- Email: [email protected]
Exciton fractional dimension in semiconductor heterostructures arising from variational principle
ARTICLE IN PRESS
Physica B 355 (2005) 255–263 www.elsevier.com/locate/physb
Exciton fractional dimension in semiconductor heterostructures arising from variational principle R.A. Escorciaa, J. Sierra-Ortegaa, I.D. Mikhailovb, F.J. Betancurb, a Departamento de Fı´sica, Universidad del Magdalena, A. A. 731 Santa Marta, Colombia Departamento de Fı´sica, Universidad Industrial de Santander, A. A. 678 Bucaramanga, Colombia
b
Received 12 December 2003; received in revised form 23 April 2004; accepted 1 November 2004
Abstract We present a simple method for calculating the ground-state energy of an exciton in quantum confined structures. We express the exciton wave function as a product of the electron and hole one-particle wave functions with a variationally determined envelope function which describes the exciton intrinsic properties. Starting from the variational principle, we derive an one-dimensional wave equation for this envelope function and show that it describes a hydrogen-like atom in an effective isotropic space with the non-integer running dimension. We establish that this dimension runs from three, as the electron–hole separation is small for all heterostructures, to two in quantum well one in quantum well-wire and to zero in quantum dot, as the separation is large. The exciton ground-state energies are calculated for different confining potential shapes. Our results for GaAs-(Ga, Al)As heterostructures with square-well potential are in an excellent agreement with those obtained previously by means of other methods. r 2004 Elsevier B.V. All rights reserved. PACS: 71.35.y; 73.61.Ey; S7.12 Keywords: Semiconductor estructures; Exciton binding energy
1. Introduction The progress in nanoscale technology has made possible the fabrication of low-dimensional heterostructures with controlled thickness and relatively sharp interfaces, where the excitons remain present even at room temperature because of the quantum confinement increases highly the electron–hole Corresponding author. Tel./fax: +57 7 63 23477.
E-mail address: [email protected] (F.J. Betancur).
attraction [1]. A great deal of attention has been devoted to experimental and theoretical studies of excitons in heterostructures based on III–V semiconductor [2–11] particularly quantum wells (QWs), superlattices (SLs), quantum-well wires (QWWs), quantum dots (QDs) and more recently quantum rings (QRs). Different methods such as the variational [2–5], matrix diagonalization [6,9,10] or finite element [8] have been used to analyze the effect of the confinement on the exciton spectrum. Up to now it has been considered the
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.11.002
ARTICLE IN PRESS 256
R.A. Escorcia et al. / Physica B 355 (2005) 255–263
confinement models with square-well [2–5], parabolic [5,9], charge image [7], exponential [10] and soft-edge-barrier [11] potentials. In spite of that these techniques give consistent results with the experimental data, in most of the cases they entails a lot of computational work and the accuracy of the variational method depends to a large extent on the form of the trial wave function. An approach that allows consider the anisotropic interactions in a three-dimensional space as isotropic ones in a lower fractional-dimensional space has been applied by He [12], who using the hydrogen-like Hamiltonian in an effective fractional space treated the interband optical transitions and bound excitons in strongly anisotropic semiconductors. Lefebvre and collaborators [13] and Oliveira and co-investigators [14] applied this method to analyze the exciton energy states and the absorption spectra in QWs and QWWs. They considered the fractional dimension as a phenomenological parameter related to the heterostructure geometry and proposed a different technique for calculating this parameter. Recently, we have proposed a simple variational procedure related to the fractal-dimensional approach [15]. In this procedure the wave equations for neutral and negatively charged donors [15] and exciton [11] in heterostructures are reduced to ones similar to hydrogenic-like atom in an isotropic effective space with fractional dimension, which is determined by using the Mandelbrot’s definition for fractal objects [16]. It has been established [15] that this procedure provides an efficient algorithm for calculating the ground state binding energies with accuracy comparable with those of sophisticated methods such as the series expansion and Monte Carlo. In this work, we extend our method to study excitons in different semiconductor heterostructures in the presence of magnetic field taking into account the anisotropy of the hole effective mass and considering different confining potential shapes.
2. Differential equation for the electron–hole correlation function
well and the barrier are neglected. The dimensionless Hamiltonian for the electron–hole pair confined in a heterostructure with an appropriate scaling for the energy and length units may be written as Hðre ; rh ; tÞ ¼ H 0 ðre ; rh Þ 2=reh ; H 0 ðre ; rh Þ ¼
Zeik
q2 qxie qxke
Zhik
q2 þ Uðre ; rh Þ: qxih qxkh
i;k¼1
3 X i;k¼1
ð1Þ
In these relations H 0 ðre ; rh Þ describes the free motion of the electron and hole confined in the heterostructure, re ¼ fx1e ; x2e ; x3e g and rh ¼ fx1h ; x2h ; x3h g are the electron and hole position vectors, Uðre ; rh Þ is the perturbation due to the confinement and the external fields and the coefficients Zeik and Zhik describe the relative values and the anisotropy of the electron and hole effective masses. 2=reh and reh ¼ jre rh j are the energy of the electron–hole interaction and the electron–hole separation, respectively. In what follows we assume that the Hamiltonian for the free electron–hole pair is separable and the solution of the corresponding eigenvalue problem for the ground state, H 0 ðre ; rh Þ f 0 ðre ; rh Þ ¼ E 0 f 0 ðre ; rh Þ;
(2)
is known. Then to solve the Schro¨dinger equation for exciton, Hðre ; rh ÞCðre ; rh Þ ¼ E ex Cðre ; rh Þ
(3)
where E ex is the exciton total energy, we choose Cðre ; rh Þ ¼ f 0 ðre ; rh ÞFðreh Þ; being Fðreh Þ a variational function, that describes the intrinsic properties of the exciton and depends only on the electron– hole separation. Starting from the Schro¨dinger’s variational principle and using the procedure of the functional derivation described in Ref. [15] we obtain the following differential equation for FðrÞ:
We consider a confinement model in which the differences between the material parameters in the
3 X
1 d dFðrÞ 2 J 1 ðrÞ FðrÞ ¼ ½E ex E 0 FðrÞ; J 0 ðrÞ dr dr r (4a)
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
