- Email: [email protected]

Physica B 398 (2007) 135–143 www.elsevier.com/locate/physb

Review

Charge distribution in quantum dot with trapped exciton J.H. Marı´ na,, I.D. Mikhailovb, L.F. Garcı´ ab a

Escuela de Fı´sica, Universidad Nacional de Colombia-Medellı´n, A.A. 3840 Medellı´n, Colombia b Escuela de Fı´sica, Universidad Industrial de Santander, A.A. 678 Bucaramanga, Colombia Received 21 February 2007; received in revised form 26 April 2007; accepted 5 May 2007

Abstract We calculate the ground state wave function of the exciton conﬁned in the In0.55Al0.45As quantum disk, lens and pyramid deposited on a wetting layer (WL) and embedded in a matrix made of Al0.35Ga0.65As material. It is shown that tunneling of the electron into the barrier is signiﬁcantly stronger than one of the holes due to the difference between their masses. In consequence, the central region of the dot with captured exciton is charged positively whereas the regions over, below and around quantum dot including the WL are charged negatively. The comparison of results obtained for quantum dots (QDs) with different geometry shows that the separation of the charges is stronger in spike-shaped quantum dots. We also ﬁnd that the increase of the WL thickness leads to a lowering of the effective barrier height for both particles enhancing the separation between them in the lateral direction provided by reinforced the particles tunneling toward the WL. r 2007 Elsevier B.V. All rights reserved. PACS: 73.21.La; 73.21.b; 71.35.y Keywords: Quantum dots; Exciton; Spatial charge distribution

Contents 1. 2. 3. 4.

Introduction . . . . . . . . Theory . . . . . . . . . . . . Results and discussion . Conclusions . . . . . . . . Acknowledgments . . . . References . . . . . . . . .

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1. Introduction The development of new semiconductor growth techniques have made possible the fabrication of the selfassembled quantum dots (QDs) whose sizes and shapes can be controlled by changing the growth conditions. The Corresponding author. Tel.: +57 7 6323095; fax: +57 7 6323477.

E-mail address: [email protected] (J.H. Marı´ n). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.05.007

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135 136 138 142 143 143

experimental characterization of the QDs structural properties suggests that they have commonly a shape of the disk-like, lens-like or cone-like layer with a small (usually less than 0.1) height-to-base radius aspect ratio [1]. The strong conﬁnement of the charge carriers inside the QD facilitates the formation of bound states such as exciton, trion, biexciton, etc. and opens a series of novel non-available in semiconductor bulk possibilities for modifying the energy spectrum and the spatial distributions

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of the carriers. One of such possibilities is associated with a superior capacity of the lighter electron for tunneling from the very thin InAs layer toward barrier regions in comparison with the corresponding capacity of the heavier hole. Such difference between the electron and hole capacities for the tunneling provides an additional variation of the energy spectrum of the exciton trapped in a ﬂat QD. As it has been established earlier in Ref. [2], in a In0.55Al0.45As/Ga0.65Al0.35As quantum disk of 3.22 nm thickness, the hole is inside the dot with a probability more than 90%, while the electron is inside the dot with a probability only about 70%. As the result, one could expect that the regions above and below the disk with captured exciton are charged negatively, whereas the central region inside the disk is charged positively. Very recently, analyzing the charge distribution in a quantum ring induced by a trapped exciton have shown that in this case the electrons are displaced mainly inside the ring hole region making available a charge distribution signiﬁcantly different from one of the disks [3]. Many researches have previously used different mathematical methods such as the ﬁnite element [2,4], variational [5–8], matrix diagonalization [9] or fractional dimensional [10] in order to analyze the effect of the conﬁnement in QDs on the exciton spectrum. In spite of the fact that these methods give consistent results with the experimental data, they entail a lot of computational effort. Recently, it has been proposed to this end a simple variational procedure [3,11] associated with the fractal-dimensional approach [12], which provides an efﬁcient algorithm whose accuracy is comparable with one of such sophisticated methods as the series expansion and Monte Carlo [10]. In the present work, we use this procedure in order to calculate the energy and the spatial charge distribution induced by a trapped exciton inside and around QD taking, besides, into account the particles tunneling toward the wetting layer (WL) which have been ignored by others authors. We also analyze the possibility of the formation of both the in-plane quadrupole momentum and the dipole momentum oriented in the crystal growth direction in QDs with captured exciton. One could expect also that the effect of the heterogeneous charge distribution should be more pronounced in spike-shaped QDs where the particles are conﬁned stronger. In order to analyze this effect, we have proposed in this paper a model for axially symmetrical QDs whose proﬁles in a plane trough the axis of symmetry are given by a simple analytical expression with a single shape-generating parameter. This model appears to be ﬂexible enough to model QDs with different morphology, look like lens, conical pyramid and disk.

