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Author’s Accepted Manuscript Control of the binding energy by tuning the single dopant position, magnetic field strength and shell thickness in ZnS/CdSe core/shell quantum dot A. Talbi, E. Feddi, A. Zouitine, M. El Haouari, M. Zazoui, A. Oukerroum, F. Dujardin, E. Assaid, M. Addou www.elsevier.com/locate/physe
PII: DOI: Reference:
S1386-9477(16)30485-4 http://dx.doi.org/10.1016/j.physe.2016.06.028 PHYSE12503
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 24 May 2016 Revised date: 29 June 2016 Accepted date: 30 June 2016 Cite this article as: A. Talbi, E. Feddi, A. Zouitine, M. El Haouari, M. Zazoui, A. Oukerroum, F. Dujardin, E. Assaid and M. Addou, Control of the binding energy by tuning the single dopant position, magnetic field strength and shell thickness in ZnS/CdSe core/shell quantum dot, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2016.06.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Control of the binding energy by tuning the single dopant position, magnetic eld strength and shell thickness in ZnS/CdSe core/shell quantum dot. A. Talbia,b, E. Feddia,∗, A. Zouitinea, M. El Haouaria,c, M. Zazouid, A. Oukerroumd, F. Dujardine, E. Assaidf , M. Addoub Groupe d'Optoélectronique des Boites Quantiques de Semiconducteurs, Ecole Normale Supérieure de l'Enseignement Technique, Mohamed V University, Rabat, Morocco. b Laboratoire Optoélectronique et Physico-chimie des Matériaux, Faculté des Sciences, Université Ibn Tofail, Kenitra, Morocco. c Centre Régional des Métiers de l'Education et de Formation (CRMEF), Tanger, Morocco. d Laboratory of Condensed Matter, Faculty of Sciences and Techniques, University of Hassan II, Mohammedia, Morocco. e LCP-A2MC, Université de Lorraine, Metz, France. f Laboratoire d'Electronique et Optique des Nanostructures de Semiconducteurs, Faculté des Sciences, Université chouaib doukali, El Jadida, Morocco. a
Abstract Recently, the new tunable optoelectronic devices associated to the inclusion of the single dopant are in continuous emergence. Combined to other eects such as magnetic eld, geometrical con nement and dielectric discontinuity, it can constitute an approach to adjusting new transitions. In this paper, we present a theoretical investigation of magnetic eld, donor position and quantum con nement eects on the ground state binding energy of single dopant con ned in ZnS/CdSe core/shell quantum dot. Within the framework of the eective mass approximation, the Schrödinger equation has numerically solved by using the Ritz variational method under the nite potential barrier. The results show that the binding energy is very aected by the core/shell sizes and by the external magnetic eld. It has been shown that the single dopant energy transitions can be controlled by tuning the dopant position ∗
Corresponding author
Email address:
[email protected]
Preprint submitted to Elsevier
(E. Feddi)
July 1, 2016
and/or the eld strength.
Keywords:
single dopant, core-shell, quantum dots, magnetic eld, donor,
binding energy, impurity
1. Introduction Quite recently, much interest has been paid to the quantum structures with dierent shapes (wells, wires, tubes, dots, etc...). Their outstanding properties have attracted much attention from several researchers [1, 2, 3, 4, 5]. In such systems, the con nement restricts the motion of the charge carriers, enhances the orbitals overlap and quanti es their electronic states. Thus, their photoluminescence can be controlled by doping and also by tuning their shapes and sizes. The rapid advance in chemical and in growth technology in semiconductors has opened the way for new generation of coated quantum dots called core/shell quantum dots (CSQD). These manufacturing processes allowed to manipulate several combinations of II − IV , III − V or IV − V I semiconductors with dierent band gap [6, 7, 8, 9, 10]. The three possible types of CSQDs ( type I , reversed type I and type II ) are dierentiated by the band alignments of the core and shell materials. In type I structure, the band gap of the shell is bigger than that of the core and vice versa for reverse type I structures. The originality of these heterostructures is that their electronic and optical properties can be controlled by changing the nature or/and the sizes of the core or the shell. This technical process so-called wave function engineering oers great possibilities to adjust the charge carriers levels which gives rise to new intra or interband transitions leading to discover new optoelectronic devices in dierent areas, such as light emitting diodes [11, 12], lasers [13], solar cells [14, 15], and medicine [16, 17, 18, 19]. The inclusion of impurities in semiconductor nanodots enriches their properties: it increases the conductivity, aects the transport phenomena and induces considerable changes in optical properties. Their binding with 2
excitons can explain some new photoluminescence transitions such as U V line in ZnO and low-threshold lasing in ZnSe [20, 21, 22, 23]. More recently, the inclusion of a single dopant in a nanocrystal reveals that some properties appear leading to a new generation of optoelectronic devices such as single-spin devices and single-dopant transistors [24, 25, 26]. In addition their sensitivity on dierent external perturbations attracts a great number of experimental and theoretical investigations, the magnetic eld is one of these external eects which can be considered as an additive con nement leading to a shrinkage of charge carriers orbitals and reinforces their densities near the donor centers [27, 28, 29, 30, 31, 32, 33, 34]. Despite the number of the studies in this way, the control of the energy levels of single dopant by an applied magnetic eld stays a challenge for many scientist mainly in the new nanosystems as CSQD [35, 36, 37]. In our last paper [38], we investigated the eect of magnetic eld on a single dopant in a core/shell nanocrystal with a simple model taking the con nement as an in nite well potential. Some interesting conclusions have been highlighted: the single dopant binding energy is very sensitive to the magnetic eld eect and the core/shell sizes. It was also found the existence of a threshold ratio (a/b)crit which represents the limit between the tridimensional and the spherical surface con nement, and nally we have demonstrated the great in uence of the dopant position on the binding energy. In order to complete our previous work, we choose in this paper a more realistic model by considering the real core/shell band oset. We focus our study to ZnS/CdSe with reverse type-I CSQD con guration, thus we investigate the changes undergone by the binding energy due to the simultaneous eects of magnetic eld, single dopant position and the shell thickness. By using the eective-mass approximation and the Ritz variational approach, we determine the binding energy for an on-center donor in ZnS/CdSe CSQD with and without the magnetic led eect and we investigate the in uence
3
of the dopant position in this structure. The paper is organized as follows: in Section 2 we describe our theoretical formalism, the Section 3 is devoted to discuss and analyze our numerical results, and nally we report our conclusion in the last part.
