DSpace at VNU: Magnetic field effects on the binding energy of hydrogen impurities in quantum dots with parabolic confinements

Physica B 292 (2000) 153}159

Magnetic "eld e!ects on the binding energy of hydrogen
impurities in quantum dots with parabolic con"nements
V. Lien Nguyen *, M. Trinh Nguyen , T. Dat Nguyen
Theoretical Department, Institute of Physics, P.O. Box 429, Bo Ho, Hanoi 10 000, Viet Nam
Physics Faculty, Hanoi National University, 90 Nguyen-Trai Str., Thanh-Xuan, Hanoi, Viet Nam
Received 8 February 2000

Abstract
Using a very simple trial function with only one variational parameter, the e!ects of parabolic con"ning potentials and
magnetic "elds on the binding energy of hydrogen impurities in quantum dots are investigated in detail. For a comparison, the perturbation calculations are also performed in the limit cases of weak and strong con"nements. The obtained
results are suggested to be used for shallow donor impurities in GaAs-type quantums dots.
2000 Elsevier Science B.V.
All rights reserved.
Keywords: Binding energy; Hydrogen impurity; Quantum dots; Magnetic "eld

1. Introduction
The aim of this work is to study the con"nement
and the magnetic "eld e!ects on the binding energy
of hydrogen impurities in quantum dots (QDs) with
parabolic con"ning potentials.
The physics of impurity states developed since
early days of the semiconductor science [1,2], has
recently received renewed attention in relation to
low-dimensional semiconductor structures such as
quantum wells, quantum wires, and quantum dots.
While for quantum wells the binding energy of
hydrogen impurities was investigated in great detail
[3}5], the problem is much less studied for quasizero-dimensional systems of QDs. Until now, almost all studies on binding energy of hydrogen
* Corresponding author. Tel.: #84-4-843-5917; fax: #84-48349050.

E-mail address: (V. Lien Nguyen).

impurities in QDs have exclusively been limited to
the e!ect of con"ning potentials: the square (in"nite
or "nite) potentials [6}11], or parabolic potentials
[12}15]. The most important feature of all these
models [6}15] is their spherical symmetry that
allows one to reduce the problem to solving a
simpler equation of the radial variable, which could
even be solved by the NUMEROV [15]. A breakage of the spherical symmetry may be caused by
di!erent factors such as the dot shapes, asymmetric
con"ning potentials, or external "elds. Recently, we
have calculated the binding energy of hydrogen
impurities in two types of QDs, spherical QDs with
parabolic con"nements and disk-like QDs with
parabolic lateral con"nements, in an external electric "eld [16]. The electric "eld destroys a symmetry of the problem (the spherical symmetry of
spherical QDs, or the cylindrical symmetry of disklike QDs), and the new behaviors of the binding
energy, depending on the relative strength of two,

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154

V. Lien Nguyen et al. / Physica B 292 (2000) 153}159

con"ning and electric "eld, potentials, have been
recognized. It should be noted here that the results

of a numerical self-consistent solution of the Poisson and Schrodinger equations in the Hartree approximation performed by Kumar et al. [17]
strongly support the parabolic form of con"ning
potentials for QDs fabricated from GaAs/AlGaAs
heterostructures.
The e!ect of an external magnetic "eld on the
energy spectrum and on the related optical
transitions is certainly the most important tool for
the study of the electronic states. This is why the
magnetic "eld e!ects were extensively studied for
the impurities in bulk semiconductors [2]. For
semiconductor QDs we do not "nd any work dealing with the problem, though the magnetic "eld
e!ects on the conduction electron energy levels
have been studied at length [18,19].
The e!ective mass Hamiltonian of a hydrogen
atom in a QD, with a con"ning potential < in an
!
external magnetic "eld B has the standard form
1
e  e
H"
p! A ! #< ,
!
c
r
2mH

(1)

where mH is the e!ective mass, e is the elementary
charge, p is the momentum, is the dielectric constant of the QD material, and A is the vector potential of the magnetic "eld B (B"rot A). The atom is

here assumed to be located at the center of QD,
which is also chosen as the origin of coordinates
system. The spin term is not included in the Hamiltonian of Eq. (1) since it simply produces a constant
shift of energies. In the model, the polarization and
image charge e!ects are also assumed to be neglected that should describe, for example, the QDs
fabricated from GaAs/AlGaAs heterostructures
[18].
The binding energy is generally de"ned as
E "E !E ,
(2)

!
where E and E are ground-state energies of the
!