where J 0 ðrÞ ¼ J 1 ðrÞ ¼
Z
Z dre Z dre
f 20 ðre ; rh Þdðreh rÞdrh ;
Z X 3 ðZeik þ Zhik Þ ðxie xih Þ ðxke xkh Þ r2eh i;k¼1
f 20 ðre ; rh Þdðreh rÞdrh ;
ð4bÞ
3. Exciton in fractal dimensional space To reveal the physical significance of Eq. (4a) let us consider a particular case as the electron and hole effective masses are isotropic Zeik ¼ Ze dik ; Zhik ¼ Zh dik and in coordinates of the center of mass Ze þ Zh ¼ 1: Therefore Eq. (4a) transforms into
1 d dFðrÞ 2 J 0 ðrÞ FðrÞ ¼ ðE ex E 0 ÞFðrÞ: J 0 ðrÞ dr dr r (5)
This equation is similar to one of a hydrogenic-like atom in an effective isotropic space with the radial part of the Jacobian given by J 0 ðrÞ: The properties of this space are related generally to J 0 ðrÞ; which in according to relation (4b), is defined by the geometry of the heterostructure. In other words, if the dependence is a power-law, J 0 ðrÞ ¼ CrD1 ; then the scaling parameter D in this dependence could be considered as the dimension of an effective space and the Eq. (5) for the electron– hole relative motion would have coincided with the eigenvalue problem for a hydrogen-like atom in a D-dimensional effective space, being the parameter D an integer o fractional. If this dependence is non-power-law, we can define the local dimension in an isotropic effective space by using the Mandelbrot’s concept for fractals [16]. The properties of this space can be described in terms of J 0 ðrÞ: This dependence is a power-law only when the space is homogeneous and it does not in other cases. Since due to the heterostructure confinement J 0 ðrÞ has no a power-law dependence on r. We assume that the parameters C and D in the above scaling equation also depends on r and consequently J 0 ðrÞ is related to the dimension D ðrÞ as J 0 ðrÞ ¼ CðrÞrD ðrÞ1 or D ðrÞ ¼ 1 þ rd½lnJ 0 ðrÞ =dr;
257
where CðrÞ and D ðrÞ are functions that vary more slowly than any power function. We perform the calculation of J 0 ðrÞ and D ðrÞ for excitons in QW, cylindrical QWW and spherical QD with square-well potentials in absence of the external field. In this case Eq. (2) is separable and one can calculate J 0 ðrÞ directly by using the relation (4b) and the well-known analytical expressions for the ground state wave functions of the electron and the hole in the heterostructure. Fig. 1(a) shows the typical behavior of J 0 ðrÞ obtained for three heterostructures. For small values of the electron–hole separations, as the electron–hole distance is essentially smaller than the size of the heterostructure, the tree curves behave as parabolic ðJ 0 r2 Þ similar to the case of the free exciton in a three-dimensional space. As the electron–hole distance increases and it becomes greater than the heterostructure size, the curves transform into a linear function ðJ 0 rÞ for a QW, a constant ðJ 0 r0 Þ for QWW and decreasing as an exponential function ðJ 0 0Þ for QD, corresponding to the free motion in two-one and zero-dimensional spaces as expected. The corresponding dependencies of the fractal dimension, D ðrÞ on the relative coordinate of the exciton are shown in Fig. 1(b). It can be seen that D ðrÞ falls from 3 for small electron–hole distances to 2 in QW, to 1 in QWW and to 0 in QD as the electron–hole separation becomes larger than the size of the corresponding heterostructure. Such dependence is typical for quasi-two, -one and –zero-dimensional heterostructures, respectively.
4. Excitons in GaAs/(Ga,Al)As QW and cylindrical QWW In order to obtain the exciton ground state energy and its wave function from Eq. (4a), we first should define explicitly the dimensionless Hamiltonian (1). If we scale in this Hamiltonian all lengths in terms of the electron Bohr radius a 0 ¼ _2 =m e e2 ; all energies in electron effective Rydberg R y ¼ e2 =2 a 0 then ignoring the GaAs(Ga,Al)As heterostructures coupling between the heavy-hole and light-hole bands, we find that Zeik is
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
258
exciton center-of-mass free motion in the xy plane, the Hamiltonian (1) in cylindrical coordinates can be written as [3]
QW
2
q2 q2 Z z qz2e qz2h 1 q q r þ Uðre ; rh Þ; ð1 þ Z? Þ r qr qr ð1 þ Z? Þg2 r2 ; ð6Þ Uðre ; rh Þ ¼ V e ðze Þ þ V h ðzh Þ þ 4 where the functions V e ðze Þ and V h ðzh Þ give the potential profiles for the electrons and holes, r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxe xh Þ2 þ ðye yh Þ2 is the in-plane distance between electron and hole and the parameter g ¼ e_B=2m e cR y is the Landau level expressed in R y : The eigenvalue problem (2) for the Hamiltonian (6) can be separated and its solution is:
J0(r)
H 0 ðre ; rh Þ ¼
1
QWW
QD 0 0.0 (a)
0.5
1.0
r/a0* 3
QW
ð0Þ 2 f 0 ðre ; rh Þ ¼ f ð0Þ e ðze Þf h ðzh Þ expðgr =4Þ;
2 D*(r)
ð0Þ E 0 ¼ E ð0Þ e þ E h þ ð1 þ Z? Þg;
QWW
ð0Þ where f ð0Þ e ðze Þ and f h ðzh Þ are solutions of the following one-dimensional equations
1 QD
Zi
q2 f ð0Þ ð0Þ ð0Þ i ðzi Þ þ V i ðzi Þf ð0Þ i ðzi Þ ¼ E i f i ðzi Þ; qz2i
i ¼ e; h; Ze ¼ 1; Zh ¼ Zz :
0 0.0 (b)
ð7aÞ
0.5
1.0
r/a0*
Fig. 1. The radial part of the Jacobian (a) and fractal dimension (b) dependences on the electron–hole separation in QW, cylindrical QWW and spherical QD.
ð7bÞ
Substituting the wave function (7a) in the general expression (4b) we obtain, after integrating over the angles, the following simplified formulae for the Jacobians [15]: Z r Z 1 2 2 2 J i ðrÞ ¼ 2pr f e ðze Þdze egðr z Þ=2 f 2h ðze þ zÞ 1
r
Ri ðz; rÞdz; i ¼ 0; 1 an identity matrix and Zhik is a diagonal matrix with two elements Z? ¼ m e =m ? corresponding to x, y directions and Zz ¼ m e =m z corresponding to z-direction, where m e ; m ? and m z are the effective masses of the electron and hole in the xy plane and z-direction, respectively. Let us consider an exciton in a QW with uniform magnetic field, B applied along the z-axis. Choosing the symmetric gauche vector potentials, Ae ¼ B ðre rh Þ=2 and Ah ¼ B ðrh re Þ=2 for electron and hole respectively and separating the
R0 ðz; rÞ ¼ 1; R1 ðz; rÞ ¼ 1 þ Z? þ ðZz Z? Þ
z2 : r2 (8)
We assume that the wire is sufficiently long, so that the motion along wire axis has translational symmetric and the confining potential is a function of the r-coordinate. Similarly one can calculate the Jacobians (4b) for a cylindrical QWW. By using the cylindrical coordinates for the electron ðre ; je ; ze Þ and hole ðrh ; jh ; zh Þ; the Hamiltonian
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
for the electron–hole-pair in the ground state without interaction between the particles can be written as H 0 ðre ; rh Þ ¼
1 q q 1 q q r Z? r re qre e qre rh qrh h qrh
þ Uðre ; rh Þ; g2 r2e Z g2 r2 þ V h ðrh Þ þ ? h : (9) 4 4 Here V e ðre Þ and V h ðrh Þ give the potential profiles for the electrons and holes. The wave equation (2) for the Hamiltonian (9) can be separated and the wave function for the electron–hole pair can be represented as Uðre ; rh Þ ¼ V e ðre Þ þ