2. Background Theory Let us consider a single dopant in a spherical reverse type-I CSQD composed by a core with radius a coated by a shell with radius b and immersed in a dielectric environment such as water or another substance with a large − → band gap (Fig. 1). The system is submitted to an uniform magnetic eld B. In the framework of the eective mass approximation, the Hamiltonian of a single dopant in a CSQD can be written as:
H=
1 e2 e 2 A − p − → − + Vw 2m∗ei c ε (re , D) re − D
(1)
the rst term is the kinetic energy, re denotes the electron position from the → − core center, D indicates the donor position and e is the electron charge. The
m∗ei are the electron eective masses in core or shell regions: ⎧ ⎨ m∗ 0 < re < a (core) e1 m∗ei = ⎩ m∗ a < r < b (shell).
(2)
e
e2
According to the positions of the electron and the ionized donor, the dielectric constant can take three dierent values: ⎧ ⎪ ⎪ ε1 0 < re < a and 0 < D < a ⎪ ⎨ ε (re , D) = ε2 a < re < b and a < D < b ⎪ ⎪ ⎪ √ ⎩ ε¯ = ε ε other cases
(3)
1 2
We have chosen ε¯ =
√
ε1 ε2 as an adequate dielectric constant when the
electron and the donor aren't in the same region [39, 40]. The magnetic eld → − − → − r . It is known that from the is described in the Coulomb gauge: A = 1 B × → 2
quantum mechanic point of view and using Landau or Coulomb gauges, the 4
term due to the magnetic eld is equivalent to that of an harmonic oscillator. For simplicity we suppose that the magnetic eld is applied along to the z-
0, B), B(0,
axis:
coordinates
H=−
thus the Hamiltonian (1) can be expressed in spherical
(r, θ, ϕ)
2 eB e2 B 2 2 2 e2 Δ(r, θ, ϕ)− r sin (θ)− +Vw Lz + ∗ ∗ ∗ 2mei 2mei c 8mei c2 ε (re , D) reD
∂ i ∂ϕ is the
Lz =
√
reD =
as:
z -component
r2 + D2 + 2rD cos θ
(4)
of the angular momentum operator and
is the electron-donor distance. The con ne-
ment potential of our system is chosen as follows:
Vw (r) =
⎧ ⎪ ⎪ V ⎪ ⎨ 0
0 < re < a
0 ⎪ ⎪ ⎪ ⎩ ∞
(5)
a < re < b elsewhere
We have neglected the interaction spin-magnetic eld described by the
; operator (g ∗ μB σ .B) σ
g∗
is the Landé factor,
μB
is the Bohr magneton and
is the spin of the electron. Indeed, for magnetic eld strengths less than
30
T,
the energy dierence between two spin states is very low compared
to the shift of Landau levels. In addition, the g-factor is important only for semiconductors with narrow gap and for diluted magnetic semiconductors. In the following, we use the donor units: the Bohr radius as unit of length and the eective Rydberg with
μ = m∗e1 m∗e2 /(m∗e1 +m∗e2 ).
RD = e2 2¯ εa D
aD = 2 ε¯μe2
as unit of energy,
In these units, the Hamiltonian of our system
can be written as:
H=
⎧ ⎪ ⎨
1 1+η
⎪ ⎩
η 1+η
−Δ − γLz + 14 γ 2 r2 sin2 θ − 2 ¯1 + Vw ; 0 < re < a |r−D|
−Δ − γLz + 14 γ 2 r2 sin2 θ − 2 ¯2 + Vw ; a < re < b |r−D|
(6) We have introduced the ratio eter
γ =
ωc 2RD ,where
η = m∗e1 /m∗e2
ωc = eBμc
and the dimensionless param-
is the eective cyclotron frequency. The
magnetic eld strength in Tesla can be obtained by using the following transformation [29]:
B(T ) = (μ/me ) μ−1 B RD γ , where μB = e/2me c is the Bohr
magneton and
me
is the mass of the free electron. 5
In the absence of impurity center and magnetic eld, the wave function has the well known form:
Ψ(r, θ, ϕ) = Rn,l (re ) Yl,m (θe , ϕe )
is the radial part and
Yl,m (θe , ϕe )
pal quantum number,
l
and
m
where
is the spherical harmonic,
n
Rn,l (re )
is the princi-
are respectively the azimuthal and magnetic
quantum numbers. In this work, we restrict our study to the ground state energy described by:
n = 1, l = 0
and
m = 0
so that the Zeeman eect
doesn't exist. The existence of the electron in the core or shell regions depends on its energy
Ee1s
and on the core/shell sizes.