Hamiltonian of Eq. (1) with and without the
Coulomb term, respectively. In this work for calculating the binding energy E of hydrogen impurities
in QDs with parabolic con"ning potentials in an
external magnetic "eld we mainly use a variational
method, but in the limit cases of weak and strong

con"nements some perturbation results are also
included for a comparison.
2. Variational calculations
2.1. Theory
Choosing the symmetric gauge in the cylindrical
coordinates: A"B[0, 0, ], and de"ning the para
bolic con"ning potential as < " ( #z), the
!
Hamiltonian of Eq. (1) for the ground state (zero

magnetic quantum number), denoted as H , could

be written in the dimensionless form
2
1
H "!
!
#  # ( #z),

( #z 4
(3)
where and are positive parameters, measuring
magnitudes of the con"ning and the magnetic potentials, respectively. In the e!ective atom units
used throughout this work the energy is measured
in units of the e!ective Rydberg R "mHe/2
 

and the length is in units of the e!ective Bohr radius
a "
 /mHe. The magnetic "eld strength is then
nothing but ,(
eB/mHc)/2R . Thus, and are

two parameters, characterizing the problem in the
study, and the binding energy will be calculated as
the functions of them. Below, conveniently, two
lengths ¸ and ¸ de"ned as ¸ "1/ and
!
+
!

¸ "1/ will also be used in equivalence to and
+
as the measures of corresponding potentials
(measuring the spatial scales of potentials, the
length ¸ is sometimes explained as the e!ective
!
radius of QD [13,14,18], while ¸ is often called
+
the magnetic length).
Without the Coulomb potential the Hamiltonian
of Eq. (3) could easily be solved exactly by separating the variables:
H "H #H ,
(4)

M
X
H "!
#( # ) ,
(5)
M
M

H "!
# z.
(6)
X
X
Each of Eqs. (4) and (5) describes a harmonic oscillator (two- and one-dimensional, respectively), and
therefore, the ground-state energy E of the total


V. Lien Nguyen et al. / Physica B 292 (2000) 153}159

Hamiltonian H is already known

E " #2( # .
(7)


In the presence of the Coulomb term, the Hamiltonian of Eq. (3) could not be solved analytically.
The variational method is widely accepted as
a good approximation for calculating the ground
state energy. In order to choose an adequate trial
function one should remark on the main features of
the Hamiltonian H : at very small distances (close

to the impurity) the Coulomb potential should be
dominant, while at large distances the harmonic
oscillator potentials play a more important role.
Reasonably, the trial function could then be suggested as the following:
"C exp[!(

# z)/2]exp[!a(( #z)],
(8)

where C is the normalization constant, a is only the
variational parameter, and (as the measure of the
total transverse potential) is de"ned by
" # .

(9)

Without magnetic "eld ( "0) the trial function of
Eq. (8) is exactly coincided with that used by Xiao
et al. [12] and Bose [13]. In Varshni's trial function
[15] the exponentially con"ning factor exp(! r)
was approximately replaced by a simple polynomial
with the introduction of one more variational parameter. Note, however, that in the model of Ref. [15]
besides the parabolic con"ning potential there
exists also an in"nite square potential well.
Substituting the trial function of Eq. (8) into the
Hamiltonian of Eq. (3) we obtain
1 "H " 2

"2 # !a
1 " 2

where



(a!2)I !2a I #2aI


#
, (10)
I




r dr exp(! r!2ar),


I "[ /( ! )] rL\ dr exp(! r!2ar)
L

;Er"[r( ! ], n"1, 2, 3
I"



155

with Er"(x) being the imaginary error function
[20,21].
For given values of the parameters
and ,
minimizing the energy of Eq. (10) with respect to
the variational parameter a, we will obtain the
energy E , and further, from the energy E of Eq.
!

