Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs
Adduced in this paper are the method and results of theoretical studying the effects of spatial confinement and exciton-phonon interaction on the position and shape of the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs. The heterojunction has been considered as unstrai...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2015
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| Цитувати: | Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs / D.V. Kondryuk, A.V. Derevyanchuk, V.M. Kramar, A.A. Kudryavtsev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 128-133. — Бібліогр.: 33 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859978193475207168 |
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| author | Kondryuk, D.V. Derevyanchuk, A.V. Kramar, V.M. Kudryavtsev, A.A. |
| author_facet | Kondryuk, D.V. Derevyanchuk, A.V. Kramar, V.M. Kudryavtsev, A.A. |
| citation_txt | Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs / D.V. Kondryuk, A.V. Derevyanchuk, V.M. Kramar, A.A. Kudryavtsev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 128-133. — Бібліогр.: 33 назв. — англ. |
| collection | DSpace DC |
| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | Adduced in this paper are the method and results of theoretical studying the effects of spatial confinement and exciton-phonon interaction on the position and shape of the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs. The heterojunction has been considered as unstrained, the nanosystem is modeled as a rectangular quantum well of a finite depth. Interaction of exciton with optical polarization phonons has been taken into account. Calculated has been the temperature dependence of the energy corresponding to transition into the background excitonic state, and determined have been temperature changes in the absorption coefficient related with this transition. It has been shown that observation of these temperature changes in the energy and absorption coefficient, caused by interaction with optical phonons, is possible in the case of exciton with heavy hole at temperatures above 100 K.
|
| first_indexed | 2025-12-07T16:24:54Z |
| format | Article |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 128-133.
doi: 10.15407/spqeo18.02.128
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
128
PACS 73.21.Fg
Temperature changes in the excitonic absorption band
in flat double nanoheterostructures GaAs/AlxGa1-xAs
D.V. Kondryuk, A.V. Derevyanchuk, V.M. Kramar, A.A. Kudryavtsev
Yuriy Fed’kovych Chernivtsi National University,
2, Kotsiubynskogo str., 58012 Chernivtsi, Ukraine
E-mail: v.kramar@chnu.edu.ua
Abstract. Adduced in this paper are the method and results of theoretical studying the
effects of spatial confinement and exciton-phonon interaction on the position and shape
of the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa1-xAs.
The heterojunction has been considered as unstrained, the nanosystem is modeled as a
rectangular quantum well of a finite depth. Interaction of exciton with optical
polarization phonons has been taken into account. Calculated has been the temperature
dependence of the energy corresponding to transition into the background excitonic state,
and determined have been temperature changes in the absorption coefficient related with
this transition. It has been shown that observation of these temperature changes in the
energy and absorption coefficient, caused by interaction with optical phonons, is possible
in the case of exciton with heavy hole at temperatures above 100 K.
Keywords: nanoheterostructure, quantum well, nanofilm, exciton, absorption coefficient.
Manuscript received 03.12.14; revised version received 13.04.15; accepted for
publication 27.05.15; published online 08.06.15.
1. Introduction
The swift development of technologies for studying and
obtaining the desired physical properties of low-
dimensional heterostructures, which is observed for
recent decades, is related with unique features of these
systems that are the base for creation of up-to-date
facilities in opto- and nanoelectronics [1-4]. The most
known structures used with this purpose are those based
on the GaAs/AlxGa1-xAs heterostructures that are already
considered as the classical ones. They serve for creation
of low-threshold lasers capable to operate at room
temperature, highly-efficient photodetectors, light
emitting diodes, re-switchers, quantum cascade lasers
and detectors [1, 5-11].