Indeed two possible cases can
be considered: the rst one corresponds to lower than a critical value
Rc
(obtained for
Ee1s < V0 , Ee1s = V0 )
the core radius
a
is
and consequently the
electron is located in the shell region, so we have:
⎧ ⎨ A1 sinh(k1 re ) re Ψe (re ) = ⎩ A sin(k2 (re −b)) 2
where
re
k1 = (1 + η) (V0 − Ee1s ) /RD
second case correspond to critical value
Rc ,
Ee1s > V0
when the core radius
⎧ ⎨ A3 sin(k3 re ) re Ψe (re ) = ⎩ A sin(k4 (re −b)) re
k3 = (1 + η) (Ee1s − V0 ) /RD
each case
Ai
(7)
a < re < b
k2 = (1 + η) Ee1s /(ηRD ). a
The
is bigger than a
then the wave function reads:
4
where
and
0 < re < a
and
0 < re < a
(8)
a < re < b
k4 = (1 + η) Ee1s /(ηRD ).
In
is a normalization constant which can be determined via the
normalization condition
ψ| ψ = 1.
The ground state energy
Ee1s
is determined by solving the transcendental
equation which is obtained via the continuity conditions and the probability current densities at the core surface:
[ΨCore (re )]re =a = [ΨShell (re )]re =a and
1 ΨCore (re ) m∗e1 dre
re =a
1 ΨShell (re ) = m∗e2 dre 6
(9)
(10)
re =a
The ground state energy of donor ED and the corresponding envelope wave function are solutions of the Schrödinger equation: 0
(11)
HΨ(r, θ, ϕ) = ED0 (γ)Ψ(r, θ, ϕ)
This eigenvalue equation has no analytical solution, then we use a variational method. The trial wave function is chosen by following the second perturbation procedure which gives a good approach to treat the problems containing more than one perturbation [38, 41, 42, 29], than we can write:
1 Ψ(r, θ, ϕ) = Ψe (re ) exp(−αreD ) exp − γr2 sin2 (θ) 4
(12)
is the one electron wave function given in Eq.(7) and Eq.(8), the rst exponential term exp(−αreD ) describes the electron-impurity correlation where α is the variational parameter and exp(− 14 γr2 sin2 θ) denotes the magnetic eld contribution on the free electron wave function. The impurity ground state energy ED (γ) can be obtained by minimizing the expectation value of H with respect to the variational parameter α: Ψe (re )
0
ED0 (γ) = min α
Ψ |H| Ψ Ψ |Ψ
(13)
As it was highlighted by Szafran et al. [43] there are two possibilities to describe the binding energy, the rst one corresponds to the interplay between the coulombic interaction and the quantum con nement. It represents the energy required to liberate the electron and move it to lowest state, it is given by: Eb = Ee (γ) − ED (γ) (14) 0
where Ee (γ) is the electron energy in the presence of magnetic eld written as: Ee (γ) = Ee1s + γ (15) The second de nition is Wb = V0 − ED (γ), which represents the energy corresponding to the dissociation process. In this study we limit ourselves to the de nition given by the relation (14). 0
7
3. Results and discussion In what follows, we will focalize on the in uence of core/shell sizes, donor position and magnetic eld strength on the single dopant fundamental state in
(ZnS/CdSe)
inverted type
materials are listed in Table
I
CSQD. The physical parameters of these
1.
m∗e /m0
ε/ε0
V0 (meV )
aD (nm)
RD (meV )
CdSe[10]
0.13
9.4
0
5.45
14.43
ZnS [10]
0.28
8.9
900
Table
2 ε¯μe2 ,
1.
Material parameters of
where
ZnS/CdSe, RD = e2 2¯ εa D , aD =
μ = m∗e1 m∗e2 /(m∗e1 + m∗e2 ).
We begin our discussion by investigating the variation of the binding energy at zero eld
(γ = 0)
versus the core/shell sizes given by the ratio
for three values of outer radius
D=
b
and for two donor positions (D
=0
a/b and
a+b 2 ). According to the Fig. 2 we can remark that when the dopant is
(D = 0)
located on the center of the structure
as the outer radius
the binding energy increases
b decreases whatever the ratio a/b, this is due to the eect
of geometrical con nement. The shape of this decrease is more pronounced in the strong con nement case
b,
(b = 1).
the binding energy decreases as
a/b
In addition, for a given outer radius increases, reaches a minimum and
then it increases. This minimum shifts to high values of radius
b
increases. The ratio
a/b
a/b
as the outer
corresponding to the energy minimum is
considered as a limit that separates
3D-spherical
geometry from a hollow
sphere considered as a narrow well [38, 44, 45]. Beyond this minimum, the impurity tends to increase its binding energy until a critical radius
Rc
from
which the electron leaves the shell to the core. As it can be seen, when the core radius
a tends to the threshold Rc , rad-
ical changes in the behavior of the binding energy occur: it undergoes a low decay and increases abruptly to its maximum value and than stabilize with a small variation. This behavior can be explained by the fact that around 8
Rc ,
the extension of the wave function in the core region causes a decrease
of the electron energy and consequently decreases the binding energy of the dopant.
By moving away from
Rc
value, the electron is more localized in
Eb
the core region near to the donor and
show that for the strong con nement aected in vicinity of
Rc ,
increases considerably. Our results
(b ≤ aD )
the binding energy is less
this is due to the fact that in strong con nement,
we have almost no dierence between the core and the shell con nement and the electron does not undergo much changes in its passage between the two regions. It is also important to underline that the threshold ing to
Ee1s = V0
correspond-
increases with increasing shell sizes because for large shell
sizes the electron needs more energy to exit the well. values of the pairs
Table
Rc
Table 2 gives some
(b, Rc ).
Shell size (aD )
0.5
1
2
3
4
Rc
0.301
0.824
1.830
2.832
3.833
2.
