- Email: [email protected]
Effects of step-potential on confinement strength of strain-induced type-I core–shell quantum dots
Superlattices and Microstructures 131 (2019) 95–103
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/yspmi
Effects of step-potential on confinement strength of strain-induced type-I core–shell quantum dots T. Shelawati b ,∗, M.S. Nurisya a,b , C. Kar Tim a,b , A.K. Mazliana b a
Laboratory of Computational Sciences & Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia b Department of Physics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
ARTICLE
INFO
Keywords: Quantum dots Colloidal quantum dots Transition energy Bessel spherical functions
ABSTRACT In this paper, the transition energy between lowest unoccupied molecular orbital (LUMO) of conduction band and highest occupied molecular orbital (HOMO) of valence band for band structures of type-I core–shell quantum dots (CSQDs) within a strong and weak confinements of charge carriers are estimated using the effective mass approximation together with single-band model. The effect of potential step at the conduction and valence bands on the confinement strength is then properly discussed. Our numerical results show that for a same size of CSQDs, the one with bigger potential steps will have stronger carriers’ confinement with more localized excitons.
1. Introduction In recent years, research field has witnessed vigorous studies in nanoscale semiconductors, ranging from quantum wells (2-dimensional), quantum wires (1-dimensional) and quantum dots (0-dimensional), where their charge carriers are spatially confined in one degree, two degrees and three degrees respectively. Quantum dots or sometimes dubbed as ‘‘artificial atoms’’ or ‘‘hyperatoms’’ [1] due to their atomic scale size, have shown great potential in agriculture and medicine fields as sensors [2], solar panels [2–4], bio-tagging materials [3–6] or light-emitting devices [3,4]. It has long established that their nano-scale size which is comparable to Bohr radius gives rise to a strong localization of charge carriers compared to their bulk counterparts. The small size property of quantum dots causes it to have a high surface to volume ratio. For instance, quantum dots with radius of 5 nm can still have 104 atoms. This has made its surface to play a significant role, which sometimes can be considered as a drawback due to possibility of having defects on its surface [6,7]. However, many research have discovered that it is possible to cap or encapsulate these quantum dots surfaces with another organic or inorganic semiconductor materials in order to enhance their optical and electrical properties as described in [4,8–10]. This novel finding, known as core–shell quantum dots (CSQDs) was further studied to distinguish their types based on regional confinement of electrons and holes as well as their band edges alignment. For type-I CSQDs, the charge carriers are confined only in core material or only in shell material (inverse type-I CSQDs). As for type-II CSQDs, the charge carriers are confined between conduction band of core material and valence band of shell material or between conduction band of shell material and valence band of core material (inverse type-II). There are three well-known ways of heteroepitaxial growth for CSQDs fabrication which are adding shell compound layer-by-layer method onto the core surface (Frank– van der Merwe), layer-by-layer followed with additional islands formation (Stranski–Krastanov) or just islands formation directly on the core surface (Volmer–Weber). All these passivating modes are governed by their interface energies and lattice mismatch [11,12].
∗ Corresponding author. E-mail addresses: [email protected] (T. Shelawati), [email protected] (M.S. Nurisya), [email protected] (C. Kar Tim), [email protected] (A.K. Mazliana).
https://doi.org/10.1016/j.spmi.2019.05.021 Received 2 January 2019; Received in revised form 4 April 2019; Accepted 10 May 2019 Available online 16 May 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
The motivations of passivating quantum dots surfaces are to provide electrical and chemical passivation [5,6] that will improve photostability of the quantum dots against oxidation [5,7] as well as to increase the confinement of charge carriers and making its amount of energy for carriers transition more tunable [2,5–7,13]. Inorganic capping materials can also reduce the chemical activity due to cationic and anionic dangling bonds [6]. However, increasing the thickness of this shell layer above its critical thickness may cause the shell to impair the whole core–shell quantum dots instead of improving its photoluminescene [12]. Hence, a study to investigate relation between shell thickness and transition energy between HOMO of conduction band and LUMO of valence band of CSQDs is very crucial. Different approaches had been considered such as single-band model [2,4–7,13,14], multiband theory (8-band model) [14–16], envelope function [5], density-functional theory (DFT) [3] and model-solid theory [2]. In this paper, instead of just depending on their core sizes as suggested in [1] and [11], we will apply single-band model onto our CSQDs by introducing two different types of confinement; strong confinement and weak confinement model that are based on strengths of potential steps on conduction and valence bands. This paper is organized as follows. First, single-band approach on energy transition in CSQDs will be briefly discussed and then applied on CSQDs with strong confinement, followed with CSQDs with weak confinement. Subsequently, numerical results is discussed by observing the radial probabilities in both strong and weak confinement cases.
2. Theory 2.1. Single-band model Consider a colloidal CSQDs as spherical quantum dots encapsulated by a layer of a different compound of semiconductor (see Fig. 1). As shell is added, there will be step-potential inducing strain on the core quantum dots. Following the usual method to solve a stationary Schödinger equation for a particle with some step-potential, one have to separate the wavefunction into radial part, 𝑅𝑛𝑙 and angular part, 𝑌𝑙𝑚 as (1)
𝛹𝑛𝑙𝑚 (⃗𝑟, 𝜃, 𝜙) = 𝑅𝑛𝑙 (𝑟)𝑌𝑙𝑚 (𝜃, 𝜙), where 𝑛, 𝑙 and 𝑚 are the principal, orbital and magnetic quantum numbers respectively.
The only interested solution is the solution of radial wavefunction for lowest transitions energy, 𝐸qd (𝑛 = 1 and 𝑙 = 𝑚 = 0), in other words 1𝑠𝑒 → 1𝑠ℎ transition. To solve energy for electrons, 𝐸𝑒 and holes, 𝐸ℎ , we could benefit from spherical Bessel functions (for the case 𝐸𝑒,ℎ > 𝑉𝑐,𝑣 ) or modified spherical Bessel functions (for the case 𝐸𝑒,ℎ < 𝑉𝑐,𝑣 ) [2].
Fig. 1. Schematic diagram of band line-up of core–shell QDs. 96
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
𝐸𝑐(𝑣) and 𝑉𝑐(𝑣) are the energy and potential step for conduction(valence) band respectively, 𝑟𝑜 and 𝑟𝑠 are radius of core quantum dots and radius of CSQDs respectively. For carriers outside the shell layer, the corresponding 𝑉 (𝑟) is infinite. ⎧ 0 ⎪ 𝑉 (𝑟) = ⎨𝑉𝑐,𝑣 ⎪∞ ⎩
0 ≤ 𝑟 ≤ 𝑟𝑜 𝑟𝑜 ≤ 𝑟 ≤ 𝑟𝑠 𝑟 ≥ 𝑟𝑠 .