ð0Þ f 0 ðre ; rh Þ ¼ f e ðre Þf h ðrh Þ; E 0 ¼ E ð0Þ e þ Eh ;
(10)
where f e ðre Þ and f h ðrh Þ are solutions of the onedimensional wave equations:
To find the wave functions for the electron and ð0Þ the hole f ð0Þ e ðze Þ; f h ðzh Þ in QW and f e ðre Þ; f h ðrh Þ in QWW we solve the respective Eqs. (7b) and (11) by using the trigonometric sweep method [17]. The same numerical procedure is used to solve Eq. (4a) for FðrÞ: In order to check the accuracy of our method, we first calculate the exciton binding energy in a GaAs-Ga0.7Al0.3As QW and cylindrical QWW considering square-well potentials. In Fig. 2a and b we display the calculation results for the exciton binding energies for heavyhole, E 1s ðhÞ; (solid lines) and for light-hole, E 1s ðlÞ; (dashed lines) excitons as function of the QW width and compare them with those from Refs. [2,3] (open circles). In both cases we intentionally choose the material parameters from corresponding references and we find an excellent concordance between the two sets of results for
Zi q qf ðr Þ Z g2 r2 ri i i þ V i ðri Þf i ðri Þ þ i i qri ri qri 4 i ¼ e; h;
Ze ¼ 1; ð11Þ
ð0Þ Here E ð0Þ e and E h are the lowest energies of the electron and hole in the wire, respectively. Substituting the wave function (10) in the general expression (4b) we obtain after integrating over angles the following simplified formulae for the Jacobians in a cylindrical QWW: Z re þr Z 1 2 J i ðrÞ ¼ 4pr f e ðre Þre dre f 2h ðrh Þ 0
Ri ðre ; rh ; rÞrh drh ;
o Ref. 2
9.0 Eb (meV)
f i ðri Þ ¼ Zh ¼ Z? :
E ð0Þ i f i ðri Þ;
7.5
Light-Hole Exciton
Heavy-Hole Exciton
6.0
0
1
(a)
2 W /a0*
3
heavy hole light hole
maxf0; re rg
20
i ¼ 0; 1;
R0 ðre ; rh ; rÞ ¼ G 0 ðre ; rh ; rÞ; R1 ðre ; rh ; rÞ ¼ ð1 þ Z? ÞG 0 ðre ; rh ; rÞ Z Z ð12Þ þ z 2 ? G A ðre ; rh ; rÞ; r where G 0 ðre ; rh ; rÞ ¼ Kðb=aÞ=a as r4re þ rh and G 0 ðre ; rh ; rÞ ¼ Kða=bÞ=b as rore þrh ; G A ðre ; rh ; rÞ ¼ aEðb=aÞ as r4re þ rh and G A ðre ; rh ; rÞ ¼ bEða=bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as rore þ rh being a ¼ r2 ðre rh Þ2 ; b ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 4re rh and KðkÞ; EðkÞ the complete elliptic integrals of the first and second kind, respectively.
4
25
Eb (meV)
259
o
Ref. 3
15 B=50 KG
10
5 0.0 (b)
B=200 KG
B=0 KG
0.5
1.0 W /a0*
1.5
2.0
Fig. 2. We compare the calculation results for the light- and heavy-hole exciton binding energies (solid and dashed line respectively) as a function of the GaAs/Ga0.7Al0.3As QW width (a) with those results from Ref. [2] and (b) from Ref. [3] (open circles) by using the similar parameters of the model.
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
260
different well widths and magnetic field strengths. As it is seen from Fig. 2a and b the binding energies of the nearly free light-hole exciton (in the limit of large QWWs radii and zero-magnetic field) is slightly larger than the corresponding values for the heavy-hole exciton and the order of these energies are inverted in the conditions of a strong confinement (the limits of the small QWW radii or large magnetic field). This effect was established in Ref. [2] and for the case of a structural confinement it was explained on the base of the anisotropic Kohn–Luttinger model for which the reduced effective mass of the light-hole exciton in the x–y plane is larger than one for heavy-hole exciton. A similar explanation is also valid for the case of a confinement produced by a strong magnetic field. We compare the calculation results of the heavyhole exciton binding energy as a function of the cylindrical QWW radius (solid lines in Fig. 3) with those from Ref. [4] (open circles). Although the isotropic hole mass approximation used in Ref. [4] for hole is applicable only for small electron–hole separations, an excellent concordance between the two sets of results can be observed in Fig. 3 for all wire radii with zero-magnetic field. Also it is seen that in the presence of the magnetic field (B ¼ 200 KG) for wire radii larger than electron effective Bohr radius our results are slightly higher than those from Ref. [4]. We attribute this small discrepancy to the additional lateral confinement
produced by the strong magnetic field that enlarges the effect of anisotropy. The motion of the particles in this case due to the strong in-plane confinement is almost quasi-one-dimensional and it is directed along the wire axis. The heavy-hole effective mass in this direction is higher than its value in the x–y plane and therefore the exciton effective Rydberg (a measure of the exciton binding energy) for anisotropic model is a slightly greater than for isotropic one. In Fig. 4 we illustrate the effect of the structural confinement on the conditional probability density for the heavy-hole exciton at zero-magnetic field, defined as Pðxe ; ze =xh ¼ yh ¼ zh ¼ ye ¼ 0Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 x2e þ z2e ; ¼ f e ðxe ÞF
ð13Þ
which corresponds to the hole and electron locations on the QWW axis and in the x– z plane, respectively. It is seen that as decrease the QWW radius the probability density becomes more anisotropic, meanwhile the peak of the distribution becomes more pronounced. It should be noted that the distribution contraction in the radial direction, due to the structural confinement is also accompanied by its weak contraction in the z-direction.
5. Excitons in cylindrical nanotubes 20 o Ref.4 Eb (meV)
15
B=200 KG
10 B=0 KG
5 0
1
2
3
4
R / a0* Fig. 3. Heavy-hole exciton binding energy as a function of the wire radius with and without magnetic field. Our results (solid lines) are compared with the theoretical data from Ref. [4] (open circles).
Finally, we apply our theory to analyze the effects of the potential shape on the exciton ground state binding energy in nanotubes. Usually the confinement potentials for an electron and hole in GaAl-(Ga,Al)As heterostructures are modeled by either the square-well or parabolic potentials with finite or infinite depth, respectively. Nevertheless, as it results from numerical solution of the Poisson equation, a realistic potential within quantum-well heterostructures is nearly parabolic at the center of the well and it becomes nonparabolic at the junctions [10]. Moreover, if we want to take into account the effects originating from a local fluctuation in the Al content (alloy clustering) in the regions of the junctions we
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
261
Fig. 4. Schematic representation of the electron conditional probability density in the x–z plane, for different QWW radii: (b) R ¼ 4a 0 ; (c) R ¼ 1a 0 ; (d) R ¼ 0:3a 0 ; for the heavy-hole exciton with the fixed hole position at the axis.
