Vertical coupling effects and transition energies in multilayer InAs/GaAs quantum dots

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Vertical coupling eﬀects and transition energies in multilayer

InAs/GaAs quantum dots

Yiming Li

a,b,*

a_{Department of Computational Nanoelectronics, National Nano Device Laboratories, Hsinchu 300, Taiwan} b_{Microelectronics and Information Systems Research Center, National Chiao Tung University, Hsinchu 300, Taiwan}

Available online 19 June 2004

Abstract

We investigate the transition energy of vertically stacked semiconductor quantum dots with a complete three-dimensional (3D) model in an external magnetic field. In this study, the model formulation includes: (1) the position-dependent effective mass Hamiltonian in non-parabolic approximation for electrons, (2) the position-position-dependent effective mass Hamiltonian in parabolic approximation for holes, (3) the finite hard-wall confinement potential, and (4) the Ben Daniel–Duke boundary conditions. To solve the nonlinear problem, a nonlinear iterative method is imple-mented in our 3D nanostructure simulator. For multilayer small InAs/GaAs quantum dots, we find that the electron– hole transition energy is dominated by the number of stacked layers. The inter-distance d plays a crucial role in the tunable states of the quantum dots. Under zero magnetic field for a 10-layer QDs structure with d¼ 1:0 nm, there is about 30% variation in the electron ground state energy. Dependence of the magnetic field on the electron–hole transition energy is weakened when the number of stacked layers is increased. Our investigation is constructive in studying the magneto-optical phenomena and quantum optical structures.

Keywords: Computer simulations; Magnetic phenomena (cyclotron resonance, phase transitions, etc.); Quantum eﬀects; Tunneling; Indium arsenide; Gallium arsenide; Heterojunctions

1. Introduction

Nanoscale semiconductor quantum dots (QDs) have been of great interest and successive investi-gation from both the experimental and theoretical point of views in recent years [1–3]. These struc-tures have diverse applications to electronic and optoelectronic devices, such as semiconductor laser, light emitting diodes, single electron

transis-tors, and single photon emitters. We can currently use the advanced nanofabrication technology to consider another degree of freedom along the growth direction for vertically coupled multilayer QDs. One of evident features in this system is the effects of dot-to-dot interactions on the electronic structure, the electronic entanglement, and charge transfer [4–6]. By considering a 2D lateral geom-etry and 2D confinement potential models, most of previous works have focused on vertically coupled 2-layer QDs [4–8]. However, it is known that these nanostructures have a 3D confinement potential for the carriers and consequently have a discrete

*

Present address: P.O. Box 25-178, Hsinchu 300, Taiwan. Tel.: +886-930-330766; fax: +886-3-5726639.

E-mail address:[email protected](Y. Li).

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energy spectrum with ideally delta-like densities of states [9–11] and ultranarrow gain spectrum. Meanwhile, the geometry configurations signifi-cantly impact their electronic structure and lumi-nescence properties. Hence, nanoscale details of the QDs topology for energy spectra as well as the evolution of the coupling effect are critical for the development of novel photonic and electronic applications. To thoroughly clarify the electronic structure and tunneling ability, it is necessary for us to investigate the vertically coupled multilayer QDs with a full 3D approach.

We study in this paper the electronic structure of vertically coupled multilayer QDs using a uni-fied 3D model under applied magnetic fields. Our model considers: (1) the position-dependent effec-tive mass Hamiltonian in non-parabolic approxi-mation for electrons, (2) the position-dependent effective mass Hamiltonian in parabolic approxi-mation for holes, (3) the finite hard-wall confine-ment potential, and (4) the Ben Daniel–Duke boundary conditions [12–17]. The problem is solved numerically with a generalized nonlinear iterative method [15–17]. This iterative method enables us to compute the vertically coupled mul-tilayer QDs electronic structure efficiently [15]. The constructed system in our investigation possesses 10-layer InAs dots embedded into the GaAs ma-trix. Each ellipsoidal-shaped QDs layer is equally separated by certain inter-distance d. The effects of the number of stacked layers N and d on the electronic structure have explored. For small dots separated by a fixed d, the transition energy is essentially dominated by N . When N increases, the electron transition energy decreases monotonically and tends eventually to a saturated value. For the 10-layer QDs structure with d¼ 1 nm, the varia-tion of ground state energy is significant and it is less dependent on N for the first excited state. We find d plays a crucial role in the tunable states of the QDs. The dependence of magnetic fields (B) on the electron transition energy is weakened when N is increased. The results presented in this work show the importance of the 3D modeling and simulation in exploring the nanoscale vertically coupled multilayer semiconductor QDs. It is con-structive in the magneto-optical studies and the development of quantum optical devices.

