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Journal Pre-proofs Dielectric environment effect on linear and nonlinear optical properties for CdS/ ZnS core/shell quantum dots N. Zeiri, A. Naifar, S. Abdi-Ben Nasrallah, M. Said PII: DOI: Reference:
S2211-3797(19)32243-0 https://doi.org/10.1016/j.rinp.2019.102661 RINP 102661
To appear in:
Results in Physics
Received Date: Accepted Date:
26 July 2019 12 September 2019
Please cite this article as: Zeiri, N., Naifar, A., Abdi-Ben Nasrallah, S., Said, M., Dielectric environment effect on linear and nonlinear optical properties for CdS/ZnS core/shell quantum dots, Results in Physics (2019), doi: https:// doi.org/10.1016/j.rinp.2019.102661
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Dielectric environment effect on linear and nonlinear optical properties for CdS/ZnS core/shell quantum dots N. Zeiri*(1), A. Naifar (1), S. Abdi-Ben Nasrallah(2), M. Said(1) *Corresponding author: N. Zeiri, E-mail address: [email protected] Tel: +216 73382656 Fax: +216 73382658 (1)Laboratoire
de la Matière Condensée et des Nanosciences (LMCN)
Département de Physique, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia (2)
Laboratoire d’Etudes des Systèmes Thermique et Energétique (LESTE), Université de Monastir
Abstract: The eigenvalues, transition energy, the linear and nonlinear dielectric functions have been numerically investigated for CdS/ZnS spherical core/shell quantum dots embedded in various dielectric matrices. The evaluation was carried out for three commonly used matrices, such as PVA, PMMA and SiO2. Calculations were done under the effective mass approximation and compact density matrix approach. Results revealed that the nonlinear optical property is strongly affected by the nature of the matrix material. By increasing core/shell radii ratio the energy states as well as the transition energy are decreasing. It is also indicated that the presence of the dielectric mismatch in the QD-matrix system can cause significant enhancement on the linear and third order nonlinear dielectric function.
Keywords: Nonlinear optic, Effective mass approximation, Core/shell, Density matrix approach, Complex dielectric function. Introduction Quantum dots (QDs) belong to the most investigated objects of nanophysics due to their wide potential applications in optoelectronic and photonic devices [1-8]. If one quantum dot is covered by a layer of another semiconductor, the resulting system is called core/shell quantum dot (CSQD). Spherical (CSQDs) have received a huge attention owing to their interesting
physical and chemical properties. Nowadays, progress in experimental techniques of growth has made it possible to synthesize high quality semiconductor QDs within a large range of sizes from spherical [9-11], pyramidal [12, 13] to cylindrical [14-16] and other shapes. The greatness of such nano-system is behind the development of lasers [17-19], light emitting diodes [20], solar concentrators [21, 22], photovoltaic cells [23, 24] and electroluminescent devices [25, 26]. They are promising functional materials applied in biological fields [27-30]. Numerous studies [31-35] are reported in the literature for the linear and nonlinear optical properties of quantum dots. In order to stabilize the QDs, matrix materials such as PVA (poly-vinyl alcohol), PMMA (poly-methyl methacrylate), PVP (poly-vinyl pyrrolidone) and SiO2 are often used [36-41]. By encapsulating the QDs into a suitable matrix material, new optical properties can be emerged. Recently, the effect of dielectric mismatch between the QDs