Stokes shift and fine structure splitting in composition-tunable ZnxCd1−xSe nanocrystals: Atomistic tight-binding theory

Author’s Accepted Manuscript Stokes shift and fine structure splitting in composition-tunable ZnxCd1-xSe Nanocrystals: Atomistic tight-binding theory Worasak Sukkabot www.elsevier.com/locate/physb

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S0921-4526(16)30554-3 http://dx.doi.org/10.1016/j.physb.2016.11.023 PHYSB309727

To appear in: Physica B: Physics of Condensed Matter Received date: 3 October 2016 Revised date: 1 November 2016 Accepted date: 17 November 2016 Cite this article as: Worasak Sukkabot, Stokes shift and fine structure splitting in composition-tunable ZnxCd1-xSe Nanocrystals: Atomistic tight-binding theory, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2016.11.023 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.

Stokes shift and fine structure splitting in composition-tunable ZnxCd1-xSe Nanocrystals: Atomistic tight-binding theory Worasak Sukkabota,* a

Department of Physics, Faculty of Science, Ubon Ratchathani University, 85

Sathollmark Rd. Warinchamrab, Ubon Ratchathani, Thailand, 34190 *

Corresponding author. Tel.: +6684535-3401-4; fax: +66845-288-381. [email protected]

Abstract

I report on the atomistic correlation of the structural properties and excitonic splitting of ternary alloy Zn xCd1-xSe wurtzite nanocrystals using the sp3s* empirical tight-binding method with the description of the first nearest neighbouring interaction and bowing effect. Based on a successful model, the computations are presented under various Zn compositions (x) and diameters of alloy ZnxCd1-xSe nanocrystals with the experimentally synthesized compositions and sizes. With increasing Zn contents (x), the optical band gaps and electron-hole coulomb energies are improved, while ground electron-hole wave function overlaps, electron-hole exchange energies, stokes shift and fine structure splitting are reduced. A composition-tunable emission from blue to yellow wavelength is obviously demonstrated. The optical band gaps, ground electron-hole wave function overlaps, electron-hole interactions, stokes shift and fine structure splitting are progressively decreased with the increasing diameters. Alloy ZnxCd1-xSe nanocrystal with Zn rich and large diameter is the best candidate to optimistically be used as a source of entangled photon pairs. The agreement with the experimental data is remarkable. Finally, the present systematic study on the structural properties and excitonic splitting predominantly opens a new perspective to understand the size- and compositiondependent properties of ZnxCd1-xSe nanocrystals with a comprehensive strategy to design the optoelectronic devices.

Keywords: Tight-binding theory, ZnxCd1-xSe, alloy, nanocrystals, stokes shift, fine structure splitting ;

1. Introduction As an exceptional realization of the semiconductor nanocrystals, semiconductor nanocrystals can act as the alternative candidates in a broad range of applications such as optoelectronic devices [1, 2, 3, 4], solar cells [5, 6, 7, 8], chemical sensors [9] and biological labels [10, 11, 12, 13] owing to their unique structural and optical properties. A general methodology to directly tune the absorption spectrum is accessible through changing the size of the

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nanocrystals. This technique has expansively been used with II-VI semiconductor nanocrystals in the framework of experimental [14, 15, 16, 17, 18, 19] and theoretical studies [20, 21, 22, 23, 24, 25, 26]. However, these semiconductor nanocrystals are still not very easy to deliver the stable blue-yellow emission between 450 and 600 nm because of the surface passivation and size control. It is expected that these regions of the wavelengths are of particular interest for the preparation of high-quality blue and blue-yellow light-emitting diodes (LEDs) for the nextgeneration displays. An alternative is the convention of alloy semiconductor nanocrystals of the type-ternary A1xBxC

with varying the size and the contents (x). Therefore, it is still very important to exploit the synthesis methods

