(Ga, Al) triple quantum dots

Superlattices and Microstructures 49 (2011) 269–274

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Combined effects of electric field and hydrostatic pressure on electron states in asymmetric GaAs/(Ga, Al) triple quantum dots Guillermo L. Miranda a,b , R.L. Restrepo a,b , C.A. Duque b,∗ a

Fisica Teórica y Aplicada, Escuela de Ingeniería de Antioquia, AA 7516, Medellín, Colombia

b

Instituto de Física, Universidad de Antioquia, AA 1226, Medellín, Colombia

article

info

Article history: Received 20 April 2010 Received in revised form 28 June 2010 Accepted 4 August 2010 Available online 15 September 2010 Keywords: Coupled quantum dots Hydrostatic pressure Electric field

abstract The combined effects of an in-growth direction applied electric field and hydrostatic pressure on the exciton binding energy and photoluminescence energy transitions are reported in this work for triple vertically coupled quantum dots. The calculations have been carried out within the effective mass approximation, and using a variational procedure. The results show that the exciton binding energy and the photoluminescence energy transitions are functions of external probes like the hydrostatic pressure and the applied electric field. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The advances in the techniques for the obtention of semiconductor crystals have allowed the fabrication of low dimensional systems like quantum wells, quantum well wires, and quantum dots (QDs). These systems have attracted interest in basic and applied research due to their phenomenology and their potential use to develop optoelectronic devices such as ultrafast optical switches and infrared waveguide devices. In QD, the electrons occupy well-defined discrete quantum states and consequently these kinds of heterostructures have been called artificial atoms. QDs can interact with each other to form coupled systems, which are then known as artificial molecules whose physical characteristics can be tuned by external probes such as hydrostatic pressure and an applied electrical field. Studies regarding different properties of coupled QDs have been put forward in some recent works. A model for the formation of artificial molecules in double QDs was proposed by Bednarek et al. [1], and

∗

Corresponding author. Tel.: +57 4 219 56 30. E-mail addresses: [email protected] (G.L. Miranda), [email protected] (R.L. Restrepo), [email protected], [email protected] (C.A. Duque). 0749-6036/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2010.08.003

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Fig. 1. (Color online) Pictorial view of the studied triple quantum dot heterostructure.

the photoluminescence (PL) of such kinds of molecules was reported by Beirne et al. [2]. In addition, Neogi et al. [3] studied the PL of multi periodic QDs. They found that the binding energy of excitons in these systems is larger than in the single period QD system. Besides, Szafran et al. [4] showed that if an external electric field is applied oriented along the in-growth direction of vertically coupled QDs, the variation in the alignment of the QD does not give rise to changes in the Stark effect. Xie studied the ground state transitions in vertically coupled QDs under a magnetic field. He found that some ground state transitions are prevented from occuring due to inter-dot correlations [5]. On the other hand, the effects of hydrostatic pressure on shallow impurities states in symmetric double QD heterostructures were reported by Liu et al. [6]. They found that for all positions of the donor ion, the binding energy shows a nonlinear behavior in the indirect-gap regime. The present work is concerned with the combined effects of hydrostatic pressure and in-growth direction applied electric field on the binding and PL-peak energies of excitons in triple quantum dots (TQDs) with cylindrical geometry. The calculations are performed with a variational scheme using the effective mass approximation. The paper is organized as follows: in Section 2 is presented the theoretical framework, in Section 3 the results and discussions are shown, and finally, our conclusions are given in Section 4. 2. Theoretical framework With the use of effective units — energies are taken as multiples of the effective Rydberg (R0 =

µ(P )e4 ) 2 h¯ 2 ε02 (P )

h2 ε (P )

and lengths are given as multiples of the effective Bohr radius (a0 = ¯µ(P0)e2 ). According to the model proposed by Le Goff and Stébé, the Hamiltonian for exciton states in a cylindrical GaAs–Ga1−x Alx As TQD within the framework of the effective mass approximation, including the effects of hydrostatic pressure (P), is given by Le Goff and Stébé [7]:

] − µ(P ) [ ∂ 2 1 ∂ ρ 2 ± (ρe2 − ρh2 ) ∂ 2 H = − 2− − m∗i (P ) ρi ∂ρi ρρi ∂ρ∂ρi ∂ρi i=e,h [ ] [ ] − µ(P ) ∂ 2 ∂2 1 ∂ 2 + − ∗ ± eFzi + Vi (ρi , zi ) − + − , (1) 2 2 m ( P ) ∂ρ ρ ∂ρ r ∂ zi i i=e,h  − → − → where r = ρ 2 + (ze − zh )2 , ρ = | ρe − ρh |, ± stands for electrons and holes, respectively. m∗e (m∗h ) is the electron’s (heavy-hole) effective mass [8], µ is the electron–hole’s reduced mass, e is the absolute value of the electron’s charge, and Vi is the cylindrical quantum dot confining potential. In Fig. 1 we present the pictorial view of the TQD system. We consider a free-standing system consisting of three cylindrical GaAs quantum dots – each one of radius R and height HD – and coupled by two Ga1−x Alx As potential barriers — each one of width Lb . It means that Vi is modeled as an infinite function when |z | > Lb + 3HD /2 or when ρ > R, whereas Vi is zero in the dot regions, and Vi = V0 (P ) in the barriers

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regions. The hydrostatic pressure dependent potential V0 is given by Elabsy [9]: V0 ( P ) = (P )



(P )

Γb

(P )

Xb (P )

− Γd(P ) , − Γd(P ) ,

P ≤ P1 , P > P1 ,

(2)

(P )

where Γd , Γb , and Xb are the minimum of the Γ and X conduction-bands in the dot (d) and barrier (b) materials. All these energies are referred to the maximum of the Γ valence-band of the dot material and considering that 60% of the dot-barrier band-offset corresponds to the conduction band. (P ) (P ) Here P1 (= 5 kbar) is the crossover pressure between the Xb and Γb conduction-bands. For the sake of simplicity, and given the relatively low value of the Al molar fraction considered in the barrier alloy, the dielectric constant, ε0 , and the electron and hole effective masses will be taken to be the same as in GaAs throughout the GaAs–Ga1−x Alx As TQD [8]. These parameters are pressure dependent. The dielectric constant and the electron and hole effective masses are given, respectively, by Raigoza et al. [8]:

ε0 = 12.83 exp(−1.67 × 10−3 kbar−1 P ), ] −1 [ 7510 15 020 m∗e = 1 + + m0 , Eg (P ) Eg (P ) + 341

(3)

m∗h = (0.34 − 0.1 × 10−3 kbar−1 P )m0 ,

(5)

(4)

and

where m0 is the free electron mass and Eg (P ) is the pressure-dependent gap for the GaAs material [8] Eg (P ) = (1519 + 10.7 kbar−1 P ) meV.

(6)

The hydrostatic-pressure-dependent dimensions of the TQD system (R, HD , Lb ) are obtained from the fractional pressure-induced change of the volume for the zinc blende GaAs material [10]. The exciton wavefunction, and the corresponding energy, can be calculated via a variational procedure [11,12]. For the trial wave function, Ψ , we use [7]

Ψ = N Υ (ρe , ρh , ze , zh ) exp[−αρ − λ(ze − zh )2 ],

(7)

where N is the normalization constant, α and λ are the variational parameters, and Υ (ρe , ρh , ze , zh ) = F (ρe )F (ρh )g (ze )g (zh ) is the eigenfunction of the Hamiltonian in Eq. (1) without the impurity potential term at the right. The z-dependent electron and hole non-correlated wave functions can be obtained as an expansion in a complete set of sine functions [13]. Furthermore, the values of the α and λ parameters are determined by minimizing the functional of energy E (α, λ) =

⟨Ψ |H |Ψ ⟩ . ⟨Ψ |Ψ ⟩

(8)

The exciton binding energy (Eb ) is obtained from the definition Eb = E0 − E ,

(9)

where E0 is the eigenvalue associated to Υ (ρe , ρh , ze , zh ). 3. Results and discussions In Fig. 2 we show the calculated exciton binding energy for a symmetric GaAs–Ga0.67 Al0.33 As TQD. Several geometries for the heterostructure have been analyzed. As a general behavior, the binding energy monotonically decreases as long as the dimensions of the structure (dot height, barrier width, and dot radius) are set to augment. This is because the increasing of the size of the heterostructure leads to a weakening of the interaction between the exciton wave function and the potential barriers. Consequently the expectation value of the electron–hole distance increases with the corresponding decrease of the Coulomb interaction. From Fig. 2(b) and (d) we can see that the binding energy is a

