Charge transfer magnetoexciton formation at vertically coupled quantum dots
 Willian Gutiérrez^{1}Email author,
 Jairo H Marin^{2} and
 Ilia D Mikhailov^{1}
DOI: 10.1186/1556276X7585
© Gutierrez et al.; licensee Springer. 2012
Received: 17 July 2012
Accepted: 15 October 2012
Published: 23 October 2012
Abstract
A theoretical investigation is presented on the properties of charge transfer excitons at vertically coupled semiconductor quantum dots in the presence of electric and magnetic fields directed along the growth axis. Such excitons should have two interesting characteristics: an extremely long lifetime and a permanent dipole moment. We show that wave functions and the lowlying energies of charge transfer exciton can be found exactly for a special morphology of quantum dots that provides a parabolic confinement inside the layers. To take into account a difference between confinement potentials of an actual structure and of our exactly solvable model, we use the Galerkin method. The density of energy states is calculated for different InAs/GaAs quantum dots’ dimensions, the separation between layers, and the strength of the electric and magnetic fields. A possibility of a formation of a giant dipolar momentum under external electric field is predicted.
Keywords
Magnetoexciton Vertically coupled quantum dots Giant dipolar momentum Galerkin method Density of energy states.Background
Over the last decades, small semiconductor systems with discrete energy spectra, known as quantum dots (QDs) and that are analogs of atoms, have fired the imagination of researchers in many fields of physics [1–3]. Unlike in atomic systems, a variety of geometries and configurations of the charge state in these artificial atoms are possible. These particular features make possible to consider QDs as building blocks for the fabrication of more complex structures, such as solidstate artificial molecules, case in which coupled QDs act similar to coupled atoms in a natural molecule [4–6]. Although a diverse range of technologies have been implemented to fabricate QDs, in the case of artificial molecules, there has been a growing interest in spontaneous formation techniques by utilizing selfassembling phenomena on crystal surfaces. One of the most interesting manifestations of this phenomenon is the process of vertical selfalignment of the stacked selfassembled quantum dots (SAQDs) [7–9]. These wonderful structures composed of two or more vertically stacked SAQDs have the advantage of possessing different morphologies such as disks, pyramids, rings or lenses with very few imperfections. Also, they are, in general, thin layers and have, for the most part, a small heighttobase aspect ratio, which is an significant advantage that allows us, on one hand, to modify essentially the energy spectrum of the particles confined within the heterostructure, making them more stable and, on the other hand, to use simple theoretical models.
Currently, there is significant interest in understanding the role of the quantum tunneling of charge carriers between vertically coupled QDs, driven not only by a fundamental nature of this phenomenon but also by their potential applications. Particularly, many efforts have been focused on the theoretical study of the simplest configuration of a QD molecule, namely, a pair of QDs coupled by tunneling [10–13]. In part, interest in such structure arises from its application as a possible gate in a quantum processor required to entangle different states of an electronhole pair created optically [14–16]. Different exciton states can be disentangled by preventing the tunneling through the application of an electric field along the growth direction. Formed in this way, one of the untangled states, chargetransfer exciton has two important characteristics: an extremely long lifetime and a permanent dipole moment [17, 18]. Additionally, its optical properties can be controlled by means of an external magnetic field.
In this work, we consider heterostructures consisting of two vertically aligned hillshaped InAs/GaAs SAQDs. The QDs’ morphology has been modeled using a special shape which provides an almost parabolic confinement, allowing us to perform a relatively simple calculation of the exciton spectrum. In our model, a twodot molecule with a single captured electronhole pair can remain in one of two possible configurations with different dipole moments. In the first case, when the electron and the hole are located at the same dot (onsite exciton), the dipole moment is small, while in the second case, as the particles are situated at different dots (charge transfer exciton), the dipole moment can be very large. In order to illustrate how the electric and magnetic fields applied along to the heterostructure growth direction can facilitate or block a transition between two possible carriers configurations and in this way control electrooptical properties of such structures, we have calculated their densities of states and the averaged values of the dipole moment for different temperatures.
Methods
Theoretical model
Here, threedimensional position vectors of the electron and the hole in the cylindrical coordinates are r_{ p } = (ρ_{ p }, ϑ_{ p }, z_{ p }); p = e, h.