and theoretical studies to assist for the requirement as demonstrating in the following. D. Mourad and G. Czycholl [27] used the multi-band sp3 empirical tight-binding model in the conjunction with the configuration interaction scheme to calculate the optical properties of CdxZn1-xSe spherical nanocrystals. A very good agreement between experimental data and their calculations was obtained. By varying the predetermined amounts of the reaction precursors, Xinhua Zhong et al. [28, 29] synthesized reproducibly and precisely the high-quality ZnxCd1-xSe nanocrystals. With increasing Zn contents, the optical spectra were adjusted in the short wavelength spectral region from 420 to 500 nm, which is of special interest for the preparation of nanocrystal-based blue LEDs and white light generation. In addition, Xinhua Zhong et al. [30] synthesized high-quality alloy ZnxCd1-xS wurtzite nanocrystals. The optical spectra were systematically across the visible spectrum from 391 to 474 nm with the increasing Zn content. As the experimental results, alloy ZnxCd1-xS nanocrystals could be very encouraging candidates as biological labels and short wavelength optoelectronic devices. The technique to synthesize a series of Zinc Oxysulfide (ZOS) (ZnO1-xSx; 0  ≤  x  ≥  1; x  =  Sulfur) alloys nanoparticles was reported by Shiv K. Pandey et al. [31]. Nilanka P. Gurusinghe et al. [32] fabricated the ternary CdSxTe1-x nanocrystals via pyrolysis of organometallic precursors. The optical band gaps of alloy CdS xTe1-x were highly nonlinear with the increasing alloy compositions. The significant applications of these alloy nanocrystals were in the areas of biomedical imaging, solar cells and quantum dot-based LEDs. In spite of the importance of the alloy systems, detailed studies of ternary alloy nanocrystals have been only theoretically reported by D. Mourad and G. Czycholl, while numerous experimental studies on ternary alloy nanocrystals have been extensively reported. On the basis of D. Mourada and G. Czycholl’s investigation, there are still lacking studied topics. The atomistic correlation among electronic structures, excitonic splitting, sizes and alloy

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compositions needs to be demonstrated. To exceptionally realize the alloy application in the field of quantum information, the splitting of the excitonic states must be computed in the combination of the ground electron-hole overlaps and atomistic electron-hole interactions. In addition, the comparison between my model and recently experimental data is presented. To achieve the purpose of the study, atomistic tight-binding approach is a practical method to describe the electronic structures of alloy semiconductors, in which the Hamiltonian matrix elements are represented by the Slater and Koster parameters with five orbitals sp3s* per atom as well as the first nearest neighbouring interaction [33]. To consider an alloy content in semiconductor nanocrystals, I theoretically implement the virtual crystal approximation (VCA). Owing to the non-linear dependent compositions, an empirical bowing parameter is utilized to compensate this effect via the s orbital on-site term [34]. By using the structural parameters from Xinhua Zhong et al. [29], ZnxCd1-xSe nanocrystals with the experimentally synthesized diameters and Zn contents (x) are modeled by the combination of atomistic tight-binding theory and configuration interaction description. To scrutinize the impact of sizes and Zn compositions (x) on the electronic structures and splitting of the excitonic states, I numerically compute single-particle spectra, optical band gaps, ground electron-hole wave function overlaps, electron-hole interactions, stokes shift and fine structure splitting. In addition, the comparison between atomistic tight-binding result and experimental data is highlighted to confirm such calculations. To examine how the dimensions and alloy compositions (x) mainly affect the structural properties and excitonic splitting of ZnxCd1-xSe nanocrystals, this paper is organized as follows. In Section 2, the numerical procedure is briefly elucidated. Using a simple but effective computational method based on the empirical tightbinding theory, I can suggest that the alloy ZnxCd1-xSe nanocrystals are structurally distinguishable. Based on the observations, alloy ZnxCd1-xSe nanocrystal with Zn rich and large diameter is the best candidate to be a source of entangled photon pairs. In addition, the agreement between the tight-binding calculations and the experimental data is notable. The present work is expected to guide the physical behaviours engineering by sizes and alloy compositions. Finally, the resulting computations are concluded in Section 4.

2. Theory For the quantitative analysis, I use a simple but effective computational method based on the empirical

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tight-binding theory to calculate the electronic properties and splitting of excitonic states in alloy ZnxCd1-xSe wurtzite nanocrystals with different Zn compositions (x) and diameters. Wurtzite structure is generated by hexagonal closely packed lattice (HCP) with four atoms per unit cell. The atomic positions of the nanocrystal with the desired diameter are built up by cutting a spherical shape out of a bulk semiconductor without the presence of the surface relaxation. The surface is theoretically passivated by dangling bonds as described by the technique of S. Lee et al. [35] in order to evade the gaps states. In the experiment, the surface of the nanocrystal is terminated by organic ligands. Owing to the flexible surrounding materials, the nanocrystals are assumed to be nearly unstrained [36]. Consequently, one s orbital, three p orbitals and one s* per spin component localized on the sites R in the alloy ZnxCd1-xSe wurtzite nanocrystals are generated as the wave functions.