The outline of this paper is as follows. Section 2 states the 3D model for the vertically coupled multilayer QDs structure and the simulation methodology. Section 3 is the results and discus-sion. Section 4 draws the conclusions.

2. 3D quantum dot model and simulation method We consider the vertically stacked QDs with the hard-wall confinement potential [12–17]. In a magnetic field the effective mass Hamiltonian for electrons (k¼ e) and for holes (k ¼ h) is given in the form b Hk¼ Pr 1 2mkðE; rÞ Prþ VkðrÞ þ1 2gkðE; rÞlBBr; ð1Þ

where Pr¼ ih»rþ eAðrÞ stands for the electron

momentum vector, »ris the spatial gradient, AðrÞ

is the vector potential (B¼ curl A), and r is the vector of the Pauli matrices. For electrons, meðE; rÞ

and geðE; rÞ are the energy- and

position-depen-dent eﬀective mass and the Lande factor, respec-tively 1 meðE; rÞ ¼2P 2 3h2 2 Eþ EgðrÞ VeðrÞ þ 1 Eþ EgðrÞ VeðrÞ þ DðrÞ ; ð2Þ geðE; rÞ ¼ 2 1 m0 meðE; rÞ DðrÞ 3½E þ EgðrÞ þ 2DðrÞ ; ð3Þ where VeðrÞ is the conﬁnement potential, EgðrÞ and

DðrÞ stand for position-dependent energy band gap and spin–orbit splitting in the valence band, P is the momentum matrix element, m0 and e are the

free electron mass and charge. For holes, mhðrÞ

and ghðrÞ are assumed to be only position

depen-dent. For both the electron and hole, the hard-wall conﬁnement potential in the inner region of each QD (1) and environmental crystal matrix (2) can be presented as: VkðrÞ ¼ 0 for all r 2 1 and

VkðrÞ ¼ Vk0 for all r2 2, respectively. The Ben

Daniel–Duke boundary conditions for the elec-tron and hole wave functions WðrÞ are given by

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Wk1ðrsÞ ¼ Wk2ðrsÞ and ð h

2

2mkðE;rsÞ»rÞjnWkðrsÞ ¼ const:,

where rs denotes the position of the system

inter-face.

Because of the cylindrical symmetry of the system, the wave function for electrons and holes can be represented as WkðrÞ ¼ UkðR; zÞ expðil/Þ

where l¼ 0; 1; 2; . . ., is the orbital quantum number. It leads to a problem in the ðR; zÞ coor-dinate, and the Schr€odinger equation for electrons (k¼ e) and holes (k ¼ h) is h 2 2mkjðEÞ o2 oz2 þ o 2 oR2þ 1 R o oR l2 R2 UkjðR; zÞ þ mkjðEÞX 2 kjðEÞR 2 8 " þ slB 2 gkjðEÞB þ hXkjðEÞ 2 l þ Vk0dj2 # UkjðR; zÞ ¼ EUkjðR; zÞ; ð4Þ

where j¼ 1; 2, XkjðEÞ ¼ eB=mkjðEÞ, and s ¼ 1

refers to the orientation of the electron spin along z-axis. The Ben Daniel–Duke boundary conditions are Uk1ðR; zÞ ¼ Uk2ðR; zÞ and 1 mk1ðEÞ oUk1ðR; zÞ oR þdfðRÞ dR oUk1ðR; zÞ oz z¼f ðRÞ ¼ 1 mk2ðEÞ oUk2ðR; zÞ oR þdfðRÞ dR oUk2ðR; zÞ oz z¼f ðRÞ ð5Þ where z¼ f ðRÞ presents the generating contour of the vertically coupled multilayer QDs on thefR; zg plane.

