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Department of Mathematics and Physics, North Carolina Central University, Durham, NC, USA

- *Corresponding Author:
- Dvoyan KG

Department of Mathematics and Physics

North Carolina Central University

Durham, NC, USA

**Tel:**+1 919-530-6100

**E-mail:**kdvoyan@nccu.edu

**Received Date**: March 20, 2017; **Accepted Date:** March 30, 2017; **Published Date**: March 31, 2017

**Citation: **Dvoyan KG, Tshantshapanyan AA, Melikyan HM, Vlahovic B (2017)
Electronic States and Optical Transitions in an Asymmetric Quantum Dot Molecule.
J Laser Opt Photonics 4: 152. doi: 10.4172/2469-410X.1000152

**Copyright:** © 2017 Dvoyan KG, et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.

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In the framework of adiabatic approximation the electronic states and direct interband absorption of light in the asymmetric double quantum dot molecule (QDM) having a shape of Cassini lemniscate revolution are discussed. Analytical expressions for the wave functions and energy spectrum of the electron in the QDM are derived. Nonmonotonic split and step-like behavior of the energy spectrum is revealed due to the possibility of the electron tunneling between quantum dots (QDs) in the molecule. The corresponding selection rules of quantum transitions for the direct interband absorption of light are obtained. The absorption edge characteristics depending on the QDM geometrical sizes are also revealed.

Quantum dot molecule; Adiabatic approximation; Light absorption; Selection rules

Advances in modern high precision methods of nanostructures
growth such as molecular beam epitaxy, laser deposition, Stranski-
Krastanov mode, and metalorganic chemical vapor deposition,
provide ample opportunities for production of not only the isolated
QDs but also two or more closely spaced coupled QDs − quantum
dot molecules (QDMs). Whereas QDs are artificial atoms, the QDMs
are artificial molecules with bonding and anti-bonding orbitals [1].
From the practical point of view, an interesting coupling between QDs
provides wide possibilities for QDMs applications in modern opto-and
nanoelectronics. In contrast to atoms in molecules, the charge carriers’
confinement in each QD and the coupling between QDs in a QDM can
be controlled separately. Redistribution of the energy spectrum and
wave functions (WFs) in the QDM leads to dramatic changes in the
optical and transport properties of QDMs compared to single QDs [2].
In real growth conditions, the obtained symmetric QDMs are rather
rare; more often, due to various physical and technical reasons, grown
lateral QDMs are asymmetric [3-5]. Similarities between the energy
spectra of symmetric molecules (e.g., O_{2}, H_{2}, or N_{2}) and symmetric
QDMs, and, correspondingly, asymmetric molecules (e.g., CO, CO_{2},
or NO_{2}) and asymmetric QDMs potentially allow one to design various
biochemical sensors and detectors with a single molecule sensitivity
[6,7]. For applications in semiconductor based photonic devices
[8,9] one needs to find the energy spectrum of charge carriers as well
as the possible optical transitions in QDMs considering quantum
confinement effects. Electronic and optical properties of QDMs largely
depend on the shape and sizes of the QDs and the confining potential
[10-14]. Significant differences in size and chemical composition of
composite QDs complicate the theoretical description of the QDM
properties. Obviously, the charge carrier is more likely to be localized
in a large QD, yet the presence of one or more small satellites changes
the charge carrier behavior fundamentally due to the rearrangement
of energy levels in a large QD. This complex problem can be described
analytically with the correct choice of the equation for the entire QDM
surface [15]. Theoretical study of semiconductor QDMs allows reducing
the number of expensive experiments to design the optimal structure
of new generation photonic devices. In this paper, the electronic states
and direct interband absorption of light are theoretically investigated
in the asymmetric QDM having Cassini lemniscate shaped surface and
the isthmus connecting two QDs **(Figure 1)**.

Let us consider an impermeable asymmetric QDM consisting of
two QDs schematically shown on **Figure 1**. The surface equation of the
QDM in cylindrical coordinates and the charge carrier (an electron or
hole) potential energy in the QDM are as follows:

(1)

(2)

where c_{1} and c_{2} are the focal length of the small and large semi-lemniscates
respectively, a_{1} and a_{2} are the product of distances from foci to any
point on the surface for each semi-lemniscate shaped QD, respectively.
Note that the relationship between QDM geometric parameters allows
one to describe the isthmus between QDs, but reduces the number of
independent parameters. Similarly, a bond between atoms reduces a
degree of freedom of a molecule. For simplicity of formulas, we will
consider the case of equal effective masses in both QDs .

