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SiO2 quantum dots
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Optical properties of an exciton bound to an ionized impurity in ZnO/SiO2 quantum dots Lobna Dallali a,n, Sihem Jaziri a, Juan Martínez-Pastor b a b
Département de Physique, Faculté des Sciences de Bizerte, Jarzouna, 7021 Bizerte, Tunisie Institut de Ciència dels Materials de la Universitat de Valencia (ICMUV), P.O. Box 2085, 46071 Valencia, Spain
art ic l e i nf o
a b s t r a c t
Article history: Received 28 May 2014 Received in revised form 2 January 2015 Accepted 17 February 2015 by V. Pellegrini
The energy of the ground and the excited states for the exciton and the binding energy of the acceptor– donor exciton complexes ðA ; XÞ and ðD þ ; XÞ as a function of the radius for an impurity position located in the center in the spherical ZnO quantum dots (QDs) embedded in a SiO2 matrix are calculated using the effective mass approximation under the diagonalzation matrix technique, including a threedimensional confinement of carrier in the QD and assuming a finite depth. Numerical results show that the binding energy of the acceptor–donor exciton complexes is very sensitive to the quantum dot size. These results could be particularly helpful since they are closely related to experiments performed on such nanoparticles. This may allow us to improve the stability and efficiency of the semiconductor quantum dot luminescence which is, in fact, considered critical. & 2015 Elsevier Ltd. All rights reserved.
Keywords: A. ZnO QDs D. Acceptor–donor complex exciton D. Binding energy
1. Introduction ZnO is normally in the hexagonal wurtzite crystal structure, showing n-type conductivity. It has a wide range of applications in optical ultraviolet devices such as light-emitting diodes, laser diodes, gas sensors, optical wave-guide and solar cells [1–2]. Nanostructures and heterostructures are made of Zinc Oxide (ZnO), which is an exceptionally important semiconductor with a wide band gap energy. Due to the large binding excitons energy of 60 meV, the exciton has long life at room temperature [3–4]. It has also been attracting much attention because of its high luminescent efficiency and non-ohmic properties. It belongs to the space group p63mc in II– VI group of semiconductors. It has been proven that impurity states play a very important role in some optoelectronic devices. Without impurity, there would be no diode, no transistor, and neither of semiconductor science and technology [5]. A deep understanding of the effect of impurities on the exciton states of semiconductor nanostructures is a fundamental question in semiconductor physics since their presence can dramatically alter the performance of quantum devices [6]. Theoretical investigations on the optical and electronic properties in spherical ZnO QDs are very recent. There have been only a few works regarding the calculation of excitons bound to impurities in QDs: Stébé et al. [7] investigated the new optical properties of the exciton and of the bound-exciton in the quantum confinement, by using a by the Ritz's variational method.
n
Corresponding author. Tel.: þ 216 94 063 634; fax: þ 216 72 590 566. E-mail address: [email protected] (L. Dallali).
Wang et al. [8,9] calculated the influence of an external charge on the electron–hole pair in spherical CdSe QDs. Baskoutas et al. [10] studied the stability of an exciton bound to an ionized donor in a parabolic two-dimensional QD. Using a variational approach in the framework of the adiabatic approximation. Avetisyan et al. [11,12] investigated the electronic and optical properties of the excition bound to charged or neutral donors in spherical QDs as a function of an external magnetic field. Recently Dujardin et al. [13] studied the effect of stark and dissociation process of an ionized donor bound exciton in spherical quantum dots, Dallali et al. [14] have theoretically investigated the energies of exciton and acceptor-bound exciton complex in ZnO QDs embedded in SiO2 matrix, Maikhuri et al. [15] has studied the optical properties of ZnO QDs dispersed in SiO2 matrix and Xie [16] studied the influence of the size in the confinement of an exciton– acceptor complex in quantum dots. This paper is organized as follow: in Section 2, we introduce our theoretical model used to describe the acceptor–donor exciton complexes confined in spherical ZnO QDs. The optical properties of Zinc Oxide QDs are presented in Section 3. Finally, we have summarized the main conclusions obtained in this paper in Section 4.
2. Theoretical model 2.1. Model of an exciton of ZnO/SiO2 nanoparticles: finite confining potentiel In the framework of the effective mass approximation, the exciton Hamiltonian with or without an ionized impurity in
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spherical parabolic quantum dot is written as ! ! ! ! ! ! H exi ¼ H e þ H h þ Eg þV int ð r e ; r h Þ þ α V int R ; r e þ V int R ; r h : Here, H eðhÞ ¼ P 2eðhÞ =2mneðhÞ þ V conf ðreðhÞ Þ is the electron (hole) Hamiltonian, mneðhÞ is the electron (hole) effective mass and V conf ðreðhÞ Þ denotes the confinement potential. ( ( 0 rrR 0 rrR ; V conf ðrh Þ ¼ V conf ðre Þ ¼ Vh r 4R Ve r 4R where ΔV e ¼ 23 Eg ðM Þ Eg and ΔV h ¼ 13 Eg ðM Þ Eg . Eg ðM Þ is the matrix energy band gap (Eg ðM Þ ¼ 9:2 eV) [2] and Eg is the ZnO band gap energy. ! ! V int r e ; r h is the Coulomb potential energy of the electron– hole system represented bythe electron–hole interaction energy, ! ! ! ! and written as V int r e ; r h ¼ e2 =ε r e r h where r - eðr- h Þ is, indeed, the position vector of the electron (hole) written in spherical coordinates, e is the absolute value of the electron charge and ε is the optical dielectric constant. !! V int R ; r eðhÞ is the Coulomb potential energy of the impurity–electron (hole) system, represented by the ionized impurity– electron (hole) interaction energy, and written as 0 1 ! ! e2 B 1 1 ! ! þ ! C: α V int R ; r e þ V int R ; r h ¼ α @! ! A ε R ! r e R r h where α ¼ 1 for a donor, α ¼ 1 for an acceptor, and α ¼ 0 when there is any impurity and R- is the radius vector of the impurity.