Calculated critical radius
Rc
of some shell sizes
b.
Let us discuss the in uence of the dopant position on the binding energy by considering arbitrarily that the ionized donor is located in the middle of the shell
(D = (a + b)/2).
The variations of the binding energy of an
o-center donor versus the ratio esting conclusions: the on-center donor
a/b
presented in Fig. 3 reveals some inter-
a reverse behavior is obtained compared to the case of
(D = 0) .
For a given
a/b
the geometrical con nement
eect leads to an increasing of the binding energy. For the weak con nement
b = 3aD
(
and
sponding to
Rc ,
2aD ),
the binding energy increases to a maximum corre-
then it decreases drastically and stabilizes when
However, for the strong con nement
b 1aD ,
a/b → 1.
the binding energy presents
another behavior, it decreases until a minimum, then it increases up to its maximum and decreases signi cantly weaker than the weak con nement case. We note that the minimum of binding energy is less remarkable for the weak con nement and it shifts toward the small values of
9
a/b
when the shell size
increases and vanishes for
b ≥ 3aD .
We note that the sharp drop of the
binding energy is due to the separation of the electron and the donor because after the critical radius the electron moves to the core region while the donor stays in the shell, so the electron feels less the eect of the Coulombic potential. In an eort to explain these behaviors of the binding energy and understand the underlying physics related to the magnetic eld in uence, we present in Fig. sus
a/b
with
(γ = 0, 0.5
4a and Fig.
b = 2aD
and
1)
4b the variations of the binding energies ver-
for dierent values of the magnetic eld strength
respectively for an on-center and o-center donor (note
that here, given the parameters of materials, we have the relation
22.1γ ).
B(T ) =
We can see in both cases that the binding energy increases as
increases whatever the ratio
a/b
γ
value, the magnetic eld diminishes the cy-
clotron radius leading to a shrinkage of the donor orbitals which increases the binding energy. To more clarify the magnetic eld eect on the binding energy, we analyze the shift center
ΔEb = Eb (γ) − Eb (γ = 0)
(D = 0)
and o-center
as function of ratio
(D = (a + b)/2)
a/b
donor. The Fig. 5 shows the
existence of two totally dierent tendencies: for the
D=0
case, it can be
seen that the impact of magnetic eld decreases as the ratio reaches its minimum at
Rc
for an on-
and increases quickly.
a/b
increases,
We can notice that the
magnetic eld eect is very weak around the critical radius
Rc
because the
electron and the donor are separated (the electron in the shell and the donor in the core).
On the other hand, for
D = (a + b)/2
that the magnetic eld eect increases as the ratio its maximum at
Rc
case, it can be seen
a/b
decreases, reaches
and decreases rapidly, we can remark that the eect of
magnetic eld is more remarkable when
a
tends to
Rc
because the electron
is more localized near to the donor. To show the possibility to control the energy transitions by tuning the
10
single dopant position and the magnetic strength, we have investigated the in uence of the ionized donor localization on the binding energy. In Fig. 6a and Fig. 6b we give the variation of the binding energy as function of the donor position
D for dierent magnetic eld strengths (γ = 0, 0.5 and 1) and
for the two sensible regions separated by the core critical value corresponding to the outer radius case where
a = 0.5aD < Rc .
b = 2aD .
Rc = 1.82aD
In Fig. 6a we analyze the rst
In this situation the electron is localized in the
shell. We can remark that the maximum of
Eb
is obtained when the ionized
donor is located in the shell region and precisely at the distance
(a + b) /2,
from this value, the binding energy undergoes a rapid decay and reaches its lowest value when the donor is located on the surface of the shell
(D = b) .
On the other hand, for a given impurity position the binding energy increases with increasing magnetic eld strength. This shift is more important when the dopant is placed in the core region. Fig. 6b gives our results for the second case when
a = 1.9aD > Rc,
in the core region.
in this situation the electron is localized
We can separate two regions: when
D < (a + b) /2,
the binding energy decreases slowly whatever the magnetic strength, while when
D > (a + b) /2
minimum of
Eb
we observe a rapid decay of the binding energy. The
is obtained when the impurity is located at the shell edge.
4. Conclusion This paper has clearly shown the importance of external magnetic eld on single dopant in spherical core/shell quantum dots. Dierent parameters were taken in consideration: core/shell sizes, magnetic eld strength and single dopant position. Our calculations have been done in the framework of the eective mass approximation using a variational-perturbative method and by considering a nite potential barrier. It has been found a remarkable sensitivity of binding energy on the core radii and shell thickness. We have also demonstrated the existence of a critical radius depending on the CSQD
11
Figure 1: Fig. 1. Schematic representation of a core/shell quantum dot and corresponding potential energies. size and on the material band oset. Our numerical calculation shows the possibility to control the energy transitions of the single dopant by tuning its position and the magnetic strength or tuning the core or/and the shell sizes. We hope that these interesting results will be useful to understand the behaviors related to magnetic eld perturbation in core/shell structures.
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Figure 2: Fig. 2. Binding energy of an on-center donor as function of the ratio a/b without magnetic eld and for three values of shell size b = 1, 2 and 3aD .
Figure 3: Fig. 3. Binding energy of an o-center donor as function of the ratio a/b without magnetic eld and for three values of shell size b = 1, 2 and 3aD .
