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Excitons in cylindrical GaAs–Ga1 xAlxAs quantum dots under applied electric ﬁeld M.E. Mora-Ramos a, C.A. Duque b,n a b

´noma del Estado de Morelos, Av. Universidad 1001, CP 62209, Cuernavaca, Morelos, Mexico Facultad de Ciencias, Universidad Auto Instituto de Fı´sica, Universidad de Antioquia, AA 1226 Medellı´n, Colombia

a r t i c l e i n f o

abstract

Article history: Received 14 January 2012 Accepted 14 March 2012 Available online 21 March 2012

The exciton binding energy and photoluminescence energy transition in a GaAs-Ga1 xAlxAs cylindrical quantum dot are studied with the use of the effective mass approximation and a variational calculation procedure. The inﬂuence of these properties on the application of an electric ﬁeld along the growth direction of the cylinder is particularly considered. It is shown that for zero applied ﬁeld the binding energy and the photoluminescence energy transition are decreasing functions of the quantum dot radius and height. Given a ﬁxed geometric conﬁguration, both quantities then become decreasing functions of the electric ﬁeld strength as well. & 2012 Elsevier B.V. All rights reserved.

Keywords: Quantum dot Excitons Electric ﬁeld

1. Introduction Investigations on excitons in quantum dot (QD) systems is relevant for the understanding of some of their optical properties. In consequence, there have been a signiﬁcant number of works that consider different aspects of excitonic effects in these structures (see, for instance, Refs. [1–8]). The work by Le Goff and Ste´be´ laid down the model for the description of excitons in cylindrical semiconducting QD, analyzing both the inﬂuences of longitudinal and lateral conﬁnement on excitonic properties [1]. A study of the excitonic effects on third-harmonic-generation coefﬁcient for disk-like parabolic QD appears in Ref. [2]. The investigation on the dissociation of excitons and trions in vertically coupled QDs with a parabolic lateral conﬁnement potential due to the inﬂuence of an external electric ﬁeld was put forward by Szafran et al. [3]. It is also possible to refer articles dealing, respectively, with Coulomb-driven Stark effect anomaly for an exciton in the same QD systems [4]; the consideration of horizontal ﬁeld orientation and nonaligned dots in the study of Stark effect on the exciton spectra of vertically coupled double disk-shaped QD with smooth walls [5]; and the calculation of the excitonic spectra in vertical stacks of perfectly aligned triple and quadruple QD with individual disk-shaped quantum well (QW) proﬁles for the conﬁnement potential [7]. Further, the energy spectra of exciton states in disk-shaped QD under growth-direction magnetic ﬁelds is obtained in Ref. [6]. We can mention, as

n

Corresponding author. Tel.: þ57 4 219 56 30; fax: þ57 4 233 01 20. E-mail addresses: [email protected]ﬁsica.udea.edu.co, [email protected] (C.A. Duque). 0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.03.036

well, an investigation on the effect of tilted electric ﬁeld on the magnetoexciton ground state of cylindrical QD [8]. One of the external probes that researchers use with the aim of modifying the spectrum of localized states in semiconducting nanostructures is the dc electric ﬁeld. The ﬁeld action reﬂects in a change in the carrier conﬁnement proﬁles of the conduction and valence band potential energies. Since the electrons and holes have opposite charges, the effect of the electric ﬁeld on them implies an increase in the average value of the electron–hole distance. As a consequence, one should expect that the excitonic properties will be affected. Very scarce reports can be found in the literature regarding the electric ﬁeld effects in QDs. For instance, there is a work from John Peter et al. on the inﬂuence of the electric ﬁeld on donor binding energies in QDs with parabolical, spherical, and rectangular conﬁnement [9]. In the case of the excitonic properties, it is possible to mention a study on excitons in parabolic QDs [10], and the experimental investigation performed by Heller and collaborators [11]. However, we have not been able to ﬁnd a report on the exciton properties in GaAs-based QDs under the effect of an external dc electric ﬁeld. In the present work, we calculate the exciton binding energy and the photoluminescence (PL) energy transition associated with the conﬁned electron and hole states in cylindrical GaAs– Ga1 xAlxAs QD whose prototypical scheme and potential energy proﬁle are those shown in Fig. 1. We particularly focus our study on the effects of applied dc electric ﬁeld onto the exciton’s properties. The paper is organized as follows. In Section 2 we describe the details of the used theoretical approach. Section 3 is dedicated to the results and discussion, and ﬁnally, our conclusions are given in Section 4.

