analysis of wetting layer effect on electronic structures of truncated-pyramid quantum dots
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8/4/2019 Analysis of Wetting Layer Effect on Electronic Structures of Truncated-pyramid Quantum Dots
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Opt Quant ElectronDOI 10.1007/s11082-011-9460-0
Analysis of wetting layer effect on electronic structures
of truncated-pyramid quantum dots
Qiuji Zhao · Ting Mei · Daohua Zhang ·
Oka Kurniawan
Received: 27 September 2010 / Accepted: 16 March 2011© Springer Science+Business Media, LLC. 2011
Abstract We carry out a theoretical analysis of wetting layer effect on band-edge profiles
and electronic structures of InAs/GaAs truncated-pyramid quantum dots, including the strain
effect. A combinationof ananalytical strainmodelandaneight-band Fourier transform-based
k · p method is adopted in the calculation. Strain modified band-edge profiles indicates that
wetting layer widens the potential well inside the dot region. Wetting layer changes ground-
state energy significantly whereas modifies probability density function only a little. The
main acting region of wetting layer is just underneath the base of the dot. Wetting layerredistributes probability density functions of the lowest electron state and probability den-
sity functions of highest hole state differently because of the different action of quantum
confinement on electrons and holes.
Keywords Electronic structures · Fourier-transform based k · p method ·
InAs/GaAs quantum dots · Wetting layer effect
1 Introduction
Self-assembled quantum dots in the application of optoelectronics are usually fabricated
by the Stranski-Krastanov (SK) epitaxial growth model (Bimberg et al. 1999). Taking the
InAs/GaAs structure shown in Fig. 1 as an example, about a 1-2ML InAs wetting layer (WL)
Q. Zhao · D. Zhang
School of Electrical and Electronic Engineering, Nanyang Technological University,
50 Nanyang Avenue, Singapore 639798, Singapore
T. Mei (B)Institute of Optoelectronic Material and Technology, South China Normal University,
51063 Guangzhou, People’s Republic of China
e-mail: ting.mei@ieee.org
O. Kurniawan
Institute of High Performance Computing, A*STAR,
Singapore 117528, Singapore
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8/4/2019 Analysis of Wetting Layer Effect on Electronic Structures of Truncated-pyramid Quantum Dots
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Q. Zhao et al.
Fig. 1 Geometric model of a
truncated-pyramid InAs/GaAs
quantum-dot structure. The
bottom dimension, the top
dimension, and the height of dot
are 12nm, 12 f nm, and
6(1– f ) nm, respectively. The
thickness of WL is 0.5nm. f is
the truncation factor
x[100]
y[010]
z[001]
GaAs
InAs
is grown on a substrate, followed by a spontaneous coherent InAs island formation process.Finally, a deposition of an additional GaAs material, which is usually the same material as
the substrate, is adopted to cover the quantum-dot island. Deposition of the thin WL leads to
misfit strain at the interface of WL and substrate, and the associated strain energy increases
rapidly as the WL thickens. As a critical thickness of WL is created, formation of the dot
island can occur to relieve the strain energy (Matthews 1975). Therefore, there must be influ-
ence of WL on strain distribution and electronic structures of quantum dots. However, WL
is usually omitted or seldom discussed throughly by considering the very thin thickness of
the WL in the past works (Pryor 1998; Andreev et al. 1999; Stier et al. 1999; Schlliwa et
al. 2007). Therefore, it is necessary to examine the effect of WL completely, especially for
those structures with high aspect ratios. WL effect on electronic structures of a very shallowInAs/GaAs lens-shape quantum dot has been investigated using the atomic strain model and
the empirical tight-binding model that are much more complicated than the continuum strain
model and the k · p model (Lee et al. 2004). Consequently, we adopt an eight-band Fou-
rier transform-based k · p method (FTM) (Mei 2007; Zhao and Mei 2011) together with an
analytical continuum strain model (Andreev et al. 1999) to study effect of WL on electronic
structures of a series of truncated-pyramid InAs/GaAs quantum dots in this article. We first
investigate effect of WL on ground-state energies of quantum dots with different heights,
and then we select half-truncated structures to find the detail information of WL effect.