20 square-well potential 16 Eb (meV)
should consider also non-abrupt confinement potential. To describe such behavior of the confining potential we use the following nonabrupt version of the step function: Yðx; x0 ; W Þ ¼ 0 for xox0 W ; Yðx; x0 ; W Þ ¼ ½ðx x0 Þ2 =W 2 1 2 for x0 W pxox0 and Yðx; x0 ; W Þ ¼ 1 for xXx0 which turns into the Heaviside function as W ! 0: The parameter W is associated with the thickness of the transition region in the junctions and varying its value one can obtain a different confining potentials. For example, V ðrÞ ¼ V 0 Yðr; R0 ; W Þ describes the potential for the electron or hole in a cylindrical QWW of radius R0 and barrier height V 0 : As W ! 0 and W ! R0 the almost rectangular (square-well) and the parabolic potentials with a finite barrier height are modeled by this function, respectively. The relation V ðrÞ ¼ V i Yðr; Ri ; W Þþ V e Yðr; Re ; W Þ describes the potential of the particles in a nanotube with internal radius, R i and barrier height, V i ; which determines the properties of the repulsive core. The external radius, Re and barrier height, V e is determined for the parameters of the wire. Fig. 5 shows the ground state binding energy of exciton as a function of the GaAs-Ga0.7Al0.3As
parabolic potential 12
8
soft-edge barrier potential
core-repulsive potential
0
1
2
3
R/a0* Fig. 5. Exciton binding energies in GaAs/Ga0.7Al0.3As cylindrical QWWs with different confinement potential shapes as a function of the radius.
cylindrical QWW radius for different potential shapes: square-well ðRi ¼ 0; W ¼ 0:01 Re Þ; softedge barrier ðRi ¼ 0; W ¼ 0:5 Re Þ; parabolic finitebarrier ðRi ¼ 0; W ¼ Re Þ; and a cylindrical nanotube-like wire with repulsive core ðRi ¼ 0:15Re ; V i ¼ V e ; W ¼ 0:01 Re Þ: As one compare the curves (Fig. 5) for cylindrical QWWs with equal finite barrier height
ARTICLE IN PRESS 262
R.A. Escorcia et al. / Physica B 355 (2005) 255–263
and different potential shapes, square-well (solid line), soft-edge barrier (dotted line) and parabolic (dashed line) potentials one can see that for large and intermediate wire radii the larger the transition region thickness the higher is the binding energy. It is apparent that for intermediate and large values of the QWW radius the parabolic shape gives the largest confinement, followed by the soft-edge barrier, whereas the rectangular potential corresponds to the smallest confinement. As the wire radius Re decreases, the exciton binding energy for all potential shapes climbs up until it reaches a maximum. As Re ; further decreases, the exciton wave function leaks into the barrier region and the exciton binding energy begins to fall off rapidly meanwhile the exciton 3D character is restored. It is evident that this leakage of the exciton wave function in QWW with parabolic confinement occurs earlier due to stronger confinement than for the other two types of potential and consequently the maximum of the binding energy for this model is lower than the maxima for the soft-edge-barrier and rectangular potentials. In Fig. 5 we also present the results for a nanotube-like structure which is modeled by a quantum wire with repulsive core of internal radius, Ri ¼ 0:5Re and barrier height V i ¼ V e (dashed–dotted line). It is seen that for large QWW radii the exciton binding energies in nanotubes are higher than those in QWWs with square-well potential (solid line) and conversely for small radii. This is due to the fact that for large outer radii, Re ; both electron and hole are mostly located within the ‘‘ring’’ of the nanotube on the same side with respect to the core and the exciton undergoes a stronger confinement than in the wire. As consequence, the exciton binding energy is enhanced as the thickness of such ‘‘ring’’ is decreased up to a value comparable with the exciton radius. On the contrary, as the ring thickness is smaller than the exciton radius and the core size ðRi Þ is comparable with the exciton size, the electron and the hole tend to be located at opposite sides of the repulsive core, the separation between them increases and the binding energy diminishes. As consequence, the exciton binding energies for small wire radii in nanotubes are lower than those in quantum wire and therefore, the
crossover of the corresponding curves in Fig. 5 is observed.
6. Conclusion We propose a simple method for calculating the ground state wave function of an exciton in semiconductor heterostructures with arbitrary potential shapes and taking into account a possible anisotropy of the effective masses of the electron and hole. Starting from the variational principle, we reduce the initial problem for an exciton confined in the heterostructure to a central force problem in an isotropic and non-homogeneous effective space with a noninteger variable dimension. An explicit expression for the fractal dimension of this effective space is deduced, by using the Mandelbrot definition.
Acknowledgements This work was partially financed by the Universidad Industrial de Santander (UIS), through the General Researches (DIF) and the Colombian Agency COLCIENCIAS (Cod 1102-05-16923). R. Escorcia and J. Sierra-Ortega wish to thank the Universidad del Magdalena for the permission to study at the UIS.
References [1] D.A. Miller, in: H. Hang, L. Banyai (Eds.), Optical Switching in Low-Dimensional Systems, Plenum, New York, 1988, pp. 1–8. [2] R.L. Greene, K.K. Bajaj, D.E. Phelps, Phys. Rev. B 29 (1984) 1807. [3] J. Cen, K.K. Bajaj, Phys. Rev. B 46 (1992) 15280; J. Cen, K.K. Bajaj, Phys. Rev. B 50 (1994) 10947. [4] Gang Li, S.V. Branis, K.K. Bajaj, J. Appl. Phys. 77 (1995) 1097. [5] S. Le Goff, B. Ste´be, Phys. Rev. B 47 (1993) 1383. [6] T. Garm, J. Condens. Matter. 8 (1996) 5725. [7] J.M. Ferreyra, C.R. Proetto, Phys. Rev. B 57 (1998) 9061; P.G. Bolcatto, C.R. Proetto, Phys. Rev. B 59 (1999) 12487. [8] S. Glutsch, F. Bechstedt, W. Wegscheider, F. Schedelbeck, Phys. Rev. B 56 (1997) 4108. [9] J. Song, S.E. Ulloa, Phys. Rev. B 63 (2001) 125302.
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263 [10] B. Szafran, J. Adamowiski and S. Bednarek. J. Condens. Matter. 14 (2002) 73; M. Ciurla, J. Adamowski, B. Szafran, S. Bednarek, Physica E 15 (2002) 261. [11] R.A. Escorcia, R. Robayo, I.D. Mikhailov, Phys. Stat. Sol. (b) 230 (2002) 431. [12] X.-F. He, Phys. Rev. B 42 (1990) 11751; X.-F. He, Phys. Rev. B 43 (1991) 2063. [13] H. Mathieu, P. Lefebvre, P. Christol, Phys. Rev. B 46 (1992) 4092; P. Lefebvre, P. Christol, H. Mathieu, Phys. Rev. B 48 (1993) 17308.
263
[14] E. Reyes-Go´mez, A. Matos-Abiague, C.A. PerdomoLeiva, M. de Dios-Leyva, L.E. Oliveira, Phys. Rev. B 61 (2000) 13104. [15] I.D. Mikhailov, F.J. Betancur, R.A. Escorcia, J. SierraOrtega, Phys. Stat. Sol. 230 (b) (2002) 469; I.D. Mikhailov, F.J. Betancur, R.A. Escorcia, J. SierraOrtega, Phys. Rev. B 67 (2003) 115317. [16] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, New york, 1983. [17] F.J. Betancur, I.D. Mikhailov, L.E. Oliveira, J. Appl. Phys. D 31 (1998) 3391.