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potential V e0 (V h0 ) is obtained from the 60% (40%) of the bandgap between the well (GaAs) and barrier (Ga1 xAlxAs) materials. The exciton wavefunction, and the corresponding energy (E), can be calculated via a Rayleigh–Ritz variational procedure. For the trial wavefunction, C, we take [1] ! !

2

Cð r e , r h Þ ¼ N Uðre , rh ,ze ,zh Þearbðze zh Þ ,

ð2Þ

where N is the normalization constant, a and b are variational parameters, and U ðre , rh ,ze ,zh Þ ¼ Fðre ÞFðrh Þgðze Þgðzh Þ is the eigenfunction of the Hamiltonian in Eq. (1) without the Coulomb term at the right hand side. The uncorrelated radial wavefunctions for single particles are given by Fðri Þ ¼ J0 ðy0 ri =RÞ,

ð3Þ

where J0 represents the Bessel function of zeroth order, and

y0 ¼ 2:4048 is its ﬁrst root.

Fig. 1. Pictorial view of the cylindrical quantum dot considered in the present work. Graphs (a) and (b) show, respectively, the xy and xz projections, deﬁning the dimensions of the heterostructure, the directions of the applied electric ﬁeld, and the electron and hole positions. Schematic representations of the radial (r) and axial (z) conﬁnement potentials are depicted in (c) and (d), respectively.

2. Theoretical framework To describe the main equations of the theoretical framework, here we use the effective units. That is, we express energies in effective Rydberg ðR0 ¼ me4 =2_2 e20 Þ, and lengths in effective Bohr radius ða0 ¼ _2 e0 =me2 Þ. According to the model proposed by Le Goff and Ste´be´, within the framework of the effective mass approximation, the Hamiltonian for exciton states in a cylindrical GaAs–Ga1 xAlxAs QD under in-growth direction applied electric ﬁeld is given by [1] ( " # X r2 7 ðr2e r2h Þ @2 m @2 1 @ n þ þ H¼ mi @r2i ri @ri rri @[email protected] i ¼ e,h ) 2 2 m @ @ 1 @ 2 ð1Þ þ , n 2 þV i ðri ,zi Þ 7eFzi r mi @zi @r2 r @r qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! ! where r ¼ r2 þ ðze zh Þ2 , r ¼ 9 re rh 9, and 7 stands for electrons and holes, respectively. In addition, mne (mnh ) labels electron (hole) effective mass, while m is the electron–hole reduced mass, e is the electron charge, e0 is the GaAs static dielectric constant, and V e ðre ,ze Þ ½V h ðrh ,zh Þ is the QD conﬁnement potential for the electron (hole) carrier. We assume that the applied electric ﬁeld is oriented along the negative z-direction. Our model considers a QD of cylindrical shape with radius R and height L (Fig. 1). For the conﬁnement potential of the carriers we have considered inﬁnite and ﬁnite conﬁnement potentials in the r- and z-directions, respectively (see Fig. 1(c) and (d)). In such an approximation is valid to consider the separation V i ðri ,zi Þ ¼ V 1i ðri Þ þ V 2i ðzi Þ. Here V 1i ðri Þ ¼ 0 for ri rR and V 1i ðri Þ-1 for ri 4 R. In the case of the z-dependent conﬁnement we have V 2i ðzi Þ ¼ 0 for 9zi 9 r L=2 and V 2i ðzi Þ ¼ V i0 for 9zi 9 4L=2. For electrons (holes) the axial conﬁnement