; ; ;
For 𝐸𝑒,ℎ > 𝑉𝑐,𝑣 ; ⎧ 𝑅1 (𝑟) = 𝐴𝑗𝑙 (𝑘1 𝑟) ⎪ 𝑅𝑖 (𝑟) = ⎨𝑅2 (𝑟) = 𝐵𝑗𝑙 (𝑘2 𝑟) + 𝐶𝑛𝑙 (𝑘2 𝑟) ⎪ 𝑅3 (𝑟) = 0 ⎩
; ; ;
0 ≤ 𝑟 ≤ 𝑟𝑜 𝑟𝑜 ≤ 𝑟 ≤ 𝑟𝑠 𝑟 ≥ 𝑟𝑠 .
(2)
; ; ;
0 ≤ 𝑟 ≤ 𝑟𝑜 𝑟𝑜 ≤ 𝑟 ≤ 𝑟𝑠 𝑟 ≥ 𝑟𝑠 .
(3)
For 𝐸𝑒,ℎ < 𝑉𝑐,𝑣 ; ⎧ 𝑅1 (𝑟) = 𝐴𝑗𝑙 (𝑘1 𝑟) ⎪ 𝑅𝑖 (𝑟) = ⎨𝑅2 (𝑟) = 𝐵ℎ𝑙 (𝑘′2 𝑟) + 𝐶𝜅𝑙 (𝑘′2 𝑟) ⎪ 𝑅3 (𝑟) = 0 ⎩
Note that 𝑗𝑙 (𝑘𝑖 ), 𝑛𝑙 (𝑘𝑖 ), ℎ𝑙 (𝑘𝑖 ) and 𝜅𝑙 (𝑘𝑖 ) are spherical Bessel function of the first kind, Neumann spherical functions, √ modified ∗ spherical Bessel function of the first kind and modified spherical Bessel function of the second kind respectively. 𝑘1 = √ ∗ √ ∗ 2𝑚2 [𝐸𝑒,ℎ −𝑉𝑐,𝑣 (𝑟)]
𝑘′2
2𝑚2 [𝑉𝑐,𝑣 (𝑟)−𝐸𝑒,ℎ ]
= 𝑘2 = , ℏ ℏ continuity of wavefunctions, hence
and
𝑚∗𝑖
is the effective mass of electrons,
𝑚𝑒𝑖
or effective mass of holes,
𝑚ℎ𝑖 .
2𝑚1 𝐸𝑒,ℎ ℏ
,
Following
𝑅𝑖 (𝑟) = 𝑅𝑖+1 (𝑟) 1 ′ 1 𝑅 (𝑟) = 𝑅′ (𝑟), 𝑚𝑖 𝑖 𝑚𝑖+1 𝑖+1 where 𝑖 = 1 for core, 𝑖 = 2 for shell and 𝑖 = 3 for outside shell layer. Following the said boundary conditions, Eqs. (2) and (3) will form three sets of equations that subsequently solve for 𝑟 normalization constants A, B and C which satisfy ∫0 4𝜋𝑟2 𝑅2𝑙 (𝑟)𝑑𝑟 = 1. For 𝐸𝑒,ℎ > 𝑉𝑐,𝑣 ; ⎡ 𝑗0 (𝑘1 𝑟𝑜 ) ⎢ 1 ′ ⎢ 𝑚∗1 𝑗0 (𝑘1 𝑟𝑜 ) ⎢ 0 ⎣
−𝑗0 (𝑘2 𝑟𝑜 ) − 𝑚1∗ 𝑗0′ (𝑘2 𝑟𝑜 ) 2
𝑗0 (𝑘2 𝑟𝑠 )
−𝑛0 (𝑘2 𝑟𝑜 ) ⎤ ⎡𝐴⎤ ⎥ − 𝑚1∗ 𝑛′0 (𝑘2 𝑟𝑜 )⎥ ⎢𝐵 ⎥ = 0. ⎢ ⎥ 2 𝑛0 (𝑘2 𝑟𝑠 ) ⎥⎦ ⎣𝐶 ⎦
(4)
−𝜅0 (𝑘′2 𝑟𝑜 ) ⎤ ⎡𝐴⎤ ⎥ − 𝑚1∗ 𝜅0′ (𝑘′2 𝑟𝑜 )⎥ ⎢𝐵 ⎥ = 0. ⎢ ⎥ 2 𝜅0 (𝑘′2 𝑟𝑠 ) ⎥⎦ ⎣𝐶 ⎦
(5)
For 𝐸𝑒,ℎ < 𝑉𝑐,𝑣 ; ⎡ 𝑗0 (𝑘1 𝑟𝑜 ) ⎢ 1 ′ ⎢ 𝑚∗1 𝑗0 (𝑘1 𝑟𝑜 ) ⎢ 0 ⎣
−ℎ0 (𝑘′2 𝑟𝑜 ) − 𝑚1∗ ℎ′0 (𝑘′2 𝑟𝑜 ) 2
ℎ0 (𝑘′2 𝑟𝑠 )
Eqs. (4) and (5) will form the transcendental equations that obey det(𝐸𝑒,ℎ ) = 0. Our transcendental equations then can be written as 𝑚∗2 𝑗0′ (𝑘1 𝑟𝑜 ) 𝑚∗1
𝑗0 (𝑘1 𝑟𝑜 )
=
𝑗0′ (𝑘2 𝑟𝑜 )𝑛0 (𝑘2 𝑟𝑠 ) − 𝑗0 (𝑘2 𝑟𝑠 )𝑛′0 (𝑘2 𝑟𝑜 ) 𝑗0 (𝑘2 𝑟𝑜 )𝑛0 (𝑘2 𝑟𝑠 ) − 𝑗0 (𝑘2 𝑟𝑠 )𝑛0 (𝑘2 𝑟𝑜 )
(6)
,
or 𝑚∗2 𝑗0′ (𝑘1 𝑟𝑜 ) 𝑚∗1
𝑗0 (𝑘1 𝑟𝑜 )
=
ℎ′0 (𝑘′2 𝑟𝑜 )𝜅0 (𝑘′2 𝑟𝑠 ) − ℎ0 (𝑘′2 𝑟𝑠 )𝜅0′ (𝑘′2 𝑟𝑜 ) ℎ0 (𝑘′2 𝑟𝑜 )𝜅0 (𝑘′2 𝑟𝑠 ) − ℎ0 (𝑘′2 𝑟𝑠 )𝜅0 (𝑘′2 𝑟𝑜 )
(7)
.