2.2. Theoretical model of an exciton bound to an impurity located at the center of a spherical ZnO/SiO2 nanoparticules We consider an impurity located on the z-axis to keep the axial symmetry of the problem. The wavefunction solutions of the electron (hole) Hamiltonian, H eðhÞ can be written as follows: eðhÞ eðhÞ ¼ Ym ψ ðn;l;mÞ l ðθ; ϕÞξn ðrÞ;
where n is the principal quantum number, l and m are the angular momentum quantum numbers, and Y m l ðθ; ϕÞ is the spherical harmonic. The normalized radial function 8 eðhÞ > jl kn;l r inside of the dot; r Z R < A1 ξeðhÞ n ðr Þ ¼ þ eðhÞ > : A2 hl iχ n;l r outside of the dot; r o R A1 and A2 are the normalized constants. The boundary conditions for the function are eðhÞ þ R jl kn;l R ¼ hl iχ eðhÞ n;l eðhÞ 1 djl kn;l r mn1 dr
r¼R
ðiÞ
þ eðhÞ 1 d hl iχ n;l r ¼ n m2 dr
ðiiÞ
r¼R
Z 0
and the Wavefunction must satisfy the normalization condition Z r2 eðhÞ eðhÞ nþ þ eðhÞ r 2 jn l kn;l r jl kn;l r dr þ r 2 hl iχ n;l r hl iχ eðhÞ r dr ¼ 1 n;l
r1
r1
ðiiiÞ eðhÞ ; iχ n;l r are the nth zero of the transcendental equation obtained from the continuity of ξenðhÞ ðrÞ and its derivative at r ¼ R. The excitonic Wavefunction, the solution of the excitonic Hamiltonian, written as the linear combination ! ! ! ! ψ je ;jh r e ; r h ¼ ψ je ð r e Þψ jh ð r h Þ;
eðhÞ kn;l r
where je ¼ ðne ; le ; me Þ and jh ¼ ðnh ; lh ; mh Þ. Using a diagonalzation matrix technique, we have decomposed the exciton complex Hamiltonian over the matrix. Only the wavefunctions with the same quantum number mex ¼ me þ mh can be coupled with the exciton complexes Hamiltonian resulting in the following selection rule: ! ! ψ je; jh j H exi j ψ j0e; j0 ¼ δmex; mex Ee þEh þ ψ je; jh j V int r e ; r h h ! ! ! ! þαðV int R ; r e þV int R ; r h Þj ψ jCCe; jCC h ! ! Ee þ Eh þ ψ je; jh j V int r e ; r h j ψ j0e; j0 ¼ δme þ m ;m0 h
e þ m0 h
h
! ! þ α ψ jh j ψ j0 ψ je j V int R ; r e j ψ j0e h ! ! þ ψ je j ψ j0e ψ jh j V int R ; r h j ψ j0 h
The exciton complex is formed when an electron and a hole are trapped by a charged impurity. Then, 8 0; ψ je; jh j H exi j ψ j0e; j0 ¼ E ðX Þ > h > < 0 0 1; ψ j H j ψ ¼ E ðA ; X Þ exi je; jh je; jh If α ¼ > > : 1; ψ j j j H exi j ψ j0 j0 ¼ E Dþ ; X e; h e; h
The calculation details for the matrix element method are explained in the Apendix. The eigenstates and the eigenenergies of the exciton and the exciton complexes are calculated numerically.
3. Results and discussions All material parameters used in our calculations are given in Table 1. The electron and the hole masses are expressed in units of free electron masses. Eg is the Band gap energy given in units of eV: ΔV e ; ΔV h are the confinement potential of the electron and the hole outside of the dot, given in units of eV. εr is the optical dielectric constant. The ZnO material is for the dot and the SiO2 is for the barrier. We see from Fig. 1 that the states energies s, p, d of the free electron increases as a function of the decreases of the radius of ZnO quantum dots, to attain the maximum rate of the confinement potential ΔV e ¼ 3:8 eV. And when R increases, the numbers of the energies rises. The energies states are more discrete for R¼10 Å than 20 Å, and for the big radius the energies states are quasi continuums.
Table 1 Values of various physical parameters used in our numerical calculation. Physical parameters
me
mh
εr
Eg
ΔVe
ΔVh
ZnO SiO2
0.24 [4,17,18] 0.5 [20]
0.45 [4,19] 1.0 [20]
3.7 [2,4,19] 3.9 [21]
3.24 [4] 9.2 [2]
3.8
2.05
Please cite this article as: L. Dallali, et al., Solid State Commun (2015), http://dx.doi.org/10.1016/j.ssc.2015.02.012i
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3
Fig. 2. States of the free carriers electron and hole of the ZnO QDs.
Fig. 1. (Color online) Free electron states energies as a function of the radius R (Å) of ZnO/SiO2 QDs.
