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NANO EXPRESSOpen Access

Double-donor complex in vertically coupled

quantum dots in a threading magnetic field

Ramón Manjarres-García1, Gene Elizabeth Escorcia-Salas1, Javier Manjarres-Torres1, Ilia D Mikhailov2

and José Sierra-Ortega1*

Abstract

We consider a model of hydrogen-like artificial molecule formed by two vertically coupled quantum dots in the

shape of axially symmetrical thin layers with on-axis single donor impurity in each of them and with the magnetic

field directed along the symmetry axis. We present numerical results for energies of some low-lying levels as

functions of the magnetic field applied along the symmetry axis for different quantum dot heights, radii, and

separations between them. The evolution of the Aharonov-Bohm oscillations of the energy levels with the increase

of the separation between dots is analyzed.

Keywords: Quantum dots, Adiabatic approximation, Artificial molecule

PACS: 78.67.-n, 78.67.Hc, 73.21.-b

Background

An important feature in low-dimensional systems is

the electron-electron interaction because it plays a crucial

role in understanding the electrical transport properties of

quantum dots (QDs) at low temperatures [1]. Such sys-

tems may involve small or large numbers of electrons as

well as being confined in one or more dimensions. The

number of electrons in a QD can be varied over a consid-

erable range. It is possible to control the size and the

number of electrons and to observe their spatial distribu-

tions in QDs. Energy spectrum of two-electron QD with a

parabolic confinement, for which two-particle wave equa-

tion can be separated completely, has been analyzed previ-

ously by using different methods [2-5].

In the present work, we propose another exactly solv-

able two-electron heterostructure in which two sepa-

rated electrons are confined in vertically coupled QDs

with a special lens-like morphology. Together with two

on-axis donors, these two electrons generate an artificial

hydrogen-like molecule whose properties can be con-

trolled by varying the geometric parameters and the

strength of the magnetic field applied along the sym-

metry axis.

Methods

The model which we analyze below consists of two

identical, axially symmetrical and vertically coupled QDs

with the on-axis donor located in each one of them (see

Figure 1). The dimension of the heterostructure is defined

by the QDs' radii R, height W, and the separation d be-

tween them along the z-axis. We assume that the QDs

have a shape of very thin layers whose profiles are given

by the following dependence of the thickness of the layers

w on the distance ρ from the axis:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Besides, for the sake of simplicity, we consider a model

with infinite barrier confinement, which is defined in cy-

lindrical coordinates as V r ð Þ ¼ 0 if 0 < z < w ρ ð Þ, and

V r ð Þ ¼ 1 otherwise.

Given that the thicknesses of the layers are much

smaller than their lateral dimensions, one can take ad-

vantage of the adiabatic approximation in order to ex-

clude from consideration the rapid particle motions

along thez-axis[6,7]and

w ρ ð Þ ¼ W=

1 þ ρ=R

ðÞ2

q

ð1Þ

obtainthe following

* Correspondence: jsierraortega@gmail.com

1Group of Investigation in Condensed Matter Theory, Universidad del

Magdalena, Santa Marta, Colombia

Full list of author information is available at the end of the article

© 2012 Manjarres-García et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative

Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Manjarres-García et al. Nanoscale Research Letters 2012, 7:531

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expression for effective Hamiltonian in polar coordi-

nates:

H ¼

X

i¼1;2

H0ρi

ð Þ þ V ρ1;ρ2

@ϑiþω2ρ2

Þ2þ γ2V ρ1;ρ2

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

ðÞ þ 2π2

W2;H0ρi

ð Þ

¼ ?Δ2D

¼ 2π=W⋅R

¼

d2þ ρ1? ρ2

2

ð

i

Þ

þ iγ

@

i

4

;ω2

ð

q

?

ðÞ

jj2

i¼1;2

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

d2þ ρi

j j2

q

þ

2

ρi

j j

64

3

75

ð2Þ

The effective Bohr radius a∗

of length, the effective Rydberg

ℏ2=2m ? a∗

as the unit of the magnetic field strength have been used

in Hamiltonian (Equation 2), with m? being the electron

effective mass and E, the dielectric constant. The polar

coordinates ρk¼ ρk;ϑk

to the first and the second electrons, respectively. It is

seen that for the selected particular profile given by

Equation 1, the Hamiltonian (Equation 2) coincides with

one which describes two particles in 2D quantum dot

with parabolic confinement and renormalized inter-

action. It is well known that such Hamiltonian may be

separated by using the center of mass, R ¼ ρ1þ ρ2

and the relative, ρ ¼ ρ1? ρ2coordinates [8]:

HR¼ ?Δ2D

Hρ¼ ?Δ2D

16

0¼ ℏ2E=m ? e2as the unit

Ry? ¼ e2=2Ea∗

0¼

02 as the energy unit, and γ ¼ eℏB=2m ? cRy?

??labeled by k ¼ 1;2 correspond

ðÞ=2,

H ¼ HRþ 2Hρ;

ð

R

2

Þ

þ1

4

2ω2R2;

ð

ρ

Þ

þω2ρ2

?3

ρ?