One will obtain 𝐸𝑒,ℎ after solving Eq. (6) or (7). As we need to solve for both electron energy and hole energy separately, the use of superscript will be handy to denote whether we are working on the electron states (𝑅𝑒𝑖 ) or hole states (𝑅ℎ𝑖 ). In case of quantum dots without shell, we can consider it to be in water environment with band gap 7.3 eV with charge carriers of free-electron mass [17]. The lowest transition energy, 𝐸qd for CSQDs will be 𝐸qd = 𝐸𝑔𝑏𝑢𝑙𝑘 + 𝐸𝑒 + 𝐸ℎ −
1.8𝑒2 . 4𝜋𝜀𝑜 𝜀𝑟 𝑟𝑜
(8) 2
1.8𝑒 Here, we treated the Coulomb interaction term ( 4𝜋𝜀 ) as first order perturbation that rises due to interactions between charge 𝑜 𝜀𝑟 𝑟𝑜 carriers. Our lowest eigenvalue is not very sensitive with this term (and often neglected) as it only has 1∕𝑟 dependence against our confinement energy term [2,18].
97
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
3. Application in CSQDs with strong confinement In this part, the above theoretical approach is applied to a CSQDs model that has a core with narrow bang gap and encapsulated with a much wider band gap shell compound. As a result, the step-potentials will be much bigger than the core bulk band gap (𝑉𝑐,𝑣 ≫ 𝐸𝑔𝑏𝑢𝑙𝑘 ), implying a strong excitons confinement within the core material. For this purpose, we chose PbS/CdS core–shell QDs with material parameters as shown in Table 1 (see Figs. 2–4). Table 1 List of parameters for PbS/CdS core–shell QDs. Materials
PbS [19]
CdS [19,20]
Band gap (eV) 𝑚𝑒 ∕𝑚0 𝑚ℎ ∕𝑚0 Dielectric constant, 𝜀𝑟
0.41 0.08 0.08 169
2.5 0.2 0.7 8.73
𝑉𝑐 (eV) 𝑉𝑣 (eV)
1.254 0.836
Fig. 2. Energy of electrons for PbS/CdS.
Fig. 3. Energy of holes for PbS/CdS.
4. Application in CSQDs with weak confinement In the second part, the single-band model is applied to a wide core QDs encapsulated with a slightly wider shell compound, where the step-potentials will be much smaller than the core bulk band gap (𝑉𝑐,𝑣 ≪ 𝐸𝑔𝑏𝑢𝑙𝑘 ) to imply a weak excitons confinement within the core material. For this purpose, we choose ZnTe/ZnSe core–shell QDs with material parameters as shown in Table 2 (see Figs. 5–7). 98
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
Fig. 4. Transition energy for PbS/CdS CSQDs.
Table 2 List of parameters for ZnTe/ZnSe core–shell QDs. Materials
ZnTe [4,19]
ZnSe [4,19]
Band gap (eV) 𝑚𝑒 ∕𝑚0 𝑚ℎ ∕𝑚0 Dielectric constant, 𝜀𝑟
2.394 0.11 0.7 8.7
2.8215 0.14 0.6 9.1
𝑉𝑐 (eV) 𝑉𝑣 (eV)
0.2565 0.171
Fig. 5. Energy of electrons for ZnTe/ZnSe.
5. Analysis on radial probability 5.1. Strong confinement case For the case of PbS quantum dots (without shell) with radius of 4.00 nm and 4.25 nm, there is no significant difference in radial probability of finding charge carriers within the cores as shown in Fig. 8, although the transition energy, 𝐸qd is higher in 4.00 nm quantum dots compared to 4.25 nm quantum dots. 99
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
Fig. 6. Energy of holes for ZnTe/ZnSe.
Fig. 7. Transition energy for ZnTe/ZnSe CSQDs.
After adding 0.25 nm of CdS shell layer onto the 4.00 nm PbS quantum dots (CSQDs radius = 4.25 nm), as shown in Fig. 9, the radial probability of charge carriers are shifted more to the center of CSQDs, signaling a stronger localization compared to the 4.25 nm PbS quantum dots. This stronger carriers localization which translate into stronger carriers confinement is proven by higher transition energy in 4.25 nm PbS/CdS CSQDs (4.00 nm core radius with 0.25 nm shell thickness) compared to transition energy in 4.25 nm PbS quantum dots. But as we increase the thickness layer of CdS, the radial probability of finding the charge carriers started to spread into the shell region, as shown in Fig. 10. This spreading of probability of radial wavefunction is caused by defects on interface area, causing delocalization of charge carriers where it becomes easier for charge carrier to tunnel through the interface and be in the shell region [12] (see Fig. 11). 5.2. Weak confinement case For ZnTe quantum dots that has wide bulk band gap, the confinement of charge carrier is not too obvious, leading to smaller increasing factor in transition energy compared to narrow band gap case. In narrow band gap, the energy for quantum dots almost doubled, while in wide band gap, the increasing in 𝐸qd is less visible. It is worthy to note that group II–VI semiconductor is less likely to have a strong carriers confinement [1,11]. However, after adding 0.25 nm of ZnSe shell layer, the transition energy of ZnTe/ZnSe CSQDs is increasing by more than 5%. From Fig. 12, it shows that the probabilities of the carriers in shell region are relatively higher compared with strong confinement case as in Fig. 9. This shows weaker localization of charge carriers. 100
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
Fig. 8. Radial probability of carriers in (top) 4.00 nm PbS quantum dots and (bottom) 4.25 nm PbS quantum dots where they show no significant difference.
Fig. 9. Drastic change in radial probability of carriers after adding 0.25 nm CdS shell layer on 4.00 nm PbS core.
Fig. 10. Radial probability of carriers in 4.50 nm PbS/CdS with 4.00 nm core radius.
The probability of charge carrier is spreading more into shell region as shown in Fig. 13 as we increase thickness shell layer. This is due to strong strain energy imposed on the core quantum dots, causing lattice mismatch to be significant in order to minimize the strain on interface area [12]. 101
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
Fig. 11. Radial probability of carriers in 4.00 nm ZnTe quantum dots.