However, despite the fact that up to date their
optical properties have been studied rather well (see, for
instance, [12]), including also the structure of excitonic
spectra in these systems [13, 14], respective investiga-
tions last till now [15-22]. In particular, using the
quantitative analysis of experimental data, it was shown
in [19] that the shape of spectral bands within the range
of excitonic reflection inherent to heterostructures with
GaAs single quantum wells (QWs) can be described in
the first approximation via Lorentz oscillators. It is
independent of temperature within the range 8 to 90 K.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 128-133.
doi: 10.15407/spqeo18.02.128
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
129
But experimental data concerning the temperature
changes in the structure of excitonic spectra
corresponding to these systems within the high-
temperature range are yet unknown. Theoretical
investigations of these questions are also absent.
The aim of this work was to study temperature
changes caused by interaction of exciton with optical
phonons, which are observed in the structure of exciton
absorption spectra inherent to the double nanohetero-
structures – nanofilms (NF) AlxGa1-xAs/GaAs/AlxGa1-xAs
of various thicknesses and concentrations of Al atoms
inside barrier material.
Assuming that interaction of exciton with phonons
takes place via participation of electron and hole with
them, we have calculated temperature changes in the
energy of transition to the background state of excitons
with heavy and light holes. It enabled us to analyze
temperature changes in positions of respective excitonic
peaks.
We have found the function describing the exciton-
phonon coupling in the case of prevailing interaction
with confined phonons inside a rectangular QW of a
finite depth, which allows investigation of the spectral
dependence for the coefficient of excitonic absorption
and its temperature changes in these nanostructures.
2. Choosing the model for investigation
and problem formulation
To study character of temperature changes in bands of
excitonic transitions in semiconductor nanohetero-
structures, usually used is approximation of effective
masses for electrons as well as the model of dielectric
continuum for phonons (see, for example, [23]).
Modeling NF with the system of rectangular QWs
(electron and hole ones) with the finite depth equal to Ve
and Vh, respectively, separated by the interval of
forbidden values for energies Eg (gap width for well
material), the authors [24] found the explicit form of
Hamiltonian for the exciton-phonon system in
representation of second quantization over all the
variables. Offered there is also the method for
determining the binding energy of quasi-two-
dimensional exciton in NFs.
We have considered the model of Wannier-Mott
exciton created as a result of direct phototransition from
the background sublevel Eh in QW of holes to the
background sublevel Ee in QW of electrons with account
of Coulomb interaction between them under availability
of spatial confinement as well as interaction with optical
polarization phonons. The energy of transition into the
background excitonic state (without any account of
interaction with phonons) can be written as the following
sum:
bheg EEEEE )0( , (1)
where E
b
is the binding energy of quasi-2D exciton.
Interaction of exciton with optical branches of the
phonon spectrum in a double nanoheterostructure (with
confined, semi-confined and interface phonons) is realized
via individual interaction of electron and hole with them. It
results in re-normalization of the energy spectrum inherent
to quasi-particles, i.e., the shift of levels to the side of
lower energies. The shift values of the background states
for electrons (Δe) and holes (Δh) appear to be dependent on
the AlxGa1-xAs/GaAs/AlxGa1-xAs NF thickness,
concentration of aluminum atoms in barrier medium and
temperature of the nanosystem [25]. In accord with (1), it
leads to the change in the exciton transition energy as
follows
heEE )0( , (2)
and can be observed as a long-wave shift of the absorption
band related with it.
The spectral dependence of absorption caused by
exciton transition with participation of phonons is
defined by the shape function of the absorption band
TTE
T
TS
,,
,
,
220
, (3)
that depends on temperature T as on parameter. Here, ħω
is the energy of exciting wave, T, and T, are
real and imaginary parts of the mass operator
),(),(),( TiTTM (4)
corresponding to the Green function of the exciton-
phonon system [24].
The main task of investigation, results of which are
described in this paper, is to ascertain the character of
temperature changes in exciton absorption bands observed
in NF AlxGa1-xAs/GaAs/AlxGa1-xAs of various
thicknesses and concentration compositions inherent to
barrier material.