2 Computation method
2.1 Strain and band-edge profiles
For the strain calculation, we utilize the analytical method developed with Fourier-transform
technique (Andreev et al. 1999). To see the detail position of conduction band (CB), HH
band, LH band, and SO band after strain modification, we investigate the band-edge profiles
with following expressions (Chuang 1995),
E c = E g + aceh E H H = −aveh − beb
E L H = −aveh + beb E S O = −aveh − , (1)
where ac, av, b, and are the CB deformation potential, hydrostatic valence band defor-
mation potential, shear deformation potential and spin-orbit splitting energy, respectively. eh
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8/4/2019 Analysis of Wetting Layer Effect on Electronic Structures of Truncated-pyramid Quantum Dots
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Analysis of wetting layer effect on electronic structures of truncated-pyramid quantum dots
and eb are the hydrostatic strain and biaxial strain expressed as,
eh = e x x + e yy + e zz
eb = e zz −
e x x + e yy/2. (2)
2.2 Electronic structures
Electronic structures of quantum dots are usually obtained by solving equation including
envelop function F(r) and position-dependent Hamiltonian matrix H(r;k) with differential
operators,
H
r; k̂ x , k̂ y, k̂ z
F(r ) = E F(r) , (3)
where E is the energy. The envelop function F(r) of a QD superlattice can be expanded in
plane waves along x, y, and z,
F(r) =1
d x d y d zei k x x +ik y y+ik z z
n x
n y
n z
cn x n y n z ei(n xκ x x+n yκ y y+n zκ z z), (4)
where d x , d y , and d z are periods along x, y, and z directions, κα = 2πd α, −π
a< k α < π
a
for α = x, y, z, cn x n y n z=
c1 c2 . . . c8
T
n x n y n z, and a is the lattice constant. Envelop func-
tion F(r) can be approximated using Fourier series and complex exponential components,
and the Fourier transform of the spatially-varying Hamiltonian H(r;k) can be obtained by
entering analytical Fourier transform of quantum dot shape functions into the Hamiltonian
matrix H(r;k) of k · p theory. Therefore, the matrix of Fourier domain Hamiltonian M for the
eigen-problem can be formulated by a bra-ket operation to (3) utilizing the orthogonal-
ity of complex exponential function in F(r), and the new eigen-problem can be written as
(Mei 2007; Zhao and Mei 2011),
[Mst ][ct ] = E [ct ], (5)
where M is constructed with Fourier series of H(r;k) as follows,
Mst =1
2
x, y, zα,β
˜ H (αβ)uv, qi = mi −ni
K K +1
2
x, y, zα
˜ H (α)uv, qi = mi −ni
K
+ ˜ H (0)uv, qi = mi −ni
, (6)
and,
K K =2k αk β + k α(mβ + nβ)κβ + k β(mα + nα)κα + (mαnβ + mβnα)κακβ
K = 2k α + (mα + nα)κα. (7)
The dimension of M is only determined by the truncated order of Fourier frequencies
of H(r;k), i.e. V (2 N x + 1)(2 N y + 1)(2 N z + 1) × V (2 N x + 1)(2 N y + 1)(2 N z + 1), V is
the number of total bands involved in the calculation, and N α(α = x, y, z) is the truncated
Fourier order. Compared the calculationusing the tight-binding model, calculationwith FTM
is much easier and more convenient. The drawback of spurious solutions in k · p theory can
also be avoided by truncating the higher order of Fourier series (Zhao and Mei 2011).
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Q. Zhao et al.
3 Results and discussion
In this section, we first discuss the influence of WL on ground-state energies of the struc-
ture shown in Fig. 1 with different truncation factors ( f = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,
0.8, 0.9). Provided the same material, critical thickness of WL is not affected by the height of the geometry. Therefore, the WL with a thickness of 0.5nm was chosen for all the structures.
To make a clear illustration, we also calculate ground-state energies of the dot (structure
without WL) with same truncation factors. Figure 2 shows ground-state energies of electron
(E1) and heavy-hole (HH1) of dots with (solid lines) and without (dashed lines) WL under
different truncation factors. It can be seen that the difference between the case with WL and
the case without WL is obvious only at higher truncation ( f ≥ 0.5), especially for HH1due
to the heavier effective mass of heavy-hole. That is to say, significant WL effect only appears
when the thickness of WL is comparable with the height of the dot.