means of the following simple relation:

2. Theory

In Eq. (2) re and rh are the electron and the hole position vectors and reh ¼ |rerh| is the separation between two particles. The parameter t in the Hamiltonian (2) is equal to zero in one-particle approximation with neglected electron–hole interaction and it is equal to one for the

We consider a model of axially symmetrical QDs in the form of a thin layer with variable thickness d, whose dependence on the distance r from the axis is modeled by

dðrÞ ¼ d b þ d 0 ½1 ðr=R0 Þn 1=n WðR0 rÞ.

(1)

Here, R0 is the base radius, db is the WL thickness, d0 is the maximum height over this layer, WðxÞ is the Heaviside step function, equal to zero for xo0 and to one for x40. The morphology is controlled in this model by means of the integer shape-generating parameter n which is equal to 1, 2 or tends to inﬁnity for conical pyramid-like, lens-like and disk-like geometrical shapes, respectively, whose correspondent 3D images are presented in Fig. 1. In accordance with the experiment and, besides, for sake of the mathematical convenience, we assume in what follows, that the height-to-base radius aspect ratio in all cases is small, i.e. (d0+db)/R051. As it was shown in Ref. [13], for sufﬁcient wide and thin self-assembled QDs, the strain ﬁeld can have a signiﬁcant impact on their electronic structure. The strain can even tune a system from type I to type II. Nevertheless, we use below the parabolic approximation for the electron and the hole, assuming that the effects caused by the strain are negligible or the QDs under consideration are strain free similarly to those which have been fabricated recently [14]. We consider for our simulations mainly the In0.55 Al0.45As/Al0.35Ga0.65As structures with the following values of physical parameters [15]: dielectric constant e ¼ 12.71, the effective masses in the In0.55Al0.45As material layer and in the Al0.35Ga0.65As matrix material for the electron mew ¼ 0.076m0 and meb ¼ 0.097m0, respectively, and for hole mhw ¼ mhb ¼ mh ¼ 0.45m0, the conduction and the valence bands offsets in junctions are V0e ¼ 358 and V0h ¼ 172 meV, respectively. Here and in below subscripts, e and h correspond to the electron and the hole, respectively, and subscripts w and b correspond to the well and the barrier regions, respectively. We scale all lengths in terms of the exciton effective Bohr radius a0 ¼ _2 =m e2 10:4 nm, all energies in the exciton effective Rydberg Ryn ¼ e2 =2an0 5:2 meV, being m the electron– hole reduced mass inside the dot, m ¼ mewmh/(mew+mh)E 0.065m0. The dimensionless Schro¨dinger equation for the electron–hole pair in the axially symmetrical QDs in the effective mass approximation can be written in cylindrical coordinates ðrp ¼ frp ; Wp ; zp g; p ¼ e; hÞ as Hðre ; rh ; tÞCðre ; rh ; tÞ ¼ EðtÞCðre ; rh ; tÞ; Hðre ; rh ; tÞ ¼ H e ðre Þ þ H h ðrh Þ 2t=reh ;

H p ðrp Þ ¼ rp Zp r þ V p ðrp ; zp Þ;

t ¼ 0; 1

Zp ¼ m=mp ;

ð2aÞ

p ¼ e; h: (2b)

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137

Fig. 1. 3D images of quantum dots.

ðd

1 0 þd b Þ

2

qf p ðr~ p ; z~p Þ q qf p ðr~ p ; z~p Þ 1 1 q Zp 2 2 Zp r~ p q~zp q~zp qr~ p R0 r~ p qr~ p

þV p ðr~ p ; z~p Þf p ðr~ p ; z~p Þ ¼ E p f p ðr~ p ; z~p Þ;

C

ð0Þ

¼ Cðre ; rh ; t ¼ 0Þ ¼ f e ðre Þf h ðrh Þ;

(3a)

E0 ¼ Ee þ Eh.