and surrounding media has been investigated by Vahdania et al [42, 43]. In ref [36], the role of dielectric environment has been examined on the third-order nonlinear optical absorption coefficient in colloidal CdS quantum dots. Authors found that QD dispersed in a matrix material having the largest dielectric constant shows the largest absorption peak intensity. A blueshift has been also obtained in the QDs embedded in PMMA and SiO2 while a redshfit is observed when they used PVA. Maikhuri et al [38] found that increasing the shell thickness redshifted the energy and enhanced the nonlinear absorption coefficients and all the refractive index changes, independently on the dielectric environments. In 2017, Dujardin et al [39] examined the impact of core/shell radii ratio and dielectric environment inside CdSe/ZnTe and ZnTe/CdSe CSQDs and they concluded that with taking into account the influence of dielectric environment, the binding energy can be size controlled. The numerical results of ref [40] revealed that in the dielectrically modulated CSQDs, the desired energy range for the impurity levels may be controlled by changing the dielectric constant of the matrix material and quantum dot parameters. In addition, in ref [42], authors have evaluated the real and imaginary parts of the dielectric function with and without considering the optical feedback effect and they found that the magnitude of the dielectric function diminishes at frequencies below the transition frequency because of the electric field concentration inside the dot. In this paper, the description of the electronic quantum confinement is based on the framework of the effective mass-approximation (E.M.A). Within the compact density matrix approach we have calculated the eigenvalues in CdS/ZnS CSQDs. Linear, nonlinear and total dielectric function have been evaluated for three different matrices (PVA, PMMA and SiO2). Theoretical analyses are presented on the base of these numerical calculations as follows. In
Section 2 we develop, in detail, the theoretical model of dielectric function. The results and a conclusion will be presented in sections 3 and 4 respectively. 2. Theory and calculation: Our model is composed of a single electron confined in an isolated CdS/ZnS CSQD with inner radius R1 corresponding to CdS and outer radius R2 corresponding to ZnS which is further embedded in a dielectric material as depicted in Fig.1. In this study, we have assumed dealing with a spherical QD We have used the potential in the core CdS material as the zero reference energy since the band gap of ZnS is wider than that of CdS thus Vc >0 [43] and within the effective mass formalism, the Schrödinger equation can be written as:
{―
ℏ2
[ (𝑟 ) +
2𝑚𝑗∗ 𝑟2
∂
2∂ ∂𝑟
∂𝑟
1
∂ sin 𝜃∂𝜃
(sin 𝜃 ) + ∂
1
∂𝜃
2
] + 𝑉 (𝑟) + ∑(𝑅 )}𝛹
∂2
𝑗
𝑠𝑖𝑛 (𝜃)∂𝜑2
𝑠
𝑛,𝑙,𝑚(𝑟,𝜃,𝜑)
= 𝐸𝛹𝑛,𝑙,𝑚 (1)
(𝑟,𝜃,𝜑)
Where ℏ represents the reduced Planck constant, mj* is the effective mass of a localized electron in the jth region, Vj(r) is the electron confining potential. Here m*j and Vj(r) both depend on the electron position in the hetero-structure and they are expressed as: 𝑚𝑗∗ =
{
∗ 𝑚𝐶𝑑𝑆 ∗ 𝑚𝑍𝑛𝑆
𝑟 < 𝑅1 ; 𝑅1 < 𝑟 ≤ 𝑅2
(2)
0 < 𝑟 ≤ 𝑅1 𝑅1 < 𝑟 ≤ 𝑅2 𝑟 > 𝑅2
(3)
And
{
0, 𝑉𝑗(𝑟) = 𝑉𝑐, ∞
Where ∑ is the self-energy term due to the polarization induced by charging the QD with dielectric environment. The analytical expression of the self-energy is [39]: ∑(𝑅𝑠 ) ≈
𝑒2
(
2𝑅𝑠
1
1
)
𝑒2
(
𝜀𝑜𝑢𝑡 ― 𝜀𝑖𝑛 +0.472𝑅𝑠𝜀𝑖𝑛
)
𝜀𝑖𝑛 ― 𝜀𝑜𝑢𝑡 𝜀𝑖𝑛 + 𝜀𝑜𝑢𝑡
(4)
Where εin and εout represents the dielectric constants of the dot and the matrix materials, respectively.
