The electronic band parameters calculated by the Tight Binding Approximation for Cd 1-x Zn x S quantum dot superlattices

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1 IOSR Journal of Applied Pysics (IOSR-JAP) e-issn: Volume 6, Issue Ver. II (Mar-Apr. 04), PP 5- Te electronic band parameters calculated by te Tigt Binding Approximation for Cd -x Zn x S quantum dot superlattices S. Marzougui, N. Safta Unité de Pysique Quantiqu Faculté des Sciences, Université de Monastir, Avenue de l Environnement, 5000 Monastir Tunisia. Abstract: Tis work reports on a teoretical study of superlattices based on Cd -x Zn x S quantum dots embedded in an insulating material. Tis system, assumed to a series of flattened cylindrical quantum dots wit a finite barrier at te boundary, is studied using te tigt binding approximation. Te electronic states of Γ miniband ave been computed as a function of zinc composition for different inter-quantum dot separations. Calculations ave been made for electrons, eavy oles and ligt oles. Tree main features were revealed: (i) in te case of electrons, te Zn composition x = 0.4 is expected to be te most favorable to give rise a superlattice beavior for te Cd -x Zn x S quantum dots studied (ii) te strong localization caracter of eavy oles is evident in te Cd -x Zn x S nanostructures (iii) te Cd 0. Zn 0.8 S system is te more appropriate to exibit a superlattice beavior for ligt oles especially wen te superlattice period is low. Keywords: Quantum dots, superlattices, Cd -x Zn x S, Tigt Binding Approximation, non volatile memories. I. Introduction From te fundamental viewpoint, te study of electronic and optical properties quantum dots (QDs) based on te Cd -x Zn x S ternary alloy is attracting a considerable interest. Tis is mainly due to teir specific caracteristics like size quantization, zero dimensional electronic states, non linear optical beaviour [-8]. On te subject of te growt metods used to prepare Cd -x Zn x S QDs, we can cite te inverted micelles [9], te selective area growt tecnique [0], te single source molecular precursors [], te colloidal metod [] and te Sol gel tecnique []. Concerning te nanostructure devices design, one of te most prominent topics is to use nanostructures containing tunneling coupled QDs as an active element. In tis framework, we ave made some teoretical investigations of superlattices based on Cd -x Zn x S quantum dots embedded in an insulating material [3-7]. To describe te QDs, we ave adopted te flattened cylindrical geometry wit a finite potential barrier at te boundary. In Ref [4], we ave used te Kronig-Penney model to illustrate te confinement potential. Tus, we ave calculated te ground and te first excited minibands for bot electrons and oles. We ave also computed te longitudinal effective mass. Calculations were carried out as a function of te Zn composition and te interquantum dot separation. In Ref [5], we ave used te sinusoidal potential to model te confinement of te carriers. Witin tis model, we ave studied te ground and te first excited minibands for electrons as a function of inter-quantum dot separation for different zinc compositions. In Ref [6], using te triangular potential model, we ave calculated, for electrons, eavy oles and ligt oles, te Γ miniband and te longitudinal effective mass versus te Zn composition and te inter-quantum dot separation as well. Te goal of te present work is to investigate systematically te coupling in superlattices made by Cd -x Zn x S QDs aving a flattened cylindrical geometry wit a finite potential barrier at te boundary. Calculations ave been carried out as a function of Zn composition going from CdS to ZnS using te Tigt Binding Approximation (TBA). Te paper is organized as follows: after a brief introduction, we present te teoretical formulation, in te following we report te numerical results obtained, conclusions derived from tis study are given in te last section. II. Teoretical Formulation In a realistic description, te electronic properties of Cd -x Zn x S QDs embedded in a dielectric matrix ave to be studied teoretically using sperical geometry. Based on tis model, two metodologies ave been proposed to describe te potential energy, a potential wit an infinite barrier [-3, 8, 9] and a potential wit a finite barrier [7, 8] at te boundary. Te latter potential as te advantage to consider te coupling between QDs. Neverteless, it presents a big difficulty concerning te determination of te band edges for coupled QDs. Fig. - a sows te geometry used to describe a cain of Cd -x Zn x S QDs. Te common confined direction is denoted by z. Te inter-quantum dot separation is labelled d wic corresponds to te period of te structure. Along a common direction of sperical Cd -x Zn x S QDs, electrons and oles see a succession of flattened cylinders of radius R and effective eigt L. According to tat reported in Ref [], te diameter D = R 5 Page