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PII: DOI: Reference:
S1386-9477(16)30485-4 http://dx.doi.org/10.1016/j.physe.2016.06.028 PHYSE12503
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 24 May 2016 Revised date: 29 June 2016 Accepted date: 30 June 2016 Cite this article as: A. Talbi, E. Feddi, A. Zouitine, M. El Haouari, M. Zazoui, A. Oukerroum, F. Dujardin, E. Assaid and M. Addou, Control of the binding energy by tuning the single dopant position, magnetic field strength and shell thickness in ZnS/CdSe core/shell quantum dot, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2016.06.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Control of the binding energy by tuning the single dopant position, magnetic eld strength and shell thickness in ZnS/CdSe core/shell quantum dot. A. Talbia,b, E. Feddia,∗, A. Zouitinea, M. El Haouaria,c, M. Zazouid, A. Oukerroumd, F. Dujardine, E. Assaidf , M. Addoub Groupe d'Optoélectronique des Boites Quantiques de Semiconducteurs, Ecole Normale Supérieure de l'Enseignement Technique, Mohamed V University, Rabat, Morocco. b Laboratoire Optoélectronique et Physico-chimie des Matériaux, Faculté des Sciences, Université Ibn Tofail, Kenitra, Morocco. c Centre Régional des Métiers de l'Education et de Formation (CRMEF), Tanger, Morocco. d Laboratory of Condensed Matter, Faculty of Sciences and Techniques, University of Hassan II, Mohammedia, Morocco. e LCP-A2MC, Université de Lorraine, Metz, France. f Laboratoire d'Electronique et Optique des Nanostructures de Semiconducteurs, Faculté des Sciences, Université chouaib doukali, El Jadida, Morocco. a
Abstract Recently, the new tunable optoelectronic devices associated to the inclusion of the single dopant are in continuous emergence. Combined to other eects such as magnetic eld, geometrical con nement and dielectric discontinuity, it can constitute an approach to adjusting new transitions. In this paper, we present a theoretical investigation of magnetic eld, donor position and quantum con nement eects on the ground state binding energy of single dopant con ned in ZnS/CdSe core/shell quantum dot. Within the framework of the eective mass approximation, the Schrödinger equation has numerically solved by using the Ritz variational method under the nite potential barrier. The results show that the binding energy is very aected by the core/shell sizes and by the external magnetic eld. It has been shown that the single dopant energy transitions can be controlled by tuning the dopant position ∗
Corresponding author
Email address:
[email protected]
Preprint submitted to Elsevier
(E. Feddi)
July 1, 2016
and/or the eld strength.
Keywords:
single dopant, core-shell, quantum dots, magnetic eld, donor,
binding energy, impurity
1. Introduction Quite recently, much interest has been paid to the quantum structures with dierent shapes (wells, wires, tubes, dots, etc...). Their outstanding properties have attracted much attention from several researchers [1, 2, 3, 4, 5]. In such systems, the con nement restricts the motion of the charge carriers, enhances the orbitals overlap and quanti es their electronic states. Thus, their photoluminescence can be controlled by doping and also by tuning their shapes and sizes. The rapid advance in chemical and in growth technology in semiconductors has opened the way for new generation of coated quantum dots called core/shell quantum dots (CSQD). These manufacturing processes allowed to manipulate several combinations of II − IV , III − V or IV − V I semiconductors with dierent band gap [6, 7, 8, 9, 10]. The three possible types of CSQDs ( type I , reversed type I and type II ) are dierentiated by the band alignments of the core and shell materials. In type I structure, the band gap of the shell is bigger than that of the core and vice versa for reverse type I structures. The originality of these heterostructures is that their electronic and optical properties can be controlled by changing the nature or/and the sizes of the core or the shell. This technical process so-called wave function engineering oers great possibilities to adjust the charge carriers levels which gives rise to new intra or interband transitions leading to discover new optoelectronic devices in dierent areas, such as light emitting diodes [11, 12], lasers [13], solar cells [14, 15], and medicine [16, 17, 18, 19]. The inclusion of impurities in semiconductor nanodots enriches their properties: it increases the conductivity, aects the transport phenomena and induces considerable changes in optical properties. Their binding with 2
excitons can explain some new photoluminescence transitions such as U V line in ZnO and low-threshold lasing in ZnSe [20, 21, 22, 23]. More recently, the inclusion of a single dopant in a nanocrystal reveals that some properties appear leading to a new generation of optoelectronic devices such as single-spin devices and single-dopant transistors [24, 25, 26]. In addition their sensitivity on dierent external perturbations attracts a great number of experimental and theoretical investigations, the magnetic eld is one of these external eects which can be considered as an additive con nement leading to a shrinkage of charge carriers orbitals and reinforces their densities near the donor centers [27, 28, 29, 30, 31, 32, 33, 34]. Despite the number of the studies in this way, the control of the energy levels of single dopant by an applied magnetic eld stays a challenge for many scientist mainly in the new nanosystems as CSQD [35, 36, 37]. In our last paper [38], we investigated the eect of magnetic eld on a single dopant in a core/shell nanocrystal with a simple model taking the con nement as an in nite well potential. Some interesting conclusions have been highlighted: the single dopant binding energy is very sensitive to the magnetic eld eect and the core/shell sizes. It was also found the existence of a threshold ratio (a/b)crit which represents the limit between the tridimensional and the spherical surface con nement, and nally we have demonstrated the great in uence of the dopant position on the binding energy. In order to complete our previous work, we choose in this paper a more realistic model by considering the real core/shell band oset. We focus our study to ZnS/CdSe with reverse type-I CSQD con guration, thus we investigate the changes undergone by the binding energy due to the simultaneous eects of magnetic eld, single dopant position and the shell thickness. By using the eective-mass approximation and the Ritz variational approach, we determine the binding energy for an on-center donor in ZnS/CdSe CSQD with and without the magnetic led eect and we investigate the in uence
3
of the dopant position in this structure. The paper is organized as follows: in Section 2 we describe our theoretical formalism, the Section 3 is devoted to discuss and analyze our numerical results, and nally we report our conclusion in the last part.