In order to determine the complete expressions of the eigenenergies and their corresponding wave functions, we have used the formalism developed in Ref [42, 43]. The wave functions must satisfy the continuity conditions at all boundaries of the studied nanostructure as follows [44, 45]: (5)
(𝑅𝑖(𝑟) = 𝑅𝑖 + 1(𝑟))𝑟 = 𝑟𝑖 1 𝑑𝑅𝑖(𝑟)
(𝑚 * 𝑖
𝑑𝑟
1 𝑑𝑅𝑖 + 1(𝑟)
= 𝑚*
𝑑𝑟
𝑖+1
(6)
)
𝑟 = 𝑟𝑖
As well as the normalization’s condition: ∞
∫0 𝑟|𝑅𝑖(𝑟)|2𝑑𝑟 = 1
(7)
The expression of the first and third-order nonlinear optical susceptibilities for two energy levels, the ground and the first excited states is given by [43,46]:
(1)
𝜒 (𝜔) = 𝐸
𝜎𝑣|𝑀21|2
21
𝜒(3)(𝜔) = 𝐸
𝜎𝑣|𝑀21|2
21
[
4|𝑀21|2
(8)
- ℏ𝜔 - 𝑖ℏ𝛤 (𝑀22 - 𝑀11)2
)2 (𝐸21 - 𝑖ℏ𝛤12)(𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤12)
- ℏ𝜔 - 𝑖ℏ𝛤 (𝐸21 - ℏ𝜔)2 + (ℏ𝛤12
]
(9)
Where 𝜎𝑣 represents the carrier density, E12= E2-E1 is the energy difference between the two states and 𝑀𝑖𝑗 = |⟨𝛹1│𝑒𝑟│𝛹2⟩| In order to investigate the linear and nonlinear dielectric functions which are related to the corresponding electric susceptibility, we have used the expressions from ref [43, 47], thus 𝜀(1)(𝜔) = 1 + 4𝜋𝜒(1)(𝜔)
𝜀(𝑛)(𝜔) = 4𝜋𝜒(𝑛)(𝜔)
(10)
(11)
Exploiting Eqs (8)-(11) the linear and third order nonlinear dielectric functions can be expressed as [47]: (1)
𝜀 (𝜔) = 1 + 4𝜋(𝐸
𝜎𝑣|𝑀21|2
21
- ℏ𝜔 - 𝑖ℏ𝛤)
(12)
And 𝜀(3)(𝜔) = 4𝜋(𝐸
[
𝜎𝑣|𝑀21|2
21
(𝑀22 - 𝑀11)2
4|𝑀21|2
- ℏ𝜔 - 𝑖ℏ𝛤) (𝐸21 - ℏ𝜔)2 + (ℏ𝛤12)2
- (𝐸21 - 𝑖ℏ𝛤12)(𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤12)
]
(13)
The total dielectric function of CSQD is: 𝜎𝑣|𝑀21|2
2 (3)
(1)
𝜀 = 𝜀 (𝜔) + |𝐸| 𝜀 (𝜔) = 1 + 4𝜋 (𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤) |𝐸|2𝜎𝑣|𝑀21|2
+ 4𝜋(𝐸
21
[
4|𝑀21|2
(𝑀22 - 𝑀11)2
)2 (𝐸21 - 𝑖ℏ𝛤12)(𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤12)
- ℏ𝜔 - 𝑖ℏ𝛤) (𝐸21 - ℏ𝜔)2 + (ℏ𝛤12
]
(14)
Finally, the effective dielectric function of CdS/ZnS CSQD, is given by [46]: 𝜀𝑒𝑓𝑓(𝜔) = 1 + 3𝑝 +9𝑝
{
3 + 4𝜋(𝐸
𝜎𝑣|𝑀21|2
21
|𝐸|2𝜎𝑣|𝑀21|2
+ 4𝜋(𝐸 - ℏ𝜔 - 𝑖ℏ𝛤)
21
[
4|𝑀21|2
- ℏ𝜔 - 𝑖ℏ𝛤) (𝐸21 - ℏ𝜔)2 + (ℏ𝛤12)2
(𝑀22 - 𝑀11)2
- (𝐸21 - 𝑖ℏ𝛤12)(𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤12)
]}
-1
(15) Where p denotes the volume concentration of the dots defined as: 𝑝 =
4𝜋𝑅3 3
𝑛. And n is the
number of dots per unit volume. 3. Results and Discussion In this work, we have considered a nanocrystal with two spherical layers embedded in a dielectric material. Our calculations were developed under the effective mass approximation and the density-matrix approach for a two level model. The different material parameters used in our calculations are taken from [38, 43]: m*e,CdS=0.18m0, m*e,ZnS=0.42m0, Eg(CdS)=2.5 eV, Eg(ZnS)=3.62 eV and Vc=0.897 eV. It should be noted that in this study, the effect of lattice mismatch between the dielectric constants of the core and the shell materials is neglected that’s why we have taken εin= 𝜀𝑟(𝐶𝑑𝑆).𝜀𝑟(𝑍𝑛𝑆) = 8.9; the dielectric constant of the different matrices is εout= 3.4, 14 and 3.9 for PMMA, PVA and SiO2 respectively [36, 38, 39]. In Fig 2, we have depicted the variation of the electron energy for the fundamental 1Se (E1), the first excited state 1Pe (E2) and the difference energy 1Se-1Pe as a function of the core-toshell radii ratio (Rc/Rs) for three different matrices denoted PVA (Fig.2a), PMMA (Fig.2b)
and SiO2 (Fig.2c) in the case E
(1)
and Imaginary Imε(1) parts of the linear dielectric function have
