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Optical properties of excitons in strained Gax In1-xAs/GaAs quan...
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Taking into account anisotropy, nonparabolicity of the conduction band, and geometrical confinement, we discuss the heavy-hole excitonic states in a strained GaxIn1-xAs/GaAs quantum dot for various Ga alloy contents. The strained quantum dot is considered as a spherical InAs dot surrounded by a GaAs barrier material. The dependence of the effective excitonic g-factor as a function of dot radius and Ga ion content is numerically measured. Interband optical energy with and without the parabolic ef
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COL 11(8), 082501(2013) CHINESE OPTICS LETTERS August 10, 2013
Optical properties of excitons in strained Ga
x
In
1−x
As/GaAs quantum dot: effect of geometrical confinement
on exciton g-factor
N. R. Senthil Kumar
1
, A. John Peter
2∗
, and Chang Woo Lee
3
1
Department of Physics, RVS School of Engineering College, Dindigul-624 401, India
2
Department of Physics, Govt. Arts College, Melur-625 106, Madurai, India
3
Department of Chemical Engineering and Green Energy Center, College of Engineering,
Kyung Hee University, 1Seochun, Gihung, Yongin, Gyeonggi 446-701, Korea
∗
Corresponding author: a.john.peter@gmail.com
Received December 7, 2012; accepted Jun 28, 2013; posted online August 2, 2013
Taking into account anisotropy, nonparabolicity of the conduction band, and geometrical confinement, we
discuss t he heavy-hole excitonic states in a strained Ga
x
In
1−x
As/GaAs quantum dot for various Ga alloy
contents. The strained qu antum dot is considered as a spherical InAs dot surrounded by a GaAs barrier
material. The dependence of the effective excitonic g-factor as a function of dot radius and Ga ion content
is numerically measured. Interband optical energy with and without the parabolic effect is computed using
structural confinement. The interband matrix element for different Ga concentrations is also calculated.
The oscillator strength of interband transitions on the dot radius is studied at different Ga concentrations in
the Ga
x
In
1−x
As/GaAs quantum dot. Heavy-hole excitonic absorption spectra are recorded for various Ga
alloy contents in the Ga
x
In
1−x
As/GaAs quantum dot. Results show that oscillator strength diminishes
when dot size decreases because of the dominance of the quantum size effect. Furthermore, exchange
enhancement and exchange splitting increase as exciton confinement increases.
OCIS codes: 250.0250, 250.5590, 160.0160, 160.4760.
doi: 10.3788/COL201311.082501.
Technological progress in the techniques for g rowing low-
dimensional semiconductor crystals ha s enabled the cus-
tomization of semiconductor structures toward desired
dimensions for specific applications. These techniques
include molecular beam epitaxy, metal organic chemi-
cal vapor deposition, and electron lithography, which en-
able the confinement of carriers in one, two, or three
dimensions (1D, 2D, or 3D). Because of the reduction in
dimensionality of carrier motion, low-dimensional semi-
conductor systems exhibit unusual properties that can
be easily accessible to experimental studies and that lend
themselves to theoretical interpretation. Quantum dots,
referred to as artificia l atoms, are characterized by elec-
tron sta tes tha t take the form of discrete e nergy levels.
The majority of research has thus far focused on the
electronic and optical properties of strained heterostruc-
tures under a strain percentage of around 1%. Con-
versely, studies on experimental data for InAs/GaAs
heterostructures that exhibit a lattice mismatch of ap-
proximately 7% are scarce. Extensively investigating
these highly strained materials is of great interest be-
cause of the large splitting between the hh and lh lower
bands; such splitting considerably modifies valence band
structure. Despite the promising prospects prese nted
by these materials, however, their wide usage is con-
strained by the limited composition of barrier materi-
als. This composition is re sponsible for the tunabil-
ity of the optical band gap of highly strained mate-
rials. Doped InAs semiconducting materials present
tremendous advantages in spintronic applications be-
cause of their large g-factor, high-mobility charge car ri-
ers, and large spin-orbit interaction
[1−3]
. The electronic,
optical, and magnetic properties of the active regions
used for high-performance long-wavelength lasers based
on InGaAs semiconducting materials are interesting be -
cause low-dimensional InGa As s e miconductors are used
on GaAs substrates to develop lasers that range from 1
200 to 1 500 nm. Low-dimensional InGaAs semico nduc-
tors are ideal materials given their tre mendo us tempera-
ture stability, high performance, br oad high-modulation
bandwidths, a nd low-threshold long wavelengths
[4−12]
.