2. Background Theory Let us consider a single dopant in a spherical reverse type-I CSQD composed by a core with radius a coated by a shell with radius b and immersed in a dielectric environment such as water or another substance with a large − → band gap (Fig. 1). The system is submitted to an uniform magnetic eld B. In the framework of the eective mass approximation, the Hamiltonian of a single dopant in a CSQD can be written as:
H=
1 e2 e 2 A − p − → − + Vw 2m∗ei c ε (re , D) re − D
(1)
the rst term is the kinetic energy, re denotes the electron position from the → − core center, D indicates the donor position and e is the electron charge. The
m∗ei are the electron eective masses in core or shell regions: ⎧ ⎨ m∗ 0 < re < a (core) e1 m∗ei = ⎩ m∗ a < r < b (shell).
(2)
e
e2
According to the positions of the electron and the ionized donor, the dielectric constant can take three dierent values: ⎧ ⎪ ⎪ ε1 0 < re < a and 0 < D < a ⎪ ⎨ ε (re , D) = ε2 a < re < b and a < D < b ⎪ ⎪ ⎪ √ ⎩ ε¯ = ε ε other cases
(3)
1 2
We have chosen ε¯ =
√
ε1 ε2 as an adequate dielectric constant when the
electron and the donor aren't in the same region [39, 40]. The magnetic eld → − − → − r . It is known that from the is described in the Coulomb gauge: A = 1 B × → 2
quantum mechanic point of view and using Landau or Coulomb gauges, the 4
term due to the magnetic eld is equivalent to that of an harmonic oscillator. For simplicity we suppose that the magnetic eld is applied along to the z-
0, B), B(0,
axis:
coordinates
H=−
thus the Hamiltonian (1) can be expressed in spherical
(r, θ, ϕ)
2 eB e2 B 2 2 2 e2 Δ(r, θ, ϕ)− r sin (θ)− +Vw Lz + ∗ ∗ ∗ 2mei 2mei c 8mei c2 ε (re , D) reD
∂ i ∂ϕ is the
Lz =
√
reD =
as:
z -component
r2 + D2 + 2rD cos θ
(4)
of the angular momentum operator and
is the electron-donor distance. The con ne-
ment potential of our system is chosen as follows:
Vw (r) =
⎧ ⎪ ⎪ V ⎪ ⎨ 0
0 < re < a
0 ⎪ ⎪ ⎪ ⎩ ∞
(5)
a < re < b elsewhere
We have neglected the interaction spin-magnetic eld described by the
; operator (g ∗ μB σ .B) σ
g∗
is the Landé factor,
μB
is the Bohr magneton and
is the spin of the electron. Indeed, for magnetic eld strengths less than
30
T,
the energy dierence between two spin states is very low compared
to the shift of Landau levels. In addition, the g-factor is important only for semiconductors with narrow gap and for diluted magnetic semiconductors. In the following, we use the donor units: the Bohr radius as unit of length and the eective Rydberg with
μ = m∗e1 m∗e2 /(m∗e1 +m∗e2 ).
RD = e2 2¯ εa D
aD = 2 ε¯μe2
as unit of energy,
In these units, the Hamiltonian of our system
can be written as:
H=
⎧ ⎪ ⎨
1 1+η
⎪ ⎩
η 1+η
−Δ − γLz + 14 γ 2 r2 sin2 θ − 2 ¯1 + Vw ; 0 < re < a |r−D|
−Δ − γLz + 14 γ 2 r2 sin2 θ − 2 ¯2 + Vw ; a < re < b |r−D|
(6) We have introduced the ratio eter
γ =
ωc 2RD ,where
η = m∗e1 /m∗e2
ωc = eBμc
and the dimensionless param-
is the eective cyclotron frequency. The
magnetic eld strength in Tesla can be obtained by using the following transformation [29]:
B(T ) = (μ/me ) μ−1 B RD γ , where μB = e/2me c is the Bohr
magneton and
me
is the mass of the free electron. 5
In the absence of impurity center and magnetic eld, the wave function has the well known form:
Ψ(r, θ, ϕ) = Rn,l (re ) Yl,m (θe , ϕe )
is the radial part and
Yl,m (θe , ϕe )
pal quantum number,
l
and
m
where
is the spherical harmonic,
n
Rn,l (re )
is the princi-
are respectively the azimuthal and magnetic
quantum numbers. In this work, we restrict our study to the ground state energy described by:
n = 1, l = 0
and
m = 0
so that the Zeeman eect
doesn't exist. The existence of the electron in the core or shell regions depends on its energy
Ee1s
and on the core/shell sizes.
Indeed two possible cases can
be considered: the rst one corresponds to lower than a critical value
Rc
(obtained for
Ee1s < V0 , Ee1s = V0 )
the core radius
a
is
and consequently the
electron is located in the shell region, so we have:
⎧ ⎨ A1 sinh(k1 re ) re Ψe (re ) = ⎩ A sin(k2 (re −b)) 2
where
re
k1 = (1 + η) (V0 − Ee1s ) /RD
second case correspond to critical value
Rc ,
Ee1s > V0
when the core radius
⎧ ⎨ A3 sin(k3 re ) re Ψe (re ) = ⎩ A sin(k4 (re −b)) re
k3 = (1 + η) (Ee1s − V0 ) /RD
each case
Ai
(7)
a < re < b
k2 = (1 + η) Ee1s /(ηRD ). a
The
is bigger than a
then the wave function reads:
4
where
and
0 < re < a
and
0 < re < a
(8)
a < re < b
k4 = (1 + η) Ee1s /(ηRD ).