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρ2þ 4d2

p

ð3Þ

The wave function is factorized into two parts,

ψ R;ρ

ðÞ ¼ Φ R

ð Þφ ρ

ð Þ, describing the center of mass and

the relative motions, respectively. Meanwhile, the total

energy splits into two terms depending on two radial

NR;nρand two azimuthal LR;lρquantum numbers:

?

where the first term represents the well-known expression

for the exact energy levels of a two-dimensional harmonic

oscillator, labeled by the radial NR¼ 0; 1; 2; ...

azimuthal LR¼ 0 ? 1 ? 2; ...

for the center of mass motion and the relative motion

energy 2Eρ nρ;lρ

must be found solving the following

one-dimensional Schrödinger equation:

E NR;LR;nρ;lρ

?¼ ERNR;LR

ðÞ þ 2Eρ nρ;lρ

Þω þ 2Eρ nρ;lρ

??

¼ 2NRþ LR

ð

??

ð4Þ

ðÞ and

ðÞ quantum numbers

??

?u00ðρÞ þ VðρÞuðρÞ ¼ Eρðnρ;lρÞuðρÞ;VðρÞ

¼ ω2ρ2=4 þ l2

ρ? 1=4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

??

=ρ2

? 3=ρ ? 4=

ρ2þ 4d2

p

ð5Þ

In our numerical, work the trigonometric sweep

method [8] is used to solve this equation.

Results and discussion

Before the results are shown and discussed, it is useful

to specify the labeling of quantum levels of the two-

electron molecular complex. According to Equation 4,

the energy levels E NR;LR;nρ;lρ

symbols NR;LR;nρ;lρ. Even and odd lpcorrespond to the

spin singlet and triplet states, respectively, consistent

with the Pauli Exclusion Principle.

We have performed numerical calculations of energy

levels of complexes with radii R between 20 and 100 nm

for different separations between layers. In all presented

calculation results, the top thickness W is taken as

0.4 nm. In order to highlight the role of the interplay be-

tween the quantum size and correlation effects in the

formation of the energy spectrum of our artificial system

different from the natural hydrogen molecular complex,

we have plotted in Figure 2 the potential curves~E d

E NR;LR;nρ;lρ

molecule in which the complex energies with the elec-

trostatic repulsion between donors included as functions

of the separation d between QDs are shown. Comparing

them with the corresponding potential curves of the

hydrogen molecule, one can to take into account that in

analyzing the structure here, the electron motion in con-

trast to hydrogen molecule is restricted inside two sepa-

rated thin layers. The energy dependencies of different

levels (labeled by four quantum numbers, NR;LR;nρ;lρ

are shown in Figure 2 for QDs with two different radii,

R=40 nm and R=100 nm. A clear difference in the be-

havior of the potential curves is readily seen. If the

curves are smooth without any crossovers for QDs of

??

can be labeled by four

ð Þ ¼

??þ 2=d, similar to those of the hydrogen

??

Figure 1 Scheme of the artificial hydrogen-like molecule.

Manjarres-García et al. Nanoscale Research Letters 2012, 7:531

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small radius, the corresponding potential curves suffer a

drastic change as the QD radius becomes large. In the

last case, the energy levels become very sensitive to the

variation of the separation between QDs, and the

quantum size effect becomes essential, providing alter-

ation of the energy gaps, multiple crossovers of levels

with the same or different spins, and the level reorder-

ing, as the distance between QDs increases from 5 to

20 nm.

We ascribe a dramatic alteration of the potential

curves with the increase of the separation between QDs

from 5 to 20 nm observed in Figure 2 to the interplay

between the structural confinement and the electron-

electron repulsion. As the QDs' radii are small(R!0),

the confinement is strong, and the kinetic energy (~1/

R2) is larger than the electron-electron repulsion energy

(~1/R), vice versa for QDs with large radii. Therefore, as

the QDs' radii increase, the arrangement of the elec-

tronic structure for different energy levels changes from

typical for the gas-like system to crystal-like one, accom-

panied by the crossovers of the curves and reordering of

the levels. As the two-electron structure arrangement

for large separation between electrons becomes almost

rigid, the relative motion of electrons is frozen out, and

the two-electron structure transforms into a rigid rotator

with practically fixed separation between electrons. The

electrons' motion in this case becomes similar to one in

1D ring, and therefore, the energy dependencies on the

external magnetic field applied along the symmetry axis

should be similar to those which exhibit the Aharonov-

Bohm effect.

0,0 0,20,40,6 0,81,0

60

80

100

120

R=40nm w = 4nm d = 6nm

Energy (Ry*)

0,00,2 0,40,60,81,0

0

5

10

Energy (Ry*)

γ

R =100nm w = 4nm d =6nm

Figure 3 Energies E NR;LR;nρ;lρ

the double-donor complex in vertically coupled QDs. As functions

of the magnetic field.

??of some low-lying levels of

05 1015 20

0

5

10

15

20

Density of States

Energies (Ry)*

γ = 0

γ = 0.5

w = 4nm R = 100nm

d = 6nm

Figure 4 Density of states for two different values of the

magnetic field. Corresponding to low-lying levels of the double-

donor complex in vertically coupled QDs.