ð4aÞ

where rp is treated as parameter (cf. electronic motion for ﬁxed nuclear position in molecular problems), and then df rp ðrÞ 1 d Zr þ E zr ðrÞf rp ðrÞ ¼ E p f rp ðrÞ; r dr p dr p ¼ e; h

ð4bÞ

(cf. nuclear motion in molecular problems). Finally, the oneparticle wave function in ﬂat QD can be written as f p ðrp Þ ¼ f zp ðzp ; rp Þf rp ðrp Þ;

p ¼ e; h:

conical pyramid n=1

lens n=2

0.6

n

disk infinity

0.4 10

20

30

40

Fig. 2. Effective potential curves for the in-plane electron motion in quantum dots with different proﬁles.

qf p ðrp ; zp Þ q Zp V p ðrp ; zp Þf zp ðrp ; zp Þ qzp qzp p ¼ e; h;

R0=40nm

0.8

ρ (nm)

Here, renormalized cylindrical coordinates z~p ¼ zp =ðd 0 þ d b Þ and r~ p ¼ rp =R0 are scaled in such way that they were of the same order inside QD. The case corresponding to QDs with a very small height-to-base radius aspect ratio, (d0+db)2/R2051 we call the adiabatic limit [16,17] for the following reason. Eq. (3a) may be regarded as the Schro¨dinger equation describing two particles, of masses (d0+db)2 and R20 interacting through the potential V p ðr~ p ; z~p Þ which are displaced along zp-axis and in plane (xp,yp), respectively. Since one of these masses is much larger than the other, we follow the well-known adiabatic procedure of ﬁrst solving the equation in original non-renormalized:

¼ E zp ðrp Þf zp ðrp ; zp Þ;

db=2nm

0

p ¼ e; h;

(3b)

d0=3nm

Eze(ρ) / V0e

actual electron–hole system. The conﬁnement potential Vp(rp,zp) is equal to zero inside the InAs layer (well) and it is equal to Vop, p ¼ e,h in the GaAs matrix (barrier). The parameter Zp for the electron is equal to ZewE0.86 in the well and to ZebE0.67 in the barrier and for the hole Zhw ¼ ZhbE0.14. In the one-particle approximation (t ¼ 0) the groundstate energy E(t ¼ 0) ¼ E0 and the corresponding wave function C(0) ¼ C(re,rh,t ¼ 0) of the Hamiltonian (2) which is independent of angular coordinate may be found by solving two following independent one-particle problems:

(4c)

Here f zp ðzp ; rp Þ is the ground-state wave functions of the electron and the hole corresponding to the one-dimensional (1D) rapid motion in zp-direction along the line at a distance rp (considered as parameter) from the symmetry axis. The functions f zp ðzp ; rp Þ as well as the corresponding energies Ezp(rp) in accordance with Eq. (4a) can be found by solving the well-known wave equation for a quantum well with rectangular barrier of the height V0p and the width equal to the thickness d(rp) of the In0.55Al0.45As layer along the particle motion line. Once the potential curves Ezp(rp) are obtained, the ground-state wave functions fpp(rp) for the inplane slow motion and the one-particle ground-state energies of the electron, Ee and Eh then can be found by solving Eq. (4b) with the frontier conditions f 0rp ð0Þ ¼ 0 and frp(N) ¼ 0. In our numerical work, we use to this aim the trigonometric sweep method [17,18]. One example of the calculated potential curves for the electron in-plane motion, Eze(re) in the disk, the lens and the conical pyramid which have the same size parameters, the base radius 40 nm, the thickness of the WL 2 nm and the greatest height over the WL 3 nm we present in Fig. 2. One can see that the shape of the in-plane conﬁnement potential strongly depends on the QD morphology being almost the mirror image of the QD contour. For our model (1), the potential curves for the particles in-plane motion have the almost triangular shape for the conical pyramid (n ¼ 1), the almost parabolic shape for the lens (n ¼ 2) and the rectangular shape for the disk (n-N) whose dependence on the distance from the