In
is a normalization constant which can be determined via the
normalization condition
ψ| ψ = 1.
The ground state energy
Ee1s
is determined by solving the transcendental
equation which is obtained via the continuity conditions and the probability current densities at the core surface:
[ΨCore (re )]re =a = [ΨShell (re )]re =a and
1 ΨCore (re ) m∗e1 dre
re =a
1 ΨShell (re ) = m∗e2 dre 6
(9)
(10)
re =a
The ground state energy of donor ED and the corresponding envelope wave function are solutions of the Schrödinger equation: 0
(11)
HΨ(r, θ, ϕ) = ED0 (γ)Ψ(r, θ, ϕ)
This eigenvalue equation has no analytical solution, then we use a variational method. The trial wave function is chosen by following the second perturbation procedure which gives a good approach to treat the problems containing more than one perturbation [38, 41, 42, 29], than we can write:
1 Ψ(r, θ, ϕ) = Ψe (re ) exp(−αreD ) exp − γr2 sin2 (θ) 4
(12)
is the one electron wave function given in Eq.(7) and Eq.(8), the rst exponential term exp(−αreD ) describes the electron-impurity correlation where α is the variational parameter and exp(− 14 γr2 sin2 θ) denotes the magnetic eld contribution on the free electron wave function. The impurity ground state energy ED (γ) can be obtained by minimizing the expectation value of H with respect to the variational parameter α: Ψe (re )
0
ED0 (γ) = min α
Ψ |H| Ψ Ψ |Ψ
(13)
As it was highlighted by Szafran et al. [43] there are two possibilities to describe the binding energy, the rst one corresponds to the interplay between the coulombic interaction and the quantum con nement. It represents the energy required to liberate the electron and move it to lowest state, it is given by: Eb = Ee (γ) − ED (γ) (14) 0
where Ee (γ) is the electron energy in the presence of magnetic eld written as: Ee (γ) = Ee1s + γ (15) The second de nition is Wb = V0 − ED (γ), which represents the energy corresponding to the dissociation process. In this study we limit ourselves to the de nition given by the relation (14). 0
7
3. Results and discussion In what follows, we will focalize on the in uence of core/shell sizes, donor position and magnetic eld strength on the single dopant fundamental state in
(ZnS/CdSe)
inverted type
materials are listed in Table
I
CSQD. The physical parameters of these
1.
m∗e /m0
ε/ε0
V0 (meV )
aD (nm)
RD (meV )
CdSe[10]
0.13
9.4
0
5.45
14.43
ZnS [10]
0.28
8.9
900
Table
2 ε¯μe2 ,
1.
Material parameters of
where
ZnS/CdSe, RD = e2 2¯ εa D , aD =
μ = m∗e1 m∗e2 /(m∗e1 + m∗e2 ).
We begin our discussion by investigating the variation of the binding energy at zero eld
(γ = 0)
versus the core/shell sizes given by the ratio
for three values of outer radius
D=
b
and for two donor positions (D
=0
a/b and
a+b 2 ). According to the Fig. 2 we can remark that when the dopant is
(D = 0)
located on the center of the structure
as the outer radius
the binding energy increases
b decreases whatever the ratio a/b, this is due to the eect
of geometrical con nement. The shape of this decrease is more pronounced in the strong con nement case
b,
(b = 1).
the binding energy decreases as
a/b
In addition, for a given outer radius increases, reaches a minimum and
then it increases. This minimum shifts to high values of radius
b
increases. The ratio
a/b
a/b
as the outer
corresponding to the energy minimum is
considered as a limit that separates
3D-spherical
geometry from a hollow
sphere considered as a narrow well [38, 44, 45]. Beyond this minimum, the impurity tends to increase its binding energy until a critical radius
Rc
from
which the electron leaves the shell to the core. As it can be seen, when the core radius
a tends to the threshold Rc , rad-
ical changes in the behavior of the binding energy occur: it undergoes a low decay and increases abruptly to its maximum value and than stabilize with a small variation. This behavior can be explained by the fact that around 8
Rc ,
the extension of the wave function in the core region causes a decrease
of the electron energy and consequently decreases the binding energy of the dopant.
By moving away from
Rc
value, the electron is more localized in
Eb
the core region near to the donor and
show that for the strong con nement aected in vicinity of
Rc ,
increases considerably. Our results
(b ≤ aD )
the binding energy is less
this is due to the fact that in strong con nement,
we have almost no dierence between the core and the shell con nement and the electron does not undergo much changes in its passage between the two regions. It is also important to underline that the threshold ing to
Ee1s = V0
correspond-
increases with increasing shell sizes because for large shell
sizes the electron needs more energy to exit the well. values of the pairs
Table
Rc
Table 2 gives some
(b, Rc ).
Shell size (aD )
0.5
1
2
3
4
Rc
0.301
0.824
1.830
2.832
3.833
2.
Calculated critical radius
Rc
of some shell sizes
b.
Let us discuss the in uence of the dopant position on the binding energy by considering arbitrarily that the ionized donor is located in the middle of the shell
(D = (a + b)/2).
The variations of the binding energy of an
o-center donor versus the ratio esting conclusions: the on-center donor
a/b
presented in Fig. 3 reveals some inter-
a reverse behavior is obtained compared to the case of
(D = 0) .