The eigenstates of the exciton ðj1; j2; j3; j4; j5; 6Þ are written as the linear combination of the states on the basis B (see Figs. 2 and 3 and appendix) The components of the eigenstates should verify this relation ( X 1; i ¼ j CiCj ¼ 0; i aj i;j
For example for R¼20 Å, radius less then Bohr radius Bohr of the ZnO (28.7 Å), the different components of the eigenstates written as j1 ¼ 0:96 Se ; Sh 0:24 p0e ; p0h þ 0:06peþ ; ph j2 ¼ 0:95 Se ; p0h þ0:28 p0e ; Sh j3 ¼ 0:04 Se ; Sh þ 0:70 pe ; phþ þ 0:70 peþ ; ph j4 ¼ 0:28 Se ; p0h þ 0:95 p0e ; Sh j5 ¼ 0:24 Se ; Sh þ 0:02 pe ; phþ þ 0:96p0e ; p0h j6 ¼ 0:04 Se ; Sh 0:70 pe ; phþ þ 0:03 p0e ; p0h þ 0:70 peþ ; ph
Fig. 3. Excitonique states of ZnO QDs via the selection rule.
and for R ¼35 Å, radiu bigger than the Bohr radius of ZnO, we found this contribution j1 ¼ 0:99 Se ; Sh þ 0:01 p0e ; p0h j2 ¼ 0:99 Se ; p0h þ 0:09 p0e ; Sh j3 ¼ 0:70 pe ; phþ 0:70 peþ ; ph j4 ¼ 0:09 Se ; p0h þ 0:99 p0e ; Sh j5 ¼ 0:999 p0e ; p0h j6 ¼ 0:70 pe ; phþ þ 0:70 peþ ; ph We can recognize that when the distance between the electron and hole became bigger then ZnO Bohr radius, the Coulomb interaction became inconsiderable. Then, the energy of the exciton states toward to the bulk exciton energy. Fig. 4 shows the variation of the excitonic energy of the ground and of the excited states as a function of ZnO QDs radius. It can be noticed from Fig. 4 that the exciton energy for the ground and for the excited states decrease in parallel with the increasing of the QDs. This behavior can be explained as such: before the Bohr radius the kinetic energy being to be more dominant than the Coulomb interaction, and for the radius being equal or bigger than the Bohr radius, the Coulomb interaction is the dominant and for the large sized ZnO QDs the distance
Fig. 4. (Color online) Exciton energy for the ground and excited states as a function of the ZnO/SiO2 QDs radius.
between the electron and the hole increases and the exciton energy tends the energy of ZnO bulk value (Table. 2). We perceive from Table. 2 that the exciton energy of the ground state decreases from 10 meV to 5 meV, for a R¼ 25 (Å)–R¼ 35 (Å) QDs.
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We remark also from Fig. 4 that the values of the three last excited states energies for the exciton ZnO QDs, Ex4 ; Ex5 ; Ex6 , are nearly similarl; the continuum domain energy (see Table 3). We can grasp from Fig. 5, that the energies of the exciton X and of the exciton bound to an ionized donor impurity ðD þ ; XÞ,which decreases monotonically, when the QDs radius of ZnO increases between 10 (Å) to 30 (Å), however the energy of the exciton bound to an ionized acceptor impurity ðA ; XÞ increases slowly, when the QDs radius of ZnO increases to 30 (Å). For example the energies of the X and of the XC for two different ZnO QDs radii are summarized in Table 4. From Fig. 5 and Table 4, we note that the EðA ; X Þ is less than the þ EðX Þ and of E D ; X . The energy of the exciton bound to the donor impurty E D þ ; X is the biggest. In fact, for the radius being less than the ZnO Bohr radius, the Coulomb interaction of the donor impurity with the electron and the
Table 2 Exciton energy of the ground state as a function of ZnO QDs radius. R (Å) E (X1) (eV)
25 3.42
30 3.32
35 3.27
Table 3 Exciton energy of the excited states as a function of ZnO QDs radius.
hole carriers which is presented by 0 1 e2 1 1 ! ! B C V int r i ; r e þ V int r i i ; r h ¼ @ - ! þ - ! A ε ri r e ri r h dominates the electron–hole Coulomb interaction. And the Coulomb effect increases the confinement of the electron and of the hole. The Coulomb interaction of the acceptor impurity with the electron and the hole carriers, presented by 0 1 2 e B 1 1 ! ! C – V int r i ; r e þV int r i ; r h ¼ @ - ! þ - ! A ε ri r e ri r h increases the electron and hole Coulomb interaction. To conclude, the system formed by the acceptor bound to the exciton complex is more stable than the sytem formed by the exciton. The latter is more stable than the system formed by the donor bound to the complex exciton. For the radius being bigger than the ZnO Bohr radius, the energy of the exciton complex, ðA ; X) and (D þ ; X) towards to the energie of the free ZnO QDs exciton. Another important quantity of interest is the acceptor–donor bound exciton binding energy Eb that indicates the electron, hole and the acceptor–donor impurity Coulomb interaction which can be computed as follows: Eb ðA ; XÞ ¼ EðXÞ EðA ; XÞ
R (Å)
E(X4) (eV)
E(X5) (eV)
E(X6) (eV)
10 20 35
5.035 3.83 3.53
5.039 3.84 3.5412
5.05 3.85 3.5413
and Eb ðD þ ; XÞ ¼ EðXÞ EðD þ ; XÞ The acceptor–donor bound exciton binding energy Eb is investigated as a function of the radius R of the system ZnO/SiO2 QDs with an impurity located in the center. We can observe from Fig. 6 that the binding energy Eb ðA ; XÞ decreases when the radius R increases. These because the radial distance between the electron and the hole is increased when the radial R increases. Thus, the Coulomb interaction between the electron and the hole decreases when increasing the radius R. Fig. 6 confirms that the system formed by the acceptor bound to the exciton complex is more stable than the system formed by the donor bound to the complex exciton. In fact for the radius less than the ZnO Bohr radius Eb ðA ; X Þ is strictly positive and the Eb ðD þ ; XÞ is strictly negative. For the ZnO large sized QD the exciton binding energy for the complex exciton tends towards the bulk ZnO Rydberg energy ð0:06 eVÞ.