Moreover, such materia ls are the preferred light source
for wideband optical communications, especially lo ng-
term telecommunications applications
[4−12]
. Zhang et
al.
[13]
recently studied the high pulse r e petition rates
(greater than 10 GHz) of diode-pumped solid-state lasers,
which were mode locked using semiconductor saturable
absorber mirror s in 1.55- µm InAs/Ga As quantum dots.
These quantum dots are ex tensively applied in laser
diodes, optical communications, o ptica l amplifier s, and
nonlinear and photonic devices.
Preisler et al.
[14]
conducted photoluminescence exci-
tation spectroscopy under a strong magnetic field to
investigate the interband transitions in se veral ensem-
bles of self-assembled InAs/GaAs qua ntum dots. The
authors determined the excitonic polaron energies and
oscillator strengths of interband transitions. Snelling
et al.
[15]
studied the Zeeman splitting of the heavy-
hole excitons confined in GaAs quantum wells, and
other scholars interpreted heavy-hole magneto excitons
through the electron-hole exchange interaction induced
by confinement
[16]
. By co ntrast, the enhancement of
the electron-hole exchange interaction induced by 2D
confinement is rarely investigated
[17]
.
1671-7694/2013/082501(7) 082501-1
c
2013 Chinese Optics Letters
COL 11(8), 082501(2013) CHINESE OPTICS LETTERS August 10, 2013
The p-like states in the valence band of InAs semi-
conductors are stro ngly coupled to the spin of such
valence ba nd; ultimately, total angular momentum is
modified, thereby drastically altering the electron g-
factor that mea sures the Ze e man splitting of electron
states in the presence of a magnetic fie ld
[18]
. Sarkar et
al.
[19]
conducted a photoluminescence study of the biex-
citon binding energy and fine-structure splitting in sin-
gle InAs/AlAs quantum dots. The authors found that
binding energy and splitting monotonically decreased
as quantum dot emission energy increas e d. Stranski–
Krastanov dots were initially suggested as pyramidal
[20]
,
prompting several groups to theoretically investigate the
structure of ideal pyramidal dots
[21−23]
. A theoretical
analysis of the mean electron and hole positions in self-
assembled InAs-Ga As quantum dot structures has been
presented, in which the plane-wave envelope-function
technique was used to determine the electronic structures
and the measured Stark effect in the structures; these
features cannot be explained by assuming a pyramidal
dot shape
[24]
. Using an eight-band strain-dependent k.p
Hamiltonian, Pryor
[25]
calculated the electronic struc-
ture of pyramidal-shaped InAs/GaAs quantum dots. The
electro-refraction in strained InAs/GaAs and InAs/InP
quantum dots has been discussed using a numerical
model based on the 4×4 Luttinger-Ko hn Hamiltonian
theoretically proposed by Prasanth
[26]
, who performed
matrix diagonalization with plane-wave basis sta tes and
exp eriments
[27]
.
In this letter, we discuss the heavy-hole excitonic states
in a strained Ga
x
In
1−x
As/GaAs qua ntum dot at var-
ious Ga alloy concentrations. The calculations per-
formed are those on anisotropy, the nonparab olicity of
the conduction band, and geometrical confinement. The
strained quantum dot is considered a spherical InAs dot
surrounded by a Ga As barrier material. The exciton
effective Lande factor is measured as a function of dot
radius and Ga a lloy content in the Ga
x
In
1−x
As/GaAs
quantum dot. Interband emission energy is computed us-
ing structural confinement, and the interband matrix el-
ement for different Ga concentrations is ca lculated. The
oscillator strength o f interband transitions on the dot ra-
dius of the InAs/GaAs quantum dot is investigated, and
heavy-hole excitonic absorption spectra for various Ga al-
loy contents in the Ga
x
In
1−x
As/GaAs quantum dot are
recorded.