(7) the binding energy E will be determined. Such
calculations have been performed for large ranges
of values of and , and the obtained results are
shown in the next sub-section.
2.2. Numerical results
It should be mentioned again that all the energies
as well as the lengths that appear in the results
shown below are measured in the atomic units. For

de"nition, taking GaAs as a typical QD material,
one has [22] mH"0.067 m , "12.9, a "

10.19 nm, and therefore R "5.478 meV. For

these values of material parameters, the value of the
strength "1 is corresponding to a magnetic "eld
of +6.68 ¹, and to the length ¸ "1 a .
+
While the magnetic "eld dependence of the energy E is well de"ned by Eq. (7), Fig. 1 shows how

the ground-state energy E depends on the mag!
netic "eld
for various values of the length
¸ : ¸ "1, 1.5, 2, 3, and 5. The most impressive
! !
feature found in the "gure is that with increasing
¸ the energy E at the beginning falls steeply, and
!
!
then ceases to fall further at ¸ +5 (all the curves of
!
E ( ) for ¸ '5 are indistinguishably close to that
!
!
for ¸ "5 shown in Fig. 1). This unambiguously
!
means that for spherical QDs with a parabolic con"ning potential the con"nement e!ect on the
ground-state energy becomes negligibly small, when
the dot `e!ective sizea ¸ is as large as 5a or more.

!
For any ¸ in the study the curve of E ( ) in
!
!
Fig. 1 follows the general behavior: in the limit of
small "elds E ( )J , while in the opposite limit
!
of high "elds E ( ) becomes linear to . The widths
!
of these limit regions depend on ¸ . For the case of
!
large ¸ "5 (it could be seen as the limit of weak
!
con"nement) a rough estimation gives the region
)0.5 for the weak "eld regime and the asymptotic behavior E ( )+(2/3) for the "eld depend!
ence of E in the high "eld regime.
!
In Fig. 2 the binding energy E is plotted as
a function of the "eld parameter for the same


156

V. Lien Nguyen et al. / Physica B 292 (2000) 153}159

Fig. 1. The variational ground-state energy E as a function of
!
the magnetic strength
for QDs with various con"nement
lengths ¸ : ¸ "1, 1.5, 2, 3, and 5 (from top). All the curves of

! !
¸ '5 (not shown) are practically coincided with that of
!
¸ "5.
!

the limit of weak con"nements the "eld dependence
of the binding energy should be linear, E ( )J , at
weak "elds. Such a linear region could really be
recognized in the curve with largest ¸ (¸ "10
! !
means "  in Eq. (7)) in Fig. 2. For other curves

of smaller ¸ two e!ects of con"ning potential and
!
of magnetic "eld are mixed with the totally e!ective
strength of Eq. (9) and a linear region of E ( ) at
weak "elds is no more seen.
We would mention that in the limit of zero "eld
our results of E ( ), describing the e!ect of con"ning potential alone on the binding energy, are in
very good agreement with those of Refs. [12}15].
For example, our calculations give for E the
values of 1.48946, 1.68020, and 1.84963 for "0.2,
0.3, and 0.4, respectively, while the corresponding
values obtained in Ref. [12] for the case of the
largest radius of hard boundary (R"7) are
1.49063, 1.68022, and 1.84963 [12,15]. The coincidence of two results for "0.4 certainly implies that
at such strong con"nements the distance of 7a
could be considered in"nite, and therefore, the hard
boundary located there does not yet a!ect the binding energy.

3. Perturbation calculations in the limit cases

Fig. 2. The variational binding energy E as a function of the
magnetic strength . The data are resulted from E of Eq. (7) and

E in Fig. 1 for the same values of the length ¸ : 1, 1.5, 2, 3, and
!
!
5 (from top). The lowest curve of ¸ "10 is added to show the
!
e!ect of con"ning potential on E for QDs of large ¸ .
!

values of the con"ning potential length ¸ as in
!
Fig. 1, except the lowest curve of ¸ "10. Though
!
the energy E ceases to depend on the length ¸ at
!
!
¸ *5, the energy E that resulted from Eq. (2)
!
with the term E depending on ,¸\, certainly

!
continues to decrease with increasing ¸ as shown
!
by this curve of ¸ "10.
!
Note that, as is well known for the bulk materials