been plotted as a function of the pump photon energy ℏ??(eV) for three different matrices: PVA, PMMA and SiO2 in the case E
The linear, nonlinear and total imaginary parts of the dielectric function have been pictured in Fig.6 as a function of pump photon energy ℏω (eV) for three different matrices: PVA, PMMA and SiO2 in the case E
4. Conclusion In this work, we have performed a theoretical study on the linear and nonlinear dielectric function for CdS/ZnS spherical CSQD embedded in different dielectric matrices under the effective mass approximation. Our numerical results indicate how the core/shell radii ratio affects the transition energy in QDs dispersed in three commonly used matrices (PVA, PMMA and SiO2). Our findings show also that the Realεeff(ω) and Imεeff(ω) parts depend on the dielectric environment. Hence, the QD capped with PVA reveals the largest linear and nonlinear dielectric function magnitude compared with those dispersed in SiO2. This work has
been developed with the purpose of serving the design of optoelectronic devices based on CdS/ZnS QDs-matrix systems.
Figures captions
Figure 1: CdS/ZnS spherical core/shell quantum dot and the potential profile with the conduction band. Figures 2: The variation of the fundamental 1Se and the first excited state 1Pe energies as well as the difference energy 1Se-1Pe as a function of the core/shell radii ratio Rc/Rs for PVA (Fig.2a), PMMA (Fig.2b) and SiO2 (Fig.2c) matrix materials in the case E
The linear and third order nonlinear part of Real εeff(ω) versus the pump photon energy ℏ𝜔 (𝑒𝑉) for different matrix materials (PVA, PMMA and SiO2) for n = 4×1016cm-1, I=0.2 MW/cm2. Figure 8: The linear and third order nonlinear part of Imεeff(ω) versus the pump photon energy ℏ𝜔 (𝑒𝑉) for different matrix materials (PVA, PMMA and SiO2) for n = 4×1016cm-1, I=0.2 MW/cm2.
R1
SiO2 PMMA PVA
R2
V(r)
CdS ZnS Vc o
R1
R2
Figure 1
r
0.8
SiO2
E2
0.7
E1 E2-E1
Energy (eV)
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
(Rc/Rs)(nm)
Figure 2.a
0.8
PMMA
E2
0.7
E1 E2-E1
Energy (eV)
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.4
0.5
0.6
0.7
(Rc/Rs)(nm)
Figure 2.b
0.8
0.9
1.0
0.8
E2
PVA
E1
0.7
E2-E1
Energy (eV)
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
(Rc/Rs) (nm)
Figure 2.c
0.8
PVA
E
PMMA
0.6
SiO2
Dielectric function
Im Real
0.4
0.2
0.0
-0.2
-0.4 0.15
0.20
0.25
0.30
0.35
Photon energy (eV)
Figure 3
0.40
0.45
0.50
3.0
PVA PMMA SiO2
2.5
1.5
2.0
1.0
0.5
0.0 0.21
0.24
0.36
0.33
0.30
0.27
0.42
0.39
Photon energy(eV)
Figure 4
0,09
PVA
E
0,06
Dielectric function
Real Im
PMMA SiO2
0,03 0,00 -0,03 -0,06 -0,09 -0,12 -0,15 0,15
0,20
0,25
0,30
0,35
Photon energy (eV)
Figure 5
0,40
0,45
0,50
0.4 0.3
PVA
E
Linear Nonlinear Total
PMMA SiO2
0.2
Im
0.1 0.0 -0.1 -0.2 -0.3 -0.4 0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Photon energy (eV)
Figure 6
4 3
PVA
E
PMMA
SiO2
eff
eff
+
eff
Real eff(
2 1 0 -1 -2 -3 -4 0.15
0.20
0.25
0.30
0.35
Photon energy (eV)
Figure 7
0.40
0.45
0.50
9 8
E
PVA
7
eff eff
+
eff
PMMA
6
Im eff( (
5
SiO2
4 3 2 1 0 0.21
0.24
0.27
0.30
0.33
0.36
Photon energy(eV)
Figure8
0.39
0.42
0.45
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S2211-3797(19)32243-0 https://doi.org/10.1016/j.rinp.2019.102661 RINP 102661
To appear in:
Results in Physics
Received Date: Accepted Date:
26 July 2019 12 September 2019