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138

symmetry axis, consistent with the numerical calculations presented in Fig. 2, is given approximately by 8 ð pÞ < E layer þ 0:4V 0p ðrp =R0 Þn ; 0orp oR0 ; V p ðrp Þ : E ð pÞ þ 0:4V 0p ; rp 4R0 ; layer p ¼ e; h:

ð5Þ pÞ E ðlayer

pÞ E ðlayer

Here and þ 0:4V 0p are the ground-state energies of the particles in the InAs layers with the thicknesses 5 and 2 nm, respectively. The actual two-particle problem (2) for exciton is not possible to solve exactly and therefore in order to ﬁnd the ground-state wave function C(re,rh,t ¼ 1) of the electron– hole pair we employ a variational principle using to this end the Bastard-type trial function in form of a product of the wave function of the neutral particles corresponding to the lowest energy, C(0) with an unknown variational function, F(reh) which takes into account the particles correlation caused by electron–hole attraction and depends only on the electron–hole separation: Cðre ; rh ; t ¼ 1Þ ¼ Fðreh ÞCð0Þ ðre ; rh ; t ¼ 0Þ ¼ Fðreh Þf e ðre Þf h ðrh Þ.

ð6Þ

We estimate the exciton ground-state energy E ¼ E ¼ (t ¼ 1) by using the Schro¨dinger variational principle that states that the best approximation (6) to the exact ground state wave function can be obtained by means of the minimization of the functional F ½F ¼ hf e f h FjH^ Ejf e f h Fi. Taking the functional derivative with respect to F(r) gives after some mathematical manipulations the following simpliﬁed wave equation [10]: 1 d dFðrÞ 2 P0 ðrÞ FðrÞ ¼ E b FðrÞ; P0 ðrÞ dr dr r E b ¼ Eðt ¼ 0Þ Eðt ¼ 1Þ ¼ E e þ E h E,

case the position vectors re and rh are almost coincident: Z 2 (9) P0 ðrÞ ¼ 4pCr ; C ¼ f 2e ðrÞf 2h ðrÞ dr. This relation manifests the three-dimensional (3D) character of exciton for very small electron–hole separation independently on the conﬁnement geometry. The wave Eq. (7) should be completed by two boundary conditions conditions, one at the points r ¼ 0 and other for r tending to inﬁnity. The ﬁrst of them arise from Coulomb cusp condition and the relation (9) F0 ð0Þ=Fð0Þ ¼ lim½2PðrÞ=rP0 ðrÞ ¼ 1. r!0

(10)

The second boundary condition one can obtain taking into account that for large electron–hole separations functions Cðre ; rh ; t ¼ 1Þ and Cð0Þ ðre ; rh ; t ¼ 0Þ are similar and therefore the function F(r) should be almost constant according to deﬁnition (6). As the consequence lim F0 ðrÞ ¼ 0.

r!1

(11)

Our numerical procedure consists of three following steps: the calculation of the one-particle functions fe(re) and fh(rh), by means of the adiabatic approximation described above; the numerical estimation of integral (8) and, ﬁnally, the ﬁnding of the exciton wave function corresponding to the lowest eigenvalue of the problem (7) with frontier conditions (10) and (11). We have estimated the values of integral (8) at 100 points on uniform lattice which we have used afterwards in order to deﬁne the values of the Jacobian, P0(r) for arbitrary electron–hole separation by using cubic spline interpolation. The wave Eq. (7) with frontier conditions (10) and (11) has been solved numerically using the trigonometric sweep method describe in Ref. [18].

ð7Þ

where Eb is the exciton binding energy and the function P0(r) is given by Z Z (8) P0 ðrÞ ¼ dre jf e ðre Þj2 jf h ðrh Þj2 dðjre rh j rÞ drh . This wave Eq. (7) is similar to one for hydrogenic atom in effective isotropic space with the radial part of the Jacobian given by P0(r). The properties of this space are generally associated with the dependence of the Jacobian on r, which in according to relation (8) is deﬁned by the geometry of the heterostructure. If, for example, for any heterostructure this dependence were the power law, P0(r) ¼ CrD1 then the scaling parameter D in this dependence could be considered as dimension of the effective space and the Eq. (7) would have coincided with eigenvalue problem for a hydrogen-like atom in D-dimensional space, being the parameter D integer o fractional [19]. One can obtain for short distances r the approximated expression the dependence of the Jacobian on r, starting from the relation (8) and taking into account that in this