[2], and as can be seen from Eqs. (2), (7), and (10), in

As was shown in the previous section, for con"ning potentials with the length ¸ '3 the e!ect of
!
con"nements on the ground-state energy E seems
!
to be very weak, and therefore one can suggest to
use a perturbation approach for calculating E in
!
this limit. Taking such an opportunity we write the
Hamiltonian of Eq. (3) in the form
H "H #H  ,



where H is the unperturbation Hamiltonian of

a hydrogen atom,
2
H "!
!

r
and the perturbation part
H  "  # r
(11)


with both e!ective con"nement and magnetic "eld
strengths and assumed to be small, ;1 and

;1.


V. Lien Nguyen et al. / Physica B 292 (2000) 153}159

The ground-state solution of the Hamiltonian
H is well known with the eigenstate and the

eigenvalue being
"(1/( )exp(!r) and E "
!1, respectively. Using these unperturbation solutions and the perturbation Hamiltonian H  of Eq.

(11) the standard and simple calculations lead to
the "rst-order approximation for the ground-state
energy
E  ,E #E  "!1#3 # /6
!
and further, from Eqs. (2), (7) the binding energy
is evaluated in the same approximation as the
following:
E  ,E !E  "1#2 # !3 ! /6,

!
(12)
where is de"ned as in Eq. (9). Thus, in the framework of the "rst-order perturbation approximation
we obtain a very simple expression for the binding
energy in the limit of weak con"nements.
In Fig. 3 the perturbation binding energy E  of
Eq. (12) is plotted (dashed lines) as a function of the
e!ective magnetic "eld strength for )0.15, and

for three values of the e!ective con"ning strength
:  ,  , and  (correspondingly, ¸ "5, 7, and
 

!
10). In this "gure the binding energies E , obtained
by the variational method in the previous section
for the same values of and are also presented for
a comparison. It is clear that even for the case of

"  two curves are very close to each other: an

estimation gives the relative di!erences between
them as (0.1%, 0.2%, 0.5%, and 1% for the "elds
"0.001, 0.05, 0.1, and 0.15, respectively. The
smaller the (weaker con"nement), the closer to
each other two corresponding curves become.
However, it should be noted that from the "eld of
+0.1, where the length 2¸ becomes smaller than
+
5, for all three cases of ¸ "5, 7, and 10 in
!
Fig. 3 the magnetic "eld potential becomes stronger
than the con"ning one in the Hamiltonian of
Eq. (3), and therefore, the relative di!erences between two results, peturbation and variational, will
be determined by the magnetic "eld rather than by
the con"ning potential that results in similar behaviors of all the curves at *0.1. It is here useful
to recall that for the GaAs-QDs the value "0.15
corresponds to a "eld of +1¹.
Thus, our calculations suggest that for QDs with

con"ning potentials of ¸ *5 the perturbation
!
method could be used for investigating the e!ect of
a magnetic "eld on the binding energy of Hydrogen
impurities at least in the range of "elds of )0.15.
Moreover, an agreement between the results, obtained by the two methods could also be seen as
a bene"t for the chosen trial function of Eq. (8).
Lastly, we would mention that Bose and Sarkar
[14] have recently used the perturbation method to
investigate the e!ect of parabolic con"ning potentials on the binding energy even in the limit of
strong con"nements. To see how two approximations, perturbation and variational, are in agreement in this limit of <1 we performed
perturbation calculations of the energy E , consid!
ering the Coulomb potential (!2/r) as a perturbation. The unperturbation Hamiltonian is then
nothing but H of Eqs. (4)}(6), and therefore, the

corresponding ground-state solution is already
known with the eigenvalue given in Eq. (7) and the
eigenfunction of the form
 "(  / )exp[!(

Fig. 3. Two binding energies E (variational, solid lines) and
E  of Eq. (12) (perturbation, dashed lines) are compared in the
weak con"nement regime, ¸ "5, 7, and 10 (from top), and in
!
weak "eld region, )0.15.