We focus here on principal consequences of a 3D approach in modeling of electron states for vertically stacked QDs. We note that any ad-vanced consideration of geometrical effects will produce further corrections to the electron and hole energy spectra. For example, the different dielectric constants (14.5/12.9) in InAs/GaAs het-erostructures may leads to a small renormalization of the electron energy (self-energy) [12,13,18,19]. However, it should not affect the main tendencies and behaviors of the state in the structure.

To compute the electron–hole energy states in a vertically stacked ellipsoidal-shaped QDs struc-ture, shown in Fig. 1, we apply the nonlinear iterative method to calculate the self-consistent

solution of the multilayer QDs structure. This computer simulation method has been developed for QDs and quantum ring simulation in our re-cent works [15–17]. Our calculation experiences show this method converges monotonically; it takes only 12–15 feedback iterative loops to meet a given convergence criterion (the maximum norm error <1012 _{eV) for all energies.}

3. Results and discussion

As shown in Fig. 1, we simulate a 10-layer vertically stacked QDs structure. All InAs dots are with the same ellipsoidal shapes and embedded into the GaAs matrix. Each layer of the ellipsoi-dal-shaped QD is separated by d equally. The base radius R0¼ 10 nm and z0¼ 2 nm are adopted for

all QDs [20–25]. The material parameters used in our investigation for InAs inside each dot are E1g¼ 0:42 eV, D1¼ 0:38 eV, and m1ð0Þ ¼ 0:024m0.

The parameters for GaAs outside of the dots are E2g¼ 1:52 eV, D2¼ 0:34 eV, m2ðV0Þ ¼ 0:067m0,

and V0¼ 0:53 eV [26]. Fig. 2 shows the calculated

transition energy versus N for the structure with d ¼ 1 nm and B ¼ 0 T. For d ¼ 1 nm the variation of ground state energy can up to 30%. The tran-sition of the ﬁrst excited state energy (jlj ¼ 1) is less dependent on N ( 14%). To clarify the dependence of N on the electron transition energy, we deﬁne the occupancy-ratio of electron wave functions W ¼R_{r2material 1}U2ðR; zÞdr3₌R

r2material 2U 2

ðR; zÞ dr3 _{for evaluating the probability in ﬁnding}

Fig. 1. Cross-sectional plot of the structure consisting of N¼ 10 embedded layers of InAs QDs, separated by a certain inter-distance in GaAs matrix. All InAs QDs are with the 3D ellipsoidal shapes as shown in the right ﬁgure.

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the electron inside and outside the QDs. The dash lines indicate the corresponding dependence of the

ratio W on N , which becomes independent on N when N > 6. As shown in Fig. 3, plots of the cal-culated wave functions significantly confirm the coupling and spreading effects for the vertically coupled QDs with N¼ 1; 2; 5, and 10. Due to the coupling effect, it is found that the increase of N results in stronger confinement and the energy converges to a stable value shown in Fig. 2. Under B¼ 0 T, Table 1 reports the variation of the electron transition energies (ðEeN¼1 EeN¼10Þ=

E_eN¼1) for the states of l¼ 0 and jlj ¼ 1 in the stacked structure with d¼ 0:5; 1; 2; 5, and 10 nm. In our investigation, it is also found that d plays a crucial role in the tunable states of the dots. For the system with N ¼ 10 under B ¼ 0 T, we have also estimated that there is more than 25% ground state energy variation when d varies from 0.5 to 10 nm. For the ﬁrst excited state the energy vari-ation is up to 12%. We have similar observvari-ations for diﬀerent number of layers.

