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CHINESE JOURNAL OF PHYSICS VOL. 46 , NO. 3 JUNE 2008 Effects of Composite InGaAs and InAlAs Layers on the Emission Wavelengths of Quantum Dots R. B. Chiou and David M. T. Kuo∗ Department of Electrical Engineering, National Central University, Jhongli, 320 Taiwan, Republic of China (Received July 10, 2007) Ground-state emission wavelengths of pyramidal InAs quantum dots (QDs) with a base length of b and a height of h capped by composite InGaAs and InAlAs layers were investigated theoretically in a one-band effective mass model framework. It was found that the ground-state emission wavelengths depend not only on the h/b ratio, but also on the thickness and position of the InGaAs and InAlAs layers. The maximum transition energy separation between the QD ground- and first-excited states appears in the composite InGaAs and InAlAs layers, but not on a single InGaAs or InAlAs overgrown layer. Such a result is attributed to the degree of delocalization for the first excited state being different from that of the ground state. PACS numbers: 73.21.La, 73.63.Kv, 73.20.At I. INTRODUCTION Recently, self-assembled quantum dots (SAQDs) grown by molecular beam epitaxy have attracted a great deal of scientific interest. The large lattice mismatch between InAs and GaAs causes the highly strained epitaxial layer of InAs on GaAs to form coherent three-dimensional islands after completion of the wetting layer [1]. These InAs islands typically take the shape of pyramids [2], disks [3], and lenses [4]. The mechanism of InAs nucleation has been investigated by Tersoff et al. [5] Many useful applications of quantum dot (QD) devices have been discussed [6, 7]. The quantum dot laser is a promising application, providing low threshold current densities and ultrahigh temperature stability at room tempearure [8, 9]. Nevertheless, the emission wavelength of InAs QDs embedded in a GaAs matrix shows a photoluminescence (PL) peak around 1 µm [10]. To obtain a wavelength of 1.3 µm for application in an optical fiber communication system, InGaAs strain-reducing layers (SRL) were used to modulate the wavelength emission of InAs QDs [11]. Although an InGaAs SRL can be used to extend the emission wavelength of InAs/GaAs QDs, an InGaAs SRL decreases the confining potential barrier and narrows the separation between the discrete energy levels of InAs QDs. Several groups proposed using a higher band-gap InAlAs-InGaAs composite layer to overcome problems resulting from the introduction of an InGaAs SRL [12–14]. Wei et al. capped the InAs QDs with various thicknesses of InAlAs, followed by InGaAs overgrowth, and demonstrated the largest energy transition separation of 108 meV [12]. Recently, Liu and Chyi experimentally inves- http://PSROC.phys.ntu.edu.tw/cjp 348 c 2008 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA VOL. 46 R. B. CHIOU AND DAVID M. T. KUO 349 FIG. 1: The schematic diagram for pyramidal InAs QDs grown on a GaAs substrate capped by composite InGaAs and InAlAs strain reducing layers. tigated the effects of combined InAlAs and InGaAs overgrown layers on InAs QDs emission wavelengths [13]. Few theoretical studies to date have considered the effects of an SRL on the optical properties of QDs [15]. This article investigates theoretically the transition energy separation between the ground- and first-excited states and the ground-state emission wavelength of InAs QDs capped by composite InGaAs and InAlAs SRLs. The schematic diagram for the system studied is shown in Figure 1. II. FORMALISM Although InAs QDs are typically shaped as pyramids, disks, and lenses, this work only considers the pyramidal InAs QDs. Several groups have studied theoretically the electronic structure of InAs QDs using different methods [15–18]. For simplicity, the electronic structure of the InAs QDs is calculated using the one-band effective mass approximation, which is adequate for studying the conduction band electrons, but is not adequate for the valence-band holes. A multiband effective mass model is required for the valence-band holes to reveal valence-band mixing [19], which is ignored in this study. The Hamiltonian for an electron (hole) is described by the equation (−∇ h̄2 2m∗e(h) (r) ∇ + V e(h) (r))ψe(h) (r) = Ee(h) ψ(r)e(h) , (1) where V (r) is the confining potential and m∗e(h) (r) is the