Please cite this article as: Zeiri, N., Naifar, A., Abdi-Ben Nasrallah, S., Said, M., Dielectric environment effect on linear and nonlinear optical properties for CdS/ZnS core/shell quantum dots, Results in Physics (2019), doi: https:// doi.org/10.1016/j.rinp.2019.102661
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Dielectric environment effect on linear and nonlinear optical properties for CdS/ZnS core/shell quantum dots N. Zeiri*(1), A. Naifar (1), S. Abdi-Ben Nasrallah(2), M. Said(1) *Corresponding author: N. Zeiri, E-mail address: [email protected] Tel: +216 73382656 Fax: +216 73382658 (1)Laboratoire
de la Matière Condensée et des Nanosciences (LMCN)
Département de Physique, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia (2)
Laboratoire d’Etudes des Systèmes Thermique et Energétique (LESTE), Université de Monastir
Abstract: The eigenvalues, transition energy, the linear and nonlinear dielectric functions have been numerically investigated for CdS/ZnS spherical core/shell quantum dots embedded in various dielectric matrices. The evaluation was carried out for three commonly used matrices, such as PVA, PMMA and SiO2. Calculations were done under the effective mass approximation and compact density matrix approach. Results revealed that the nonlinear optical property is strongly affected by the nature of the matrix material. By increasing core/shell radii ratio the energy states as well as the transition energy are decreasing. It is also indicated that the presence of the dielectric mismatch in the QD-matrix system can cause significant enhancement on the linear and third order nonlinear dielectric function.
Keywords: Nonlinear optic, Effective mass approximation, Core/shell, Density matrix approach, Complex dielectric function. Introduction Quantum dots (QDs) belong to the most investigated objects of nanophysics due to their wide potential applications in optoelectronic and photonic devices [1-8]. If one quantum dot is covered by a layer of another semiconductor, the resulting system is called core/shell quantum dot (CSQD). Spherical (CSQDs) have received a huge attention owing to their interesting
physical and chemical properties. Nowadays, progress in experimental techniques of growth has made it possible to synthesize high quality semiconductor QDs within a large range of sizes from spherical [9-11], pyramidal [12, 13] to cylindrical [14-16] and other shapes. The greatness of such nano-system is behind the development of lasers [17-19], light emitting diodes [20], solar concentrators [21, 22], photovoltaic cells [23, 24] and electroluminescent devices [25, 26]. They are promising functional materials applied in biological fields [27-30]. Numerous studies [31-35] are reported in the literature for the linear and nonlinear optical properties of quantum dots. In order to stabilize the QDs, matrix materials such as PVA (poly-vinyl alcohol), PMMA (poly-methyl methacrylate), PVP (poly-vinyl pyrrolidone) and SiO2 are often used [36-41]. By encapsulating the QDs into a suitable matrix material, new optical properties can be emerged. Recently, the effect of dielectric mismatch between the QDs and surrounding media has been investigated by Vahdania et al [42, 43]. In ref [36], the role of dielectric environment has been examined on the third-order nonlinear optical absorption coefficient in colloidal CdS quantum