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b

c

d

Lb

Lb Lb HD H D HD HD

Fig. 2. (Color online) Exciton binding energy in a GaAs–Ga0.67 Al0.33 As TQD for F = 0 and P = 0. In (a) the results for this quantity are presented as a function of the dot height with Lb = 20 Å for three distinct values of R. In (b) the results are shown as a function of the barrier width with HD = 50 Å using three values of R. In (c) the results are depicted as a function of the radius with Lb = 20 Å for two different values of HD . Graphics (d) show the calculated exciton binding energy as a function of the dot height with R = 100 Å. In this case, three values of Lb were considered. The inset in Fig. (b) is a zoom of the minimum for the case of R = 100 Å. The inset in Fig. (c) shows the z-dependent confinement potential of the coupled symmetric TQD.

complicated function of the barrier width. Comparing Fig. 2(a) and (c), it can be noticed that variations of radius have stronger effects on the binding energy than variations of the dot height. This is due to the fact that modifying the value of the radius is equivalent to make changes affecting two spatial dimensions. Fig. 2(b) presents the calculated binding energy as a function of barrier width (Lb ). In this figure, it can be seen that for values of Lb < 5 Å the binding energy decreases. This means that these narrow barriers do not affect the electron and hole wavefunctions noticeably. Such a configuration just makes the system larger in dimensions with a lower confinement on the carriers. On the other hand, for values of Lb > 5 Å, the binding energy increases; which means that the barriers affect the electron and hole wavefunctions — they are confined within the central region by effects of the central potential barriers. Indeed, the minimum of each curve in Fig. 2(b) indicates the width for which the barrier begins to decouple the QD. The Eb is shown in Fig. 2(d) for small values of Lb such as 5 and 10 Å, and in the regime of large HD (such as 50 Å); the three QD are strongly coupled and the system behaves like a single QD of radius R and length 3HD — the dashed line in Fig. 2(d) corresponds to the binding energy obtained for an exciton in a cylindrical single QD of radius 100 Å and height 150 Å, i.e., Eb = 19.19 meV. When Lb goes larger (see the curve for Lb = 20 Å), the three dots are uncoupled and the exciton is essentially confined in a QD of radius R and height HD , with the consequent increasing in the binding energy. Our results for the exciton binding energy in a GaAs–Ga0.67 Al0.33 As TQD are presented in Fig. 3. It is displayed as a function of hydrostatic pressure for two fixed values of the applied electric field (Fig. 3(a)); and as a function of the applied electric field for two fixed values of the hydrostatic pressure (Fig. 3(b)). From Fig. 3(a) it can be seen that when F = 20 kV/cm the effect of the hydrostatic pressure is always to increase the binding energy. This behavior can be explained as follows: (i) the rise in the reduced electron–hole effective mass due to the influence of the hydrostatic pressure – see Eqs. (4)–(6) – reveals in a higher localization of the exciton wavefunctions into the QD regions with the consequent increasing of the binding energy and (ii) the decrease of the static dielectric constant for growing hydrostatic pressure – see Eq. (3) – leads to a reduction in the screening of the Coulomb interaction, and therefore to the augment of the binding energy. These two effects are additive and for this reason it is always found that Eb is an increasing function of the hydrostatic pressure. It is important to notice that because of the applied electric field, the electron and hole are pushed in opposite directions — the hole is pushed towards the QD locate at the left-hand side of the structure, whereas the electron essentially remains in the central QD, provided it has a small effective mass (see Fig. 3(c)). The lowering of the central potential barriers – for the influence of hydrostatic pressure – does not affect the degree of confinement on the hole. In the case of electrons the effects of the barrier lowering can be apparent only when the hydrostatic pressure is high enough — for example around 30 kbar (see Fig. 3(d)). On the other hand, for F = 0 and in the regime of P ≤ 5 kbar, the barrier potentials remain constant and the situation is similar to the case F = 20 kV/cm. If P > 5 kbar, the height

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c

d

Fig. 3. (Color online) Exciton binding energy in a GaAs–Ga0.67 Al0.33 As TQD as a function of hydrostatic pressure for F = 0 and F = 20 kV/cm (a) and as a function of the in-growth direction applied electric field for P = 0 and P = 20 kbar (b). The amplitude of probability of the z-dependent electron (solid lines) and hole (dashed lines) wavefunctions are depicted for F = 20 kV/cm with P = 0 (c) and P = 30 kbar (d). The dimensions of the TQD system considered in this case are HD = 50 Å, Lb = 20 Å, and R = 100 Å.