Physica B 355 (2005) 255–263 www.elsevier.com/locate/physb
Exciton fractional dimension in semiconductor heterostructures arising from variational principle R.A. Escorciaa, J. Sierra-Ortegaa, I.D. Mikhailovb, F.J. Betancurb, a Departamento de Fı´sica, Universidad del Magdalena, A. A. 731 Santa Marta, Colombia Departamento de Fı´sica, Universidad Industrial de Santander, A. A. 678 Bucaramanga, Colombia
b
Received 12 December 2003; received in revised form 23 April 2004; accepted 1 November 2004
Abstract We present a simple method for calculating the ground-state energy of an exciton in quantum confined structures. We express the exciton wave function as a product of the electron and hole one-particle wave functions with a variationally determined envelope function which describes the exciton intrinsic properties. Starting from the variational principle, we derive an one-dimensional wave equation for this envelope function and show that it describes a hydrogen-like atom in an effective isotropic space with the non-integer running dimension. We establish that this dimension runs from three, as the electron–hole separation is small for all heterostructures, to two in quantum well one in quantum well-wire and to zero in quantum dot, as the separation is large. The exciton ground-state energies are calculated for different confining potential shapes. Our results for GaAs-(Ga, Al)As heterostructures with square-well potential are in an excellent agreement with those obtained previously by means of other methods. r 2004 Elsevier B.V. All rights reserved. PACS: 71.35.y; 73.61.Ey; S7.12 Keywords: Semiconductor estructures; Exciton binding energy
1. Introduction The progress in nanoscale technology has made possible the fabrication of low-dimensional heterostructures with controlled thickness and relatively sharp interfaces, where the excitons remain present even at room temperature because of the quantum confinement increases highly the electron–hole Corresponding author. Tel./fax: +57 7 63 23477.
E-mail address: [email protected] (F.J. Betancur).
attraction [1]. A great deal of attention has been devoted to experimental and theoretical studies of excitons in heterostructures based on III–V semiconductor [2–11] particularly quantum wells (QWs), superlattices (SLs), quantum-well wires (QWWs), quantum dots (QDs) and more recently quantum rings (QRs). Different methods such as the variational [2–5], matrix diagonalization [6,9,10] or finite element [8] have been used to analyze the effect of the confinement on the exciton spectrum. Up to now it has been considered the
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.11.002
ARTICLE IN PRESS 256
R.A. Escorcia et al. / Physica B 355 (2005) 255–263
confinement models with square-well [2–5], parabolic [5,9], charge image [7], exponential [10] and soft-edge-barrier [11] potentials. In spite of that these techniques give consistent results with the experimental data, in most of the cases they entails a lot of computational work and the accuracy of the variational method depends to a large extent on the form of the trial wave function. An approach that allows consider the anisotropic interactions in a three-dimensional space as isotropic ones in a lower fractional-dimensional space has been applied by He [12], who using the hydrogen-like Hamiltonian in an effective fractional space treated the interband optical transitions and bound excitons in strongly anisotropic semiconductors. Lefebvre and collaborators [13] and Oliveira and co-investigators [14] applied this method to analyze the exciton energy states and the absorption spectra in QWs and QWWs. They considered the fractional dimension as a phenomenological parameter related to the heterostructure geometry and proposed a different technique for calculating this parameter. Recently, we have proposed a simple variational procedure related to the fractal-dimensional approach [15]. In this procedure the wave equations for neutral and negatively charged donors [15] and exciton [11] in heterostructures are reduced to ones similar to hydrogenic-like atom in an isotropic effective space with fractional dimension, which is determined by using the Mandelbrot’s definition for fractal objects [16]. It has been established [15] that this procedure provides an efficient algorithm for calculating the ground state binding energies with accuracy comparable with those of sophisticated methods such as the series expansion and Monte Carlo. In this work, we extend our method to study excitons in different semiconductor heterostructures in the presence of magnetic field taking into account the anisotropy of the hole effective mass and considering different confining potential shapes.
2. Differential equation for the electron–hole correlation function
well and the barrier are neglected. The dimensionless Hamiltonian for the electron–hole pair confined in a heterostructure with an appropriate scaling for the energy and length units may be written as Hðre ; rh ; tÞ ¼ H 0 ðre ; rh Þ 2=reh ; H 0 ðre ; rh Þ ¼
Zeik
q2 qxie qxke
Zhik
q2 þ Uðre ; rh Þ: qxih qxkh
i;k¼1
3 X i;k¼1
ð1Þ
In these relations H 0 ðre ; rh Þ describes the free motion of the electron and hole confined in the heterostructure, re ¼ fx1e ; x2e ; x3e g and rh ¼ fx1h ; x2h ; x3h g are the electron and hole position vectors, Uðre ; rh Þ is the perturbation due to the confinement and the external fields and the coefficients Zeik and Zhik describe the relative values and the anisotropy of the electron and hole effective masses. 2=reh and reh ¼ jre rh j are the energy of the electron–hole interaction and the electron–hole separation, respectively. In what follows we assume that the Hamiltonian for the free electron–hole pair is separable and the solution of the corresponding eigenvalue problem for the ground state, H 0 ðre ; rh Þ f 0 ðre ; rh Þ ¼ E 0 f 0 ðre ; rh Þ;
(2)
is known. Then to solve the Schro¨dinger equation for exciton, Hðre ; rh ÞCðre ; rh Þ ¼ E ex Cðre ; rh Þ
(3)
where E ex is the exciton total energy, we choose Cðre ; rh Þ ¼ f 0 ðre ; rh ÞFðreh Þ; being Fðreh Þ a variational function, that describes the intrinsic properties of the exciton and depends only on the electron– hole separation. Starting from the Schro¨dinger’s variational principle and using the procedure of the functional derivation described in Ref. [15] we obtain the following differential equation for FðrÞ:
We consider a confinement model in which the differences between the material parameters in the
3 X
1 d dFðrÞ 2 J 1 ðrÞ FðrÞ ¼ ½E ex E 0 FðrÞ; J 0 ðrÞ dr dr r (4a)
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
where J 0 ðrÞ ¼ J 1 ðrÞ ¼
Z
Z dre Z dre
f 20 ðre ; rh Þdðreh rÞdrh ;