dots. Authors found that QD dispersed in a matrix material having the largest dielectric constant shows the largest absorption peak intensity. A blueshift has been also obtained in the QDs embedded in PMMA and SiO2 while a redshfit is observed when they used PVA. Maikhuri et al [38] found that increasing the shell thickness redshifted the energy and enhanced the nonlinear absorption coefficients and all the refractive index changes, independently on the dielectric environments. In 2017, Dujardin et al [39] examined the impact of core/shell radii ratio and dielectric environment inside CdSe/ZnTe and ZnTe/CdSe CSQDs and they concluded that with taking into account the influence of dielectric environment, the binding energy can be size controlled. The numerical results of ref [40] revealed that in the dielectrically modulated CSQDs, the desired energy range for the impurity levels may be controlled by changing the dielectric constant of the matrix material and quantum dot parameters. In addition, in ref [42], authors have evaluated the real and imaginary parts of the dielectric function with and without considering the optical feedback effect and they found that the magnitude of the dielectric function diminishes at frequencies below the transition frequency because of the electric field concentration inside the dot. In this paper, the description of the electronic quantum confinement is based on the framework of the effective mass-approximation (E.M.A). Within the compact density matrix approach we have calculated the eigenvalues in CdS/ZnS CSQDs. Linear, nonlinear and total dielectric function have been evaluated for three different matrices (PVA, PMMA and SiO2). Theoretical analyses are presented on the base of these numerical calculations as follows. In
Section 2 we develop, in detail, the theoretical model of dielectric function. The results and a conclusion will be presented in sections 3 and 4 respectively. 2. Theory and calculation: Our model is composed of a single electron confined in an isolated CdS/ZnS CSQD with inner radius R1 corresponding to CdS and outer radius R2 corresponding to ZnS which is further embedded in a dielectric material as depicted in Fig.1. In this study, we have assumed dealing with a spherical QD We have used the potential in the core CdS material as the zero reference energy since the band gap of ZnS is wider than that of CdS thus Vc >0 [43] and within the effective mass formalism, the Schrödinger equation can be written as:
{―
ℏ2
[ (𝑟 ) +
2𝑚𝑗∗ 𝑟2
∂
2∂ ∂𝑟
∂𝑟
1
∂ sin 𝜃∂𝜃
(sin 𝜃 ) + ∂
1
∂𝜃
2
] + 𝑉 (𝑟) + ∑(𝑅 )}𝛹
∂2
𝑗
𝑠𝑖𝑛 (𝜃)∂𝜑2
𝑠
𝑛,𝑙,𝑚(𝑟,𝜃,𝜑)
= 𝐸𝛹𝑛,𝑙,𝑚 (1)
(𝑟,𝜃,𝜑)
Where ℏ represents the reduced Planck constant, mj* is the effective mass of a localized electron in the jth region, Vj(r) is the electron confining potential. Here m*j and Vj(r) both depend on the electron position in the hetero-structure and they are expressed as: 𝑚𝑗∗ =
{
∗ 𝑚𝐶𝑑𝑆 ∗ 𝑚𝑍𝑛𝑆
𝑟 < 𝑅1 ; 𝑅1 < 𝑟 ≤ 𝑅2
(2)
0 < 𝑟 ≤ 𝑅1 𝑅1 < 𝑟 ≤ 𝑅2 𝑟 > 𝑅2
(3)
And
{
0, 𝑉𝑗(𝑟) = 𝑉𝑐, ∞
Where ∑ is the self-energy term due to the polarization induced by charging the QD with dielectric environment. The analytical expression of the self-energy is [39]: ∑(𝑅𝑠 ) ≈
𝑒2
(
2𝑅𝑠
1
1
)
𝑒2
(
𝜀𝑜𝑢𝑡 ― 𝜀𝑖𝑛 +0.472𝑅𝑠𝜀𝑖𝑛
)
𝜀𝑖𝑛 ― 𝜀𝑜𝑢𝑡 𝜀𝑖𝑛 + 𝜀𝑜𝑢𝑡
(4)
Where εin and εout represents the dielectric constants of the dot and the matrix materials, respectively.