The way of obtaining the uncorrelated axial wavefunctions for single particles relies in a method developed by Xia and Fan [13], which was posteriorly used in the calculation of optical absorption in superlattices under magnetic ﬁelds by de Dios Leyva and Galindo [14]. Such an approach is based on the expansion of the electron (hole) states over a complete orthogonal basis of sine functions associated with a QW of inﬁnite potential barriers whose width in this work was taken as L1 ¼ 50 nm. Consequently, the zi-dependent eigenfunctions of the Hamiltonian in Eq. (1) without the Coulomb potential term are written as 12 X 1 2 npzi np gðzi Þ ¼ : ð4Þ C n sin þ L1 n ¼ 1 L1 2 Of course, the number of terms included in the calculation cannot be inﬁnite. The convergence of Eq. (4), for the speciﬁc height of the QD considered, is ensured until 10 3 meV with the incorporation of 200 terms in the expansion of the g(zi) wavefunctions. The trial wavefunction in Eq. (2) takes into account the anisotropy of the dot via the two variational parameters, which may be important for different R and L values. It is clear that this trial wavefunction cannot reproduce the proper hydrogen-like 3D exciton wavefunction, but this function is expected to be a good approximation in both the strong conﬁnement regime and in the 2D dimensional limit. The energy of the exciton, E, is obtained after minimizing the Hamiltonian with the trial wavefunction. The exciton binding energy (Eb) is obtained from the deﬁnition Eb ¼ E0 E,

ð5Þ

where E0 is the eigenvalue associated to U ðre , rh ,ze ,zh Þ. 3. Results and discussion In order to provide a more explicit presentation of the numerical outcome of the work, we go back to giving the energy values in units of meV, and the lengths in nanometers. The chosen QD composition is GaAs–Ga0.67Al0.33As. For the effective mass parameters we have considered mne ¼ 0:067m0 , mnh ¼ 0:34m0 (where m0 is the free electron mass), e0 ¼ 12:65, V e0 ¼ 284 meV, and V0h ¼ 189 meV [15]. With these values R0 ¼4.76 meV and a0 ¼12 nm. Fig. 2 presents the dependence of exciton binding energy upon the QD height in the situation of ﬁxed radius (R¼ 10 nm), for three different applied ﬁeld strengths. Fig. 2(a) depicts the variation of Eb in the QD, whereas Fig. 2(b) shows the corresponding variations of the variational parameters a and b. The temperature of the system is set to 4 K. On the other hand, Fig. 3 depicts the variations of the exciton binding energy (Fig. 3(a)), and the parameters a, b (Fig. 3(b)) on

M.E. Mora-Ramos, C.A. Duque / Physica B 407 (2012) 2351–2357

25

E (meV)

20

15

F=0 20 kV/cm 40 kV/cm

R = 10 nm

10 5

10

15

20

15

20

variational parameters (effective units)

0.8 0.7

α

0.6 0.5 0.4 0.3 0.2 0.1

β

F=0 20 kV/cm 40 kV/cm 5

R = 10 nm 10

L (nm) Fig. 2. (a) Exciton binding energy in a cylindrical GaAs–Ga0.67Al0.33As QD as a function of the height of the dot for R¼ 10 nm and several values of the applied electric ﬁeld. In (b) are shown the a and b variational parameters in effective length units.

the QD radius for a ﬁxed height (L¼10 nm) and the same set of electric ﬁeld intensities. One can observe that the general feature of the exciton binding energy is to exhibit a decreasing dependence as a function of the enlargement of the QD, either by augmenting its radius or its vertical size. There is only a narrow range of the variation in the QD height, when L is around 3 nm, in which it is possible to detect a weak increase of Eb (Fig. 2(a)). Given the presence of the ﬁnite potential barriers at 7L/2 (Ga0.67 Al0.33As), one readily detects that, as long as the value of L decreases, the wavefunction begins to spread over the barrier region with the consequent reduction of the quantum conﬁnement in the axial direction. Thus, the binding energy experiences a decreasing. In the limit L-0 – not shown in Fig. 2(a) – the binding energy exactly corresponds to that of an exciton conﬁned within a cylindrical quantum dot of radius R and height L1 , with rigid barriers both in the axial and radial directions. Clearly, by taking a large enough value for L1 ð 10a0 Þ, the limit L-0 will lead to the binding energy of a conﬁned exciton in a quantum wire with radius R and inﬁnite potential barrier [16]. In the range 2 nm rL r 5 nm there is a strong quantum conﬁnement on the wavefunction in the axial direction, which is rather strong for small values of the height. As a consequence, the expected value of the distance 9ze zh 9 reduces and the strength of the electron–hole interaction becomes larger. This is true even in the case of non-zero applied ﬁeld. The ﬁeld-induced spatial electron–hole charge separation along the z-direction has very little effect in the case of very low values of the QD height. When the vertical dimensions (L) go larger than L ¼ 3 nm, the exciton binding energy decreases monotonically. This is a consequence of the reduction in electron and hole conﬁnement. Both