position-dependent electron (hole) effective mass. The values of V (r) and m∗e(h) (r) depend on the underlying material, as sketched in Fig. 1. Equation (1) means that the particle interactions U are ignored due to U/h̄ω << 1, where h̄ω is the ground-state emission photon energy. This investigation only considers the heavy-hole band (with Jz = ±3/2), and ignores its coupling with a lighthole band caused by the QD potential due to the large strain-induced splitting between the heavy-hole and light-hole bands for typical SAQDs. The current work places the system in a large confining cubic box with a length of R for the purpose of constructing the approximate 350 EFFECTS OF COMPOSITE INGAAS . . . VOL. 46 wave functions and adopts R=40 nm. The wave functions are expanded in a set of basis functions, chosen as sine waves √ 8 R R R ψn,l,m(r) = √ sin kl (x + ) sin km (y + ) sin kn (z + ) , (2) 2 2 2 L3 where kn = nπ/R, km = mπ/R, kl = `π/R, and n, m, ` are positive integers. Matrix elements of the Hamiltonian of Equation (1) can be readily obtained. In our calculation, states with n ≤ 20, m ≤ 10, and ` ≤ 10 are used in solving Equation (1). Table 1 gives the calculation parameters. Meanwhile, we adopt the InAs energy gap as Eg=0.72 eV [17]. TABLE I me ∗ mh ∗ InAs GaAs InGaAs InAlAs 0.04 0.34 0.067 0.35 0.0627 0.3483 0.1324 0.466 ∆Ec (eV) – 0.5 0.42 0.655 ∆Ev (eV) – 0.3 0.252 0.622 III. RESULTS AND DISCUSSION This work uses Equations (1) and (2) as a basis, and calculates the ground-stateemission wavelength of InAs QDs with a height h=6 nm for four different composite SRL configurations and a fixed thickness d=6 nm. Figure 2 shows the emission wavelengths as a function of QD size for four different composite strain reducing layers: the solid line (GaAs/GaAs), the dashed line (InGaAs/InAlAs), the dotted line (InAlAs/InGaAs), and the dot-dashed line (InGaAs/InGaAs). The notation InGaAs/InAlAs means that InGaAs and InAlAs denote the upper layer and lower layer, respectively. This convention is used throughout this article. The upper layer thickness is notably adopted as dU =3 nm in the dashed and dotted lines. We observe that the emission wavelengths are proportional to the QD size. In addition, the dashed, dotted, and dot-dashed lines exhibit redshifts with respect to the solid line. The redshift magnitude depends on the QD size. Moreover, the dashed and dotted lines exhibit blueshifts with respect to the dot-dashed line, due to the stronger confinement potential of the InAlAs layer. Such blueshifts are stronger for InAlAs located at the lower layer than at the upper layer. However position-dependent influences become very weak in the h/b < 0.5 range. This feature can be understood by particle wave function analysis. The Geometric effect localizes the wave functions at the center and bottom of the QDs. Consequently, the wave functions are not sensitive to the InAlAs SRLs locations with decreased h/b ratio. The emission wavelengths near 1.3 µm are provided in the h/b=0.3 to h/b=0.2 regime, from Figure 2. This work therefore considers InAs QDs with a height of h=6 nm and a base length of b=24 nm for studying the composite InAlAs/InGaAs SRL thickness effects VOL. 46 R. B. CHIOU AND DAVID M. T. KUO 351 FIG. 2: Emission wavelengths of InAs QDs as a function of QD size at a QD height h=6 nm for four different composite strain reducing layers. on the energy level splitting between the ground and first excited states. Particle Coulomb interactions are strong in small quantum dots with a high potential barrier. However, large quantum dots are considered in our case due to the requirement of a 1.3 µm emission wavelength. Therefore, the particle Coulomb interactions are negligible [6]. We show the lowest two energy levels of the InAs QDs as a function of the thickness of the InAlAs layer (dU =dInAlAs) in Figure 3; the solid line denotes the ground state (E0e(h) ), and the dashed line denotes the first excited state (E1e(h) ). Diagrams (a) and (b) exhibit, respectively, the energy levels for electrons and holes. We observe that both energy levels are insensitive to dInAlAs variation when dInAlAs is less than a threshold value of R0 , which is around 2 nm (1.5 nm) for electrons (holes). This result is because the wave functions of the ground state and first excited states are localized at the bottom of the QDs. However, as