We consider an exciton in a Ga
x
In
1−x
As quantum dot
surrounded by a s pherical potential barrier with GaAs
material. We assume a uniform strain in the quantum dot
with the ge neral complex of large 3D spatial variations,
as in Ref. [28]. The potential inside the dot is assumed
to be 0 and that outside is assumed to be V
0
. O n the
basis of the approximation of single-band effective mass,
the Hamiltonian of the electron-hole pair in the spherica l
quantum dot of Ga
x
In
1−x
As/GaAs can b e written as
b
H
exc
=
X
j=e,h
~p
2
j
2m
∗
j(i)
+ V
j
!
−
e
2
ε |¯r
e
− ¯r
h
|
, (1)
where j = e, h refer to the electron and hole, respec-
tively; m
∗
j(i)
is the effective mass o f the electron (hole),
with i referring to the mass of the inner and outer ma-
terials; ε is the dielectric constant of the ma terial in the
quantum dot; e represents the absolute electro n charge;
and |¯r
e
− ¯r
h
| denotes the relative distance between the
electron and hole. The s train effects induce an extra p o-
tential field, V
strain
, which we r e gard as the problem in
this letter. For the nanos tructure of the s trained quan-
tum dot, the confinement potential is considered the sum
of the energy offsets of the conduction band (o r valence
band) and the strain-induced potential in all our calcu-
lations.
The electron (hole) confinement potential (V
j
conf
) re-
sulting from the band offset in the Ga
x
In
1−x
As/GaAs
quantum dot structure is given by
V
j
conf
=
0 r
j
6 R
V
0
r
j
> R
, (2)
where V
0
represents the barrier height. For x =0.2, the
barrier heights of the conduction and valence bands ar e
288 and 192 meV, respectively. The parameter tha t in-
dicates the offset between the conduction and valence
bands is 60:40
[29]
. The band g ap difference between the
quantum dot and bar rier at Γ -point is given by
∆E
Γ
g
(x) = 0.359 + 0.491x + 0.58x
2
. (3)
For any low-dimensional semiconductor system, conduc-
tion and valence edge energies are affected primarily by
strain contribution, which is introduced in the Hamil-
tonian through band offset values. Hydrostatic s train
influences lattice volume, ca us ing changes in the energy
levels of materials. Mea nwhile, uniaxial strain causes lat-
tice constant mismatch between two materials.
The stra in-induced po tential of the conduction band
can be expressed as
[30]
V
e−strian
= a
c
(ε
xx
+ ε
yy
+ ε
zz
), (4)
where a
c
is the deformation potential constant of the con-
duction band. The strain tensor components are defined
as ε
xx
= ε
yy
=
a
0
−a
a
, where a
0
and a ar e the lattice
parameters of bulk InAs and GaAs, resp e c tively, and
ε
zz
= −2
C
12
C
11
ε
xx
, where C
ij
represents the elastic stiffness
constants, whose values are given in Table 1.
The strain-induced potential of the valence band can
be written as
[31]
V
v−strain
= a
v
(ε
xx
+ε
yy
+ε
zz
)−
b
2
(ε
xx
+ε
yy
−2ε
zz
), (5)
where a
v
and b are the defo rmation potential constants
of the valence band for the hydrostatic components of the
strain along h001i.
Internal strain influences electronic parameters and the
effective Lande factor . The effective g-factor (which de-
pends on Ga alloy content) of the e lec tron (hole) with an
energy E
c
(E
v
) measured from the bottom (top) of the
conduction (valence) band is provided thus
[32]
g(E, x) = 2 −
2E
p
(x)∆(x)
3(E
g
(x) + E)(E
g
(x) + E + ∆(x))
, (6)
where E
g
(x) is the Ga-dependent fundamental band gap
energy, ∆(x) is the spin-orbit splitting energy, and E
P
describes the coupling energy between conduction band
Γ
6
and valence bands Γ
7
and Γ
8
. The corresponding val-
ues are shown in Table 1.
082501-2
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