2353

ground state z-dependent uncorrelated densities of probability become broadly distributed, leading to larger values of the expected electron–hole distance. In Fig. 2(b) the reader can observe that the variational parameter a, associated with the in-plane correlation, shows the same behavior – as a function of the dot’s height – exhibited by the different curves corresponding to the binding energy for the distinct values of the dc ﬁeld strength chosen in the calculation. Notice that while L decreases until reaching the value of 5 nm, the a parameter augments and the b parameter diminishes. This can be understood as a strengthening of the in-plane correlation and a weakening of the correlation along the axial direction [1]. When L o5 nm, a reaches a maximum and then decreases, which means that there is a weakening of the in-plane correlation and a corresponding increment in the axial correlation, observed via the ﬁnal increase of the b parameter. Here, is worth highlighting that the expected value of the inplane electron–hole distance is proportional to a1 , whilst the expected value of the electron–hole distance along the axial 1 direction is proportional to b . This implies that, if L becomes smaller: (1) The increase of a gives a diminishing of a1 and, as a consequence, a reduction of the expected value of the in-plane distance between both carriers together with a higher in-plane correlation. (2) The diminishing of b would lead to a larger value 1 of b which translates into an increase of the axial expected value of the inter-carrier distance, associated with a reduction of the axial correlation. The degree of exponential decay of the exciton wavefunction reduces for sufﬁciently large L. We can readily notice that by observing the decreasing behavior of the variational parameters for such values of the QD height. This means that the contribution to the diminishing of the binding energy coming from this component is less important than the one associated with the uncorrelated electron and hole ground states in the z-oriented quantum well. Moreover, even in the case of zero applied ﬁeld – for which we observe an increment in the conﬁnement of the z-dependent exponential term due to the increase of the b parameter – the result of the decrease in Eb leads us to conclude that the main reason for the fall of the exciton binding energy as a function of the dot’s height is the loss of conﬁnement related with the uncorrelated electron and hole ground states. The application of an electric ﬁeld along the vertical direction of the QD implies the spatial separation of electrons and holes towards opposite sides of the system. As a consequence, there will be an additional increment in the expected electron–hole distance, with the corresponding decrease of the exciton binding energy. Of course, this effect is more noticeable when the QD size along the z direction becomes larger. The previously given arguments are valid for the physical interpretation of the results presented in Figs. 3 and 4 as well. The decreasing dependence of Eb as a function of the QD radius for ﬁxed height (Fig. 3(a)) is a result of the lesser conﬁnement of the exciton wavefunction that leads to the increase in the expected value of the electron–hole distance. Thus, the strength of the Coulombic attraction reduces as the exciton binding energy does. It is worth noticing that this effect comes from both the uncorrelated radial wavefunctions and the variational axial part, given the diminishing of the variational b parameter, as one observes from Fig. 3(b). Likewise, the variation of the exciton binding energy with the electric ﬁeld strength shows an overall decreasing dependence. Fig. 4 contains the results for Eb obtained for different geometric conﬁgurations of the GaAs–Ga0.67Al0.33As cylindrical QD. One notices that by ﬁxing the dot’s height, the binding energy grows with the reduction of the cylinder radius, due to the rise in the radial electron and hole conﬁnement. In the same situation, we

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30

E (meV)

25

20

F=0 20 kV/cm 40 kV/cm

15 5

L = 10 nm

10

15

20

variational parameters (effective units)

0.8 0.7 α

0.6

region. Therefore, there is an increase of expected value of the z-distance between electron and hole. In Fig. 5 we present the expectation value for the axial electron–hole distance (left hand panel) and the in-plane electron–hole distance (right hand panel) for exciton states in GaAs– Ga0.67Al0.33As QD, taking into account several conﬁgurations of the dimensions of the structure and the applied electric ﬁeld. The results shown here conﬁrm our previous comments on the inverse proportionality of the variational parameter a with respect to the expected value of the in-plane electron–hole distance ðreh ¼ ½/C9r2 9CS1=2 Þ, and that b is inversely proportional to the axial expected inter-carrier distance (zeh ¼ ½/C9ðze zh Þ2 9CS1=2 ). It can be seen that: (1) as long as the QD height augments, (a,b), zeh increases and reh remains essentially constant (notice that while zeh has variations of the order of 10 nm, the changes in reh are only of the order of 1 nm); (2) when the value of the QD radius becomes larger, (c,d), there is a growth in reh whilst zeh remains almost constant (it can be noticed that the variation of zeh is approximately equals to 0.5 nm, whereas the change of reh is of about 7 nm); (3) regarding the effect of the axially applied electric dc ﬁeld one may