3. Results and discussion We ﬁrst check the accuracy of our numerical procedure, comparing our results for the exciton binding energy in GaAs/Al0.3Ga0.7As and In0.55Al0.45As/Al0.35Ga0.65As cylindrical QDs with those from Refs. [2,5–7] obtained previously for a simpliﬁed model with the zero thickness of the WL (db ¼ 0) and with experimental data from Ref. [15] in the weak magnetic ﬁeld limit. In all cases, we ﬁnd a good accordance between our results and those of other authors. In Fig. 3(a), we present the exciton binding energy as a function of the radius of the GaAs/ Al0.3Ga0.7As dot with thickness d0 ¼ 7 nm, where solid lines are our results, open triangles, circles and rectangles are those obtained previously by using hydrogen-like [6], two-parameter [7] and separable [8] trial functions, respectively. As the variational calculation gives a lower estimation of the binding energies, it is seen from Fig. 3(a) that for small disk radii our results are better than those obtained by using other trial functions, although for largedisk radii our energies are similar to those obtained by

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using hydrogen-like and two-parameter trial functions. It is due to the fact that our trial function, which has not variational parameters, is more ﬂexible allowing us to take into account the change of symmetry as the disk radius becomes less than one Bohr radius. On the other hand, the comparison of our results with 3D full treatment presented in Fig. 3(b) in In0.55Al0.45As/Al0.35Ga0.65As disk with thickness d0 ¼ 3.22 nm shows that the binding energies obtained by using our procedure are slightly lower than those from Ref. [2], nevertheless the agreement between two sets of the energies is excellent, considering the simplicity of our procedure. The charge distribution inside and around the SAQD induced by the captured electron–hole pair results from the competition between the conﬁnement inside the thin

a GaAs/Ga0.7Al0.3As disk

30

Binding energy (meV)

Ref.[5] Ref.[6] Ref.[7]

20

d0=7 nm db=0 nm 10 10

20

30

Radius (nm)

b

35

139

In0.55Al0.45As layer and the tunneling toward the barrier region of the heavier hole and the lighter electron. One could expect a considerable separation of charges in SAQD with trapped exciton caused by a signiﬁcantly stronger tunneling of the lighter electron toward the GaAs matrix above and below the InAs layer than one of the heavier hole, as the layer thickness becomes sufﬁciently small. In order to evaluate dimensions of the SAQD and the WL, when such effects might be appreciable, let us to use the uncertainty relation. As the thickness of the InAs layer, d decreases the ground-state particles energy levels lift displacing in the upper part of the well whereas the density of the electron distribution for the motion in the crystal growth direction becomes almost homogeneous for which the squared coordinate uncertainty is d2/12. The kinetic energy of particles for the corresponding motion can be estimated by using uncertainty relation as 3_2 =ð2mpw d 2 Þ; p ¼ e; h. The tunneling through the barrier becomes signiﬁcant as the value of the kinetic energy approaches to the value of the barrier height Vp. It happens when the layer ﬃ thickness d fulﬁls the condition pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dpa0 3mRy =ðmpw V p Þ. It gives for In0.55Al0.45As/Al0.35 Ga0.65As nanostructures the InAs layer thickness 3 nm for the electron and 1 nm for the hole. Therefore, one can expect a considerable charging of the QD with captured exciton due to the considerable stronger electron tunneling in the crystal growth direction above and below the InAs layer as its thickness is of the order 3–5 nm. Relevant characteristics of the separation between the electron and the hole trapped in QD are multipole momenta, which can be expressed in terms of the twodimensional (2D) probability densities of ﬁnding of the electron, Pe ðr; zÞ and of the hole, Ph ðr; zÞ at the point with the cylindrical coordinates (r,z) which in according to the relations (4c) and (6) can be calculated as follows: Pe ðr; zÞ ¼ rf 2re ðrÞf 2ze ðz; rÞC h ðz; rÞ,

Exciton binding energy (meV)

In0.55Al0.45As/Al0.35Ga0.65As disk 30

Ph ðr; zÞ ¼ rf 2rh ðrÞf 2zh ðz; rÞC e ðz; rÞ, d0=3.22nm

Z

db=0

C p ðz; rÞ ¼ 25

Ref. [2]

