. Publisher Full Text

Song J, Ulloa SE: Magnetic field effects on quantum ring excitons.

Wojs A, Hawrylak P: Negatively charged magnetoexcitons in quantum dots.
Phys Rev B 1995, 51:1088010885. Publisher Full Text

Janssen KL, Peeters FM, Schweigert VA: Magneticfield dependence of the exciton energy in a quantum disk.

Escorcia R, Robayo R, Mikhailov ID: Renormalized Schrödinger equation for excitons in graded quantum dot.
Phys Stat Sol (B) 2002, 230:431. Publisher Full Text

Mikhailov ID, Betancur FJ, Escorcia R, SierraOrtega J: Offcenter neutral and negatively charged donor impurities in semiconductor heterostructures: fractal dimension method.
Phys Stat Sol (B) 2002, 234:590. Publisher Full Text

Mikhailov ID, Betancur FJ, Escorcia R, SierraOrtega J: Shallow donors in semiconductor heterostructures: fractal dimension approach and the variational principle.

He HF: Fractional dimensionality and fractional derivative spectra of interband optical transitions.
Phys Rev B 1990, 42:11751. Publisher Full Text

Mikhailov ID, Betancur FJ, Marín J, Oliveira LE: Model structure for Dstates in GaAs(Ga, Al)As quantum wells.
Phys Stat Sol (B) 1998, 210:605608. Publisher Full Text

Singh J, Birkedal D, Lyssenko VG, Hvam JM: Binding energy of twodimensional biexcitons.
Phys Rev B 1996, 53:1590915913. Publisher Full Text

Betancur FJ, Mikhailov ID, Oliveira LE: Shallow donor states in GaAs(Ga, Al)As quantum dots with different potential shapes.
J Appl Phys D 1998, 31:3391. Publisher Full Text