Fig. 12. Radial probability of carriers in 4.25 nm ZnTe/ZnSe with 4.00 nm core radius.
Fig. 13. Radial probability of carriers in 4.50 nm ZnTe/ZnSe with 4.00 nm core radius.
6. Conclusion In this work, we have shown that by varying the potential step on a same size CSQDs, we can produce different types of confinement; a strong confinement and a weak confinement. This is important factor to consider for fabrication of CSQDs as instead of relying on size factor to tune the confinement strength, we can also control it by manipulating their potential step by introducing a capping material. Increasing shell thickness will increase photoluminescene of CSQDs as this will increase confinement of electrons and holes [6]. However, as it reaching its critical thickness, the strain on interface will be significant enough to induce lattice mismatch in order to minimize the strain [12]. The high energy excitons now can tunnel easily through the core–shell interface, causing further delocalization, hence slowly reducing the confinement strength. As the delocalization of carriers keeps decreasing, they become less influenced by the shell thickness [6], as shown in Figs. 4 and 7. Acknowledgment We would like to thank Ministry of Education Malaysia for supporting this research through the Fundamental Research Grant Scheme (FRGS: 5540125). 102
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
References [1] U. Woggon, S.V. Gaponenko, Excitons in quantum dots, Phys. Status Solidi b 189 (1995) 285–343, http://dx.doi.org/10.1515/ling.1995.33.1.123. [2] T. Cheche, V. Barna, I. Stamatin, Theoretical approach for type-i semiconductor spherical core-shell quantum dots heterostructure with wide band gaps, J. Optotoelectron. Adv. M. 15 (7–8) (2013) 615–620. [3] C. Vantakhah, A. Ebadi, Quantum size effects on effective mass and band gap of semiconductor quantum dots, Res. J. Recent Sci. 2 (1) (2013) 21–24. [4] G. Jia, B. Hao, X. Lu, J. Yao, Size effect of heterostructure type in core/shell quantum dot, Int. J. Electrochem. Sci 8 (2013) 8167–8174. [5] F. Poulsen, T. Hansen, Band gap energy of gradient core-shell quantum dots, J. Phys. Chem. C 121 (25) (2017) 13655–13659, http://dx.doi.org/10.1021/ acs.jpcc.7b01792. [6] R. Kostic, D. Stojanovic, Optical properties of CdTe/ZnTe core/shell quantum dots suitable for targeted bioimaging, Optoelectron. Adv. Mat. 6 (1–2) (2012) 121–125. [7] A. Chafai, F. Dujardin, I. Essaoudi, A. Ainane, Energy spectrum of an exciton in a CdSe/ZnTe type-II core/shell spherical quantum dot, Superlattices Microstruct. 101 (2017) 40–48, http://dx.doi.org/10.1016/j.spmi.2016.11.017. [8] L.K. Sagar, W. Walravens, Q. Zhao, A. Vantomme, P. Geiregat, Z. Hens, PbS/CdS Core/shell quantum dots by additive, layer-by-layer shell growth, Chem. Mater. 28 (19) (2016) 6953–6959, http://dx.doi.org/10.1021/acs.chemmater.6b02638. [9] K. Boldt, S. Jander, K. Hoppe, H. Weller, Characterization of the organic ligand shell of semiconductor quantum dots by fluorescence quenching experiments, ACS Nano 5 (10) (2011) 8115–8123, http://dx.doi.org/10.1021/nn202748v. [10] J. Vela, H. Htoon, Y. Chen, Y.-S. Park, Y. Ghosh, P.M. Goodwin, J.H. Werner, N.P. Wells, J.L. Casson, J.A. Hollingsworth, Effect of shell thickness and composition on blinking suppression and the blinking mechanism in ‘giant’ CdSe/CdS nanocrystal quantum dots, J. Biophotonics 3 (10–11) (2010) 706–717, http://dx.doi.org/10.1002/jbio.201000058. [11] D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures, John Wiley & Sons, 1999. [12] K. Gong, D.F. Kelley, A predictive model of shell morphology in CdSe-CdS core-shell quantum dots, J. Chem. Phys. 141 (2014) 194704, http: //dx.doi.org/10.1063/1.4901428. [13] D. Schooss, A. Mews, A. Eychmuller, H. Weller, Quantum dot quantum well CdS/HgS/CdS: Theory and experiment, Phys. Rev. B 49 (24) (1994) 17072, http://dx.doi.org/10.1103/PhysRevB.49.17072. [14] E.J. Tyrrell, J.M. Smith, Effective mass modeling of excitons in type-II quantum dot heterostructures, Phys. Rev. B 84 (16) (2011) http://dx.doi.org/10. 1103/PhysRevB.84.165328. [15] C. Pryor, Eight-band calculations of strained inas/gaas quantum dots compared with one-, four-, and six-band approximations, Phys. Rev. B 57 (12) (1998) 7190–7195, http://dx.doi.org/10.1103/PhysRevB.57.7190. [16] W. Sheng, S.-J. Cheng, P. Hawrylak, Multiband theory of multi-exciton complexes in self-assembled quantum dots, Phys. Rev. B 71 (3) (2005) 17, http://dx.doi.org/10.1103/PhysRevB.71.035316. [17] C. Fang, W.-F. Li, R.S. Koster, J. Klimeš, A. van Blaaderen, M.A. van Huis, The accurate calculation of the band gap of liquid water by means of gw corrections applied to plane-wave density functional theory molecular dynamics simulations, Phys. Chem. Chem. Phys. 17 (2015) 365–375, http://dx.doi.org/10.1039/C4CP04202F. [18] L. Brus, Electronic wave functions in semiconductor clusters: Experiment and theory, J. Phys. Chem. C 90 (12) (1986) 2555–2560, http://dx.doi.org/10. 1021/j100403a003. [19] U. K. Mishra, J. Singh, Semiconductor Device Physics and Design, Springer Netherlands, Dordrecht, 2008, http://dx.doi.org/10.1007/978-1-4020-6481-4. [20] O. Madelung, U. Rossler, M. Schulz, II-VI And I-VII compounds; semimagnetic compounds: Cadmium sulfide (CdS) dielectric constants, Springer Berlin Heidelberg, Berlin, Heidelberg, 1999, http://dx.doi.org/10.1007/10681719_568.