In order to determine the complete expressions of the eigenenergies and their corresponding wave functions, we have used the formalism developed in Ref [42, 43]. The wave functions must satisfy the continuity conditions at all boundaries of the studied nanostructure as follows [44, 45]: (5)
(𝑅𝑖(𝑟) = 𝑅𝑖 + 1(𝑟))𝑟 = 𝑟𝑖 1 𝑑𝑅𝑖(𝑟)
(𝑚 * 𝑖
𝑑𝑟
1 𝑑𝑅𝑖 + 1(𝑟)
= 𝑚*
𝑑𝑟
𝑖+1
(6)
)
𝑟 = 𝑟𝑖
As well as the normalization’s condition: ∞
∫0 𝑟|𝑅𝑖(𝑟)|2𝑑𝑟 = 1
(7)
The expression of the first and third-order nonlinear optical susceptibilities for two energy levels, the ground and the first excited states is given by [43,46]:
(1)
𝜒 (𝜔) = 𝐸
𝜎𝑣|𝑀21|2
21
𝜒(3)(𝜔) = 𝐸
𝜎𝑣|𝑀21|2
21
[
4|𝑀21|2
(8)
- ℏ𝜔 - 𝑖ℏ𝛤 (𝑀22 - 𝑀11)2
)2 (𝐸21 - 𝑖ℏ𝛤12)(𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤12)
- ℏ𝜔 - 𝑖ℏ𝛤 (𝐸21 - ℏ𝜔)2 + (ℏ𝛤12
]
(9)
Where 𝜎𝑣 represents the carrier density, E12= E2-E1 is the energy difference between the two states and 𝑀𝑖𝑗 = |⟨𝛹1│𝑒𝑟│𝛹2⟩| In order to investigate the linear and nonlinear dielectric functions which are related to the corresponding electric susceptibility, we have used the expressions from ref [43, 47], thus 𝜀(1)(𝜔) = 1 + 4𝜋𝜒(1)(𝜔)
𝜀(𝑛)(𝜔) = 4𝜋𝜒(𝑛)(𝜔)
(10)
(11)
Exploiting Eqs (8)-(11) the linear and third order nonlinear dielectric functions can be expressed as [47]: (1)
𝜀 (𝜔) = 1 + 4𝜋(𝐸
𝜎𝑣|𝑀21|2
21
- ℏ𝜔 - 𝑖ℏ𝛤)
(12)
And 𝜀(3)(𝜔) = 4𝜋(𝐸
[
𝜎𝑣|𝑀21|2
21
(𝑀22 - 𝑀11)2
4|𝑀21|2
- ℏ𝜔 - 𝑖ℏ𝛤) (𝐸21 - ℏ𝜔)2 + (ℏ𝛤12)2
- (𝐸21 - 𝑖ℏ𝛤12)(𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤12)
]
(13)
The total dielectric function of CSQD is: 𝜎𝑣|𝑀21|2
2 (3)
(1)
𝜀 = 𝜀 (𝜔) + |𝐸| 𝜀 (𝜔) = 1 + 4𝜋 (𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤) |𝐸|2𝜎𝑣|𝑀21|2
+ 4𝜋(𝐸
21
[
4|𝑀21|2
(𝑀22 - 𝑀11)2
)2 (𝐸21 - 𝑖ℏ𝛤12)(𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤12)
- ℏ𝜔 - 𝑖ℏ𝛤) (𝐸21 - ℏ𝜔)2 + (ℏ𝛤12
]
(14)
Finally, the effective dielectric function of CdS/ZnS CSQD, is given by [46]: 𝜀𝑒𝑓𝑓(𝜔) = 1 + 3𝑝 +9𝑝
{
3 + 4𝜋(𝐸
𝜎𝑣|𝑀21|2
21
|𝐸|2𝜎𝑣|𝑀21|2
+ 4𝜋(𝐸 - ℏ𝜔 - 𝑖ℏ𝛤)
21
[
4|𝑀21|2
- ℏ𝜔 - 𝑖ℏ𝛤) (𝐸21 - ℏ𝜔)2 + (ℏ𝛤12)2
(𝑀22 - 𝑀11)2
- (𝐸21 - 𝑖ℏ𝛤12)(𝐸21 - ℏ𝜔 - 𝑖ℏ𝛤12)
]}
-1
(15) Where p denotes the volume concentration of the dots defined as: 𝑝 =
4𝜋𝑅3 3
𝑛. And n is the
number of dots per unit volume. 3. Results and Discussion In this work, we have considered a nanocrystal with two spherical layers embedded in a dielectric material. Our calculations were developed under the effective mass approximation and the density-matrix approach for a two level model. The different material parameters used in our calculations are taken from [38, 43]: m*e,CdS=0.18m0, m*e,ZnS=0.42m0, Eg(CdS)=2.5 eV, Eg(ZnS)=3.62 eV and Vc=0.897 eV. It should be noted that in this study, the effect of lattice mismatch between the dielectric constants of the core and the shell materials is neglected that’s why we have taken εin= 𝜀𝑟(𝐶𝑑𝑆).𝜀𝑟(𝑍𝑛𝑆) = 8.9; the dielectric constant of the different matrices is εout= 3.4, 14 and 3.9 for PMMA, PVA and SiO2 respectively [36, 38, 39]. In Fig 2, we have depicted the variation of the electron energy for the fundamental 1Se (E1), the first excited state 1Pe (E2) and the difference energy 1Se-1Pe as a function of the core-toshell radii ratio (Rc/Rs) for three different matrices denoted PVA (Fig.2a), PMMA (Fig.2b)
and SiO2 (Fig.2c) in the case E
(1)
and Imaginary Imε(1) parts of the linear dielectric function have
been plotted as a function of the pump photon energy ℏ??(eV) for three different matrices: PVA, PMMA and SiO2 in the case E
The linear, nonlinear and total imaginary parts of the dielectric function have been pictured in Fig.6 as a function of pump photon energy ℏω (eV) for three different matrices: PVA, PMMA and SiO2 in the case E
4. Conclusion In this work, we have performed a theoretical study on the linear and nonlinear dielectric function for CdS/ZnS spherical CSQD embedded in different dielectric matrices under the effective mass approximation. Our numerical results indicate how the core/shell radii ratio affects the transition energy in QDs dispersed in three commonly used matrices (PVA, PMMA and SiO2). Our findings show also that the Realεeff(ω) and Imεeff(ω) parts depend on the dielectric environment. Hence, the QD capped with PVA reveals the largest linear and nonlinear dielectric function magnitude compared with those dispersed in SiO2. This work has
been developed with the purpose of serving the design of optoelectronic devices based on CdS/ZnS QDs-matrix systems.