R ,  {s , px , p y , pz ,s* ,s , px , p y , pz ,s* } The tight-binding matrix elements of the Hamiltonian are then given by. RR ' ' E '  R  H TB R '

The empirical tight-binding Hamiltonian ( H TB ) is described by the operator c†R (cR ) creates (annihilates) the particle on the orbital  of atom R with the on-site orbital energies  R , the spin-orbit coupling constant  R ' and the hopping matrix elements t R,R '  ' as given by. Nat 10

Nat 10

R 1 1

R 1 1  ' 1

10

Nat Nat

10

10

HTB   R c†R cR   R ' c†R cR '    t R,R '  ' c†R cR '  ' R 1 R ' 1 1  ' 1

In order to fit the bulk band structure, the bulk band gap, the effective mass and the spin-orbit splitting, the parameterization scheme presented by D. Olguin [34] is implemented. To include alloy composition in ZnxCd1-xSe nanocrystals, the virtual crystal approximation (VCA) is proposed to be used. By means of the virtual crystal approximation, the tight-binding parameters of a ternary alloy ZnxCd1-xSe semiconductor are given by the weighted averages of the corresponding end-point parameters.

PZnx Cd1xSe  xPZnSe  (1  x)PCdSe Where P is a tight-binding parameter (either on-site or off-site). It is well known that in the ternary alloys the dependence of the band gap value is a non-linear function of compositions. To reimburse this effect, the s orbital

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on-site term is responsible for the correction by introducing an empirical bowing parameter into the virtual crystal approximation as described in more detail in Ref. 27 and 34. The on-site parameter of s orbital is given by.

EsZnx Cd1xSe  xEsZnSe  (1  x)EsCdSe  bv x(1  x) Where v is cation or anion atom. For alloy ZnxCd1-xSe semiconductor, b a of 0.00 and b a of 0.0370 [34] are employed. This model has been successful to widely describe the electronic structures and optical properties of alloy nanostructures [27, 37, 38, 39]. In addition, the other theoretical calculations apply the virtual crystal approximation (VCA) to study the alloy semiconductors. [40, 41, 42] Therefore, with the ability to atomistically investigate the natural properties of ZnxCd1-xSe nanocrystals with the experimentally synthesized sizes and contents, the atomistic tight-binding theory in the conjunction with virtual crystal approximation (VCA) is appropriate. To achieve the single-particle states and energies, the matrix of the empirical tight-binding Hamiltonian is computationally diagonalized by ARPACK [43]. After obtaining the single-particle spectra of alloy ZnxCd1-xSe nanocrystals, the splitting of excitonic states is computed by means of a configuration interaction technique (CI). This methodology has been successfully used to study the splitting of the excitonic states as shown in Refs. 44, 45, 46, 47 and 48 as well as in my previous papers [49, 50, 51]. The single excitonic Hamiltonian built-up from singleparticle spectra is defined as: eh, coul † † eh,exch † † H   Ei ei†ei  E j h †j h j   Vijkl h i e j ek h l   Vijkl h i e j ek h l i

j

ijkl

ijkl

Here, the first two terms are the single-particle energies of electron and hole states, respectively. The third term presents the coulomb interaction between electron and hole. The splitting of the degenerate excitonic states due to the electron-hole exchange interaction is described in the fourth term.