3. Results and discussion
The mass operator (4) defines influence of exciton-
phonon interaction (EPI) on the exciton energy spectrum
in the system, position and width of the exciton
absorption band. To find it, EPI Hamiltonian can be
represented in the following form
k q
qkqk
BссqH
,, ,
,
)(
int
ˆˆˆ)(ˆ , (5)
where
k
ñ ˆ and
k
ñ
ˆ are operators of creation and
annihilation for exciton states, while
qb ˆ and
qb ˆ –
for phonon states in NF,
qqq bbB ˆˆˆ ; symbol
distinguishes the transversal (inside QW plane) and α –
the longitudinal component of the momentum of the
phonon respective type; ν = (ne, nh, n, m), where n and m
are quantum numbers defining the state of 2D-exciton,
while ne and nh – the same for the states of electron and
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 128-133.
doi: 10.15407/spqeo18.02.128
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
130
hole that form it. The function of exciton-phonon
coupling is as follows
,)()(
)()()(
)(
,
)(
,
)(
qIqF
qIqFq
h
mnnmnn
e
mnnmnn
hh
ee
(6)
and it is defined here for the case of interaction with
confined phonons in infinitely deep QW, like to that
adduced in [26]. It can be also used for QW of a finite
depth if assuming a respective change in the look of the
function for electron-phonon coupling )(
qF nn
; its
explicit form is represented in [27].
Using the EPI function defined in this manner, in
accord to [26] we find the one-phonon mass operator
that takes into account interaction of the background
exciton state with confined phonons and obtain the
explicit form of its real
),()(
),()(
)](1[),(
),(2)(
1
)()(
1
),(2)(
1
)()(
1
1
2
0
22
awJXCC
awJXCC
T
a
e
Tw
hh
n
h
n
h
ee
n
e
n
e
n
N
),())[((
),(2
),(2)(
1
)()(
1
),()(
1
)()(
1
)(
1
)()(
1
awJXCCT
awJXCCXCC
ee
n
e
n
e
ehh
n
h
n
he
n
e
n
e
(7)
)],(2
),()(
),()(
1
)()(
1
)(
1
)()(
1
),(2)(
1
)()(
1
awJXCCXCC
awJXCC
ehh
n
h
n
he
n
e
n
e
hh
n
h
n
h
and imaginary
22
0
2)(
0
2)(
1
)()(
1
)(
0
2)(
1
)()(
1
1
2
0
23
0
)/()/)((
))](()())(()[(
])(1[),(
Nwqa
wqIXCCwqIXCC
T
a
e
Tw
hh
n
h
n
hee
n
e
n
e
n
N
)(
)/()/)((
))](()())(()[(
22
0
2)(
0
2)(
1
)()(
1
)(
0
2)(
1
)()(
1 T
Nwqa
wqIXCCwqIXCC
hh
n
h
n
hee
n
e
n
e
(8)
parts. Here, a0 and ε0 are the lattice parameter and static
dielectric permittivity of well material, respectively;
N = a/a0 is the number of well material layers in NF with
the thickness a; )( p
nC – normalization constant for the
wave function of electron (p = e) or hole (p = h) in the
state with quantum number n in its QW; )(
1
p
nX –
adduced in [27] value that defines the coupling function
for a carrier in its background state with confined
phonon in the state with the quantum number λ with
participation of the n-th level of carrier in QW; ν(T) are
the numbers of filling the phonon states; w = (ħω –
E
(0)
)/ΩLO – normalized by the value of confined phonon
energy (equal to the energy of longitudinal optical
phonon ΩLO in QW material) shift of the exciton
absorption peak relatively to its position E
(0)
calculated
without account of EPI; Γ0 – phenomenological constant
taking into account relaxation of excitons caused by
processes non-included to the accepted model. The rest
auxiliary values and functions in Exps (7) and (8) were
earlier adduced in [26].
