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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.
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EFFECTS OF COMPOSITE INGAAS . . .
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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.
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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.
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