dInAlAs increases beyond R0 , the energy level variation for the first excited state is more serious than that of the ground state. This variation indicates that the degree of delocalization for the first excited state is different from that of the ground state, even though their wave functions are distributed at the bottom of the QDs. Based on Figure 3, we plot the transition energy separation between the ground and first excited states ∆E(Al) = ∆Ee + ∆Eh , where we define the energy level separation, ∆Ee(h) = |E1e(h) − E0e(h) |, as a function of dInAlAs in Figure 4. The transition energy separation ∆E(Al) is mainly attributed to the electron energy level separation ∆Ee due to the smaller electron effective mass. ∆E(Al) reaches a maximum value (around 111.5 meV) at dInAlAs =6 nm, as expected. The minimum value of ∆E(Al) occurs around dInAlAs =1.8 nm, not at dInAlAs =0 nm, due to the delocalization degree difference between the ground state and the first excited state. 352 EFFECTS OF COMPOSITE INGAAS . . . VOL. 46 FIG. 3: Energy levels of InAs QDs as a function of dInAlAs . Diagrams (a) and (b) denote, respectively, the energy levels for electrons and holes. Despite a large ∆E can create QD laser high temperature stability. However, emission wavelengths also shift toward shorter wavelengths, when InAs QDs are capped by a 6 nm InAlAs overgrown layer (dInAlAs =6 nm). The current work next considers InAs QDs with composite InGaAs/InAlAs SRLs, in contrast to the Figure 3 case. The transition energy separation ∆E(Ga) as a function of InGaAs (dInGaAs ) thickness is shown in Figure 5. The maximum of ∆E(Ga) appears at 113 meV for dInGaAs around 1.8 nm, larger than the maximum value of ∆E(Al) shown in Figure 4. ∆E(Ga) is clearly a nonlinear function of dInGaAs . Such an interesting phenomenon can be experimentally investigated. Plotting the ground state and first excited state emission wavelengths as a function of dInGaAs in Figure 6 gives further understanding of the emission wavelength variation of InAs QDs with respect to dInGaAs . We particularly focus on the emission wavelengths at dInGaAs =3 nm. The VOL. 46 R. B. CHIOU AND DAVID M. T. KUO 353 FIG. 4: Transition energy separation ∆E(Al) and electron energy level separation ∆Ee as a function of dInAlAs . FIG. 5: Transition energy separation ∆E(Ga) and electron energy level separation ∆Ee as a function of dInGaAs . Es and ∆E(Ga) are, respectively, 1297 nm and 108 meV. Although such a result achieves a good agreement with the experimental measurement of Ref. [13], where Es =1292 nm and ∆E(Ga)=100 meV, a strain effect has been ignored in the above analysis. Therefore, further investigation is needed for designing a more efficient structure by controlling the SRL composition and thickness. 354 EFFECTS OF COMPOSITE INGAAS . . . VOL. 46 FIG. 6: Emission wavelengths for the ground state and first excited state as a function of dInGaAs . IV. CONCLUSION We theoretically calculated the emission wavelengths of pyramidal InAs QDs capped by composite InGaAs/InAlAs layers in a one-band effective mass model framework and investigated the effects of composite InAlAs/InGaAs and InGaAs/InAlAs SRLs on the emission wavelengths. Our findings show that the height and base length ratio significantly influences the emission wavelengths of the InAs QDs capped with composite layers. Furthermore, InAs QDs overgrown with combined InGaAs and InAlAs layers are much better than QDs capped by only a InGaAs or InAlAs layer for obtaining longer wavelengths of QDs near 1.3 µm and a larger transition energy separation between the ground and first excited states. Acknowledgments This work was supported by National Science Council of the Republic of China under Contract No. NSC 96-2221-E-008-108. References Electronic address: [email protected] [1] J. Y. Marzin, J. M. Gerard, A. Izrael, D. Barrier, and G. Bastard, Phys. Rev. Lett. 73, 716 (1994). [2] M. Grundman, O. Stier, and D. Bimberg, Phys. Rev. B 52, 11969 (1995). [3] F. M. Peeter and A. Schweigert, Phys. Rev. B 53, 1460 (1996). [4] H. Drexel, D. Leonard, W. Hansen, J. P. Kotthaus, and P. M. Petroff, Phys. Rev. Lett. 73, 2252 (1994). ∗ VOL. 46 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] R. B. CHIOU AND DAVID M. T. KUO 355 J. Tersoff, C. Teichert, and M. G. Lagally, Phys. Rev. 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