The values of the physical parameters pertaining to InAs used in our calculations are dielectric constant ε = 15.2, the effective masses in the InAs material layer for the electron m_{e} = 0.04m_{0} and for hole m_{h} = 0.34m_{0}, the conduction and the valence bands offsets in junctions are V_{0e} = 450 meV and V_{0h} = 316 meV, respectively [19].
In these equations H_{0e} and H_{0h} represent the Hamiltonians of the unbound electron and hole respectively, confined inside their heterostructures. The following units are used in the dimensionless Hamiltonian (4), the exciton effective Bohr radius a_{0}^{∗} = ℏ^{2}ε/μ e^{2} as the unit of length, the effective Rydberg Ry * = e^{2}/2ε a_{0}^{∗} = ℏ^{2}/2μ a_{0}^{∗}^{2} as the energy unit, and γ = eℏB/2μ c Ry * and α = ea_{0}^{*}F/Ry * as the units of the magnetic and electric field strengths respectively, with μ = m_{e}m_{h}/(m_{e} + m_{h}) being the reduced mass. The parameters η_{e}, η_{h} satisfy the relation η_{e} + η_{h} = 1; in our calculations, we assume that η_{e} > η_{h}.
Once Equation (11) is solved and the set of wave functions (10) is found then it can be used as the basis to calculate the energy corrections due to the presence of the perturbation U in the Hamiltonian (7) in the framework of the socalled exact diagonalization or Galerkin method.
Results and discussion
In our numerical work, we solve Equation (11) by using the trigonometric sweep method [20] initially for d = 0 and later for d ≠ 0. The eigenfunctions and eigenvalues found in the first calculation were used for calculating the energy levels E_{ k }^{(0)}, k = 1, 2, 3… of the on site exciton (the electron and the hole are mainly situated at the same QD), while the results of the second calculation were used to find the energy levels E_{ k }^{(t)}, k = 1, 2, 3… of the charge transfer exciton (the electron and the hole are mainly situated at different QDs).
One can see other peculiarity in Figure 3, the curves for different temperatures all have intersections at the same point. In other words, when the electric field F becomes equal to a critical value F_{c}, the value of the dipole moment is the same for different temperatures. One can explain this result, taking into account that densities of states both for the onsite exciton and for the charge transfer exciton close to their thresholds are almost linear. Therefore, when F = F_{c}, the densities of states of two types of excitons satisfy the relation ρ^{(t)}(E − eF_{c}d) = αρ^{(0)}(E). Here, α is the ratio of the slopes of the linear parts of curves close to their thresholds. Substituting this relation to Equation (13), one can obtain 〈p〉 = edα / (1 + α), i.e., dipole moment does not depend on temperature.
Conclusions
In conclusion, our results demonstrate that electrooptical properties of an exciton confined in two vertically coupled quantum dots can be changed remarkably by electric and magnetic fields applied along the growth direction. Especially, we find that through the electric field, a strong dipole moment can be induced; this is due to the tunneling of charge carriers across the potential barrier between dots, which leads to a charge redistribution of electronhole pair in the structure, passing from a configuration: onsite exciton (the electron and the hole are mainly situated at the same QD) to a configuration charge transfer exciton (the electron and the hole are mainly situated at different QDs).
Authors’ information
IDM received the Ph.D. degree from the Physical Technical Institute from Moscow and the D. Sc. degree from the same institute. Currently, IDM is a titular professor at the School of Physics, Universidad Industrial de Santander, Bucaramanga, Colombia. JM received the Ph.D. degree from the Universidad Industrial de Santander; currently, he is an associate professor at the School of Physics, Universidad Nacional de Colombia, Medellín, Colombia.
WG received the Ph.D. degree from the Universidad Industrial de Santander; currently, he is an assistant professor at the School of Physics, Universidad Industrial de Santander, Bucaramanga, Colombia.
Abbreviations
 QDs:

quantum dots
 SAQDs:

selfassembled quantum dots.
Declarations
Acknowledgments
This work was financed by the Universidad Industrial de Santander (UIS) through the Vicerrectoria de Investigación y Extensión (VIE), DIEF de Ciencias (Cod. 5124), and by the Patrimonio Autonomo del Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco Jose De Caldas under contract RC–no. 275–2011 and cod. no. 11020516923 subscribed with Colciencias.
Authors’ Affiliations
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