0.5 0.4

F=0 20 kV/cm 40 kV/cm

0.3 0.2 5

8

L = 10 nm

10

6

15

6.0

4

20

R (nm)

2 5.5

0

Fig. 3. (a) Exciton binding energy in a cylindrical GaAs–Ga0.67 Al0.33As QD as a function of the radius of the dot for L¼ 10 nm and several values of the applied electric ﬁeld. In (b) are shown the a and b variational parameters in effective length units.

5

10 15 L (nm)

5

20

3.5

10 15 L (nm)

20

10

3.4 (nm)

3.2

(nm)

3.1

ρ

21

3.3

z

24

E (meV)

6.5

10

β

2.9 5

15 L = 10 nm

R = 10 nm

L = 15 nm

R = 8 nm

12 0

10

20

30

40

F (kV/cm) Fig. 4. Exciton binding energy in a cylindrical GaAs–Ga0.67 Al0.33As QD as a function of the applied electric ﬁeld for several dimensions (radius and length) of the heterostructure.

detect that the effect of the in-axis applied electric ﬁeld is very weak, independently of the value of R, for small L, and becomes more inﬂuential for higher values of the QD vertical size. The decrease of Eb seen for large values of F cannot be related with the electron–hole spatial separation, which reaches a limiting value after the ﬁeld goes above a certain value – for a given value of L. In fact, the decreasing behavior is associated to the contribution of the variational part of the exciton wavefunction. Observing the variation of the variational parameter b in Fig. 2(b), it is possible to conclude that, when the ﬁeld augments, the spatial distribution of the exciton density of probability spreads over a larger vertical

6 4

3.0 18

8

10 15 R (nm)

20

2

6

6.5

5

6.0

4

5.5

3

5.0

2 0

10 20 30 F (kV/cm)

40

5

10 15 R (nm)

20

0

10 20 30 F (kV/cm)

40

4.5

Fig. 5. Expectation value of the axial electron–hole distance (left hand panel) and in-plane electron–hole distance (right hand panel) of exciton states in GaAs– Ga0.67Al0.33As QD for several conﬁgurations of the dimensions of the structure and the applied electric ﬁeld. In (a) and (b) the results are the function of the height for ﬁxed values of the radius (R ¼10 nm) and the applied electric ﬁeld (F ¼0, solid lines; 20 kV/cm, dashed lines; 40 kV/cm, dotted lines). In (c) and (d) the results are the function of the radius for ﬁxed height (L ¼10 nm) and the applied electric ﬁeld (F ¼0, solid lines; 20 kV/cm, dashed lines; 40 kV/cm, dotted lines). In (e) and (f) the results are the function of the applied electric ﬁeld for ﬁxed values of the radius and height (L ¼10 nm, solid lines; L ¼15 nm, dashed lines; R ¼8 nm, solid squares; R¼ 10 nm, solid circles). Exciton binding energy in a cylindrical GaAs–Ga0.67Al0.33As QD as a function of the applied electric ﬁeld for several dimensions (radius and length) of the heterostructure.

M.E. Mora-Ramos, C.A. Duque / Physica B 407 (2012) 2351–2357

150

-E (GaAs)] (meV)

120

90

F=0 20 kV/cm

30

[PL

[PL

-E (GaAs)] (meV)

150

60

2355

40 kV/cm R = 10 nm

120

90 L = 10 nm F=0

60

20 kV/cm 40 kV/cm

30

0 5

10

15

5

20

10

15

20

15

20

0.15

0.08

F=0

overlap integral

overlap integral

20 kV/cm 0.07

0.06

40 kV/cm L = 10 nm

0.10

F=0 20 kV/cm 0.05

40 kV/cm R = 10 nm 0.05

5 5

10

15

20

L (nm) Fig. 6. (a) Normalized photoluminescence energy transition in a cylindrical GaAs– Ga0.67Al0.33As QD as a function of the height of the dot for R ¼10 nm and several values of the applied electric ﬁeld. (b) The electron–hole overlap integral as a function of the QD height for the same geometric and applied ﬁeld conﬁgurations.

