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Physica B 358 (2005) 109–113 www.elsevier.com/locate/physb
Exciton states trapped by a parabolic quantum dot Wenfang Xie Department of Physics, Guangzhou University, Guangzhou 510405, PR China Received 8 December 2004; received in revised form 22 December 2004; accepted 22 December 2004
Abstract In this paper, an exciton trapped by a parabolic quantum dot has been investigated. The energy spectra of the exciton in a three-dimensional spherical quantum dot and a two-dimensional disc-like quantum dot are calculated as a function of the conﬁnement strength by means of matrix diagonalization. The results for the three-dimensional spherical quantum dot and the two-dimensional disc-like quantum dot are compared with each other. We found that the energies of the states and the interval between states decrease with reducing space dimensions. r 2005 Elsevier B.V. All rights reserved. PACS: 71.35.y; 71.15.m Keywords: Exciton; Quantum dot; Semiconductor
The investigation of artiﬁcial few-body systems is an attractive topic for both application and interest in basic research. In these systems usually an external ﬁeld is introduced so that the particles can be trapped and the quantum states can be manipulated. Quantum dots (QDs) are structures in which the charge carriers (electrons or holes) are conﬁned in all three dimensions [1,2]. In such nanometer QDs, some novel physics phenomena and potential electronic device applications have generated a great deal of interest. Three-dimensional (3D) quantum conﬁnement of charge carriers makes it possible to observe quantum size Tel.: +8620 86231902; fax: +8620 86230002.
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effect in QDs . The natural length scales in a QD are of the order of a few nanometers and because of this reduced dimensionality it contains a ﬁnite number of the charge carriers and has a discrete energy spectrum. QDs, in fact, offer an excellent ground for testing quantum mechanics and, therefore, they are intrinsically appealing from the point of view of fundamental physics. Because of the quantum size effect, QD structures exhibit many new physical properties that are very interesting and are quite different from those of bulk systems. Modern technologies, like selforganized growth or molecular beam epitaxy, allow scientists to fabricate QDs as small as 10 nm and yet their size, shape, and other properties can be controlled in the experiments. Thus
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increasing attention has been focused on fewparticle conﬁned Coulomb states mainly because of their importance in the optical properties of these QDs. Excitonic effects play a central role in the optical properties of semiconductors. Recently, very ﬁne structures were observed in exciton photoluminescence from GaAs quantum wells [4–6]. These sharp lines are interpreted as luminescence from localized excitons at island structures in the quantum wells. These island structures can be regarded as zero-dimensional QDs. In semiconductor QDs the electron–hole interaction is enhanced owing to the conﬁnement effect. Excitons in QDs have attracted considerable attention in recent years. A number of theoretical investigations of exciton levels in QDs have been published. Most of them are related to variational studies of the excitonic ground state in spherical dots or to calculations in some limiting cases [7–15]. Usually, inﬁnite barriers are considered. A full numerical analysis of this problem has been done by Hu and co-workers [16,17], expanding the excitonic wave functions in terms of solutions of the single-particle Schro¨dinger equations. There are also papers studying nonspherical dots, e.g., boxes , square ﬂat plates , and cylindrical QDs [20–22]. Effects such as dielectric