2 Te electronic band parameters calculated by te Tigt Binding Approximation for Cd -x Zn x S varies from 9 nm to 4 nm going from CdS to ZnS. Tus, if we consider L = nm wic corresponds to te value reported in Ref [3], one can note tat L is lower tan D and terefore te quantum confinement along transversal direction can be disregarded. Consequently, te Cd -x Zn x S multi quantum dot system being studied can be considered as a QDs superlattice along te longitudinal confined direction. Tus, te system to investigate is a Cd -x Zn x S QD superlattice were te Cd -x Zn x S flattened cylinders QDs beave as wells wile te ost dielectric lattice forms a barrier of eigt U 0. For te sake of simplicity, te electron and ole states are assumed to be uncorrelated. Te problem to solve is, ten, reduced to tose of one particle in a one dimensional potential. In tis work, we consider te potential depicted in Fig. - b. Suc a potential can be expressed as: V wit z U z - nd n 0; - L/ zl/ U z ;L/ z d-l/ U0 Te subscripts e and refer to electrons and oles respectively and n is te nt period. For tis potential, te electron and ole states can be calculated using te effective Hamiltonian: d H V z () m dz were is te Plank s constant and m is te effective mass of carriers. In deriving te Hamiltonian H, we ave adopted te effective mass teory (EMT) and te band parabolicity approximation (BPA). Te mismatc of te effective mass between te well and te barrier as been neglected. Values of te electron and ole effective masses, as adopted for CdS and ZnS [4,7], are listed in Table. Tese two parameters for Cd -x Zn x S wit different Zn compositions ave been deduced using te Vegard s law. () Fig.: (a) A scematic diagram of Cd -x Zn x S QD superlattices according to te flattened cylindrical geometry (b) Te barrier potential in te framework of te Tigt Binding Approximation. sows tat te E e, were wit U We ave resolved te Scrodinger equation using te Tigt Binding Approximation. Our calculation = 4β 0 d L Γ B miniband widt is given by: L d d A, B, e, d L B e e e k L L k L, L e e e cos, s k, sin c (4) e e k (3) 6 Page

3 Miniband widt (ev) Te electronic band parameters calculated by te Tigt Binding Approximation for Cd -x Zn x S A B kl cos L sink k kl A cos L (5) (6) k ρ m m E U 0 E (7) (8) Her E e corresponds to te lowest energy of electrons, E is te lowest energy of eavy oles and E l is te lowest energy of ligt oles. All tese energies are associated wit an isolated flattened cylindrical quantum dot of Cd -x Zn x S. III. Results In a first pas we ave calculated, for electrons, te widt of te Γ E e miniband as a function of te ZnS molar fraction. Values of parameters used in tese calculations are summarized in Table. All tese parameters are taken from Refs [4, 7]. X Table. : Parameters used to calculate te Γ e -, Γ - and Γ l - minibands for Cd -x Zn x S QD superlattices. m e m m U ev l 0e U 0 ev L nm E e ev E ev E l m 0 m 0 m 0 ev Typical results are depicted in Fig.. In addition, tese results were fitted by polynomial laws as a function of x for te different inter-qd separations studied and summarized in Table e - miniband d=.5 nm d=.7 nm d=.9 nm Fig.: Te Γ Composition x d=. nm d=.3 nm d=.5 nm - miniband widt, as calculated for electrons versus te ZnS molar fraction for different inter-qd separations. 7 Page