observe that as long as the strength of the electric ﬁeld increases, (e,f), the value of zeh augments and that of reh behaves as an extremely insensitive quantity. The inﬂuence of the ﬁeld is more noticeable in those structures having larger sizes along the z-direction (observe that while zeh varies in the order of 2 nm when L¼15 nm, the corresponding variations in zeh are only of the order of 0.5 nm for L¼10 nm). In addition, from the graphs (e,f) one sees that the changes in reh are just around the order of 0.25 nm. On the other hand, the minima shown by both zeh and reh in Fig. 5(a) and (b) strictly associate with the fact that the wavefunction penetrates deeper inside the barrier region, when the height of the QD becomes small enough. In Fig. 6(a), Fig. 7(a), and Fig. 8(a) we are presenting the calculated results for the normalized PL energy transition (NPLET) for different geometric and/or electric ﬁeld conﬁgurations. This quantity is depicted as a function of the QD height, for a ﬁxed value of the radius, R¼10 nm, and three different electric ﬁeld intensities in the case of Fig. 6(a). Fig. 7(a) contains the NPLET as a function of the radius of the dot in the case of a ﬁxed QD height and the same set of values for F. In Fig. 8(a), this quantity is shown as a function of the electric ﬁeld strength for a number of different geometries of the cylindrical QD. We deﬁne the normalized PL energy transition by subtracting the GaAs energy gap from the quantity Ee þ Eh þEgap Eb (the PL-peak energy), where Ee and Eh are, respectively, the conﬁned ground state energy of the electron and hole and Egap is the GaAs bandgap.

10 R (nm)

Fig. 7. (a) Normalized photoluminescence energy transition in a cylindrical GaAs– Ga0.67Al0.33As QD as a function of the radius of the dot for L¼ 10 nm and several values of the applied electric ﬁeld. (b) The electron–hole overlap integral as a function of the QD height for the same geometric and applied ﬁeld conﬁgurations.

The NPLET is a decreasing function of the dot’s height – for ﬁxed radii – as well as of the dot’s radius – for ﬁxed height – and this behavior is somewhat reinforced with the application of a dc electric ﬁeld. Despite the decrease of the exciton binding energy, one obtains that the PL-peak diminishes for larger values of L because the conﬁned ground state energies associated with the electron and hole motion along the z-direction also acquire lower values. This happens with and without applied electric ﬁeld. That is, the energy position varies inversely with the quantum well width, and the electric ﬁeld modiﬁes the conﬁning potential proﬁle by making the well bottom to descend, and pushing the carrier position towards the lower part of the quantum well. Likewise, the variations of the NPLET depicted in Figs. 7 and 8 as functions of dot’s radius and electric ﬁeld intensity, respectively, show behaviors that are consistent with the physical justiﬁcation given above for the dependencies of the exciton binding energy upon the same quantities. That is the enlargement of the dot’s size in the radial direction reﬂects in the rise of the average electron–hole distance which leads to the diminishing of the exciton energy. The increase in the strength of the z-oriented applied electric ﬁeld, for a ﬁxed geometric conﬁguration implies the selective spatial separation of conﬁned electrons and holes and the consequent reduction in the intensity of the Coulombic interaction between both types of carriers. We can see from Fig. 8(a) that, as it should be expected, the effect of the electric ﬁeld is more pronounced for larger vertical QD size, because in

2356

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[PL

-E (GaAs)] (meV)

90 80 R = 8 nm

70

R = 10 nm

60

L = 10 nm L = 15 nm

50 40 30 0

10

20

30

40

20

30

40

0.070

overlap integral

0.068 0.066 R = 8nm R = 10 nm

0.064 0.062 0.060

L = 10 nm L = 15 nm

0.058 0

10

F (kV/cm) Fig. 8. (a) Normalized photoluminescence energy transition in a cylindrical GaAs– Ga0.67Al0.33As QD as a function of the applied electric ﬁeld for several dimensions (radius and length) of the heterostructure. (b) The electron–hole overlap integral as a function of the QD height for the same geometric and applied ﬁeld conﬁgurations.