Z X 3 ðZeik þ Zhik Þ ðxie xih Þ ðxke xkh Þ r2eh i;k¼1
f 20 ðre ; rh Þdðreh rÞdrh ;
ð4bÞ
3. Exciton in fractal dimensional space To reveal the physical significance of Eq. (4a) let us consider a particular case as the electron and hole effective masses are isotropic Zeik ¼ Ze dik ; Zhik ¼ Zh dik and in coordinates of the center of mass Ze þ Zh ¼ 1: Therefore Eq. (4a) transforms into
1 d dFðrÞ 2 J 0 ðrÞ FðrÞ ¼ ðE ex E 0 ÞFðrÞ: J 0 ðrÞ dr dr r (5)
This equation is similar to one of a hydrogenic-like atom in an effective isotropic space with the radial part of the Jacobian given by J 0 ðrÞ: The properties of this space are related generally to J 0 ðrÞ; which in according to relation (4b), is defined by the geometry of the heterostructure. In other words, if the dependence is a power-law, J 0 ðrÞ ¼ CrD1 ; then the scaling parameter D in this dependence could be considered as the dimension of an effective space and the Eq. (5) for the electron– hole relative motion would have coincided with the eigenvalue problem for a hydrogen-like atom in a D-dimensional effective space, being the parameter D an integer o fractional. If this dependence is non-power-law, we can define the local dimension in an isotropic effective space by using the Mandelbrot’s concept for fractals [16]. The properties of this space can be described in terms of J 0 ðrÞ: This dependence is a power-law only when the space is homogeneous and it does not in other cases. Since due to the heterostructure confinement J 0 ðrÞ has no a power-law dependence on r. We assume that the parameters C and D in the above scaling equation also depends on r and consequently J 0 ðrÞ is related to the dimension D ðrÞ as J 0 ðrÞ ¼ CðrÞrD ðrÞ1 or D ðrÞ ¼ 1 þ rd½lnJ 0 ðrÞ =dr;
257
where CðrÞ and D ðrÞ are functions that vary more slowly than any power function. We perform the calculation of J 0 ðrÞ and D ðrÞ for excitons in QW, cylindrical QWW and spherical QD with square-well potentials in absence of the external field. In this case Eq. (2) is separable and one can calculate J 0 ðrÞ directly by using the relation (4b) and the well-known analytical expressions for the ground state wave functions of the electron and the hole in the heterostructure. Fig. 1(a) shows the typical behavior of J 0 ðrÞ obtained for three heterostructures. For small values of the electron–hole separations, as the electron–hole distance is essentially smaller than the size of the heterostructure, the tree curves behave as parabolic ðJ 0 r2 Þ similar to the case of the free exciton in a three-dimensional space. As the electron–hole distance increases and it becomes greater than the heterostructure size, the curves transform into a linear function ðJ 0 rÞ for a QW, a constant ðJ 0 r0 Þ for QWW and decreasing as an exponential function ðJ 0 0Þ for QD, corresponding to the free motion in two-one and zero-dimensional spaces as expected. The corresponding dependencies of the fractal dimension, D ðrÞ on the relative coordinate of the exciton are shown in Fig. 1(b). It can be seen that D ðrÞ falls from 3 for small electron–hole distances to 2 in QW, to 1 in QWW and to 0 in QD as the electron–hole separation becomes larger than the size of the corresponding heterostructure. Such dependence is typical for quasi-two, -one and –zero-dimensional heterostructures, respectively.
4. Excitons in GaAs/(Ga,Al)As QW and cylindrical QWW In order to obtain the exciton ground state energy and its wave function from Eq. (4a), we first should define explicitly the dimensionless Hamiltonian (1). If we scale in this Hamiltonian all lengths in terms of the electron Bohr radius a 0 ¼ _2 =m e e2 ; all energies in electron effective Rydberg R y ¼ e2 =2 a 0 then ignoring the GaAs(Ga,Al)As heterostructures coupling between the heavy-hole and light-hole bands, we find that Zeik is
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
258
exciton center-of-mass free motion in the xy plane, the Hamiltonian (1) in cylindrical coordinates can be written as [3]
QW
2
q2 q2 Z z qz2e qz2h 1 q q r þ Uðre ; rh Þ; ð1 þ Z? Þ r qr qr ð1 þ Z? Þg2 r2 ; ð6Þ Uðre ; rh Þ ¼ V e ðze Þ þ V h ðzh Þ þ 4 where the functions V e ðze Þ and V h ðzh Þ give the potential profiles for the electrons and holes, r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxe xh Þ2 þ ðye yh Þ2 is the in-plane distance between electron and hole and the parameter g ¼ e_B=2m e cR y is the Landau level expressed in R y : The eigenvalue problem (2) for the Hamiltonian (6) can be separated and its solution is:
J0(r)
H 0 ðre ; rh Þ ¼
1
QWW
QD 0 0.0 (a)
0.5
1.0
r/a0* 3
QW
ð0Þ 2 f 0 ðre ; rh Þ ¼ f ð0Þ e ðze Þf h ðzh Þ expðgr =4Þ;
2 D*(r)
ð0Þ E 0 ¼ E ð0Þ e þ E h þ ð1 þ Z? Þg;
QWW
ð0Þ where f ð0Þ e ðze Þ and f h ðzh Þ are solutions of the following one-dimensional equations
1 QD
Zi
q2 f ð0Þ ð0Þ ð0Þ i ðzi Þ þ V i ðzi Þf ð0Þ i ðzi Þ ¼ E i f i ðzi Þ; qz2i
i ¼ e; h; Ze ¼ 1; Zh ¼ Zz :
0 0.0 (b)
ð7aÞ
0.5
1.0
r/a0*
Fig. 1. The radial part of the Jacobian (a) and fractal dimension (b) dependences on the electron–hole separation in QW, cylindrical QWW and spherical QD.
ð7bÞ
Substituting the wave function (7a) in the general expression (4b) we obtain, after integrating over the angles, the following simplified formulae for the Jacobians [15]: Z r Z 1 2 2 2 J i ðrÞ ¼ 2pr f e ðze Þdze egðr z Þ=2 f 2h ðze þ zÞ 1
r
Ri ðz; rÞdz; i ¼ 0; 1 an identity matrix and Zhik is a diagonal matrix with two elements Z? ¼ m e =m ? corresponding to x, y directions and Zz ¼ m e =m z corresponding to z-direction, where m e ; m ? and m z are the effective masses of the electron and hole in the xy plane and z-direction, respectively. Let us consider an exciton in a QW with uniform magnetic field, B applied along the z-axis. Choosing the symmetric gauche vector potentials, Ae ¼ B ðre rh Þ=2 and Ah ¼ B ðrh re Þ=2 for electron and hole respectively and separating the
R0 ðz; rÞ ¼ 1; R1 ðz; rÞ ¼ 1 þ Z? þ ðZz Z? Þ
z2 : r2 (8)
We assume that the wire is sufficiently long, so that the motion along wire axis has translational symmetric and the confining potential is a function of the r-coordinate. Similarly one can calculate the Jacobians (4b) for a cylindrical QWW. By using the cylindrical coordinates for the electron ðre ; je ; ze Þ and hole ðrh ; jh ; zh Þ; the Hamiltonian
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
for the electron–hole-pair in the ground state without interaction between the particles can be written as H 0 ðre ; rh Þ ¼
1 q q 1 q q r Z? r re qre e qre rh qrh h qrh
þ Uðre ; rh Þ; g2 r2e Z g2 r2 þ V h ðrh Þ þ ? h : (9) 4 4 Here V e ðre Þ and V h ðrh Þ give the potential profiles for the electrons and holes. The wave equation (2) for the Hamiltonian (9) can be separated and the wave function for the electron–hole pair can be represented as Uðre ; rh Þ ¼ V e ðre Þ þ