103
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/yspmi
Effects of step-potential on confinement strength of strain-induced type-I core–shell quantum dots T. Shelawati b ,∗, M.S. Nurisya a,b , C. Kar Tim a,b , A.K. Mazliana b a
Laboratory of Computational Sciences & Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia b Department of Physics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
ARTICLE
INFO
Keywords: Quantum dots Colloidal quantum dots Transition energy Bessel spherical functions
ABSTRACT In this paper, the transition energy between lowest unoccupied molecular orbital (LUMO) of conduction band and highest occupied molecular orbital (HOMO) of valence band for band structures of type-I core–shell quantum dots (CSQDs) within a strong and weak confinements of charge carriers are estimated using the effective mass approximation together with single-band model. The effect of potential step at the conduction and valence bands on the confinement strength is then properly discussed. Our numerical results show that for a same size of CSQDs, the one with bigger potential steps will have stronger carriers’ confinement with more localized excitons.
1. Introduction In recent years, research field has witnessed vigorous studies in nanoscale semiconductors, ranging from quantum wells (2-dimensional), quantum wires (1-dimensional) and quantum dots (0-dimensional), where their charge carriers are spatially confined in one degree, two degrees and three degrees respectively. Quantum dots or sometimes dubbed as ‘‘artificial atoms’’ or ‘‘hyperatoms’’ [1] due to their atomic scale size, have shown great potential in agriculture and medicine fields as sensors [2], solar panels [2–4], bio-tagging materials [3–6] or light-emitting devices [3,4]. It has long established that their nano-scale size which is comparable to Bohr radius gives rise to a strong localization of charge carriers compared to their bulk counterparts. The small size property of quantum dots causes it to have a high surface to volume ratio. For instance, quantum dots with radius of 5 nm can still have 104 atoms. This has made its surface to play a significant role, which sometimes can be considered as a drawback due to possibility of having defects on its surface [6,7]. However, many research have discovered that it is possible to cap or encapsulate these quantum dots surfaces with another organic or inorganic semiconductor materials in order to enhance their optical and electrical properties as described in [4,8–10]. This novel finding, known as core–shell quantum dots (CSQDs) was further studied to distinguish their types based on regional confinement of electrons and holes as well as their band edges alignment. For type-I CSQDs, the charge carriers are confined only in core material or only in shell material (inverse type-I CSQDs). As for type-II CSQDs, the charge carriers are confined between conduction band of core material and valence band of shell material or between conduction band of shell material and valence band of core material (inverse type-II). There are three well-known ways of heteroepitaxial growth for CSQDs fabrication which are adding shell compound layer-by-layer method onto the core surface (Frank– van der Merwe), layer-by-layer followed with additional islands formation (Stranski–Krastanov) or just islands formation directly on the core surface (Volmer–Weber). All these passivating modes are governed by their interface energies and lattice mismatch [11,12].
∗ Corresponding author. E-mail addresses: [email protected] (T. Shelawati), [email protected] (M.S. Nurisya), [email protected] (C. Kar Tim), [email protected] (A.K. Mazliana).
https://doi.org/10.1016/j.spmi.2019.05.021 Received 2 January 2019; Received in revised form 4 April 2019; Accepted 10 May 2019 Available online 16 May 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
The motivations of passivating quantum dots surfaces are to provide electrical and chemical passivation [5,6] that will improve photostability of the quantum dots against oxidation [5,7] as well as to increase the confinement of charge carriers and making its amount of energy for carriers transition more tunable [2,5–7,13]. Inorganic capping materials can also reduce the chemical activity due to cationic and anionic dangling bonds [6]. However, increasing the thickness of this shell layer above its critical thickness may cause the shell to impair the whole core–shell quantum dots instead of improving its photoluminescene [12]. Hence, a study to investigate relation between shell thickness and transition energy between HOMO of conduction band and LUMO of valence band of CSQDs is very crucial. Different approaches had been considered such as single-band model [2,4–7,13,14], multiband theory (8-band model) [14–16], envelope function [5], density-functional theory (DFT) [3] and model-solid theory [2]. In this paper, instead of just depending on their core sizes as suggested in [1] and [11], we will apply single-band model onto our CSQDs by introducing two different types of confinement; strong confinement and weak confinement model that are based on strengths of potential steps on conduction and valence bands. This paper is organized as follows. First, single-band approach on energy transition in CSQDs will be briefly discussed and then applied on CSQDs with strong confinement, followed with CSQDs with weak confinement. Subsequently, numerical results is discussed by observing the radial probabilities in both strong and weak confinement cases.
2. Theory 2.1. Single-band model Consider a colloidal CSQDs as spherical quantum dots encapsulated by a layer of a different compound of semiconductor (see Fig. 1). As shell is added, there will be step-potential inducing strain on the core quantum dots. Following the usual method to solve a stationary Schödinger equation for a particle with some step-potential, one have to separate the wavefunction into radial part, 𝑅𝑛𝑙 and angular part, 𝑌𝑙𝑚 as (1)
𝛹𝑛𝑙𝑚 (⃗𝑟, 𝜃, 𝜙) = 𝑅𝑛𝑙 (𝑟)𝑌𝑙𝑚 (𝜃, 𝜙), where 𝑛, 𝑙 and 𝑚 are the principal, orbital and magnetic quantum numbers respectively.
The only interested solution is the solution of radial wavefunction for lowest transitions energy, 𝐸qd (𝑛 = 1 and 𝑙 = 𝑚 = 0), in other words 1𝑠𝑒 → 1𝑠ℎ transition. To solve energy for electrons, 𝐸𝑒 and holes, 𝐸ℎ , we could benefit from spherical Bessel functions (for the case 𝐸𝑒,ℎ > 𝑉𝑐,𝑣 ) or modified spherical Bessel functions (for the case 𝐸𝑒,ℎ < 𝑉𝑐,𝑣 ) [2].
Fig. 1. Schematic diagram of band line-up of core–shell QDs. 96
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
𝐸𝑐(𝑣) and 𝑉𝑐(𝑣) are the energy and potential step for conduction(valence) band respectively, 𝑟𝑜 and 𝑟𝑠 are radius of core quantum dots and radius of CSQDs respectively. For carriers outside the shell layer, the corresponding 𝑉 (𝑟) is infinite. ⎧ 0 ⎪ 𝑉 (𝑟) = ⎨𝑉𝑐,𝑣 ⎪∞ ⎩
0 ≤ 𝑟 ≤ 𝑟𝑜 𝑟𝑜 ≤ 𝑟 ≤ 𝑟𝑠 𝑟 ≥ 𝑟𝑠 .