Fig. 5. (Color online) Energies of the exciton and the exciton complex, XC, in ZnO/ SiO2 QDs as a function of the dot radius. Impurity being located in the center.
Table 4 The comparison between the energies of the X and of the XC. R (Å)
10 30
E (eV) E(D þ , X)
E(X)
E(A , X)
5442 3346
4110 3349
2779 3343
Fig. 6. (Color online) Binding energy of the complex ecxiton with an impurity located in the center of the QDs as a function of the dot radius.
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4. Conclusions
5
But we only need to calculate the energies of the central bloc, because it is embedded with the elements of the two others blocs.
To summarize, within the effective-mass approximation and the diagonalization matrix approach, we have shown that the influence of the ionized impurity fixed charge on the acceptor and the donor binding energy is significant when the QD radius is smaller than the exciton Bohr radius 28.7 Å, we have proven that in this domain the system formed by the acceptor bound to the exciton complex is more stable than the system formed by the donor bound to the complex exciton, above this value, E b ðA ; XÞ and Eb ðD þ ; XÞ are just the value of the free bulk exciton binding energy. Finally, we have demonstrated a strong tunability of the optical properties (optical transition energies and binding energy) of ZnO nanoparticles not only with the size, but also with the presence of the ionized impurities inside them. This can be of potential use for different applications such as magnetic storage or optoelectronic and photovoltaic devices. Uncited references [17–21]. Acknowledgments
2
D1 6 6 D2 6 6 D3 6 And the matrix becomes : 6 6 D2 6 6 D4 4
D2
D3
D2
D4
D5
D6
D5
D7
D6
D8
D6
D9
D5
D6
D5
D7
D7
D9
D7
D10
D3
D6
D8
D6
D9
D3
3
7 D6 7 7 D8 7 7 7 D6 7 7 D9 7 5 D8
This matrix is diagonally symmetric. With D1 ¼ ψ e0;0 ψ h0;0 j H xc j ψ e0;0 ψ h0;0 ;
D2 ¼ ψ e0;0 ψ h0;0 j H xc j ψ e0;0 ψ h1;0
D3 ¼ ψ e0;0 ψ h0;0 j H xc j ψ e1 1 ψ h1;1 ;
D4 ¼ ψ e0;0 ψ h0;0 j H xc j ψ e1;0 ψ h1;0
D5 ¼ ψ e0;0 ψ h1;0 j H xc j ψ e0;0 ψ h1;0 ;
D6 ¼ ψ e0;0 ψ h1;0 j H xc j ψ e1; 1 ψ h1;1
D7 ¼ ψ e0;0 ψ h1;0 j H xc j ψ e1;0 ψ h1;0 ;
D8 ¼ ψ e1; 1 ψ h1;1 j H xc j ψ e1; 1 ψ h1;1
D9 ¼ ψ e1; 1 ψ h1;1 j H xc j ψ e1;0 ψ h1;0 ;
D10 ¼ ψ e1;0 ψ h1;0 j H xc j ψ e1;0 ψ h1;0
The authors are grateful to professor Mr. Imade Hethli for fruitful discussions. Appendix
References
Due to the selection rule, the basis B of the states composed has dimension of 16 and it is composed of these elements ψ ðl;mÞe ψ ðl;mÞh l ¼ 0;1 and m ¼ 0; 8 1 eðhÞ
eðhÞ
We will then eliminate the quantum number principal n to light the notation. 9 8 e ψ ψ h ; ψ e0;0 ψ h1; 1 ; ψ e1; 1 ψ h0;0 ; ψ e1; 1 ψ h1;0 > > > > > > 1; 1 1; 1 > > > = < ψ e1;0 ψ h1; 1 ; ψ e0;0 ψ h0;0 ; ψ e0;0 ψ h1;0 ; ψ e1; 1 ψ h1;1 > B¼ e h e h e h e h > > ψ ψ ; ψ ψ ; ψ ψ ; ψ ψ > 1;0 0;0 1;0 1;0 1;1 1; 1 0;0 1;1 > > > > > > > e h e h e h : ψe ψh ; ψ 1;1 ψ 0;0 ; ψ 1;1 ψ 1;0 ; ψ 1;1 ψ 1;1 ; 1;0 1;1 In this basis, H xc decomposes in a matrix and through the selection rule, relation ðnÞ n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n n n n 0 0 0 0 0 0 0 0 0 0 n n n n 0 0 0 0 0 0 0 0 0 0 n n n n 0 0 0 0 0 0 0 0 0 0 n n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n n n 0 0 0 0 0 0 0 0 0 0 0 n n n 0 0 0 0 0 0 0 0 0 0 0 n n n 0 0 0 0 0 0 0 0 0 0 0 n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0
can be simplified 0 0 0 0 0 0 0 0 0 0 0 n n n n
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Optical properties of an exciton bound to an ionized impurity in ZnO/SiO2 quantum dots Lobna Dallali a,n, Sihem Jaziri a, Juan Martínez-Pastor b a b
Département de Physique, Faculté des Sciences de Bizerte, Jarzouna, 7021 Bizerte, Tunisie Institut de Ciència dels Materials de la Universitat de Valencia (ICMUV), P.O. Box 2085, 46071 Valencia, Spain
art ic l e i nf o
a b s t r a c t
Article history: Received 28 May 2014 Received in revised form 2 January 2015 Accepted 17 February 2015 by V. Pellegrini
The energy of the ground and the excited states for the exciton and the binding energy of the acceptor– donor exciton complexes ðA ; XÞ and ðD þ ; XÞ as a function of the radius for an impurity position located in the center in the spherical ZnO quantum dots (QDs) embedded in a SiO2 matrix are calculated using the effective mass approximation under the diagonalzation matrix technique, including a threedimensional confinement of carrier in the QD and assuming a finite depth. Numerical results show that the binding energy of the acceptor–donor exciton complexes is very sensitive to the quantum dot size. These results could be particularly helpful since they are closely related to experiments performed on such nanoparticles. This may allow us to improve the stability and efficiency of the semiconductor quantum dot luminescence which is, in fact, considered critical. & 2015 Elsevier Ltd. All rights reserved.