ð0Þ f 0 ðre ; rh Þ ¼ f e ðre Þf h ðrh Þ; E 0 ¼ E ð0Þ e þ Eh ;
(10)
where f e ðre Þ and f h ðrh Þ are solutions of the onedimensional wave equations:
To find the wave functions for the electron and ð0Þ the hole f ð0Þ e ðze Þ; f h ðzh Þ in QW and f e ðre Þ; f h ðrh Þ in QWW we solve the respective Eqs. (7b) and (11) by using the trigonometric sweep method [17]. The same numerical procedure is used to solve Eq. (4a) for FðrÞ: In order to check the accuracy of our method, we first calculate the exciton binding energy in a GaAs-Ga0.7Al0.3As QW and cylindrical QWW considering square-well potentials. In Fig. 2a and b we display the calculation results for the exciton binding energies for heavyhole, E 1s ðhÞ; (solid lines) and for light-hole, E 1s ðlÞ; (dashed lines) excitons as function of the QW width and compare them with those from Refs. [2,3] (open circles). In both cases we intentionally choose the material parameters from corresponding references and we find an excellent concordance between the two sets of results for
Zi q qf ðr Þ Z g2 r2 ri i i þ V i ðri Þf i ðri Þ þ i i qri ri qri 4 i ¼ e; h;
Ze ¼ 1; ð11Þ
ð0Þ Here E ð0Þ e and E h are the lowest energies of the electron and hole in the wire, respectively. Substituting the wave function (10) in the general expression (4b) we obtain after integrating over angles the following simplified formulae for the Jacobians in a cylindrical QWW: Z re þr Z 1 2 J i ðrÞ ¼ 4pr f e ðre Þre dre f 2h ðrh Þ 0
Ri ðre ; rh ; rÞrh drh ;
o Ref. 2
9.0 Eb (meV)
f i ðri Þ ¼ Zh ¼ Z? :
E ð0Þ i f i ðri Þ;
7.5
Light-Hole Exciton
Heavy-Hole Exciton
6.0
0
1
(a)
2 W /a0*
3
heavy hole light hole
maxf0; re rg
20
i ¼ 0; 1;
R0 ðre ; rh ; rÞ ¼ G 0 ðre ; rh ; rÞ; R1 ðre ; rh ; rÞ ¼ ð1 þ Z? ÞG 0 ðre ; rh ; rÞ Z Z ð12Þ þ z 2 ? G A ðre ; rh ; rÞ; r where G 0 ðre ; rh ; rÞ ¼ Kðb=aÞ=a as r4re þ rh and G 0 ðre ; rh ; rÞ ¼ Kða=bÞ=b as rore þrh ; G A ðre ; rh ; rÞ ¼ aEðb=aÞ as r4re þ rh and G A ðre ; rh ; rÞ ¼ bEða=bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as rore þ rh being a ¼ r2 ðre rh Þ2 ; b ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 4re rh and KðkÞ; EðkÞ the complete elliptic integrals of the first and second kind, respectively.
4
25
Eb (meV)
259
o
Ref. 3
15 B=50 KG
10
5 0.0 (b)
B=200 KG
B=0 KG
0.5
1.0 W /a0*
1.5
2.0
Fig. 2. We compare the calculation results for the light- and heavy-hole exciton binding energies (solid and dashed line respectively) as a function of the GaAs/Ga0.7Al0.3As QW width (a) with those results from Ref. [2] and (b) from Ref. [3] (open circles) by using the similar parameters of the model.
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
260
different well widths and magnetic field strengths. As it is seen from Fig. 2a and b the binding energies of the nearly free light-hole exciton (in the limit of large QWWs radii and zero-magnetic field) is slightly larger than the corresponding values for the heavy-hole exciton and the order of these energies are inverted in the conditions of a strong confinement (the limits of the small QWW radii or large magnetic field). This effect was established in Ref. [2] and for the case of a structural confinement it was explained on the base of the anisotropic Kohn–Luttinger model for which the reduced effective mass of the light-hole exciton in the x–y plane is larger than one for heavy-hole exciton. A similar explanation is also valid for the case of a confinement produced by a strong magnetic field. We compare the calculation results of the heavyhole exciton binding energy as a function of the cylindrical QWW radius (solid lines in Fig. 3) with those from Ref. [4] (open circles). Although the isotropic hole mass approximation used in Ref. [4] for hole is applicable only for small electron–hole separations, an excellent concordance between the two sets of results can be observed in Fig. 3 for all wire radii with zero-magnetic field. Also it is seen that in the presence of the magnetic field (B ¼ 200 KG) for wire radii larger than electron effective Bohr radius our results are slightly higher than those from Ref. [4]. We attribute this small discrepancy to the additional lateral confinement
produced by the strong magnetic field that enlarges the effect of anisotropy. The motion of the particles in this case due to the strong in-plane confinement is almost quasi-one-dimensional and it is directed along the wire axis. The heavy-hole effective mass in this direction is higher than its value in the x–y plane and therefore the exciton effective Rydberg (a measure of the exciton binding energy) for anisotropic model is a slightly greater than for isotropic one. In Fig. 4 we illustrate the effect of the structural confinement on the conditional probability density for the heavy-hole exciton at zero-magnetic field, defined as Pðxe ; ze =xh ¼ yh ¼ zh ¼ ye ¼ 0Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 x2e þ z2e ; ¼ f e ðxe ÞF
ð13Þ
which corresponds to the hole and electron locations on the QWW axis and in the x– z plane, respectively. It is seen that as decrease the QWW radius the probability density becomes more anisotropic, meanwhile the peak of the distribution becomes more pronounced. It should be noted that the distribution contraction in the radial direction, due to the structural confinement is also accompanied by its weak contraction in the z-direction.
5. Excitons in cylindrical nanotubes 20 o Ref.4 Eb (meV)
15
B=200 KG
10 B=0 KG
5 0
1
2
3
4
R / a0* Fig. 3. Heavy-hole exciton binding energy as a function of the wire radius with and without magnetic field. Our results (solid lines) are compared with the theoretical data from Ref. [4] (open circles).
Finally, we apply our theory to analyze the effects of the potential shape on the exciton ground state binding energy in nanotubes. Usually the confinement potentials for an electron and hole in GaAl-(Ga,Al)As heterostructures are modeled by either the square-well or parabolic potentials with finite or infinite depth, respectively. Nevertheless, as it results from numerical solution of the Poisson equation, a realistic potential within quantum-well heterostructures is nearly parabolic at the center of the well and it becomes nonparabolic at the junctions [10]. Moreover, if we want to take into account the effects originating from a local fluctuation in the Al content (alloy clustering) in the regions of the junctions we
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263
261
Fig. 4. Schematic representation of the electron conditional probability density in the x–z plane, for different QWW radii: (b) R ¼ 4a 0 ; (c) R ¼ 1a 0 ; (d) R ¼ 0:3a 0 ; for the heavy-hole exciton with the fixed hole position at the axis.