; ; ;
For 𝐸𝑒,ℎ > 𝑉𝑐,𝑣 ; ⎧ 𝑅1 (𝑟) = 𝐴𝑗𝑙 (𝑘1 𝑟) ⎪ 𝑅𝑖 (𝑟) = ⎨𝑅2 (𝑟) = 𝐵𝑗𝑙 (𝑘2 𝑟) + 𝐶𝑛𝑙 (𝑘2 𝑟) ⎪ 𝑅3 (𝑟) = 0 ⎩
; ; ;
0 ≤ 𝑟 ≤ 𝑟𝑜 𝑟𝑜 ≤ 𝑟 ≤ 𝑟𝑠 𝑟 ≥ 𝑟𝑠 .
(2)
; ; ;
0 ≤ 𝑟 ≤ 𝑟𝑜 𝑟𝑜 ≤ 𝑟 ≤ 𝑟𝑠 𝑟 ≥ 𝑟𝑠 .
(3)
For 𝐸𝑒,ℎ < 𝑉𝑐,𝑣 ; ⎧ 𝑅1 (𝑟) = 𝐴𝑗𝑙 (𝑘1 𝑟) ⎪ 𝑅𝑖 (𝑟) = ⎨𝑅2 (𝑟) = 𝐵ℎ𝑙 (𝑘′2 𝑟) + 𝐶𝜅𝑙 (𝑘′2 𝑟) ⎪ 𝑅3 (𝑟) = 0 ⎩
Note that 𝑗𝑙 (𝑘𝑖 ), 𝑛𝑙 (𝑘𝑖 ), ℎ𝑙 (𝑘𝑖 ) and 𝜅𝑙 (𝑘𝑖 ) are spherical Bessel function of the first kind, Neumann spherical functions, √ modified ∗ spherical Bessel function of the first kind and modified spherical Bessel function of the second kind respectively. 𝑘1 = √ ∗ √ ∗ 2𝑚2 [𝐸𝑒,ℎ −𝑉𝑐,𝑣 (𝑟)]
𝑘′2
2𝑚2 [𝑉𝑐,𝑣 (𝑟)−𝐸𝑒,ℎ ]
= 𝑘2 = , ℏ ℏ continuity of wavefunctions, hence
and
𝑚∗𝑖
is the effective mass of electrons,
𝑚𝑒𝑖
or effective mass of holes,
𝑚ℎ𝑖 .
2𝑚1 𝐸𝑒,ℎ ℏ
,
Following
𝑅𝑖 (𝑟) = 𝑅𝑖+1 (𝑟) 1 ′ 1 𝑅 (𝑟) = 𝑅′ (𝑟), 𝑚𝑖 𝑖 𝑚𝑖+1 𝑖+1 where 𝑖 = 1 for core, 𝑖 = 2 for shell and 𝑖 = 3 for outside shell layer. Following the said boundary conditions, Eqs. (2) and (3) will form three sets of equations that subsequently solve for 𝑟 normalization constants A, B and C which satisfy ∫0 4𝜋𝑟2 𝑅2𝑙 (𝑟)𝑑𝑟 = 1. For 𝐸𝑒,ℎ > 𝑉𝑐,𝑣 ; ⎡ 𝑗0 (𝑘1 𝑟𝑜 ) ⎢ 1 ′ ⎢ 𝑚∗1 𝑗0 (𝑘1 𝑟𝑜 ) ⎢ 0 ⎣
−𝑗0 (𝑘2 𝑟𝑜 ) − 𝑚1∗ 𝑗0′ (𝑘2 𝑟𝑜 ) 2
𝑗0 (𝑘2 𝑟𝑠 )
−𝑛0 (𝑘2 𝑟𝑜 ) ⎤ ⎡𝐴⎤ ⎥ − 𝑚1∗ 𝑛′0 (𝑘2 𝑟𝑜 )⎥ ⎢𝐵 ⎥ = 0. ⎢ ⎥ 2 𝑛0 (𝑘2 𝑟𝑠 ) ⎥⎦ ⎣𝐶 ⎦
(4)
−𝜅0 (𝑘′2 𝑟𝑜 ) ⎤ ⎡𝐴⎤ ⎥ − 𝑚1∗ 𝜅0′ (𝑘′2 𝑟𝑜 )⎥ ⎢𝐵 ⎥ = 0. ⎢ ⎥ 2 𝜅0 (𝑘′2 𝑟𝑠 ) ⎥⎦ ⎣𝐶 ⎦
(5)
For 𝐸𝑒,ℎ < 𝑉𝑐,𝑣 ; ⎡ 𝑗0 (𝑘1 𝑟𝑜 ) ⎢ 1 ′ ⎢ 𝑚∗1 𝑗0 (𝑘1 𝑟𝑜 ) ⎢ 0 ⎣
−ℎ0 (𝑘′2 𝑟𝑜 ) − 𝑚1∗ ℎ′0 (𝑘′2 𝑟𝑜 ) 2
ℎ0 (𝑘′2 𝑟𝑠 )
Eqs. (4) and (5) will form the transcendental equations that obey det(𝐸𝑒,ℎ ) = 0. Our transcendental equations then can be written as 𝑚∗2 𝑗0′ (𝑘1 𝑟𝑜 ) 𝑚∗1
𝑗0 (𝑘1 𝑟𝑜 )
=
𝑗0′ (𝑘2 𝑟𝑜 )𝑛0 (𝑘2 𝑟𝑠 ) − 𝑗0 (𝑘2 𝑟𝑠 )𝑛′0 (𝑘2 𝑟𝑜 ) 𝑗0 (𝑘2 𝑟𝑜 )𝑛0 (𝑘2 𝑟𝑠 ) − 𝑗0 (𝑘2 𝑟𝑠 )𝑛0 (𝑘2 𝑟𝑜 )
(6)
,
or 𝑚∗2 𝑗0′ (𝑘1 𝑟𝑜 ) 𝑚∗1
𝑗0 (𝑘1 𝑟𝑜 )
=
ℎ′0 (𝑘′2 𝑟𝑜 )𝜅0 (𝑘′2 𝑟𝑠 ) − ℎ0 (𝑘′2 𝑟𝑠 )𝜅0′ (𝑘′2 𝑟𝑜 ) ℎ0 (𝑘′2 𝑟𝑜 )𝜅0 (𝑘′2 𝑟𝑠 ) − ℎ0 (𝑘′2 𝑟𝑠 )𝜅0 (𝑘′2 𝑟𝑜 )
(7)
.