of the barriers starts to decrease and it reaches the zero value when P ≃ 30 kbar. In this case, the wave function confinement diminishes because the exciton wavefunction – which in the first regime of hydrostatic pressure is mainly located at the central QD – largely spreads towards the two external QDs. As a result, the binding energy decreases as it is shown in the second regime for the upper curve of Fig. 3(a). Here, the effect on the binding energy associated to the lowering of the potential barrier with P is dominant over the effects due to the diminishing of dielectric constant and the increasing of reduced electron–hole effective mass [14]. It should be noticed that for a large enough hydrostatic pressure the limit of the TQD is that of single QD with the same radius but with HD = 190 Å. From Fig. 3(b) it can be observed that, as a general trend, the electric field causes the diminishing of the binding energies. This is due to the field-induced polarization of the exciton pair. As a consequence, the strength of the electron–hole interaction becomes smaller, as was explained above in relation with the Fig. 3(a). Notice that for large enough electric field intensity, the limit of the expectation value of the electron–hole distance along the z-direction is ∼140 Å. For this reason, in the regime of large applied electric fields, the binding energy should reach a constant behavior. In Fig. 4 we show the calculations for the PL-peak energy transitions in a GaAs–Ga0.67 Al0.33 As TQD as a function of hydrostatic pressure for two values of the applied electric field (Fig. 4(a)). In addition, the same quantity is presented as a function of the applied electric field for two values of the hydrostatic pressure (Fig. 4(b)). The curves for the two applied electric fields in Fig. 4(a) are superimposed because of the strong confinement associated with the small dimensions of the heterostructure. The PL-peak energy grows almost linearly with the hydrostatic pressure with linear pressure coefficient of 9.7 meV/kbar, which is less than the corresponding one for the GaAs bulk pressure coefficient. This is in agreement with experimental findings by Venkateswaran et al. [15]. Here, due to the strong confinement regime, all the PL-peak energy transitions are related to spatially direct excitons. In Fig. 4(b), where the PL-peak energy transition is given as a function of applied electric field, it can be noticed that the curves present a rigid blue shift around 174 meV, when the hydrostatic pressure goes from zero to 20 kbar. This is due to the linear dependence of the GaAs energy band gap, and of the electron effective masses with the hydrostatic pressure. Since the applied electric field polarizes the exciton and tends to put the hole and electron in different dots, two well defined regimes are observed for both values of hydrostatic pressure: for electric fields up to ∼20 kV/cm the transitions are related to spatially direct excitons – see Fig. 4(c) – whereas for larger values of electric field the transitions are related to spatially indirect excitons — see Fig. 4(d). 4. Conclusions In this work, a theoretical study of the effects of hydrostatic pressure and an applied electric field on the exciton binding energies and PL-peak energy transitions in TQD was performed. The calculations

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b

c

d

Fig. 4. (Color online) PL-peak energy transitions in a GaAs–Ga0.67 Al0.33 As TQD as a function of hydrostatic pressure for F = 0 and F = 20 kV/cm (a) and as a function of the in-growth direction applied electric field for P = 0 and P = 20 kbar (b). The amplitude of probability of the z-dependent electron (solid lines) and hole (dashed lines) wavefunctions are depicted for P = 0 with F = 5 kV/cm (c) and F = 80 kV/cm (d). The dimensions of the TQD system are HD = 50 Å, Lb = 20 Å, and R = 100 Å. The dashed line in (a) corresponds to the hydrostatic pressure dependent GaAs bandgap.

were made in the effective mass approximation, using a variational procedure. Our main findings can be summarized as follows: (1) the effects of the hydrostatic pressure on the exciton binding energy are very low due to the strong confinement regime considered for the heterostructures, (2) the radial confinement is responsible for the more noticeable changes in the exciton binding energy and in the PL-peak energy transitions compared with the changes associated with the longitudinal confinement, (3) depending on the strength of the applied electric field, the lowest PL-peak energy transition can be tuned from the spatially direct to the spatially indirect exciton regime, and (4) the PL-peak energy transition has a linear blue-shift dependence with the hydrostatic pressure, and the corresponding pressure coefficient is smaller than the corresponding one in the bulk material. Acknowledgements This research was partially supported by Dirección de Investigación de la Escuela de Ingeniería de Antioquia (cc: 1540155), COLCIENCIAS, CODI-Universidad de Antioquia (Estrategia de Sostenibilidad Grupo de Materia Condensada-UdeA, 2009–2010), Facultad de Ciencias Exactas y NaturalesUniversidad de Antioquia (CAD-exclusive dedication project 2009–2010), and the Excellence Center for Novel Materials/COLCIENCIAS (Contract No. 043-2005). CAD is grateful to Dr. Anna Kurczyńska for useful suggestions and discussions. The authors are grateful to Dr. M.E. Mora-Ramos for the critical reading of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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