Keywords: A. ZnO QDs D. Acceptor–donor complex exciton D. Binding energy
1. Introduction ZnO is normally in the hexagonal wurtzite crystal structure, showing n-type conductivity. It has a wide range of applications in optical ultraviolet devices such as light-emitting diodes, laser diodes, gas sensors, optical wave-guide and solar cells [1–2]. Nanostructures and heterostructures are made of Zinc Oxide (ZnO), which is an exceptionally important semiconductor with a wide band gap energy. Due to the large binding excitons energy of 60 meV, the exciton has long life at room temperature [3–4]. It has also been attracting much attention because of its high luminescent efficiency and non-ohmic properties. It belongs to the space group p63mc in II– VI group of semiconductors. It has been proven that impurity states play a very important role in some optoelectronic devices. Without impurity, there would be no diode, no transistor, and neither of semiconductor science and technology [5]. A deep understanding of the effect of impurities on the exciton states of semiconductor nanostructures is a fundamental question in semiconductor physics since their presence can dramatically alter the performance of quantum devices [6]. Theoretical investigations on the optical and electronic properties in spherical ZnO QDs are very recent. There have been only a few works regarding the calculation of excitons bound to impurities in QDs: Stébé et al. [7] investigated the new optical properties of the exciton and of the bound-exciton in the quantum confinement, by using a by the Ritz's variational method.
n
Corresponding author. Tel.: þ 216 94 063 634; fax: þ 216 72 590 566. E-mail address: [email protected] (L. Dallali).
Wang et al. [8,9] calculated the influence of an external charge on the electron–hole pair in spherical CdSe QDs. Baskoutas et al. [10] studied the stability of an exciton bound to an ionized donor in a parabolic two-dimensional QD. Using a variational approach in the framework of the adiabatic approximation. Avetisyan et al. [11,12] investigated the electronic and optical properties of the excition bound to charged or neutral donors in spherical QDs as a function of an external magnetic field. Recently Dujardin et al. [13] studied the effect of stark and dissociation process of an ionized donor bound exciton in spherical quantum dots, Dallali et al. [14] have theoretically investigated the energies of exciton and acceptor-bound exciton complex in ZnO QDs embedded in SiO2 matrix, Maikhuri et al. [15] has studied the optical properties of ZnO QDs dispersed in SiO2 matrix and Xie [16] studied the influence of the size in the confinement of an exciton– acceptor complex in quantum dots. This paper is organized as follow: in Section 2, we introduce our theoretical model used to describe the acceptor–donor exciton complexes confined in spherical ZnO QDs. The optical properties of Zinc Oxide QDs are presented in Section 3. Finally, we have summarized the main conclusions obtained in this paper in Section 4.
2. Theoretical model 2.1. Model of an exciton of ZnO/SiO2 nanoparticles: finite confining potentiel In the framework of the effective mass approximation, the exciton Hamiltonian with or without an ionized impurity in
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spherical parabolic quantum dot is written as ! ! ! ! ! ! H exi ¼ H e þ H h þ Eg þV int ð r e ; r h Þ þ α V int R ; r e þ V int R ; r h : Here, H eðhÞ ¼ P 2eðhÞ =2mneðhÞ þ V conf ðreðhÞ Þ is the electron (hole) Hamiltonian, mneðhÞ is the electron (hole) effective mass and V conf ðreðhÞ Þ denotes the confinement potential. ( ( 0 rrR 0 rrR ; V conf ðrh Þ ¼ V conf ðre Þ ¼ Vh r 4R Ve r 4R where ΔV e ¼ 23 Eg ðM Þ Eg and ΔV h ¼ 13 Eg ðM Þ Eg . Eg ðM Þ is the matrix energy band gap (Eg ðM Þ ¼ 9:2 eV) [2] and Eg is the ZnO band gap energy. ! ! V int r e ; r h is the Coulomb potential energy of the electron– hole system represented bythe electron–hole interaction energy, ! ! ! ! and written as V int r e ; r h ¼ e2 =ε r e r h where r - eðr- h Þ is, indeed, the position vector of the electron (hole) written in spherical coordinates, e is the absolute value of the electron charge and ε is the optical dielectric constant. !! V int R ; r eðhÞ is the Coulomb potential energy of the impurity–electron (hole) system, represented by the ionized impurity– electron (hole) interaction energy, and written as 0 1 ! ! e2 B 1 1 ! ! þ ! C: α V int R ; r e þ V int R ; r h ¼ α @! ! A ε R ! r e R r h where α ¼ 1 for a donor, α ¼ 1 for an acceptor, and α ¼ 0 when there is any impurity and R- is the radius vector of the impurity.