Using the following GaAs parameters (a0 =
5.653 Ǻ, ε0 = 13.18, Eg = 1.424 eV, ΩLO = 36.25 meV)
and the AlAs ones (a0 = 5.661 Ǻ, ε0 = 10.06, Eg =
2.168 еV, ΩLO = 50.09 mеV) as well as approximation
of parameters for AlxGa1-xAs solid solution offered in
[28], we calculated temperature dependences of
energies inherent to transition to the background
exciton state in AlxGa1-xAs/GaAs/AlxGa1-xAs NF of
various thicknesses with aluminum concentration
values 0.2, 0.3 and 0.4. As it follows from the
calculation results adduced in Fig. 1, the energy of
transition to the background exciton state with heavy
hole (mh = 0.45m0) is a monotonic temperature
function, the character and value of which are defined
by efficiency of interaction with interface, confined and
semi-confined phonons [29], and therefore depend both
on the NF thickness and composition of well material.
Accordingly to the results of our calculations,
changes in the energy of exciton possessing heavy hole
are weak in the studied nanostructures and can be
observed at temperatures above 100 K. In NF with the
thickness higher than 2.5 nm (N > 4), increase in
temperature from 100 up to 300 K causes non-linear
decrease of the transition energy in dependence on the
values x and a by the value 2 to 3 meV. In ultra-thin NF
(N < 3), its value (also in dependence on the values x
and a) can be lowered by the value 1 to 1.5 meV, or
enhanced, or remains uncharged (see curves N = 2 in
Fig. 1), which is explained by peculiarities of interaction
with interface phonons [29].
The energy of transition to the background state of
exciton possessing light hole (mh = 0.08m0) considerably
weaker depends on temperature: the value of its
temperature changes does not exceed 0.5 meV.
Availability of the weak temperature dependence
for the exciton energy causes the similar changes in
spectra of exciton absorption. In particular, the
coefficient of exciton absorption α becomes dependent
on temperature. In a homogeneous absorbing system, it
is related with the shape function of the absorption band
via the following relation
TSDT ,2),( 2
0 ,
where D0 is the value of the moment corresponding to
electro-dipole transition to the respective exciton state.
Non-uniformity in the thickness of NF (roughness)
causes unhomogeneous broadening of the band.
Therefore, the absorption coefficient should be found as
a convolution of two functions – S(ω,T) and density of
normal distribution (Gauss function)
2
II
2ln4exp
12ln
2)(
E
G
, (9)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 128-133.
doi: 10.15407/spqeo18.02.128
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
131
where ГI is the value of unhomogeneous broadening
[30, 31]. Thus,
dGTS
D
T )(,
2ln4
),(
I
2
0 . (10)
a)
b)
c)
Fig. 1. Temperature dependences of the heavy-hole-exciton
energy in AlxGa1-xAs/GaAs/AlxGa1-xAs nanofilms of various
thicknesses (N) and different concentrations of aluminum
atoms (x): (a) 0.2, (b) 0.3 and (c) 0.4.
Fig. 2. Temperature changes in absorption bands related with
transitions to the background state of exciton with heavy (hh)
and light holes (lh) in the nanofilm of the thickness 73.5 nm
(N = 130).
Using the value Γ0 = 0.33 meV typical for bulk
GaAs samples [32] and the value D0 determined by
comparison with experimental data for the absorption
coefficient in QWs based on GaAs/Al0.3Ga0.7As
heterojunctions [33], we calculated spectral dependences
of the exciton absorption coefficient for various
temperatures with account of interaction with confined
phonons. In these cases, we considered random
divergences in the NF thickness at the level of one GaAs
layer to be possible. The results of calculations indicate
that temperature changes in the exciton absorption
spectrum at T < 100 K caused by interaction with optical
polarization phonons are negligibly small. The position,
height of the peak and half-width of the exciton
absorption band are defined only by values of the NF
thickness and composition of well material.
When the temperature increases from 100 K up to
the room one in NF with the thickness 73.5 nm
(N = 130), one can observe a weak non-linear shift of the
absorption peak (related with transition to the
background state of exciton possessing heavy hole) to
the long-wave range. The value of this shift reaches 2 to
3 meV, and it is the larger, the higher temperature is
(Fig. 2). In this case, the height of the peak is lowered,
while the half-width grows from approximately 2 to
almost 3 meV in all the studied nanoheterostructures. As
it follows from Fig. 1, in NF with a lower thickness the
temperature shift value depends both on the thickness
and concentration of Al in AlxGa1-xAs solid solution.