One will obtain 𝐸𝑒,ℎ after solving Eq. (6) or (7). As we need to solve for both electron energy and hole energy separately, the use of superscript will be handy to denote whether we are working on the electron states (𝑅𝑒𝑖 ) or hole states (𝑅ℎ𝑖 ). In case of quantum dots without shell, we can consider it to be in water environment with band gap 7.3 eV with charge carriers of free-electron mass [17]. The lowest transition energy, 𝐸qd for CSQDs will be 𝐸qd = 𝐸𝑔𝑏𝑢𝑙𝑘 + 𝐸𝑒 + 𝐸ℎ −
1.8𝑒2 . 4𝜋𝜀𝑜 𝜀𝑟 𝑟𝑜
(8) 2
1.8𝑒 Here, we treated the Coulomb interaction term ( 4𝜋𝜀 ) as first order perturbation that rises due to interactions between charge 𝑜 𝜀𝑟 𝑟𝑜 carriers. Our lowest eigenvalue is not very sensitive with this term (and often neglected) as it only has 1∕𝑟 dependence against our confinement energy term [2,18].
97
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
3. Application in CSQDs with strong confinement In this part, the above theoretical approach is applied to a CSQDs model that has a core with narrow bang gap and encapsulated with a much wider band gap shell compound. As a result, the step-potentials will be much bigger than the core bulk band gap (𝑉𝑐,𝑣 ≫ 𝐸𝑔𝑏𝑢𝑙𝑘 ), implying a strong excitons confinement within the core material. For this purpose, we chose PbS/CdS core–shell QDs with material parameters as shown in Table 1 (see Figs. 2–4). Table 1 List of parameters for PbS/CdS core–shell QDs. Materials
PbS [19]
CdS [19,20]
Band gap (eV) 𝑚𝑒 ∕𝑚0 𝑚ℎ ∕𝑚0 Dielectric constant, 𝜀𝑟
0.41 0.08 0.08 169
2.5 0.2 0.7 8.73
𝑉𝑐 (eV) 𝑉𝑣 (eV)
1.254 0.836
Fig. 2. Energy of electrons for PbS/CdS.
Fig. 3. Energy of holes for PbS/CdS.
4. Application in CSQDs with weak confinement In the second part, the single-band model is applied to a wide core QDs encapsulated with a slightly wider shell compound, where the step-potentials will be much smaller than the core bulk band gap (𝑉𝑐,𝑣 ≪ 𝐸𝑔𝑏𝑢𝑙𝑘 ) to imply a weak excitons confinement within the core material. For this purpose, we choose ZnTe/ZnSe core–shell QDs with material parameters as shown in Table 2 (see Figs. 5–7). 98
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
Fig. 4. Transition energy for PbS/CdS CSQDs.
Table 2 List of parameters for ZnTe/ZnSe core–shell QDs. Materials
ZnTe [4,19]
ZnSe [4,19]
Band gap (eV) 𝑚𝑒 ∕𝑚0 𝑚ℎ ∕𝑚0 Dielectric constant, 𝜀𝑟
2.394 0.11 0.7 8.7
2.8215 0.14 0.6 9.1
𝑉𝑐 (eV) 𝑉𝑣 (eV)
0.2565 0.171
Fig. 5. Energy of electrons for ZnTe/ZnSe.
5. Analysis on radial probability 5.1. Strong confinement case For the case of PbS quantum dots (without shell) with radius of 4.00 nm and 4.25 nm, there is no significant difference in radial probability of finding charge carriers within the cores as shown in Fig. 8, although the transition energy, 𝐸qd is higher in 4.00 nm quantum dots compared to 4.25 nm quantum dots. 99
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
Fig. 6. Energy of holes for ZnTe/ZnSe.
Fig. 7. Transition energy for ZnTe/ZnSe CSQDs.
After adding 0.25 nm of CdS shell layer onto the 4.00 nm PbS quantum dots (CSQDs radius = 4.25 nm), as shown in Fig. 9, the radial probability of charge carriers are shifted more to the center of CSQDs, signaling a stronger localization compared to the 4.25 nm PbS quantum dots. This stronger carriers localization which translate into stronger carriers confinement is proven by higher transition energy in 4.25 nm PbS/CdS CSQDs (4.00 nm core radius with 0.25 nm shell thickness) compared to transition energy in 4.25 nm PbS quantum dots. But as we increase the thickness layer of CdS, the radial probability of finding the charge carriers started to spread into the shell region, as shown in Fig. 10. This spreading of probability of radial wavefunction is caused by defects on interface area, causing delocalization of charge carriers where it becomes easier for charge carrier to tunnel through the interface and be in the shell region [12] (see Fig. 11). 5.2. Weak confinement case For ZnTe quantum dots that has wide bulk band gap, the confinement of charge carrier is not too obvious, leading to smaller increasing factor in transition energy compared to narrow band gap case. In narrow band gap, the energy for quantum dots almost doubled, while in wide band gap, the increasing in 𝐸qd is less visible. It is worthy to note that group II–VI semiconductor is less likely to have a strong carriers confinement [1,11]. However, after adding 0.25 nm of ZnSe shell layer, the transition energy of ZnTe/ZnSe CSQDs is increasing by more than 5%. From Fig. 12, it shows that the probabilities of the carriers in shell region are relatively higher compared with strong confinement case as in Fig. 9. This shows weaker localization of charge carriers. 100
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
Fig. 8. Radial probability of carriers in (top) 4.00 nm PbS quantum dots and (bottom) 4.25 nm PbS quantum dots where they show no significant difference.
Fig. 9. Drastic change in radial probability of carriers after adding 0.25 nm CdS shell layer on 4.00 nm PbS core.
Fig. 10. Radial probability of carriers in 4.50 nm PbS/CdS with 4.00 nm core radius.
The probability of charge carrier is spreading more into shell region as shown in Fig. 13 as we increase thickness shell layer. This is due to strong strain energy imposed on the core quantum dots, causing lattice mismatch to be significant in order to minimize the strain on interface area [12]. 101
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
Fig. 11. Radial probability of carriers in 4.00 nm ZnTe quantum dots.