**Strong confinement regime**

In the strong confinement (SC) regime, Coulomb interaction between the electron and hole can be neglected, and the problem reduces to the determination of their energy states independently. As it follows from the geometrical shape of the QDM, the electron motion in z direction is faster than in perpendicular direction. Hereafter all units are dimensionless, if otherwise is not specifically noted. The geometric adiabatic approximation allows one to represent the system Hamiltonian as a sum of the fast and slow subsystem Hamiltonians,

where

(3)

and the following notations are introduced:

is the effective Rydberg energy of an electron, is the effective Bohr radius of an electron, is a dielectric permittivity, e and are the charge and effective mass of an electron, respectively. WFs of the problem are sought in the form

(4)

where C is a normalization constant. At a fixed value of the coordinate z of the slow subsystem the particle motion is localized in a two-dimensional potential well with an effective width, where . The fast subsystem Schrödinger equation solutions are given by the first kind

Bessel functions

(5)

and A(z) is a normalization constant. Taking into account boundary conditions, one obtains the following expression for the effective twodimensional motion energy:

(6)

where α_{n,m} are roots of the first kind Bessel functions. The energy
(6) plays role of an effective potential in the slow subsystem
Schrödinger equation. For lower energy levels, the electron is localized
in the geometrical center of one of the QDs, with coordinates . Expanding (6) in a power series around these
points, one gets the expression for the small and large QDs respectively **(Figure 2)**

(7)

with depends on the QDM geometric parameters and ensures the continuity of the approximated effective confinement potential function at the point z=0. Usually, in the problems of QDs energy spectra determination, the effective potential of the slow subsystem turns out parabolic due to the adiabatic approximation application [15].

In contrast, in current problem the electron is influenced by the asymmetric double parabolic effective potential due to the specific geometric form of the QDM. Further, using the geometric adiabatic approximation technique, one solves the Schrödinger equation with the effective potential (7):

(8)

which solutions are the parabolic cylinder functions D_{v}(t):

(9)

and Finally, the electron WFs in the QDM can be written

(10)

Total energy of electron in the QDM for the SC regime is determined from the sewing of WFs (9) at the point z=0:

(11)

**Weak confinement regime**

In this case, the exciton binding energy (in all three geometrical directions) prevails over the confinement energy; Coulomb interaction predominates, and the weak influence of the QDM walls appears as a small correction in the Hamiltonian: equation (12)

(12)

Where is the mass of the exciton, and is the reduced mass of the exciton.

The system WFs in the weak confinement (WC) regime can be represented as

(13)

where describes the relative motion of the electron and hole, whereas describes the motion of the center of gravity of the exciton. For the exciton center of gravity, one gets the energy Gr analogous to the (11), with the exciton mass M instead of the relative motion,

(14)

where is the exciton effective radius. Finally, for the total energy and WFs one obtains

(15)

(16)

Where Y_{l,m}(θ,ϕ) are spherical functions.

Let us consider now the direct interband absorption of light in the asymmetric QDM in the SC regime. The absorption coefficient for the heavy hole approximation is determined by the expression [16]:

(17)

where α and α′ are the quantum number (QN) sets corresponding to
the electron and a heavy hole, E_{g} is the bandgap of a bulk semiconductor,
is an incident light frequency, and A is a quantity proportional to the
square of the matrix element taken by the Bloch functions. For the
absorption edge (AE), one gets the following expression describing the
dependence of the effective band gap on the parameters a_{1} and c_{1}:

(18)

Where and AE shifts to the long wave
region with an increase in the parameter a1, whereas the increase in the
c_{1} shifts AE to the short wave region. Quantum transitions are allowed
for the energy levels with the magnetic QNs m=-m, and for the fast
subsystem QNs n=n′. Slow subsystem selection rules are completely
removed due to the continuity of logarithmic

Derivatives of the WFs (11), and the analytical expression (18) is given taking into account above mentioned selection rules.

For the WC regime, in view of the exciton localization in a vicinity of the QD geometric center, the absorption coefficient is given by the following expression:

(19)

where E represents the energy [17] in dimensional units. It should be noted, that equation only for the ground state, when l=m=0 (l is the orbital QN of the exciton). In this regime one obtains the following expression for the AE:

(20)

where Here, the shift of the exciton energy level with changing parameters of the QDM is determined by the total mass of the exciton.

In general, the geometric adiabatic approximation application for the QDs energy spectra calculations leads to the forming of the slow subsystem energy levels families for each energy level of the fast subsystem. Also, in most cases, the slow subsystem energy levels turn out equidistant, since the effective confinement potential is parabolic. As it is seen from (7), the slow subsystem confinement potential has the form of an asymmetric double parabola, in contrast to the other cases of the geometric adiabatic approximation application. Obviously, now the slow subsystem energy spectra are non-equidistant. Moreover, some levels are double split due to the tunneling possibility between two QDs, and the magnitude of the splitting depends on the geometrical parameters of the QDM and the fast subsystem QN.