Figures captions
Figure 1: CdS/ZnS spherical core/shell quantum dot and the potential profile with the conduction band. Figures 2: The variation of the fundamental 1Se and the first excited state 1Pe energies as well as the difference energy 1Se-1Pe as a function of the core/shell radii ratio Rc/Rs for PVA (Fig.2a), PMMA (Fig.2b) and SiO2 (Fig.2c) matrix materials in the case E
The linear and third order nonlinear part of Real εeff(ω) versus the pump photon energy ℏ𝜔 (𝑒𝑉) for different matrix materials (PVA, PMMA and SiO2) for n = 4×1016cm-1, I=0.2 MW/cm2. Figure 8: The linear and third order nonlinear part of Imεeff(ω) versus the pump photon energy ℏ𝜔 (𝑒𝑉) for different matrix materials (PVA, PMMA and SiO2) for n = 4×1016cm-1, I=0.2 MW/cm2.
R1
SiO2 PMMA PVA
R2
V(r)
CdS ZnS Vc o
R1
R2
Figure 1
r
0.8
SiO2
E2
0.7
E1 E2-E1
Energy (eV)
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
(Rc/Rs)(nm)
Figure 2.a
0.8
PMMA
E2
0.7
E1 E2-E1
Energy (eV)
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.4
0.5
0.6
0.7
(Rc/Rs)(nm)
Figure 2.b
0.8
0.9
1.0
0.8
E2
PVA
E1
0.7
E2-E1
Energy (eV)
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.4
0.5
0.6
0.7
0.8
0.9
1.0
(Rc/Rs) (nm)
Figure 2.c
0.8
PVA
E
PMMA
0.6
SiO2
Dielectric function
Im Real
0.4
0.2
0.0
-0.2
-0.4 0.15
0.20
0.25
0.30
0.35
Photon energy (eV)
Figure 3
0.40
0.45
0.50
3.0
PVA PMMA SiO2
2.5
1.5
2.0
1.0
0.5
0.0 0.21
0.24
0.36
0.33
0.30
0.27
0.42
0.39
Photon energy(eV)
Figure 4
0,09
PVA
E
0,06
Dielectric function
Real Im
PMMA SiO2
0,03 0,00 -0,03 -0,06 -0,09 -0,12 -0,15 0,15
0,20
0,25
0,30
0,35
Photon energy (eV)
Figure 5
0,40
0,45
0,50
0.4 0.3
PVA
E
Linear Nonlinear Total
PMMA SiO2
0.2
Im
0.1 0.0 -0.1 -0.2 -0.3 -0.4 0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Photon energy (eV)
Figure 6
4 3
PVA
E
PMMA
SiO2
eff
eff
+
eff
Real eff(
2 1 0 -1 -2 -3 -4 0.15
0.20
0.25
0.30
0.35
Photon energy (eV)
Figure 7
0.40
0.45
0.50
9 8
E
PVA
7
eff eff
+
eff
PMMA
6
Im eff( (
5
SiO2
4 3 2 1 0 0.21
0.24
0.27
0.30
0.33
0.36
Photon energy(eV)
Figure8
0.39
0.42
0.45
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