2.2. Theoretical model of an exciton bound to an impurity located at the center of a spherical ZnO/SiO2 nanoparticules We consider an impurity located on the z-axis to keep the axial symmetry of the problem. The wavefunction solutions of the electron (hole) Hamiltonian, H eðhÞ can be written as follows: eðhÞ eðhÞ ¼ Ym ψ ðn;l;mÞ l ðθ; ϕÞξn ðrÞ;
where n is the principal quantum number, l and m are the angular momentum quantum numbers, and Y m l ðθ; ϕÞ is the spherical harmonic. The normalized radial function 8 eðhÞ > jl kn;l r inside of the dot; r Z R < A1 ξeðhÞ n ðr Þ ¼ þ eðhÞ > : A2 hl iχ n;l r outside of the dot; r o R A1 and A2 are the normalized constants. The boundary conditions for the function are eðhÞ þ R jl kn;l R ¼ hl iχ eðhÞ n;l eðhÞ 1 djl kn;l r mn1 dr
r¼R
ðiÞ
þ eðhÞ 1 d hl iχ n;l r ¼ n m2 dr
ðiiÞ
r¼R
Z 0
and the Wavefunction must satisfy the normalization condition Z r2 eðhÞ eðhÞ nþ þ eðhÞ r 2 jn l kn;l r jl kn;l r dr þ r 2 hl iχ n;l r hl iχ eðhÞ r dr ¼ 1 n;l
r1
r1
ðiiiÞ eðhÞ ; iχ n;l r are the nth zero of the transcendental equation obtained from the continuity of ξenðhÞ ðrÞ and its derivative at r ¼ R. The excitonic Wavefunction, the solution of the excitonic Hamiltonian, written as the linear combination ! ! ! ! ψ je ;jh r e ; r h ¼ ψ je ð r e Þψ jh ð r h Þ;
eðhÞ kn;l r
where je ¼ ðne ; le ; me Þ and jh ¼ ðnh ; lh ; mh Þ. Using a diagonalzation matrix technique, we have decomposed the exciton complex Hamiltonian over the matrix. Only the wavefunctions with the same quantum number mex ¼ me þ mh can be coupled with the exciton complexes Hamiltonian resulting in the following selection rule: ! ! ψ je; jh j H exi j ψ j0e; j0 ¼ δmex; mex Ee þEh þ ψ je; jh j V int r e ; r h h ! ! ! ! þαðV int R ; r e þV int R ; r h Þj ψ jCCe; jCC h ! ! Ee þ Eh þ ψ je; jh j V int r e ; r h j ψ j0e; j0 ¼ δme þ m ;m0 h
e þ m0 h
h
! ! þ α ψ jh j ψ j0 ψ je j V int R ; r e j ψ j0e h ! ! þ ψ je j ψ j0e ψ jh j V int R ; r h j ψ j0 h
The exciton complex is formed when an electron and a hole are trapped by a charged impurity. Then, 8 0; ψ je; jh j H exi j ψ j0e; j0 ¼ E ðX Þ > h > < 0 0 1; ψ j H j ψ ¼ E ðA ; X Þ exi je; jh je; jh If α ¼ > > : 1; ψ j j j H exi j ψ j0 j0 ¼ E Dþ ; X e; h e; h
The calculation details for the matrix element method are explained in the Apendix. The eigenstates and the eigenenergies of the exciton and the exciton complexes are calculated numerically.
3. Results and discussions All material parameters used in our calculations are given in Table 1. The electron and the hole masses are expressed in units of free electron masses. Eg is the Band gap energy given in units of eV: ΔV e ; ΔV h are the confinement potential of the electron and the hole outside of the dot, given in units of eV. εr is the optical dielectric constant. The ZnO material is for the dot and the SiO2 is for the barrier. We see from Fig. 1 that the states energies s, p, d of the free electron increases as a function of the decreases of the radius of ZnO quantum dots, to attain the maximum rate of the confinement potential ΔV e ¼ 3:8 eV. And when R increases, the numbers of the energies rises. The energies states are more discrete for R¼10 Å than 20 Å, and for the big radius the energies states are quasi continuums.
Table 1 Values of various physical parameters used in our numerical calculation. Physical parameters
me
mh
εr
Eg
ΔVe
ΔVh
ZnO SiO2
0.24 [4,17,18] 0.5 [20]
0.45 [4,19] 1.0 [20]
3.7 [2,4,19] 3.9 [21]
3.24 [4] 9.2 [2]
3.8
2.05
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Fig. 2. States of the free carriers electron and hole of the ZnO QDs.
Fig. 1. (Color online) Free electron states energies as a function of the radius R (Å) of ZnO/SiO2 QDs.