To further investigate the detail information about effect of WL, we calculate ground-state
energy of the half-truncated dot ( f = 0.5) with a cube just underneath its bottom (denoted
as QDC). The thickness of cube is also 0.5nm. Ground-state energy E1 and HH1 of QDC
are tabulated in Table 1 with those of half-truncated structure shown in Fig. 1 (denoted as
QDW) and half-truncated dot (denoted as QD). Figure 3 plots the strain components e x x and
e zz of QD (dashed lines) and QDW (solid lines) along the [001] direction. Corresponding
band-edge profiles of QD (dashed lines) and QDW (solid lines) along the [001] direction after
strain modification are also shown in Fig. 4. Since WL just causes the change of the initial
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
HH1
G r o u n d - s t a t e e n e r g y E 1 a n d H H 1 ( e V )
Truncation factor f
E1
Solid lines: with WL
Dashed lines: without WL
Fig. 2 Ground-state energies E1 and HH1 of a series of truncated-pyramid InAs/GaAs dots with (solid lines)
and without (dashed lines) inclusion of WL
Table 1 Ground-state energy E1
and HH1 of QD, QDC, and QDW Ground-state energy Structures in calculation
QD QDC QDW
E1 (eV) 0.5472 0.5099 0.4972
HH1 (eV) –0.6437 –0.6308 –0.6223
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Analysis of wetting layer effect on electronic structures of truncated-pyramid quantum dots
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-0.04
-0.02
0.00
0.02
0.04
-0.06
-0.04
-0.02
0.00
0.02
0.04
ezz
S t r a i n a l o n g [ 0
0 1 ] d i r e c t i o n
exx
Distance(nm)
Fig. 3 Strain components e x x and e zz of QD (dashed lines) and QDW (solid lines) along the [001] direction
-6 -4 -2 0 2 4 6-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Distance (nm)
B a n d - e d g e p r o f i l e s a l o n g [ 0
0 1 ] (
e V )
CB
HH
LH
SO
Fig. 4 Band-edge profiles of QD (dashed lines) and QDW (solid lines) along [001] direction
lattice mismatch at the interface of dot and WL, WL just redistributes the strain profile in the
WL region other than changes strain magnitude inside the dot area so much. Consequently,
WL widens potential wells but keeps positions of band edges unchanged almost inside the
dot region. As a result, ground-state energy of QDW in Table 1 varies a lot from that of QD,especially for ground-state energy of electron owing to the smaller electron mass.
Additionally, it canbe seen that discrepancyof ground-stateenergy betweenQDandQDW
is much more than that between QDW and QDC. This indicates that the main influence of
WL on ground-state energy comes from the area just beneath the bottom of the dot. Figure 5a
describes probability density functions (PDFs) corresponding to E1 of QD and QDW. After
the hybrid 2D/0D WL confinement (Cornet 2005), the center of the PDF of QDW shifts
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Fig. 5 PDFs corresponding to ground-state energy E1 of QD and QDW (Fig. 5a) and PDFs corresponding
to ground-state energy HH1 of QD and QDW. The black lines make up the shape section of QD and QDW
towards the WL region and a very small portion of PDF in the barrier region moves into the
region just beneath the bottom of the dot but not the whole WL region. Consequently, effect
of WL to electronic structures mainly comes from the region just beneath the bottom of the
dot.
PDFs corresponding to HH1 of QD and QDW are shown in Fig. 5b. Similar as that in
the Fig. 5a, the center of the PDF of QDW also moves towards the WL region. The most
significant difference in the PDFs of E1 and HH1 is that that heavy-hole are confined more
obvious than electron, this can be inferred from the smaller PDF volume and bigger PDF
value on the colorbar. As a result, very small portion of PDF of HH1 lies in the barrier regionand hence the PDF portion that locates in the WL is mainly from the dot region.
4 Conclusions
In summary, we have adopted a continuum strain model and eight-band FTM to investigate
the effect of WL on electronic structures of truncated-pyramid InAs/GaAs quantum dots.
Compared with the tight-bind method, FTM with a simple and neat Hamiltonian matrix sim-
plifies the computation process, and reduces the computation time as well. FTM also canavoid spurious solutions in the k · p theory handily.
WL effect becomes obvious only as the thickness of WL is comparable with the height of
the dot. Confinement of WL changes the ground-state electron energy up to 0.05 eV. Due to
the different quantum confinement acting on electron and hole, WL drives a small portion of
PDF of E1from barrier region into WL but drives a small portion of PDF of HH1 from the dot
region in the WL. WL also widens the potential well and thus reduces the energy band-gap,
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Analysis of wetting layer effect on electronic structures of truncated-pyramid quantum dots
the main influence of WL on electronic structures is just from the region underneath the
bottom of the truncated-pyramid dot. We therefore can conclude that our work has supported
the previous study done by Lee et al. very well and also has supplemented the their study
from a very different point of view.
Acknowledgments This work is supported by the Singapore National Research Foundation under CRP
Award No. NRF-G-CRP 2007-01. Q. J. Zhao would like to thank the institute above for financial support.
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