this case, this selective spatial separation is less affected by the presence of the Ga0.67Al0.33As potential barriers. The graphics labeled as (b) in Figs. 6–8 contain the correspondR ! ! ! 2 ing results for the overlap integral J ¼ 9 Cð r e , r e Þd r e 9 . This integral is evaluated between 0 and R in the radial coordinates and between L1 =2 and L1 =2 in the z-direction. The features of J as a function of the vertical size L are governed mostly by the single electron and hole ground state functions of the type (4). For zero applied ﬁeld, the increase in L – with a ﬁxed value of L1 – makes these functions to extend over larger intervals within the integration interval with amplitudes signiﬁcantly different from zero. Therefore, the evaluation of J will give larger values of this quantity for increasing L as shown in the corresponding curve in Fig. 6(b) when the dot’s height goes above 15 nm. For smaller L, the decreasing behavior of the zero-ﬁeld overlap integral relates with the different regimes of conﬁnement of the electron and hole wavefunctions. The conﬁning barrier for holes is lower – due to the valence band-offset. Thus, the hole ground state wavefunction extends largely into the barrier region whilst the electron one locates mostly within the quantum well. As a consequence, the overlap of both wavefunctions becomes reduced and the integral would give a lower value. The application of an electric ﬁeld affects the above discussed behavior. If the ﬁeld strength is not very large, the form of the curve for J deviates slightly from the zero ﬁeld one, and the main difference appears for larger L. Augmenting the ﬁeld intensity makes the curve of the overlap integral to become strictly

decreasing over the entire range of dot’s height under consideration. The reason for this variation lies in the selective spatial separation induced by the ﬁeld, which causes the convolution of ge(z) and gh(z) to practically vanish for sufﬁciently large axial size. Since the selective spatial separation is of lesser inﬂuence in a shorter QD, we readily observe that the curves of J for ﬁnite F practically coincide with the zero ﬁeld one. The slight increase of the zero ﬁeld J observed for 2 nm o Lo 5 nm has to do with the reduction of zeh previously discussed. It is a well known fact that a very narrow quantum well has the energy positions of its conﬁned levels closer to the top of the barrier, and the states are less localized in the well. Consequently, the single particle envelope wavefunctions have a signiﬁcant penetration within the conﬁning potential barrier region. These arguments regarding ﬁeld-induced charge separation can be also used to explain the more pronounced decreasing variation of J as a function of the ﬁeld intensity for larger values if L for a ﬁxed dot radius, as shown in Fig. 8(b). However, Fig. 7(b) shows J to be a monotonically increasing function of the QD radius R. This is not difﬁcult to understand if we look at the form of the radial part of the integrand (Eq. (3)). There, the quantity R appears in the denominator of the argument. This means that augmenting the dot’s radius makes Fðre,h Þ to be evaluated at a value closer to 0. Reminding the behavior of the Bessel function of the ﬁrst kind J0 in the region of its small arguments, immediately justiﬁes the variation of J reported. We will use our results of the overlap integral to evaluate the so-called exciton oscillator strength through the expression f osc ¼ 2Q 2 J=ðm0 Eb Þ , where Q describes all intracell matrix-element effects and m0 is the bare electron mass. We use for Q 2 =m0 a value of 1000 meV, which is a typical value in the case of III–V zincblende semiconductors (see, for instance, Ref. [12]). The results for this quantity are presented in Fig. 9 for the three distinct sets of conﬁgurations reported in the evaluation of the exciton binding energy and the NPLET. The explanation for the functional variations shown by the oscillator strength in each graph of Fig. 9 follows the same arguments used above, related with the corresponding changes of the exciton binding energy and the overlap integral. That is, the basic physical reason is the behavior of the averaged electron– hole distance and the associated intensity of the Coulombic interaction. This will be a consequence of the changes in the carrier conﬁnement due to the variations of the QD geometry and/ or the application of the electric ﬁeld. While the curves in Fig. 6(b) are all essentially decreasing, in Fig. 9(a), the entire set of ﬁgures represent absolutely increasing variations. This means that the dominant contribution to the oscillator strength comes from the properties of the exciton binding energy – in the denominator in fosc – as a function of the QD height. This is also the reason for the exchange in the positions of dashed and solid lines in both ﬁgures. On the other hand, by observing Fig. 7(b) and Fig. 9(b) we readily notice that all the curves represent increasing functions of the dot’s radius. So, the dominant contribution to the value of the oscillator strength is given by the overlap integral, with respect to that coming from the binding energy. Besides, one can detect once again that the electric ﬁeld has almost no inﬂuence on this variation. Now, by comparing Fig. 8(b) and Fig. 9(c), it is seen that the binding energy has the dominant role in the variation of fosc as a function of the electric ﬁeld strength for ﬁxed QD geometries. We observe that the functional dependence of Eb acts in a way that changes the monotony of the curves. When there is a ﬁxed height, L, a larger value of R always leads to a higher result of the oscillation strength. Notice also that there is an electric ﬁeld intensity (30 kV/cm) which can tune the same value of the