The eigenstates of the exciton ðj1; j2; j3; j4; j5; 6Þ are written as the linear combination of the states on the basis B (see Figs. 2 and 3 and appendix) The components of the eigenstates should verify this relation ( X 1; i ¼ j CiCj ¼ 0; i aj i;j
For example for R¼20 Å, radius less then Bohr radius Bohr of the ZnO (28.7 Å), the different components of the eigenstates written as j1 ¼ 0:96 Se ; Sh 0:24 p0e ; p0h þ 0:06peþ ; ph j2 ¼ 0:95 Se ; p0h þ0:28 p0e ; Sh j3 ¼ 0:04 Se ; Sh þ 0:70 pe ; phþ þ 0:70 peþ ; ph j4 ¼ 0:28 Se ; p0h þ 0:95 p0e ; Sh j5 ¼ 0:24 Se ; Sh þ 0:02 pe ; phþ þ 0:96p0e ; p0h j6 ¼ 0:04 Se ; Sh 0:70 pe ; phþ þ 0:03 p0e ; p0h þ 0:70 peþ ; ph
Fig. 3. Excitonique states of ZnO QDs via the selection rule.
and for R ¼35 Å, radiu bigger than the Bohr radius of ZnO, we found this contribution j1 ¼ 0:99 Se ; Sh þ 0:01 p0e ; p0h j2 ¼ 0:99 Se ; p0h þ 0:09 p0e ; Sh j3 ¼ 0:70 pe ; phþ 0:70 peþ ; ph j4 ¼ 0:09 Se ; p0h þ 0:99 p0e ; Sh j5 ¼ 0:999 p0e ; p0h j6 ¼ 0:70 pe ; phþ þ 0:70 peþ ; ph We can recognize that when the distance between the electron and hole became bigger then ZnO Bohr radius, the Coulomb interaction became inconsiderable. Then, the energy of the exciton states toward to the bulk exciton energy. Fig. 4 shows the variation of the excitonic energy of the ground and of the excited states as a function of ZnO QDs radius. It can be noticed from Fig. 4 that the exciton energy for the ground and for the excited states decrease in parallel with the increasing of the QDs. This behavior can be explained as such: before the Bohr radius the kinetic energy being to be more dominant than the Coulomb interaction, and for the radius being equal or bigger than the Bohr radius, the Coulomb interaction is the dominant and for the large sized ZnO QDs the distance
Fig. 4. (Color online) Exciton energy for the ground and excited states as a function of the ZnO/SiO2 QDs radius.
between the electron and the hole increases and the exciton energy tends the energy of ZnO bulk value (Table. 2). We perceive from Table. 2 that the exciton energy of the ground state decreases from 10 meV to 5 meV, for a R¼ 25 (Å)–R¼ 35 (Å) QDs.
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We remark also from Fig. 4 that the values of the three last excited states energies for the exciton ZnO QDs, Ex4 ; Ex5 ; Ex6 , are nearly similarl; the continuum domain energy (see Table 3). We can grasp from Fig. 5, that the energies of the exciton X and of the exciton bound to an ionized donor impurity ðD þ ; XÞ,which decreases monotonically, when the QDs radius of ZnO increases between 10 (Å) to 30 (Å), however the energy of the exciton bound to an ionized acceptor impurity ðA ; XÞ increases slowly, when the QDs radius of ZnO increases to 30 (Å). For example the energies of the X and of the XC for two different ZnO QDs radii are summarized in Table 4. From Fig. 5 and Table 4, we note that the EðA ; X Þ is less than the þ EðX Þ and of E D ; X . The energy of the exciton bound to the donor impurty E D þ ; X is the biggest. In fact, for the radius being less than the ZnO Bohr radius, the Coulomb interaction of the donor impurity with the electron and the
Table 2 Exciton energy of the ground state as a function of ZnO QDs radius. R (Å) E (X1) (eV)
25 3.42
30 3.32
35 3.27
Table 3 Exciton energy of the excited states as a function of ZnO QDs radius.
hole carriers which is presented by 0 1 e2 1 1 ! ! B C V int r i ; r e þ V int r i i ; r h ¼ @ - ! þ - ! A ε ri r e ri r h dominates the electron–hole Coulomb interaction. And the Coulomb effect increases the confinement of the electron and of the hole. The Coulomb interaction of the acceptor impurity with the electron and the hole carriers, presented by 0 1 2 e B 1 1 ! ! C – V int r i ; r e þV int r i ; r h ¼ @ - ! þ - ! A ε ri r e ri r h increases the electron and hole Coulomb interaction. To conclude, the system formed by the acceptor bound to the exciton complex is more stable than the sytem formed by the exciton. The latter is more stable than the system formed by the donor bound to the complex exciton. For the radius being bigger than the ZnO Bohr radius, the energy of the exciton complex, ðA ; X) and (D þ ; X) towards to the energie of the free ZnO QDs exciton. Another important quantity of interest is the acceptor–donor bound exciton binding energy Eb that indicates the electron, hole and the acceptor–donor impurity Coulomb interaction which can be computed as follows: Eb ðA ; XÞ ¼ EðXÞ EðA ; XÞ
R (Å)
E(X4) (eV)
E(X5) (eV)
E(X6) (eV)
10 20 35
5.035 3.83 3.53
5.039 3.84 3.5412
5.05 3.85 3.5413
and Eb ðD þ ; XÞ ¼ EðXÞ EðD þ ; XÞ The acceptor–donor bound exciton binding energy Eb is investigated as a function of the radius R of the system ZnO/SiO2 QDs with an impurity located in the center. We can observe from Fig. 6 that the binding energy Eb ðA ; XÞ decreases when the radius R increases. These because the radial distance between the electron and the hole is increased when the radial R increases. Thus, the Coulomb interaction between the electron and the hole decreases when increasing the radius R. Fig. 6 confirms that the system formed by the acceptor bound to the exciton complex is more stable than the system formed by the donor bound to the complex exciton. In fact for the radius less than the ZnO Bohr radius Eb ðA ; X Þ is strictly positive and the Eb ðD þ ; XÞ is strictly negative. For the ZnO large sized QD the exciton binding energy for the complex exciton tends towards the bulk ZnO Rydberg energy ð0:06 eVÞ.