ð12Þ

Z þ1 f 2rp ðrp Þ drp f 2zp ðzp ; rp Þ dzp 0 1 Z 2p qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F2 r2 þ r2p 2rrp cos W þ ðz zp Þ2 dW. 1

0

ð13Þ 20 8

10

12

14

16

18

20

Radius (nm) Fig. 3. Comparison of our calculations for the exciton ground state binding energy in (a) GaAs/Al0.3Ga0.7As and (b) In0.55Al0.45As/Al0.35Ga0.65As structures with theoretical results obtained previously. The disks parameters are shown in ﬁgures. Our results (solid lines) for exciton ground-state binding energy as a function of the quantum disk radius are compared with (a) the variational calculations (different symbols) and (b) full 3D treatment (open triangles).

The correspondent density of charge distribution Q(r,z) in SAQD with trapped exciton is equal to the difference between the hole and the electron probability densities: Qðr; zÞ ¼ Ph ðr; zÞ Pe ðr; zÞ.

(14)

In Fig. 4 we present contour plots, which correspond to the level lines of the charge density distributions along a cross section in the middle of the In0.55Al0.45As/ Ga0.65Al0.35As quantum disk, lens and conical pyramid with trapped exciton. The left- and right-side parts of the ﬁgure correspond to the models of SAQDs without and

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Fig. 4. Contour plots of the radial charge distribution density induced by captured exciton in a plane trough the axis of symmetry of the In0.55Al0.45As/ Al0.35Ga0.65As quantum disk, lens and conical pyramid with the base radii 20 nm and the heights 3 nm. The left-side plots are for a model without WL and right-side plots correspond to the structures with the WL of 2 nm thickness. The shadowed parts of ﬁgures indicate the positively charged regions inside the In0.55Al0.45As material. Thick solid lines show the proﬁles of the corresponding In0.55Al0.45As layer.

with WL, respectively. The shadowed parts of ﬁgures indicate the positively charged regions inside the In0.55Al0.45As material. It is seen that the density of the radial charge distribution inside the QDs close to the axis is positive and it is negative in the upper and lower parts of QDs in peripheral regions close to junctions. Also, one can see that the electron tunneling in plane of the layer on the left-side plots for the models of the SAQDs without WL is depreciable. In this case, both particles remain in their inplane motion mainly inside the quantum dot but the heavier hole is situated mainly nearer to the axis. The existence of the WL leads to a lowering of the effective barrier height of the effective 2D potential (see Fig. 2) for both particles in-plane motion reinforcing signiﬁcantly their tunneling in the lateral direction. As the result, a stronger charging of the peripheral regions in the lateral direction is observed in right-side plots of Fig. 4 for models with the WL. Comparing in Fig. 4 contour plots for the disk, the lens and the pyramid one can also observe the variation of the

charge distribution inside and around their InAs layers, which is similar to one in a charged metallic surface when its geometry varies from the ﬂat to the spiked-type one. Such variation of the charge distribution is a consequence of the stronger conﬁnement in structures with spiked-type QD geometry that is manifested in Fig. 4 in a displacement of both positively charged core inside QD and negatively charged cloud around the QD toward the axis conserving almost invariable the charge distribution in the vertical direction as the geometry becomes more spike shaped. Due to axial symmetry the in-plane component of the dipole moment in all structures is equal to zero and the charge distribution in these QDs induced by a captured exciton is characterized by a non-zero dipole momentum in crystal growth direction and a non-zero in-plane quadrupole momentum. In order to clarify the nature of the vertical component of the dipole momentum induced by captured exciton in a QD and to explain why it is the largest in the structures with spiked-type geometry we present in Fig. 5 the charge distribution in two conical QDs

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141

Fig. 5. The same as Fig. 4, but now only for two conical pyramids with heights 2 and 5 nm.