**Figure 3** represents the dependence of the first family of the
energy levels in the QDM on the parameter a_{1} at a given value of the
parameter a2 and semi-lemniscate focal length c_{1}. With an increase in
the parameter a1 the energy curves decrease, as a consequence of the
increase in the isthmus connecting two QDs. At the same time, the
magnitude of the splitting of the levels varies due to the fact that the
probability of the electron tunneling from one QD to another increases
for large values of a1. An increase in the parameter a1 leads to the electron
spectrum modification and new energy levels arise in an increased
quantum well. But the dependence of energy levels splitting on a_{1} is
non-monotone, since the higher levels existence affects probability of
tunneling of lower levels. In other words, inter-level distances decrease
at first, then increase and decrease again **(Figure 4)**. The lower levels
demonstrate step-like arrangement. It should be noted that for small
values of the isthmus width, the tunneling probability decreases, the
curves of the singlet and triplet states merge, while for the larger values
of the isthmus width situation changes, and the splitting is more for
the higher positioned levels. Thus, at a_{2}=2.4 and N=0, the singlet-triplet
energy levels splitting is ΔE≈0.25E_{R}, whereas at N=1 the splitting is ΔE≈
1.12ER. All numerical calculations were performed for the GaAs QDM
with the following parameters: , where m_{e} is a free electron mass, k=13.8, E_{R}=5.275 meV, a_{B} =104 A_{0} and a_{h}=15 A_{0} are
the effective Bohr radii for the electron and the hole, respectively, and
E_{g}=1.43 eV is a bandgap of a bulk semiconductor. Opposite behavior of
the energy levels is observed in **Figure 4**., which shows the dependence
of the first family of the electron energy levels in the QDM on the
semi-lemniscate focal length c_{1} at a given values of the parameters a_{1} and a_{2}. Increase in the parameter c_{1} leads to the increase of the energy
levels, conditioned by the increase in the distance between two QDs
and narrowing of the isthmus connecting them. At the same time, the
increase in the semi-lemniscate focal length also leads to the decrease
in the electron tunnelling probability. As a consequence, the energy
levels splitting magnitude varies. Note, that at the value a_{1} c_{1}, Cassini
lemniscate transforms to Bernoulli lemniscate, corresponding to the
case of the zero width isthmus between closely spaced QDs. However,
the absence of the isthmus does not result in a complete removal of
the splitting. Thus, at the value of a_{1}=c_{1}=2.2 the excited levels are still
split due to a non-zero probability of the electron tunnelling. Further
increase in the focal parameter leads to the final isolation of QDs.

**Figure 5** plots the dependence of the electron energy on the
parameter a_{2}, for the given values of the parameters a_{1} and c_{1}. As it can
be seen from the figure, the zone of a step-like splitting is also observed
in this case. With an increase in the parameter a_{2}, the lower energy
level (N=0) increases monotonically at first, then starting at the value
a_{2}=a_{1}=2.2, it almost does not respond to the further increase in the
parameter, since the small QD becomes larger than the initially large
QD after this point. In other words, the particle more likely localizes in
a large QD and reacts weakly to the further increase in the parameter.
A similar dependence is observed for the levels with N 1 also. For
the higher energy levels outside the splitting zone, the increase in the
parameter a_{2} does not introduce significant changes.

**Figure 6** plots the AE dependence on the parameter a1 at the
given values of the parameters c_{1} and a_{2} in the SC regime. As it is seen
from the figure, the increase in the isthmus results in the decrease
in the AE due to the reduction of the confinement effects (red shift).
The energy spectrum step-like behaviour manifests itself also in
the AE dependences. The curve corresponding to the small value of
the parameter c_{1} is positioned lower, since in this case the electron
localization area in the QDM becomes wider. The opposite pattern
is observed in the **Figure 6b**, which shows the dependence of the AE
on the focal length distance c_{1}. As expected, increase in c_{1} results in
the blue shift of the edge frequencies, due to the confinement impact
increase. For this reason, the curve corresponding to the small value
of the a1 is positioned higher. Note also, that at the small values of the
focal length the AE curves are positioned closer for the different values of the parameter a_{1}. The increase in the parameter c_{1} leads to increase
of the curves discrepancy.

From the above follows that during the growth it is possible to
control the electron energy value and the splitting of the energy levels
by manipulating sizes and the distance between QDs. It may promote
the design of a new generation of the highly sensitive and selective
sensors. In case when there is the coincidence in the energy split of
the levels of the QDM and the potentially detectable analyte molecules,
it is possible to achieve a successful tunneling of electrons from the
analyte molecules to the QDM. It is important also that this tunneling
is possible (most probable) only with the full concurrence of interlevel
distances of the QDM and a detectable molecule. This ensures
high selectivity of the device, which will not detect another molecule **(Figure 6)** by mistake. For the design of high-precision devices the most
accurate measurements of the splitting of the energy levels are needed.
For this purpose the easiest is to measure the frequencies of the light
absorption in the QDM. Knowing the specific absorption frequencies,
it is not difficult to establish the difference between the energy levels
and the amount of the splitting.

This work is supported by the NSF: HRD-1345219 and DMR-15236 17, and NASA: NNX09AV07A.

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