157

# z)/2].

(13)

In this case since the unperturbation energy is
exactly coincided with the energy E the bind
ing energy seems entirely to be de"ned by the
perturbation correction (with opposite signs). This
correction could be evaluated by the standard procedure of calculating perturbation energies, using


158

V. Lien Nguyen et al. / Physica B 292 (2000) 153}159

the unperturbation solution of Eqs. (7) and (13).
Thus, to the "rst-order approximation we obtain
the binding energy in the strong con"nement limit
E  "4





 exp(! r)
Er"[r( ! ].
dr
(
!

(14)

In Fig. 4, the binding energy E  of Eq. (14) for
the zero-"eld case is presented (dashed lines) as
a function of the con"ning length ¸ in comparison
!
with the corresponding variational binding energy
E , calculated in the previous section. The relative
di!erences between two energies seem to be as large
as +1.4%, 5%, 7%, and 11% for ¸ "0.1, 0.3, 0.5,
!
and 0.8, respectively. The discrepancy between two
approximations certainly increases with increasing
¸ . Note that it is not interesting to study the
!
magnetic "eld e!ect in this limit since for such
strong con"nements as those in Fig. 4 (¸ )0.8)
!
the e!ect of magnetic "elds realized usually in experiments is relatively small.
Originally, the concept of strong and weak con"nement regimes was introduced by Efros and Efros [23] in relation to the exiton problems. Ekimov
et al. [24] suggested that in calculating the exiton
binding energy in the strong con"nement regime,
the perturbation method could be used with an

Fig. 4. The zero-"eld binding energies E (variational, solid
line), and E  of Eq. (14) (perturbation, dashed line) are in
comparison plotted versus the length ¸ for the strong con"ne!
ment regime of ¸ ranging from 0.1 to 0.8.
!

error of +1% for square-well QDs of sizes as large
as 2a . This estimation, certainly, is not related to

the present problem of impurities in parabolic con"ning potentials. Our results in Fig. 4 suggest that
a perturbation approach could perhaps be used for
calculating the impurity binding energy in the regime of such strong con"nements as of ¸ (0.2.
!
Thus, while Fig. 3 could really be seen as a support for using the perturbation method as a good
approximation in the weak con"nement regime,
in the opposite regime of strong con"nements of
Fig. 4 we, however, have no such belief, except the
limit region of very small con"ning lengths.

4. Conclusion
We calculated the e!ects of both the parabolic
con"ning potential and the magnetic "eld on the
binding energy of hydrogen impurities in parabolic
con"nement QDs, using mainly a variational
method. A very simple trial function with only one
variational parameter was suggested that gives for
the binding energy a con"ning potential dependence, which agrees very well with those of previous
publications in the limit of zero "eld. In the presence of a magnetic "eld both the ground-state energy E and the binding energy E are increased,
!
but the e!ect depends on the con"nement strength:
the weaker the con"nement, the stronger the magnetic "eld e!ect. While the con"nement e!ect on
E could be considered negligibly small, when the
!
con"ning length ¸ *5, the binding energy E still
!
decreases with increasing ¸ and the magnetic "eld
!
dependence behavior of E at large ¸ "10 is
!

similar to that for the bulk semiconductors. The
perturbation calculations are also performed in the
limit regimes of weak and strong con"nements.
A good coincidence of two results, variational and
perturbation, gives a con"dence in the chosen trial
function, and therefore in the variational results
presented.
In reality, the etched GaAs/AlGaAs QDs often
have the shape of cubes. However, since the size of
cubes is always much greater than the length ¸
!
of parabolic con"ning potentials [18], the e!ect of
cubical boundaries on the binding energy E is
relatively small, and the present model of QDs with


V. Lien Nguyen et al. / Physica B 292 (2000) 153}159

parabolic con"nements should then be applied.
Concerning the impurity position, we believe that
for a given magnetic "eld the binding energy decreases as the impurity moves away from the dot
center in the same way as is shown in Ref. [13] for
the case of zero "eld. Thus, it is hoped that our
results might provide useful insights on experimental investigations of shallow donor impurities
in GaAs-type QDs and stimulate further theoretical interest in the problem.
Acknowledgements
One of the authors (NVL) thanks Professor Peter
Thomas for the kind hospitality at Physics Department, Philipps-University Marburg, where this
work was "nally completed. This work was partly
supported by the collaboration fund from the Solid

State Group of Lund University, Sweden.
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