M.E. Mora-Ramos, C.A. Duque / Physica B 407 (2012) 2351–2357

oscillator strength

12 F=0 20 kV/cm 40 kV/cm

9

R = 10 nm

6 5

10 L (nm)

15

20

oscillator strength

20 F=0 20 kV/cm 40 kV/cm

15 10

L = 10 nm

5

2357

energy induced by the selective spatial separation of conﬁned electrons and holes, and the corresponding weakening of the Coulombic interaction. We have also calculated the dependence of the exciton-related normalized photoluminescence energy transition on quantum dot size and dc applied ﬁeld. The same decreasing behavior is detected for this property, and can be explained as well by the properties of the electron and hole conﬁnement, via the corresponding modiﬁcations of the expected value of the distance between electron and hole, due to the changes in the dot’s geometry and/or the applied ﬁeld intensity. This explanation is also valid to discuss the changes of the exciton-related oscillator strength in this kind of low-dimensional structures. Such quantity is reported for varying QD geometric conﬁgurations with or without the inﬂuence of the external ﬁeld. Its monotonic properties as a function of dot’s height, radius, as well as of the intensity of the dc ﬁeld are straightforwardly explained by analyzing the corresponding behaviors of the electron–hole overlap integral and the exciton binding energy.

0 5

10

15

20

R (nm)

oscillator strength

8.8

Acknowledgments

L = 10 nm L = 15 nm

8.0 7.2 6.4

R = 8 nm R = 10 nm

5.6 4.8 0

10

20 F (kV/cm)

30

40

Fig. 9. The exciton-related oscillator strength in a cylindrical GaAs–Ga0.67Al0.33As QD as a function of: (a) the QD’s vertical size for several values of the applied electric ﬁeld and ﬁxed radius, (b) the radius of the QD for several values of the applied electric ﬁeld for ﬁxed dot’s height, and (c) the applied electric ﬁeld for several dimensions (radius and length) of the heterostructure.

oscillator strength in two very different QD geometric conﬁgurations: L¼15 nm, R¼8 nm, and L¼ 10 nm, R¼ 10 nm. Interestingly, the extrapolation of the ﬁeld intensity to larger values shows that this coincidence does not appear further.

4. Conclusions In this work we present the results of the calculation of the exciton binding energy in cylindrical GaAs–Ga0.67Al0.33As quantum dots with an external in-axis dc electric ﬁeld. In general, augmenting the dimensions of the quantum dot leads to a decrease of this quantity due to the reduction of carrier conﬁnement. The application of the electric ﬁeld implies an additional diminishing of the binding

This research was partially supported by Colombian Agencies: CODI-Universidad de Antioquia, Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD-exclusive dedication project 2011–2012), and ‘‘El Patrimonio Auto´nomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnologı´a y la Innovacio´n Francisco Jose´ de Caldas’’ Contract RC-No. 275-2011. The work was developed with the help of CENAPAD-SP, Brazil. In addition, the authors acknowledge support from Mexico– Colombia bilateral collaboration Grant ‘‘Estudio de propiedades ´pticas y electro ´nicas en nanoestructuras y sistemas semiconductores o ´n’’. M.E.M.R. also thanks support from Mexican de baja dimensio CONACYT through research Grant CB-2008/No. 101777.

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