5

10

15

20

20

15

10

(2, 3, 1, 0

(1, 1, 1, 1)

(1, 3, 1, 0)

(1, 0, 1, 1)

(1, 2, 1, 0)

(1, 1, 1, 0)

(1, 0, 1, 0)

(2, 2, 1, 0)

R = 100nm, w = 4nm

Energy (Ry*)

d (nm)

5

60

80

100

120

20

15

10

5

(2, 2, 1, 0)

(1, 1, 1, 1)

(1, 3, 1, 0)

(1, 0, 1, 1)

(1, 2, 1, 0)

(1, 1, 1, 0)

(1, 0, 1, 0)

R = 40nm, w = 4nm

Energy (Ry*)

Figure 2 Energies~E d

corresponding to some low-lying levels in vertically coupled

QDs. As functions of the distance between them.

ð Þ of the double-donor complex

Manjarres-García et al. Nanoscale Research Letters 2012, 7:531

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In order to verify this hypothesis, we present in Figure 3

the calculated molecular complex energies E NR;LR;nρ;lρ

of some lower levels as functions of the magnetic field

strength for QDs with small R=40 nm (upper curves) and

large R=100 nm radii (lower curves).

It is seen that for QD of small radius, the energies are

increased smoothly with very few intersections. Such de-

pendence is typical for gas-like systems where the para-

magnetic term contribution is depreciable in comparison

with the diamagnetic one. On the contrary, the energy de-

pendency curves for QD of large radius present multiple

crossovers and level-ordering inversion as the magnetic

field strength increases from 0 to 1. It is due to a competi-

tion between diamagnetic (positive) and paramagnetic

(negative) terms of the Hamiltonian whose contributions

in total two-electron energy in QDs of large radii are of

the same order while the electron arrangement is similar

to a rigid rotator. In other words, the correlation in this

case becomes as strong as the electrons are mainly located

on the opposite sides within a narrow ring-like region.

Finally, in Figures 4 and 5, we present results of the

calculation of the density of electronic states for double-

donor molecular complex confined in vertically coupled

QDs. It is clear from the discussion above that the pres-

ence of the magnetic field should provide a significant

change of the density of the electronic states as the QDs'

radii are sufficiently large. Indeed, it is seen from Figure 4

that under relatively weak magnetic field (γ=0.5), as the

molecular complex is confined in QDs of 100-nm with

6-nm separation between them, the density of states

becomes essentially more homogeneous since the widths

of individual lines are broadened and the gaps between

them are reduced. Such change of the density of states is

observed due to a splitting and displacement of the indi-

vidual lines accompanied by their crossovers and the

reordering of the energy levels.

??

In Figure 5, we present similar curves of the molecular

complex density of states for three different separations

between QDs. It is seen that the curves of the density of

states are modified only slightly, essentially less than

under variation of the magnetic field. Particularly, the

lower energy peak positions are almost insensitive to any

change of the distance between dots, while the upper en-

ergy peaks are noticeably displaced toward higher energy

regions.

Conclusions

In short, we propose a simple numerical procedure for

calculating the energies and wave functions of a molecu-

lar complex formed by two separated on-axis donors

located at vertically coupled quantum dots with a par-

ticular lens-type morphology which produces in-plane

parabolic confinement. We show that in the adiabatic

approximation, the Hamiltonian of this two-electron sys-

tem included in the presence of the external magnetic

field is separable. The curves of the energy dependencies

on the external magnetic field and the separation be-

tween quantum dots are presented. Analyzing the curves

of the low-lying energies as functions of the magnetic

field applied along the symmetry axis, we find that the

two-electron configuration evolves from one similar to a

rigid rotator to gas-like as the dot radii decrease. This

quantum size effect is accompanied by a significant

modification of the density of the energy states and the

energy dependencies on the external magnetic field and

geometric parameters of the structure.

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, RMG, and JMT. GES performed the numerical

calculations and wrote the manuscript. All authors discussed the results and

implications, commented on the manuscript at all stages, and read and

approved the final manuscript.

Acknowledgments

This work was financed by the Universidad del Magdalena through the

Vicerrectoría de Investigaciones (Código 01).

Author details

1Group of Investigation in Condensed Matter Theory, Universidad del

Magdalena, Santa Marta, Colombia.2Universidad Industrial de Santander, A.

A. 678, Bucaramanga, Colombia.

Received: 10 July 2012 Accepted: 29 August 2012

Published: 26 September 2012

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05

Energies (Ry)*

1015

0

5

10

15

20

25

Density of States

d = 0nmB

d = 50nm

d = 100nm

w = 4nm

R = 100nm

γ = 0

Figure 5 Density of states for three different distances between

layers. Corresponding to low-lying levels of the double-donor

complex in vertically coupled QDs.

Manjarres-García et al. Nanoscale Research Letters 2012, 7:531

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doi:10.1186/1556-276X-7-531

Cite this article as: Manjarres-García et al.: Double-donor complex in

vertically coupled quantum dots in a threading magnetic field.

Nanoscale Research Letters 2012 7:531.

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