The position and halfwidth of the band related with
transition to the background state of exciton with light
hole does not practically change with temperature. Like
to the case of exciton with heavy hole, the band width is
predominantly defined by the processes of unhomo-
geneous broadening and is approximately 9 meV.
4. Conclusions
Obtained in this work have been analytical expressions
allowing investigation of temperature changes in the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 128-133.
doi: 10.15407/spqeo18.02.128
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
132
exciton absorption bands observed in flat nanohetero-
structures with a single rectangular quantum well of a
finite depth. The results of calculations performed for
the nanofilms AlxGa1-xAs/GaAs/AlxGa1-xAs (x = 0.2, 0.3,
0.4) have shown that noticeable temperature changes in
the band position and shape, which are caused by
interaction with confined, semi-confined and interface
phonons in these nanosystems, are possible at
temperatures above 100 K. Increasing the NF
temperature causes a monotonic non-linear shift of the
exciton peak to the long-wave range.
In the case of exciton with heavy hole, the
temperature shift value reaches 1 to 3 meV, in
dependence on the NF thickness and composition of
barrier material. The band width is also slightly
increased and reaches 2…3 meV. The position and half-
width of the absorption band corresponding to exciton
with light hole is not considerably changed. The results
of our calculations agree with the experimental data
described in [19, 33].
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| id | nasplib_isofts_kiev_ua-123456789-120721 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-12-07T16:24:54Z |
| publishDate | 2015 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Kondryuk, D.V. Derevyanchuk, A.V. Kramar, V.M. Kudryavtsev, A.A. 2017-06-12T17:54:01Z 2017-06-12T17:54:01Z 2015 Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs / D.V. Kondryuk, A.V. Derevyanchuk, V.M. Kramar, A.A. Kudryavtsev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 128-133. — Бібліогр.: 33 назв. — англ. 1560-8034 DOI: 10.15407/spqeo18.02.128 PACS 73.21.Fg https://nasplib.isofts.kiev.ua/handle/123456789/120721 Adduced in this paper are the method and results of theoretical studying the effects of spatial confinement and exciton-phonon interaction on the position and shape of the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs. The heterojunction has been considered as unstrained, the nanosystem is modeled as a rectangular quantum well of a finite depth. Interaction of exciton with optical polarization phonons has been taken into account. Calculated has been the temperature dependence of the energy corresponding to transition into the background excitonic state, and determined have been temperature changes in the absorption coefficient related with this transition. It has been shown that observation of these temperature changes in the energy and absorption coefficient, caused by interaction with optical phonons, is possible in the case of exciton with heavy hole at temperatures above 100 K. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs Article published earlier |
| spellingShingle | Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs Kondryuk, D.V. Derevyanchuk, A.V. Kramar, V.M. Kudryavtsev, A.A. |
| title | Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs |
| title_full | Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs |
| title_fullStr | Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs |
| title_full_unstemmed | Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs |
| title_short | Temperature changes in the excitonic absorption band in flat double nanoheterostructures GaAs/AlxGa₁₋xAs |
| title_sort | temperature changes in the excitonic absorption band in flat double nanoheterostructures gaas/alxga₁₋xas |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120721 |
| work_keys_str_mv | AT kondryukdv temperaturechangesintheexcitonicabsorptionbandinflatdoublenanoheterostructuresgaasalxga1xas AT derevyanchukav temperaturechangesintheexcitonicabsorptionbandinflatdoublenanoheterostructuresgaasalxga1xas AT kramarvm temperaturechangesintheexcitonicabsorptionbandinflatdoublenanoheterostructuresgaasalxga1xas AT kudryavtsevaa temperaturechangesintheexcitonicabsorptionbandinflatdoublenanoheterostructuresgaasalxga1xas |