3. Results and Discussions

In this section, I discuss the atomistic calculations of alloy ZnxCd1-xSe nanocrystals as obtained from the atomistic tight-binding theory in order to attain more realistic band structures. As already discussed in Ref. 29, spherical anion-centred nanocrystals with wurtzite structure are assumed to be the computational models. The diameters of ZnxCd1-xSe nanocrystals are increased proportionally from 5.2 to 7.5 nm with the increasing Zn

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compositions (x) from 0.00 to 0.67. The aim is to uncover the underlying mechanism that bridges the natural properties and compositions of alloy ZnxCd1-xSe nanocrystals with the experimentally synthesized compositions (x) and sizes. It is expected that alloy nanocrystals are the potential ideal light emitting devices or semiconductor lasers in the field of the optoelectronic applications. In addition, the excitonic splitting is the main key to be observed with the aim to realize the alloy ZnxCd1-xSe nanocrystals in the potential application of quantum information. To do so, in the beginning Figure 1 shows the numbers of the atoms in alloy ZnxCd1-xSe nanocrystals with different compositions (x) and diameters. The number of atoms is increased with the increasing of the Zn contents because the lattice constants of alloy ZnxCd1-xSe materials are reduced with the increasing Zn compositions. Normally, the quantity of atoms is increased with the increasing diameters of alloy ZnxCd1-xSe nanocrystals. To physically scrutinize the effect of compositions (x) and diameters, I compute the single-electron and -hole spectra in Figure 2 and 3, respectively. With the increasing Zn contents (x), the single-electron energies are progressively increased because the lowest electron energy of bulk ZnSe is greater than one of bulk CdSe, while the single-hole energies remain almost constant. With the increasing diameters of alloy ZnxCd1-xSe nanocrystals, the decrease and increase in electron energies and hole energies are reported, respectively. Besides, the calculations of the optical band gaps with different Zn compositions (x) are observed as a function of the diameters in Figure 4. The improvement of the optical band gaps is presented with the increasing Zn contents because the band gap of bulk ZnSe is greater than one of bulk CdSe. In particular, a composition-tunable emission from blue (450 nm.) to yellow (600 nm.) wavelength is obviously exposed to unlock the new opportunities to design the optical nanodevices within these regions. The reduction of the optical band gaps with the increasing diameters is realized as caused by the quantum confinement. To authenticate these calculations, the comparison between atomistic tight-binding model and experimental data is acquired. Xinhua Zhong et al. [29] successfully prepared high-quality ZnxCd1-xSe nanocrystals by integrating stoichiometric amounts of Zn and Se into pre-prepared CdSe nanocrystals. The evolution of absorption spectra of ZnxCd1-xSe nanocrystals under various Zn contents (x) and diameters was offered. The optical band gaps of atomistic tight-binding theory and experiment are compared in Figure 5. The discrepancy between atomistic tightbinding theory and experiment is reduced with the increasing diameters and Zn contents. As the comparison, a good agreement is obtained between the tight-binding values and the experimental data.

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With a deeper understanding, I also calculate the ground electron-hole wave function overlaps in each of the Zn components and diameters in the alloy ZnxCd1-xSe nanocrystals with the aim to observe how changes in carrier localization are. The ground electron-hole wave function overlaps of alloy ZnxCd1-xSe nanocrystals as a function of Zn compositions (x) and diameters are represented in Figure 6. The reduction of the ground electronhole wave function overlaps is revealed with the increasing Zn contents. On the basis of the calculations, electron and hole are weakly localized in Zn rich of alloy ZnxCd1-xSe nanocrystals. In term of dimension, the ground electron-hole wave function overlaps are decreased as the increasing diameters. It is from the fact that the carriers extensively blowout throughout the alloy ZnxCd1-xSe nanocrystals. To theoretically investigate the effects of Zn compositions (x) and diameters on the atomistic electron-hole interactions of alloy ZnxCd1-xSe nanocrystals, Figure 7 and 8 depict the ground-state coulomb and exchange energies, respectively. With the increasing Zn contents, the ground-state coulomb energies are increased, while the ground-state exchange energies are reduced. On the basis of the ground-state coulomb energies, this finding means that electron and hole are strongly bound in Zn rich of alloy ZnxCd1-xSe nanocrystals. As caused by the ground-state exchange energies, the electron-hole exchange interaction tends to be decreased in Zn pronounced ZnxCd1-xSe nanocrystals, consequently inducing the reduction of the stokes shift and fine structure splitting. As is apparent from the Figures, the electron-hole interactions, coulomb and exchange term, are mainly reduced with the increasing diameters. This implies that the coulomb interaction between electron and hole is decreased with the increasing diameters. Also, the reduction of the stokes shift and fine structure splitting is clearly reported with the increasing sizes of ZnxCd1-xSe nanocrystals. From these observations, Zn compositions and diameters are the alternative way used to manipulate the atomistic electron-hole interactions in alloy ZnxCd1-xSe nanocrystals. In the end, the splittings of the excitonic states are demonstrated with different Zn compositions (x) and diameter using the configuration interaction description (CI). It is well known that the excitonic splitting is naturally convinced by the electron-hole exchange interaction. For the numerical analysis, the configuration interaction matrix is constructed by considering 12 electron and hole levels in order to obtain a warranting convergence of the excitonic energies in the order of meV. The stokes shift and fine structure splitting of alloy ZnxCd1-xSe nanocrystals are illustrated as a function of Zn compositions (x) and diameters in Figure 9 and 10, respectively. To describe the tendency of the stokes shift and fine structure splitting, the overlaps of ground electron-hole wave functions are