9 d0=3nm db=2nm

8 Eb(Ry*)

with different heights. The arrows in these ﬁgures show the direction of the dipole momentum along corresponding cross sections. It is seen that the higher the structure the larger is the dipole momentum component in the crystal growth direction. It is due to the fact that the electron with major probability is situated in peripheral region of the dot where the thickness of the layer is small and the electron is mainly conﬁned nearly the base of the structure. Conversely, the hole is situated mainly in the middle of layer apart from the base in the region nearly the axis with the greatest layer thickness. Therefore, the existence of the non-zero dipole momentum in the crystal growth direction can be attributed to a non-homogeneous conﬁnement in vertical direction in QDs with wedged-shaped morphology. In contrast to the dipole momentum, the in-plane quadrupole momentum can be ascribed to the difference between the mean tunneling paths of the particles toward the WL which, as it is seen from Fig. 4, reinforces signiﬁcantly as the thickness of the WL growing becomes comparable with the height of the QD over the WL. The value of the quadrupole momentum also is sensitive to the alteration of the QD morphology. One can see from Fig. 4 that the electron–hole separation in the lateral direction is stronger in the spike-shaped QDs and it increases as the QD’s morphology changes from disk like to lens like and further to pyramid like. Thus, the dipole and quadrupole momenta of QDs with trapped exciton are important characteristics related to their morphology, which undergoes a remarkable change as the shape of the QD is modiﬁed. Other important characteristic which together with the electron–hole separation may suffer a notable variation due to the change of the QD morphology is the exciton binding energy, Eb deﬁned by the relation (7). In Fig. 6, we display the exciton binding energy dependence on the base radius in the disk, the lens and the conical pyramid with the height 3 nm over WL of the thickness 2 nm. One can see that the base radius dependencies in QDs with different shapes are rather similar being the binding energies in the conical pyramid (the shape-generating parameter n ¼ 1) for all base radii higher than those in the lens (n ¼ 2) and in the lens higher than in the disk (n-N). In other words, the

conical pyramid

7 n=1 lens

6

n=2 5

disk n=∞

4 10

20

30

40

50

60

70

R(nm) Fig. 6. Exciton binding energy as a function of the base radius in quantum dots with different proﬁles obtained by solving Eq. (7) (solid lines) and estimated by using the uncertainty relation (dashed lines).

smaller the shape-generating parameter n in Eq. (1) the higher is the exciton binding energy. In all cases, as the base radius increases the conﬁnement becomes weaker and the binding energies fall down tending to the 2D exciton binding energy limit equal to 4Ry . In order to explain this tendency, let us establish the dependence of the exciton binding energy on the shape-generating parameter n by using the uncertainty relation for two particles motion in 2D central ﬁeld with the potential given by the expression (5). The ground-state one-particle energies in 2D central ﬁeld with the potential given by the expression (5) can be estimated by using the uncertainty relation as 2=ðnþ2Þ E p E ðpÞ layer þ Ry ð0:4V 0p =Ry Þ

ðm=4mpw Þn=ðnþ2Þ ða0 =R0 Þ2n=ðnþ2Þ ;

p ¼ e; h:

ð15Þ

The ﬁrst term in Eq. (15) gives the potential energy of the particle located nearly the axis and the second term corresponds to the kinetic energy for the in-plane motion.

ARTICLE IN PRESS J.H. Marı´n et al. / Physica B 398 (2007) 135–143

142

As the base radius grows, the kinetic energy falls down and tends to zero but the rapidity of the convergence to zero depends on shape-generating parameter n. For example, in conical pyramid this term in Eq. (15) decreases as qﬃﬃﬃﬃﬃﬃ 1= 3 R20 , in lens as 1=R0 and in disk as 1=R20 . In order to estimate the electron–hole pair energy, one should take into account that the kinetic energy of this system is equal approximately to kinetic energy of the only heavier hole and in addition the term originated by the Coulomb attraction between particles should be included. The energy of the electron–hole interaction in QD with radius larger than 10 nm is about 4Ry , the binding energy of the nonconﬁned 2D exciton (the averaged electron–hole separation in 2D exciton is about 5 nm). Therefore, the estimated energy of the exciton in QD is E E ðeÞ layer þ E h 4Ry . Finally, according to the deﬁnition (6), we have obtained the following semi-empiric expressions for the exciton binding energy as a function of the base radius R0 in SAQDs with different shape-generating parameters n: E b ½4 þ ð2 þ 2=nÞða0 =R0 Þ2n=ðnþ2Þ Ry ;

n ¼ 1; 2; 1. (16)