4 Te electronic band parameters calculated by te Tigt Binding Approximation for Cd -x Zn x S Table : Te fit of te Superlattice period (nm) Γ e - miniband widt versus te Zn composition for different inter-qd separations. e miniband x-.780x +0.58x x-.80x +0.85x x-3.663x +0.95x x-4.097x +0.59x x-3.634x +.489x x-4.495x +.387x 3 It was revealed tat (i) for any composition x, te widt of te Γ E e miniband decreases wit te increase of te SL period d. Te coupling between nanoparticles sows a significant drop as te inter quantum dot separation increases. Tis beaviour can be attributed to te fact tat tis coupling is governed by te tunnelling effect for sorter SL periods (ii) for larger ZnS molar fractions, te coupling is low especially for iger SL periods. In tis cas Cd -x Zn x S nanocrystallites tend to beave as isolated QDs. Suc a trend is due mainly to te largeness of barrier eigts (iii) practically, for all te inter quantum dot separations studied, E e is sown to increase wit Zn composition up to x = 0.4 and it decreases as x increases from 0.4 to.0. Since te effective mass m e remains practically uncanged for all Zn compositions, tis result is, presumably related to te barrier potential eigtu 0 and te energy E e (iv) for Cd -x Zn x S QDs wit x = 0.4, te order of magnitude of te widt E e e is important especially for sorter inter-quantum dot separations and sows te strong degree of coupling between te QDs. For comparison wit results obtained by using te Kronig-Penney, te sinusoidal and te triangular potential models [4-6], we report in Table.3, widts of Γ miniband as calculated in te present work and tose obtained in Refs. [4-6]. As can be seen, practically, for all te compositions and inter- QD separations studied, te miniband widts of te present work are iger compared to tose obtained by te oter potential models. Te reason consists on te ypotesis adopted witin te Tigt Binding Approximation. Table 3: Widts of te Γ e (ev) miniband for te present work (a) and tose obtained by te triangular potential (b), te sinusoidal potential (c) and te kronig-penney potential (d). x d (nm) (a) (a) 0.78 (a) 0.86 (a) 0.70 (a) 0.50 (a) (b) 0.30 (b) 0.89 (b) 0.54 (b) 0.36 (b) 0.0 (b) 0.7 (c) (c) 0.5 (c) 0.40 (c) 0.9 (c) 0.6 (c) 0.77 (d) (d) (d) (d) 0.33 (d) 0.34 (d) (a) (a) (a) (a) (a) 0.38 (a) 0.86 (b) 0.5 (b) 0.4 (b) 0.98 (b) 0.46 (b) 0.63 (b) 0.53 (c) (c) (c) (c) 0.48 (c) 0.05 (c) (d) (d) 0.44 (d) 0.35 (d) 0.9 (d) 0.30 (d) (a) (a) (a) (a) (a) 0. (a) 0.40 (b) 0.00 (b) 0.73 (b) 0.58 (b) 0.46 (b) 0.3 (b) (c) (c) 0.9 (c) 0.30 (c) 0.09 (c) (c) (d) (d) (d) 0.4 (d) 0.0 (d) 0.05 d). 0.3 (a) (a) 0.64 (a) (a) 0.43 (a) 0.0 (a) 0.0 (b) 0.63 (b) 0.45 (b) 0.30 (b) 0.0 (b) 0.07 (b) 8 Page

5 Miniband widt (ev) Miniband widt (ev) Te electronic band parameters calculated by te Tigt Binding Approximation for Cd -x Zn x S (c) 0.8 (c) 0.3 (c) 0.65 (c) (c) (c) (d) (d) 0.34 (d) 0.4 (d) (d) (d) (a) 0.59 (a) (a) (a) 0.64 (a) (a) 0.50 (b) 0.36 (b) 0.0 (b) 0.08 (b) 0.00 (b) (b) 0.76 (c) 0.6 (c) 0.78 (c) 0.0 (c) (c) 0.03 (c) 0.3 (d) 0.50 (d) 0.75 (d) 0. (d) (d) 0.0 (d) (a) (a) (a) 0.34 (a) 0.0 (a) 0.05 (a) 0.30 (b) 0.5 (b) 0.0 (b) (b) (b) (b) 0.30 (c) 0.90 (c) 0.36 (c) (c) 0.03 (c) (c) 0.67 (d) 0.60 (d) 0.53 (d) (d) 0.05 (d) 0.0 (d) In a second pas we ave calculated, for te eavy oles and ligt oles, te Tis parameter is denoted for tese carriers by E and E l Γ miniband widt. respectively. Calculations were also carried out for te inter-seet separations and Zn compositions studied. Te parameters used are reported in Table. Te results are depicted in Fig.3 and Fig.4. Besides, tese results were fitted by polynomial laws and reported in Table 4 and Table miniband d=.5 nm 0.00 d=.7 nm d=.9 nm d=. nm d=.3 nm d=.5 nm Fig.3: Te Γ Composition x - miniband widt, as calculated for eavy oles versus te ZnS molar fraction for different inter-qd separations. 0.7 l - miniband d=. 5nm d=. 7nm d=. 9nm d=. nm d=.3 nm d=.5 nm Fig.4 Te Γ Composition x - miniband widt, as calculated for ligt oles versus te ZnS molar fraction for different inter-qd separations. 9 Page