conﬁnement and electron–hole exchange interaction on excitonic states in semiconductor QDs have been also studied . Marfn et al. performed a variational calculation of the ground state energy of excitons conﬁned in spherical QDs with a ﬁnite-height potential wall . Consider an electron and a hole moving in a spherical (or disc-like) QD under the conﬁned parabolic potential well, we make the generalization that the Hamiltonian in this system within the effective-mass approach is given by X _2 r 2 1 e2 i H¼ þ mi o20 r2i ; (1) 2 2mi reh i¼e;h where me (mh ), and ~ re ( ~ rh ), respectively, denote the effective mass, and the position vector of the electron (hole), reh ¼ j~ re ~ rh j is the distance between the electron and the hole, o0 is the strength of the conﬁnement, and is the dielectric
constant of the medium in which the electron and the hole are moving. The Hamiltonian has spherical (or cylindrical) symmetry which implies that the total orbital angular momentum, L, is a conserved quantity, i.e., the corresponding quantum number a good one. Hence, the eigenstates of the excitons in spherical (cylindrical) QDs can be classiﬁed according to the total orbital angular momentum, i.e., after solving our Hamiltonian, a series of energy levels which we indicate by the quantum number L. On the other hand, for spherical QDs, the parity p is also a good quantum number. To obtain the eigenfunction and the eigenenergy associated with the excitons in QDs, we diagonalized H. The better the basis describes the Hamiltonian, the faster the convergence. The most common basis chosen is the one that describes the Hamiltonian at zero order. rh In terms of the relative coordinate ~ r ¼~ re ~ ~ ¼ ðme~ and the center-of-mass coordinate R re þ mh~ re Þ=ðme þ mh Þ; the Hamiltonian can be rewritten as H ¼ H0 þ V ;
with H0 ¼
_2 r2r 1 2 2 _2 r2R 1 þ mo r þ Mo2 R2 ; 2 2 2m 2M
1 e2 1 V 1 ¼ mðo20 o2 Þr2 þ Mðo20 o2 ÞR2 ; 2 r 2
where m ¼ me mh =M is the electron–hole reduced mass and M ¼ me þ mh is the total mass. To obtain the eigenenergies and eigenstates, H is diagonalized in model space spanned by a set of translational invariant harmonic product basis functions with a total orbital angular momentum L and a parity p as X ~ FLp ½jo rÞjo (5) ½K ¼ n1 ‘1 ð~ n2 ‘2 ðRÞ L ; ½K
where jo n‘ is a 3D harmonic oscillator state with a frequency o and an energy ð2n þ ‘ þ 3=2Þ _o; and [K] denotes the set quantum numbers (n1 ; ‘1 ; n2 ; ‘2 ) in brevity. The angular momenta of the electron, the hole, and the pair are denoted by ‘1 ; ‘2 ; and L ¼ ‘1 þ ‘2 ; respectively. In practical calculation,
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Lp hFLp ½K jH 0 jF½K 0 i ¼ ½2ðn1 þ n2 Þ þ ‘ 1
þ ‘2 þ 3 _od½K ;½K 0 ;
Lp I 0 0 0 hFLp ½K jV jF½K 0 i ¼ U n1 n0 d‘1 ;‘1 dn2 ;n2 d‘2 ;‘2 1
0 0 0 þ U II n2 n0 d‘2 ;‘2 dn1 ;n1 d‘1 ;‘1 :
with U Inn0
1 e2 Rn‘ ðrÞ mðo20 o2 Þr2 2 r
Rn0 ‘ ðrÞr dr; U II nn0 ¼
1 Rn‘ ðrÞ Mðo20 o2 Þr2 Rn0 ‘ ðrÞr dr; 2
where Rn‘ ðrÞ is the radial part of 3D harmonic oscillator function. It is similar for 2D disc-like QDs, let H be diagonalized in a model space spanned by translational invariant 2D harmonic product bases P o ~ FL½K ¼ ½K ½jo rÞjo n1 ‘1 ð~ n2 ‘2 ðRÞ L ; fn‘ is a 2D harmonic oscillator wave function with frequency o (o is considered as an adjustable variational parameter), an energy ð2n þ j‘j þ 1Þ_o: The accuracy of the solutions depends on how large the model space is. Since we are interested only in the ground and low-excited states and in qualitative aspect, the model space adopted is neither very large to facilitate numerical calculation, nor very small to assure the qualitative accuracy. This is achieved by expanding the dimension of the model space step by step, in each step the new results are compared with the previous results from a smaller space, until satisfactory convergence is achieved.