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theoretically employed. The stokes shift is the energy difference between the lowest optically forbidden excitonic state and the lowest optically allowed excitonic state. Fine structure splitting is energy gap between two optically allowed excitonic states. As is apparent from the Figure, the stokes shift and fine structure splitting are progressively reduced with the increasing Zn compositions and diameters as can be described by the fashion of the ground electron-hole wave function overlaps. In particular, it is expected that alloy ZnxCd1-xSe nanocrystals with Zn rich and immense sizes are the good candidates to optimistically offer the way toward a source of entangled photon pairs in the application of quantum information. From these observations, I can suggest that the stokes shift and fine structure splitting in alloy ZnxCd1-xSe nanocrystals are monotonically controlled by Zn contents (x) and diameters.

4. Conclusions In conclusion, I present the atomistic tight-binding studies on the electronic properties and excitonic splitting of alloy ZnxCd1-xSe wurtzite nanocrystals with the experimentally synthesized Zn compositions (x) and sizes. By changing Zn compositions and diameters, the single-particle energies, optical band gaps, overlaps of ground electron and hole wave functions, electron-hole interactions, stokes shift and fine structure splitting are numerically computed. For instance, with the increasing Zn contents, the single-electron energies are progressively increased, while the single-hole energies are approximately unaffected. The reduction and escalation in electron energies and hole energies are conveyed as the diameters are increased, respectively. The enhancement of the optical band gaps is presented with the increasing Zn contents with a composition-tunable emission in range from blue (450 nm.) to yellow (600 nm.) wavelength. The reduction of the optical band gaps with the increasing diameters is recognized because of the quantum confinement. In addition, the optical band gaps are in a good agreement with available information of the alloy ZnxCd1-xSe nanocrystals. The reduction of the ground electron and hole wave function overlaps is established with the increasing Zn components and diameters. The stokes shift and fine structure splitting are reported under different Zn compositions and diameter using the configuration interaction description (CI). The resulting computations highlight that the reduction of stokes shift and fine structure splitting is gradually demonstrated with the increasing Zn compositions and diameters as can be described by the trend of the ground electron-hole wave function overlaps. Based on these observations, the alloy ZnxCd1-xSe nanocrystals with Zn rich content and vast diameters are enthusiastically a source of entangled photon pairs in the application of quantum

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information. Finally, to use a diversity of alloy contents and sizes in the nanocrystals, the present work may provide more accurate and effective computations correlated to design the potential applications.

Acknowledgements The author would like to acknowledge the financial support from the Thailand Research Fund Grants (TRG58880072) and Department of Physics, Faculty of Science, Ubon Ratchathani University, Thailand.

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Figure 1: The number of atoms in alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.

Worasak Sukkabot/ Physica B

Figure 2: Single-electron states of alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.

Worasak Sukkabot/Physica B

Figure 3: Single-hole states of alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.

Figure 4: Optical band gaps of alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.

Worasak Sukkabot/ Physica B

Figure 5: The comparison of optical band gaps in alloy ZnxCd1-xSe nanocrystals between atomistic tight-binding theory and experiment [29] as a function of compositions (x) and diameters.

Figure 6: Ground electron-hole wave function overlaps of alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.

Worasak Sukkabot/Physica B

Figure 7: Electron-hole coulomb energies of alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.

Figure 8: Electron-hole exchange energies of alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.

Figure 9: Stokes shift of alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.

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Figure 10: Fine structure splitting of alloy ZnxCd1-xSe nanocrystals as a function of compositions (x) and diameters.