6 Te electronic band parameters calculated by te Tigt Binding Approximation for Cd -x Zn x S Table 4: Te fit of te Superlattice period (nm) Γ - miniband widt versus te Zn composition for different inter-qd separations. miniband x+0.5x -0.67x x+0.05 x -0.07x x+0.06x x x+0.07x -0.0x x x x x+0.00x 0.00x 3 Table 5: Te fit of te Superlattice period (nm) Γ l - miniband widt versus te Zn composition for different inter-qd separations. l miniband x+0.764x -0.37x x+.603x -.0x x+.04x -.465x x+.7x -.57x x+.040x -.5x x+.834x -.374x 3 E An analysis of te obtained results led to two main observations: (i) concerning te eavy oles, is insignificant in terms of magnitude order wic means tat te strong localization caracter of eavy oles in te SL systems is igly preserved going from CdS to ZnS independently to te inter-qd separation (ii) for te ligt oles, practically for all te cases studied, E l is sligtly lower tan E e E l presents a maximum at x = 0.8. Moreover,. As a consequenc te superlattice beavior affects not only te conduction electrons but also te ligt oles especially for sort SL periods. Tese results are mainly due to te difference in effective masses between te electrons and ligt oles on te one and and te eavy oles on te oter and. IV. Conclusion We investigated te electronic properties of nanostructure based on Cd -x Zn x S embedded in a dielectric matrix for compositions ranging from CdS to ZnS. To describe te QDs, we ave adopted te flattened cylindrical geometry wit a finite potential barrier at te boundary. Using te Tigt Binding Approximation, we ave calculated, in a first pas te Γ - miniband for electrons. Calculations ave been made as a function of Zn composition for different inter quantum dot separations. An analysis of te obtained results as evidenced tat te Zn composition x = 0.4 is expected to be te most favorable to give rise a superlattice beavior for te Cd -x Zn x S quantum dots studied. For eavy oles, te Γ - miniband is sown to be lower in comparison wit te electron miniband. Tis result reflects te strong localization caracter of eavy oles in te Cd -x Zn x S nanostructures. As for te ligt oles, te widt magnitude order of Γ - miniband is sligtly lower to te one of electrons. Moreover, as as been demonstrated, te Cd 0. Zn 0.8 S system is te more appropriate to exibit a superlattice beavior for ligt oles especially wen te superlattice period is low. In te applied pysics, tis study can open a way for designing a new family of nanocrystal devices based on Cd -x Zn x S QDs particularly te non volatile memories. References [] B. Battacarje S. K. Mandal, K. Cakrabarti, D. Ganguli and S. Caudui, J. Pys. D: Appl. Pys. 35 (00) [] K.K Nanda, S.N. Sarangi, S. Moanty, S.N. Sau, Tin Solid Films 3 (998) -7. [3] H. Yükselici, P. D. Persans and T. M. Hayes, Pys. Rev. B. 5 (995) 763. [4] Y. Kayanuma, Pys. Rev. B 38 (988) 9797 [5] L Zou, Y Xing and Z P Wang, Eur. Pys. J. B. 85 (0) [6] A. Jon Peter and C. W. Le Cinese Pys. B (0) [7] N. Safta, A. Sakly, H. Mejri and Y. Bouazra, Eur. Pys. J. B 5 (006) [8] A. Sakly, N. Safta, A. Mejri, H. Mejri, A. Ben Lamin J. Nanomater (00), ID [9] L. Cao, S. Huang and S. E, Journal of colloid and interface scienc 73 (004) [0] H. Kumano, A. Ueta and I. Suemun Pysica E 3 (00) [] Y. Cai Zang, W. Wei Cen and X. Ya Hu, Materials Letters 6 (007) Page

7 Te electronic band parameters calculated by te Tigt Binding Approximation for Cd -x Zn x S [] K. Tomiira, D. Kim and M. Nakayama, Journal of Luminescence (005) [3] N. Safta, A. Sakly, H. Mejri and M. A. Zaïdi, Eur. Pys. J. B 53 (006) [4] A. Sakly, N. Safta, H. Mejri, A. Ben Lamin J. Alloys Compd. 476 (009) [5] A. Sakly, N. Safta, H. Mejri, A. Ben Lamin J. Alloys Compd.509 (0) [6] S. Marzougui, N. Safta, IOSR-JAP 5 (5) (04) 36-4 [7] S. Marzougui, A study of te excited states in te flattened cylindrical quantum dots of Cd -xzn xs, Master, University of Monastir (0). [8] Q. Pang, B.C. Guo, C.L. Yang, S. H. Yang, M.L. Gong, W.K. Ge and J. N. Wang, Journal of Crystal Growt, 69 (004) 3. [9] M.C. Klein, F. ac D. Ricard and C. Flytzanis, Pys Rev. B. 4 (990) 3. Page