Our numerical computation is carried out for GaAs for which the dielectric constant is taken to be ¼ 12:4 and the electron effective mass me ¼ 0:067m0 (m0 is the single electron bare mass). The energy is measured in the unit of the effective Rydberg constant Ryn ¼ me e4 =ð2_2 Þ: We set mh ¼ 0:095m0 and plotted in Fig. 1 the energy spectrum of the four lowest quantum-conﬁned states (i.e. Lp ¼ 0þ ; 1 ; 2þ and 3 states) of the exciton in the 3D spherical parabolic potentials as function of the conﬁnement strength _o0 which corresponds to the QDs with the strong and intermediate conﬁnement, i.e., 0:02Ryn p_o0 p2:0Ryn : Although the L ¼ 0 excitons have been extensively discussed in the literature, there exists little experimental or theoretical information on LX1 exciton states. The L ¼ 1 excitons are especially interesting since these may be excited in infrared spectroscopy as well as in two-photon spectroscopy. Both these phenomena have received some experimental attention [25,26]. From Fig. 1 we ﬁnd that, in general, the correlation energies of the ground and low-excited states increase with increasing _o0 : The larger the orbital momentum L is, the faster the energy increases, i.e., the exciton energies are proportional to the orbital momentum. It is only the energies of the ground and
4 3D spherical QD Correlation Energy (Ry*)
o serves as an adjustable variational parameter around o0 to minimize the eigenvalues. The exact diagonalization method consists of spanning the Hamiltonian for a given basis and extract the lowest eigenvalues (energies) of the matrix generated. The better the basis describes the Hamiltonian, the faster the convergence will be. The most common basis chosen consists of oscillator harmonics which are the eigenfunctions of the Hamiltonian when the electron–hole interaction is switched off. The matrix elements of H are then given by the following expressions:
0 -1 0.02
1.0 hω0 (Ry*)
Fig. 1. The correlation energies of the ground and low-excited states of an exciton trapped by a 3D spherical parabolic QD are plotted as function of the conﬁnement strength _o0 potential. The levels are labeled by quantum number L.
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qﬁrst-excited states decrease, at ﬁrst, with increasing _o0 ; but they quickly reach the minimum. On the other hand, the energy interval between states increases with increasing _o0 : This can be interpreted as follows. It is the competition between the single-particle energy and Coulomb interaction energy that ﬁnally determines the energies of lowlying states of an exciton in QDs. The stronger the conﬁnement strength _o0 is, the higher the singleparticle energy is. On the other hand, we know that the orbit radius of the electron and the hole trapped by a QD is inversely proportional to the conﬁnement _o0 : When _o0 increases, the dot radius will decrease and the spatial overlap between an electron and a hole will increase, leading to the increase in the Coulomb binding energy. In addition, when _o0 increases, the rotational inertia of an exciton reduces and the rotational energy of an exciton increases. Hence, the exciton energies and the energy intervals between states trapped by a QD are proportional to orbital momentum L and the minimum value position decreases with increasing L. The energy of the ground state increases slowly with increasing _o0 because of L ¼ 0: However, the increase in the rotation energy becomes predominant and cannot be compensated by the increase of the electron– hole interaction for La0: The energies of the lowexcited states increase fast with increasing the conﬁnement strength _o0 : To further our investigation of the feature of the ground and low-excited states of an exciton trapped a parabolic QD, in Fig. 2, we plot the correlation energies with Lp3 as a function of the conﬁnement strength _o0 for a disc-like parabolic QD. The Fig. 2 shows that the qualitative properties of energy levels for the disc-like parabolic QD and the corresponding spherical parabolic QD are similar. However, the quantitative differences are also obvious. We ﬁnd, at ﬁrst, that the energy of the ground state decreases always with increasing _o0 ; i.e., the stronger the conﬁnement strength is, the lower the energy is. This is obviously different from 3D spherical QDs. We ﬁnd, secondly, that the energies of the states and the interval between states decrease with reducing the space dimensions as expected. The energy levels for 2D disc-like QDs are located below the
4 2D disc-like QD Correlation Energy (Ry*)
1.0 hω0 (Ry*)
Fig. 2. The correlation energies of the ground and low-excited states of an exciton trapped by a 2D disc-like parabolic QD are plotted as function of the conﬁnement strength _o0 potential. The levels are labeled by quantum number L.
corresponding levels for 3D spherical QDs. The energy difference between states in 2D disc-like QDs is also obviously lower than the corresponding levels for 3D spherical QD. The physical origin of this effect is related to the enhanced effective conﬁnement of an electron and a hole in 2D disclike QDs. In conclusion, we have applied the spherical parabolic conﬁnement potential to a description of the excitons in semiconductor QDs. We have calculated the energy levels of the ground and low-excited states for Lp3 as function of the conﬁnement strength _o0 : The same calculations performed with the disc-like parabolic conﬁnement potential lead to the results, which are qualitatively and quantitatively different. It may be important in the quantitative understanding of future experimental work involving excitons trapped by parabolic QDs. The present results are useful to understand the optical and magnetic properties of QD material. This work is ﬁnancially supported by the Natural Science Foundation of Guangdong Province, China (Grant No. 04009519) and the National Natural Science Foundation of China under Grant No. 10275014.
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