For a given
a/b
the geometrical con nement
eect leads to an increasing of the binding energy. For the weak con nement
b = 3aD
(
and
sponding to
Rc ,
2aD ),
the binding energy increases to a maximum corre-
then it decreases drastically and stabilizes when
However, for the strong con nement
b 1aD ,
a/b → 1.
the binding energy presents
another behavior, it decreases until a minimum, then it increases up to its maximum and decreases signi cantly weaker than the weak con nement case. We note that the minimum of binding energy is less remarkable for the weak con nement and it shifts toward the small values of
9
a/b
when the shell size
increases and vanishes for
b ≥ 3aD .
We note that the sharp drop of the
binding energy is due to the separation of the electron and the donor because after the critical radius the electron moves to the core region while the donor stays in the shell, so the electron feels less the eect of the Coulombic potential. In an eort to explain these behaviors of the binding energy and understand the underlying physics related to the magnetic eld in uence, we present in Fig. sus
a/b
with
(γ = 0, 0.5
4a and Fig.
b = 2aD
and
1)
4b the variations of the binding energies ver-
for dierent values of the magnetic eld strength
respectively for an on-center and o-center donor (note
that here, given the parameters of materials, we have the relation
22.1γ ).
B(T ) =
We can see in both cases that the binding energy increases as
increases whatever the ratio
a/b
γ
value, the magnetic eld diminishes the cy-
clotron radius leading to a shrinkage of the donor orbitals which increases the binding energy. To more clarify the magnetic eld eect on the binding energy, we analyze the shift center
ΔEb = Eb (γ) − Eb (γ = 0)
(D = 0)
and o-center
as function of ratio
(D = (a + b)/2)
a/b
donor. The Fig. 5 shows the
existence of two totally dierent tendencies: for the
D=0
case, it can be
seen that the impact of magnetic eld decreases as the ratio reaches its minimum at
Rc
for an on-
and increases quickly.
a/b
increases,
We can notice that the
magnetic eld eect is very weak around the critical radius
Rc
because the
electron and the donor are separated (the electron in the shell and the donor in the core).
On the other hand, for
D = (a + b)/2
that the magnetic eld eect increases as the ratio its maximum at
Rc
case, it can be seen
a/b
decreases, reaches
and decreases rapidly, we can remark that the eect of
magnetic eld is more remarkable when
a
tends to
Rc
because the electron
is more localized near to the donor. To show the possibility to control the energy transitions by tuning the
10
single dopant position and the magnetic strength, we have investigated the in uence of the ionized donor localization on the binding energy. In Fig. 6a and Fig. 6b we give the variation of the binding energy as function of the donor position
D for dierent magnetic eld strengths (γ = 0, 0.5 and 1) and
for the two sensible regions separated by the core critical value corresponding to the outer radius case where
a = 0.5aD < Rc .
b = 2aD .
Rc = 1.82aD
In Fig. 6a we analyze the rst
In this situation the electron is localized in the
shell. We can remark that the maximum of
Eb
is obtained when the ionized
donor is located in the shell region and precisely at the distance
(a + b) /2,
from this value, the binding energy undergoes a rapid decay and reaches its lowest value when the donor is located on the surface of the shell
(D = b) .
On the other hand, for a given impurity position the binding energy increases with increasing magnetic eld strength. This shift is more important when the dopant is placed in the core region. Fig. 6b gives our results for the second case when
a = 1.9aD > Rc,
in the core region.
in this situation the electron is localized
We can separate two regions: when
D < (a + b) /2,
the binding energy decreases slowly whatever the magnetic strength, while when
D > (a + b) /2
minimum of
Eb
we observe a rapid decay of the binding energy. The
is obtained when the impurity is located at the shell edge.
4. Conclusion This paper has clearly shown the importance of external magnetic eld on single dopant in spherical core/shell quantum dots. Dierent parameters were taken in consideration: core/shell sizes, magnetic eld strength and single dopant position. Our calculations have been done in the framework of the eective mass approximation using a variational-perturbative method and by considering a nite potential barrier. It has been found a remarkable sensitivity of binding energy on the core radii and shell thickness. We have also demonstrated the existence of a critical radius depending on the CSQD
11
Figure 1: Fig. 1. Schematic representation of a core/shell quantum dot and corresponding potential energies. size and on the material band oset. Our numerical calculation shows the possibility to control the energy transitions of the single dopant by tuning its position and the magnetic strength or tuning the core or/and the shell sizes. We hope that these interesting results will be useful to understand the behaviors related to magnetic eld perturbation in core/shell structures.
5. Figures captions 6. References [1] D. Bimberg, M. Grundmann, N. N. Ledentsov, Quantum Dot Heterostructures, Wiley, Chichester, New York (1999). [2] A.D. Yoe, Adv. Phys.
42, 173 (1993).
[3] A. P. Alivisatos, Science, New Series,
271, 5251, 933-937 (1996).
[4] P. Harrison, Quantum Wells, Wires and Dots, Wiley, NY (2006).
12
Figure 2: Fig. 2. Binding energy of an on-center donor as function of the ratio a/b without magnetic eld and for three values of shell size b = 1, 2 and 3aD .
Figure 3: Fig. 3. Binding energy of an o-center donor as function of the ratio a/b without magnetic eld and for three values of shell size b = 1, 2 and 3aD .
13
Figure 4: Fig. 4a. Binding energy of an on-center donor versus the ratio a/b for b = 2aD and for three values of magnetic eld strength γ = 0, 0.5 and 1.Fig. 4b. Binding energy of an o-center donor versus the ratio a/b for b = 2aD and for three values of magnetic eld strength γ = 0, 0.5 and 1.
Figure 5: Fig. 5. The shift ΔEb = Eb (γ = 0.5) − Eb (γ = 0) as function of the ratio a/b for both on-center and o-center donor and for a = 0.5aD and b = 2aD . 14
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