20 square-well potential 16 Eb (meV)
should consider also non-abrupt confinement potential. To describe such behavior of the confining potential we use the following nonabrupt version of the step function: Yðx; x0 ; W Þ ¼ 0 for xox0 W ; Yðx; x0 ; W Þ ¼ ½ðx x0 Þ2 =W 2 1 2 for x0 W pxox0 and Yðx; x0 ; W Þ ¼ 1 for xXx0 which turns into the Heaviside function as W ! 0: The parameter W is associated with the thickness of the transition region in the junctions and varying its value one can obtain a different confining potentials. For example, V ðrÞ ¼ V 0 Yðr; R0 ; W Þ describes the potential for the electron or hole in a cylindrical QWW of radius R0 and barrier height V 0 : As W ! 0 and W ! R0 the almost rectangular (square-well) and the parabolic potentials with a finite barrier height are modeled by this function, respectively. The relation V ðrÞ ¼ V i Yðr; Ri ; W Þþ V e Yðr; Re ; W Þ describes the potential of the particles in a nanotube with internal radius, R i and barrier height, V i ; which determines the properties of the repulsive core. The external radius, Re and barrier height, V e is determined for the parameters of the wire. Fig. 5 shows the ground state binding energy of exciton as a function of the GaAs-Ga0.7Al0.3As
parabolic potential 12
8
soft-edge barrier potential
core-repulsive potential
0
1
2
3
R/a0* Fig. 5. Exciton binding energies in GaAs/Ga0.7Al0.3As cylindrical QWWs with different confinement potential shapes as a function of the radius.
cylindrical QWW radius for different potential shapes: square-well ðRi ¼ 0; W ¼ 0:01 Re Þ; softedge barrier ðRi ¼ 0; W ¼ 0:5 Re Þ; parabolic finitebarrier ðRi ¼ 0; W ¼ Re Þ; and a cylindrical nanotube-like wire with repulsive core ðRi ¼ 0:15Re ; V i ¼ V e ; W ¼ 0:01 Re Þ: As one compare the curves (Fig. 5) for cylindrical QWWs with equal finite barrier height
ARTICLE IN PRESS 262
R.A. Escorcia et al. / Physica B 355 (2005) 255–263
and different potential shapes, square-well (solid line), soft-edge barrier (dotted line) and parabolic (dashed line) potentials one can see that for large and intermediate wire radii the larger the transition region thickness the higher is the binding energy. It is apparent that for intermediate and large values of the QWW radius the parabolic shape gives the largest confinement, followed by the soft-edge barrier, whereas the rectangular potential corresponds to the smallest confinement. As the wire radius Re decreases, the exciton binding energy for all potential shapes climbs up until it reaches a maximum. As Re ; further decreases, the exciton wave function leaks into the barrier region and the exciton binding energy begins to fall off rapidly meanwhile the exciton 3D character is restored. It is evident that this leakage of the exciton wave function in QWW with parabolic confinement occurs earlier due to stronger confinement than for the other two types of potential and consequently the maximum of the binding energy for this model is lower than the maxima for the soft-edge-barrier and rectangular potentials. In Fig. 5 we also present the results for a nanotube-like structure which is modeled by a quantum wire with repulsive core of internal radius, Ri ¼ 0:5Re and barrier height V i ¼ V e (dashed–dotted line). It is seen that for large QWW radii the exciton binding energies in nanotubes are higher than those in QWWs with square-well potential (solid line) and conversely for small radii. This is due to the fact that for large outer radii, Re ; both electron and hole are mostly located within the ‘‘ring’’ of the nanotube on the same side with respect to the core and the exciton undergoes a stronger confinement than in the wire. As consequence, the exciton binding energy is enhanced as the thickness of such ‘‘ring’’ is decreased up to a value comparable with the exciton radius. On the contrary, as the ring thickness is smaller than the exciton radius and the core size ðRi Þ is comparable with the exciton size, the electron and the hole tend to be located at opposite sides of the repulsive core, the separation between them increases and the binding energy diminishes. As consequence, the exciton binding energies for small wire radii in nanotubes are lower than those in quantum wire and therefore, the
crossover of the corresponding curves in Fig. 5 is observed.
6. Conclusion We propose a simple method for calculating the ground state wave function of an exciton in semiconductor heterostructures with arbitrary potential shapes and taking into account a possible anisotropy of the effective masses of the electron and hole. Starting from the variational principle, we reduce the initial problem for an exciton confined in the heterostructure to a central force problem in an isotropic and non-homogeneous effective space with a noninteger variable dimension. An explicit expression for the fractal dimension of this effective space is deduced, by using the Mandelbrot definition.
Acknowledgements This work was partially financed by the Universidad Industrial de Santander (UIS), through the General Researches (DIF) and the Colombian Agency COLCIENCIAS (Cod 1102-05-16923). R. Escorcia and J. Sierra-Ortega wish to thank the Universidad del Magdalena for the permission to study at the UIS.
References [1] D.A. Miller, in: H. Hang, L. Banyai (Eds.), Optical Switching in Low-Dimensional Systems, Plenum, New York, 1988, pp. 1–8. [2] R.L. Greene, K.K. Bajaj, D.E. Phelps, Phys. Rev. B 29 (1984) 1807. [3] J. Cen, K.K. Bajaj, Phys. Rev. B 46 (1992) 15280; J. Cen, K.K. Bajaj, Phys. Rev. B 50 (1994) 10947. [4] Gang Li, S.V. Branis, K.K. Bajaj, J. Appl. Phys. 77 (1995) 1097. [5] S. Le Goff, B. Ste´be, Phys. Rev. B 47 (1993) 1383. [6] T. Garm, J. Condens. Matter. 8 (1996) 5725. [7] J.M. Ferreyra, C.R. Proetto, Phys. Rev. B 57 (1998) 9061; P.G. Bolcatto, C.R. Proetto, Phys. Rev. B 59 (1999) 12487. [8] S. Glutsch, F. Bechstedt, W. Wegscheider, F. Schedelbeck, Phys. Rev. B 56 (1997) 4108. [9] J. Song, S.E. Ulloa, Phys. Rev. B 63 (2001) 125302.
ARTICLE IN PRESS R.A. Escorcia et al. / Physica B 355 (2005) 255–263 [10] B. Szafran, J. Adamowiski and S. Bednarek. J. Condens. Matter. 14 (2002) 73; M. Ciurla, J. Adamowski, B. Szafran, S. Bednarek, Physica E 15 (2002) 261. [11] R.A. Escorcia, R. Robayo, I.D. Mikhailov, Phys. Stat. Sol. (b) 230 (2002) 431. [12] X.-F. He, Phys. Rev. B 42 (1990) 11751; X.-F. He, Phys. Rev. B 43 (1991) 2063. [13] H. Mathieu, P. Lefebvre, P. Christol, Phys. Rev. B 46 (1992) 4092; P. Lefebvre, P. Christol, H. Mathieu, Phys. Rev. B 48 (1993) 17308.
263
[14] E. Reyes-Go´mez, A. Matos-Abiague, C.A. PerdomoLeiva, M. de Dios-Leyva, L.E. Oliveira, Phys. Rev. B 61 (2000) 13104. [15] I.D. Mikhailov, F.J. Betancur, R.A. Escorcia, J. SierraOrtega, Phys. Stat. Sol. 230 (b) (2002) 469; I.D. Mikhailov, F.J. Betancur, R.A. Escorcia, J. SierraOrtega, Phys. Rev. B 67 (2003) 115317. [16] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, New york, 1983. [17] F.J. Betancur, I.D. Mikhailov, L.E. Oliveira, J. Appl. Phys. D 31 (1998) 3391.