The corresponding results for the exciton binding energies estimated by using this relation are shown in Fig. 6 (dashed lines). The agreement with our variational calculations is gratifying considering the simplicity of the derivation. In order to analyze the effect of the variation of the QD height on the exciton properties, we have calculated the exciton binding energy dependence on the base radius of conical pyramids with three different heights, d0 over WL of the thickness db ¼ 2 nm. Some of these results we display in Fig. 7. At ﬁrst sight, the decrease of the QD height should always provide a reduction of the separation between the electron and the hole and, as a consequence,

db=2nm conical pyramid

Eb(Ry*)

8.0

7.5

d0=3nm

7.0

d0=2nm d0=8nm

6.5 15

20 R0(nm)

Fig. 7. The exciton binding energy in pyramid-like quantum dots with different heights as a function of the base radius.

a rise of the exciton binding energy. Nevertheless, one can observe in Fig. 6 that the exciton binding energies in QDs with the height 2 nm is lower than those in QDs with the height 3 nm for all QD’s radii. The reason for such unusual behavior of the exciton binding energy consists in the fact that as the QD height reduces from 3 to 2 nm the kinetic energy of the electron increasing becomes comparable with the conﬁnement potential barrier height. The leakage of the electron wave function toward the barrier region along the z-direction in these conditions becomes so signiﬁcant that the averaged separation between light electron and heavy hole increasing becomes almost as large as one for weakly conﬁned particles in the 8 nm height QD. As have been estimated above by using the uncertainty relation, the threshold value for the well width corresponding to the electron wave leakage into barrier region is about 3 nm whereas the similar value for the hole is about 1 nm. It is reason of an unusual behavior of the exciton binding energy in the QDs whose heights are lower than 3 nm. Other interesting feature of the dependencies presented in Fig. 7 is the intersection of the curves corresponding to the QDs with heights 2 and 8 nm when their radii are about 12 nm. It can be explained only by a weaker conﬁnement of the electron in QDs with height 2 nm than one in QDs with height 8 nm as their radii are less than 12 nm and it can be considered as a manifestation of the leakage of the electron wave function as the QD’s dimension decreasing becomes smaller than a critical value not only in z-direction but also in the lateral direction. 4. Conclusions In order to study the effect of the morphology on the charge distribution in self-assembled QDs with trapped exciton, we calculate the ground-state wave function and the energy of the exciton in ﬂat disk-shaped, lensshaped and cone-shaped In0.55Al0.45As/Ga0.65Al0.35As QDs with different base radii, heights and WL thicknesses. A signiﬁcant difference is found between the electron and the hole distributions in the crystal growth direction owing to the difference between their masses. Provided that the QD height is of order 2–3 nm, the tunneling of the lighter electron toward the WL and the barrier region in crystal growth direction over and below the quantum dot is considerably stronger than the corresponding tunneling of the heavier hole. As a result, a signiﬁcant separation of the positive and negative charges in crystal growth and lateral directions provided by the captured exciton can be expected in In0.55Al0.45As/Ga0.65Al0.35As QDs. The comparison of the results obtained for QDs with different geometry shows that the separation of the charges in these directions is stronger in spike-shaped QDs, i.e. the separation in the conical pyramid is stronger than one in the lens and in the lens it is stronger than in the disk. In short, we ﬁnd that the effect of the in-plane conﬁnement on the electron–hole separation is stronger in spike-shaped QDs and therefore the exciton binding energy,

ARTICLE IN PRESS J.H. Marı´n et al. / Physica B 398 (2007) 135–143

which depends stronger on the in-plane electron–hole separation than one in the growth direction, in the conical pyramid is larger than one in the lens and in the lens it is larger than in the disk. By using the uncertainty relation, we derive simple formulae for estimating the exciton binding energies in disk-shaped, lens-shaped and cone-shaped QDs, which are in a good accordance with our variational calculations as the base radius is larger than 20 nm.

[10]

Acknowledgments

[11]

This work was ﬁnanced by the Industrial University of Santander (UIS) through the Direccio´n General de Investigaciones (DIF Ciencias, Cod. 5124) and the Excellence Centre of Novel Materials—ECNM, under Contract no. 043-2005 and the Cod. no. 1102-05-16923 subscribed with COLCIENCIAS. J.H. Marı´ n wishes to thank the Universidad Nacional, Sede Medellı´ n for the permission to study at the UIS.

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