analysis of wetting layer effect on electronic structures of truncated-pyramid quantum dots

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: [email protected]

O. Kurniawan

Institute of High Performance Computing, A*STAR,

Singapore 117528, Singapore

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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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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).

123



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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-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

123



Q. Zhao et al.

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,

123




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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