Furthermore, for the structure with d¼ 1 nm the electron transition energy among the states is calculated systematically as a function of magnetic fields with arbitrary strength. As shown in Fig. 4, we observed that the dependence of B on the electron transition energy for the case of l¼ 0 is weakened when N increases. The magnetic fields play significant effects on the vertically coupled 2-layer QDs than that on other stacked configura-tions. For the system consists of only 2-layer QDs, it demonstrates 30 meV energy transition when B varies from 0 to 20 T. For the 10-layer QDs structure the transition is about 5 meV. Fig. 5 shows the ratio W versus N and B for the structure with d¼ 1 nm. For those structures with the small N the electron wave function spreads out of the QDs (W 1) and the property of the transition energy is dominated by the band parameters of GaAs matrix. With the same calculation method we have computed the hole energy states, where

Fig. 3. Contour plots of the electron wave functions for the vertically stacked QDs with N¼ 1; 2; 5, and 10.

Fig. 2. Electron ground state energy versus N for the structure with d¼ 1 nm and B ¼ 0 T. The dash lines indicate the cor-responding dependence of the ratio W on N .

Table 1

Variation of the electron transition energies for the cases of l¼ 0 and jlj ¼ 1 in a multilayer QDs structure with diﬀerent inter-distance under B¼ 0 T

Inter-distance d (nm) 0.5 1 2 5 10

Variation of the ground state energyðl ¼ 0Þ 36% 30% 26% 18% 13% Variation of the ﬁrst excited state energyðjlj ¼ 1Þ 17% 14% 10% 7% 5%

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the hole eﬀective mass was taken as m1h¼ 0:4m0

and m2h¼ 0:5m0, respectively; the hole band oﬀset

is taken as Vh0¼ 0:33 eV [12,13]. The system

energy band gap DEðBÞ is equal to DEgeðBÞ þ

DEghðBÞ þ DEgRðBÞ, where Ege and Egh are the

electron–hole ground state energies, and EgR is the

energy gap in the QDs system. The calculated re-sult of the energy gap between the lowest electron and hole states for the stacked 10-layer InAs/GaAs QDs system with the fixed d¼ 1 nm is shown in Fig. 6. For a fixed B we find the energy gap decreases when N increases. The decrease of en-ergy gap flats off when N > 6 due to the stable localization effect (weak coupling effect). For N¼ 1 the coupled system becomes a single InAs/GaAs

QD and the energy gap variation is mainly from diamagnetic shift [16]. For the system with each N >2 the shift of energy gap is weakened. It is a result of the strong conﬁnement eﬀect for elec-trons and holes with our correct 3D description. This phenomenon is useful in magnetic-photolu-minescence spectra of the nanoscale vertically stacked QDs. However, more detail estimations will be necessary by including the electron–hole Coulomb interaction for those structures with small N and d.

4. Conclusions

In this paper the coupling effects and transition energies have been investigated for the vertically stacked InAs/GaAs QDs under magnetic fields. For small QDs, our 3D model and simulation has shown that the transition energy was directly dominated by the number of stacked layers. The inter-distance between layers also plays a crucial role in the tunable states of the dots. The depen-dence of magnetic fields on the electron transition energy was depressed when the number of verti-cally coupled layers is increased. To verify the calculated transition energies of the stacked QDs quantitatively, transition energies can be measured experimentally from the photoluminescence spec-tra [27–32]. It is believed that this study is useful for the study of quantum optical structures and

Fig. 5. The structure ratio W versus N and B, where d¼ 1 nm.

Fig. 6. The energy gap versus N and B for the stacked InAs/ GaAs QDs with d¼ 1 nm.

Fig. 4. Electron ground state energy versus N and B for the structure with d¼ 1 nm.

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design of advanced semiconductor photonic and electronic devices.

Acknowledgements

This work was supported in part by the Na-tional Science Council (NSC) of Taiwan under contract nos. 92-2112-M-429-001 and NSC-93-2572-E-009-002-PAE and the Ministry of Eco-nomic Aﬀairs, Taiwan under contract no. PSOC 92-EC-17-A-07-S1-0011.

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