Fig. 12. Radial probability of carriers in 4.25 nm ZnTe/ZnSe with 4.00 nm core radius.
Fig. 13. Radial probability of carriers in 4.50 nm ZnTe/ZnSe with 4.00 nm core radius.
6. Conclusion In this work, we have shown that by varying the potential step on a same size CSQDs, we can produce different types of confinement; a strong confinement and a weak confinement. This is important factor to consider for fabrication of CSQDs as instead of relying on size factor to tune the confinement strength, we can also control it by manipulating their potential step by introducing a capping material. Increasing shell thickness will increase photoluminescene of CSQDs as this will increase confinement of electrons and holes [6]. However, as it reaching its critical thickness, the strain on interface will be significant enough to induce lattice mismatch in order to minimize the strain [12]. The high energy excitons now can tunnel easily through the core–shell interface, causing further delocalization, hence slowly reducing the confinement strength. As the delocalization of carriers keeps decreasing, they become less influenced by the shell thickness [6], as shown in Figs. 4 and 7. Acknowledgment We would like to thank Ministry of Education Malaysia for supporting this research through the Fundamental Research Grant Scheme (FRGS: 5540125). 102
Superlattices and Microstructures 131 (2019) 95–103
T. Shelawati et al.
References [1] U. Woggon, S.V. Gaponenko, Excitons in quantum dots, Phys. Status Solidi b 189 (1995) 285–343, http://dx.doi.org/10.1515/ling.1995.33.1.123. [2] T. Cheche, V. Barna, I. Stamatin, Theoretical approach for type-i semiconductor spherical core-shell quantum dots heterostructure with wide band gaps, J. Optotoelectron. Adv. M. 15 (7–8) (2013) 615–620. [3] C. Vantakhah, A. Ebadi, Quantum size effects on effective mass and band gap of semiconductor quantum dots, Res. J. Recent Sci. 2 (1) (2013) 21–24. [4] G. Jia, B. Hao, X. Lu, J. Yao, Size effect of heterostructure type in core/shell quantum dot, Int. J. Electrochem. Sci 8 (2013) 8167–8174. [5] F. Poulsen, T. Hansen, Band gap energy of gradient core-shell quantum dots, J. Phys. Chem. C 121 (25) (2017) 13655–13659, http://dx.doi.org/10.1021/ acs.jpcc.7b01792. [6] R. Kostic, D. Stojanovic, Optical properties of CdTe/ZnTe core/shell quantum dots suitable for targeted bioimaging, Optoelectron. Adv. Mat. 6 (1–2) (2012) 121–125. [7] A. Chafai, F. Dujardin, I. Essaoudi, A. Ainane, Energy spectrum of an exciton in a CdSe/ZnTe type-II core/shell spherical quantum dot, Superlattices Microstruct. 101 (2017) 40–48, http://dx.doi.org/10.1016/j.spmi.2016.11.017. [8] L.K. Sagar, W. Walravens, Q. Zhao, A. Vantomme, P. Geiregat, Z. Hens, PbS/CdS Core/shell quantum dots by additive, layer-by-layer shell growth, Chem. Mater. 28 (19) (2016) 6953–6959, http://dx.doi.org/10.1021/acs.chemmater.6b02638. [9] K. Boldt, S. Jander, K. Hoppe, H. Weller, Characterization of the organic ligand shell of semiconductor quantum dots by fluorescence quenching experiments, ACS Nano 5 (10) (2011) 8115–8123, http://dx.doi.org/10.1021/nn202748v. [10] J. Vela, H. Htoon, Y. Chen, Y.-S. Park, Y. Ghosh, P.M. Goodwin, J.H. Werner, N.P. Wells, J.L. Casson, J.A. Hollingsworth, Effect of shell thickness and composition on blinking suppression and the blinking mechanism in ‘giant’ CdSe/CdS nanocrystal quantum dots, J. Biophotonics 3 (10–11) (2010) 706–717, http://dx.doi.org/10.1002/jbio.201000058. [11] D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures, John Wiley & Sons, 1999. [12] K. Gong, D.F. Kelley, A predictive model of shell morphology in CdSe-CdS core-shell quantum dots, J. Chem. Phys. 141 (2014) 194704, http: //dx.doi.org/10.1063/1.4901428. [13] D. Schooss, A. Mews, A. Eychmuller, H. Weller, Quantum dot quantum well CdS/HgS/CdS: Theory and experiment, Phys. Rev. B 49 (24) (1994) 17072, http://dx.doi.org/10.1103/PhysRevB.49.17072. [14] E.J. Tyrrell, J.M. Smith, Effective mass modeling of excitons in type-II quantum dot heterostructures, Phys. Rev. B 84 (16) (2011) http://dx.doi.org/10. 1103/PhysRevB.84.165328. [15] C. Pryor, Eight-band calculations of strained inas/gaas quantum dots compared with one-, four-, and six-band approximations, Phys. Rev. B 57 (12) (1998) 7190–7195, http://dx.doi.org/10.1103/PhysRevB.57.7190. [16] W. Sheng, S.-J. Cheng, P. Hawrylak, Multiband theory of multi-exciton complexes in self-assembled quantum dots, Phys. Rev. B 71 (3) (2005) 17, http://dx.doi.org/10.1103/PhysRevB.71.035316. [17] C. Fang, W.-F. Li, R.S. Koster, J. Klimeš, A. van Blaaderen, M.A. van Huis, The accurate calculation of the band gap of liquid water by means of gw corrections applied to plane-wave density functional theory molecular dynamics simulations, Phys. Chem. Chem. Phys. 17 (2015) 365–375, http://dx.doi.org/10.1039/C4CP04202F. [18] L. Brus, Electronic wave functions in semiconductor clusters: Experiment and theory, J. Phys. Chem. C 90 (12) (1986) 2555–2560, http://dx.doi.org/10. 1021/j100403a003. [19] U. K. Mishra, J. Singh, Semiconductor Device Physics and Design, Springer Netherlands, Dordrecht, 2008, http://dx.doi.org/10.1007/978-1-4020-6481-4. [20] O. Madelung, U. Rossler, M. Schulz, II-VI And I-VII compounds; semimagnetic compounds: Cadmium sulfide (CdS) dielectric constants, Springer Berlin Heidelberg, Berlin, Heidelberg, 1999, http://dx.doi.org/10.1007/10681719_568.
103