Fig. 5. (Color online) Energies of the exciton and the exciton complex, XC, in ZnO/ SiO2 QDs as a function of the dot radius. Impurity being located in the center.
Table 4 The comparison between the energies of the X and of the XC. R (Å)
10 30
E (eV) E(D þ , X)
E(X)
E(A , X)
5442 3346
4110 3349
2779 3343
Fig. 6. (Color online) Binding energy of the complex ecxiton with an impurity located in the center of the QDs as a function of the dot radius.
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4. Conclusions
5
But we only need to calculate the energies of the central bloc, because it is embedded with the elements of the two others blocs.
To summarize, within the effective-mass approximation and the diagonalization matrix approach, we have shown that the influence of the ionized impurity fixed charge on the acceptor and the donor binding energy is significant when the QD radius is smaller than the exciton Bohr radius 28.7 Å, we have proven that in this domain the system formed by the acceptor bound to the exciton complex is more stable than the system formed by the donor bound to the complex exciton, above this value, E b ðA ; XÞ and Eb ðD þ ; XÞ are just the value of the free bulk exciton binding energy. Finally, we have demonstrated a strong tunability of the optical properties (optical transition energies and binding energy) of ZnO nanoparticles not only with the size, but also with the presence of the ionized impurities inside them. This can be of potential use for different applications such as magnetic storage or optoelectronic and photovoltaic devices. Uncited references [17–21]. Acknowledgments
2
D1 6 6 D2 6 6 D3 6 And the matrix becomes : 6 6 D2 6 6 D4 4
D2
D3
D2
D4
D5
D6
D5
D7
D6
D8
D6
D9
D5
D6
D5
D7
D7
D9
D7
D10
D3
D6
D8
D6
D9
D3
3
7 D6 7 7 D8 7 7 7 D6 7 7 D9 7 5 D8
This matrix is diagonally symmetric. With D1 ¼ ψ e0;0 ψ h0;0 j H xc j ψ e0;0 ψ h0;0 ;
D2 ¼ ψ e0;0 ψ h0;0 j H xc j ψ e0;0 ψ h1;0
D3 ¼ ψ e0;0 ψ h0;0 j H xc j ψ e1 1 ψ h1;1 ;
D4 ¼ ψ e0;0 ψ h0;0 j H xc j ψ e1;0 ψ h1;0
D5 ¼ ψ e0;0 ψ h1;0 j H xc j ψ e0;0 ψ h1;0 ;
D6 ¼ ψ e0;0 ψ h1;0 j H xc j ψ e1; 1 ψ h1;1
D7 ¼ ψ e0;0 ψ h1;0 j H xc j ψ e1;0 ψ h1;0 ;
D8 ¼ ψ e1; 1 ψ h1;1 j H xc j ψ e1; 1 ψ h1;1
D9 ¼ ψ e1; 1 ψ h1;1 j H xc j ψ e1;0 ψ h1;0 ;
D10 ¼ ψ e1;0 ψ h1;0 j H xc j ψ e1;0 ψ h1;0
The authors are grateful to professor Mr. Imade Hethli for fruitful discussions. Appendix
References
Due to the selection rule, the basis B of the states composed has dimension of 16 and it is composed of these elements ψ ðl;mÞe ψ ðl;mÞh l ¼ 0;1 and m ¼ 0; 8 1 eðhÞ
eðhÞ
We will then eliminate the quantum number principal n to light the notation. 9 8 e ψ ψ h ; ψ e0;0 ψ h1; 1 ; ψ e1; 1 ψ h0;0 ; ψ e1; 1 ψ h1;0 > > > > > > 1; 1 1; 1 > > > = < ψ e1;0 ψ h1; 1 ; ψ e0;0 ψ h0;0 ; ψ e0;0 ψ h1;0 ; ψ e1; 1 ψ h1;1 > B¼ e h e h e h e h > > ψ ψ ; ψ ψ ; ψ ψ ; ψ ψ > 1;0 0;0 1;0 1;0 1;1 1; 1 0;0 1;1 > > > > > > > e h e h e h : ψe ψh ; ψ 1;1 ψ 0;0 ; ψ 1;1 ψ 1;0 ; ψ 1;1 ψ 1;1 ; 1;0 1;1 In this basis, H xc decomposes in a matrix and through the selection rule, relation ðnÞ n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n n n n 0 0 0 0 0 0 0 0 0 0 n n n n 0 0 0 0 0 0 0 0 0 0 n n n n 0 0 0 0 0 0 0 0 0 0 n n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 n n n n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n n n 0 0 0 0 0 0 0 0 0 0 0 n n n 0 0 0 0 0 0 0 0 0 0 0 n n n 0 0 0 0 0 0 0 0 0 0 0 n n n 0 0 0 0 0 0 0 0 0 0 0 0 0 0
can be simplified 0 0 0 0 0 0 0 0 0 0 0 n n n n
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n
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