Excitons in Nanosystems Consisting of Semiconductor Quantum Dots
As found, within the band gap of the quantum dot of zinc selenide, a zone of exciton states located at the bottom of the conduction band appears. As shown, a decrease in the band gap width for this nanosystem is conditioned by transition of the electron from quantum-dimensional level within the vale...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2017
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irk-123456789-1300302018-02-05T03:02:47Z Excitons in Nanosystems Consisting of Semiconductor Quantum Dots Pokutnyi, S.I. Gorbyk, P.P. Makhno, S.M. Prokopenko, S.L. Mazurenko, R.V. As found, within the band gap of the quantum dot of zinc selenide, a zone of exciton states located at the bottom of the conduction band appears. As shown, a decrease in the band gap width for this nanosystem is conditioned by transition of the electron from quantum-dimensional level within the valence band of the quantum dot to the levels of the zone of exciton states. The dependence of the energy of a ground state of an exciton on the radius of quantum dot is obtained using a modified method of the effective mass. Было установлено, что в пределах ширины запрещённой зоны квантовой точки из селенида цинка появляется зона экситонных состояний, расположенная в нижней части зоны проводимости. Было показано, что уменьшение ширины запрещённой зоны этой наносистемы обусловлено переходом электрона из квантово-размерного уровня в валентной зоне квантовой точки к уровням зоны экситонных состояний. Зависимость энергии основного состояния экситона от радиуса квантовой точки была получена с использованием модифицированного метода эффективной массы. Було встановлено, що в межах ширини забороненої зони квантової точки з селеніду цинку з'являється зона екситонних станів, розташована в нижній частині зони провідности. Було показано, що зменшення ширини забороненої зони цієї наносистеми зумовлене переходом електрона з квантово-розмірного рівня у валентній зоні квантової точки до рівнів зони екситонних станів. Залежність енергії основного стану екситона від радіюса квантової точки було одержано з використанням модифікованої методи ефективної маси. 2017 Article Pokutnyi S.I., Gorbyk P.P., Makhno S.M., Prokopenko S.L., Mazurenko R.V. / S.I. Pokutnyi, P.P. Gorbyk, S.M. Makhno, S.L. Prokopenko, R.V. Mazurenko // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2017. — Т. 15, № 1. — С. 37-46. — Бібліогр.: 12 назв. — англ. 1816-5230 PACS: 71.35.-y, 73.20.Mf, 73.21.La, 73.22.Lp, 78.67.Hc, 81.07.Ta http://dspace.nbuv.gov.ua/handle/123456789/130030 en Наносистеми, наноматеріали, нанотехнології Інститут металофізики ім. Г.В. Курдюмова НАН України |
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As found, within the band gap of the quantum dot of zinc selenide, a zone of exciton states located at the bottom of the conduction band appears. As shown, a decrease in the band gap width for this nanosystem is conditioned by transition of the electron from quantum-dimensional level within the valence band of the quantum dot to the levels of the zone of exciton states. The dependence of the energy of a ground state of an exciton on the radius of quantum dot is obtained using a modified method of the effective mass. |
| format |
Article |
| author |
Pokutnyi, S.I. Gorbyk, P.P. Makhno, S.M. Prokopenko, S.L. Mazurenko, R.V. |
| spellingShingle |
Pokutnyi, S.I. Gorbyk, P.P. Makhno, S.M. Prokopenko, S.L. Mazurenko, R.V. Excitons in Nanosystems Consisting of Semiconductor Quantum Dots Наносистеми, наноматеріали, нанотехнології |
| author_facet |
Pokutnyi, S.I. Gorbyk, P.P. Makhno, S.M. Prokopenko, S.L. Mazurenko, R.V. |
| author_sort |
Pokutnyi, S.I. |
| title |
Excitons in Nanosystems Consisting of Semiconductor Quantum Dots |
| title_short |
Excitons in Nanosystems Consisting of Semiconductor Quantum Dots |
| title_full |
Excitons in Nanosystems Consisting of Semiconductor Quantum Dots |
| title_fullStr |
Excitons in Nanosystems Consisting of Semiconductor Quantum Dots |
| title_full_unstemmed |
Excitons in Nanosystems Consisting of Semiconductor Quantum Dots |
| title_sort |
excitons in nanosystems consisting of semiconductor quantum dots |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/130030 |
| citation_txt |
Pokutnyi S.I., Gorbyk P.P., Makhno S.M., Prokopenko S.L., Mazurenko R.V. / S.I. Pokutnyi, P.P. Gorbyk, S.M. Makhno, S.L. Prokopenko, R.V. Mazurenko // Наносистеми, наноматеріали, нанотехнології: Зб. наук. пр. — К.: РВВ ІМФ, 2017. — Т. 15, № 1. — С. 37-46. — Бібліогр.: 12 назв. — англ. |
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Наносистеми, наноматеріали, нанотехнології |
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2025-07-09T12:43:41Z |
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37
PACS numbers: 71.35.-y, 73.20.Mf, 73.21.La, 73.22.Lp, 78.67.Hc, 81.07.Ta
Excitons in Nanosystems Consisting of Semiconductor
Quantum Dots
S. I. Pokutnyi, P. P. Gorbyk, S. M. Makhno, S. L. Prokopenko, and
R. V. Mazurenko
O. O. Chuiko Institute of Surface Chemistry, N.A.S. of Ukraine,
17 General Naumov Str.,
UA-03164 Kyyiv, Ukraine
As found, within the band gap of the quantum dot of zinc selenide, a zone
of exciton states located at the bottom of the conduction band appears. As
shown, a decrease in the band gap width for this nanosystem is condi-
tioned by transition of the electron from quantum-dimensional level with-
in the valence band of the quantum dot to the levels of the zone of exciton
states. The dependence of the energy of a ground state of an exciton on
the radius of quantum dot is obtained using a modified method of the ef-
fective mass.
Було встановлено, що в межах ширини забороненої зони квантової то-
чки з селеніду цинку з'являється зона екситонних станів, розташована
в нижній частині зони провідности. Було показано, що зменшення ши-
рини забороненої зони цієї наносистеми зумовлене переходом електро-
на з квантово-розмірного рівня у валентній зоні квантової точки до рі-
внів зони екситонних станів. Залежність енергії основного стану екси-
тона від радіюса квантової точки було одержано з використанням мо-
дифікованої методи ефективної маси.
Было установлено, что в пределах ширины запрещённой зоны кванто-
вой точки из селенида цинка появляется зона экситонных состояний,
расположенная в нижней части зоны проводимости. Было показано,
что уменьшение ширины запрещённой зоны этой наносистемы обу-
словлено переходом электрона из квантово-размерного уровня в ва-
лентной зоне квантовой точки к уровням зоны экситонных состояний.
Зависимость энергии основного состояния экситона от радиуса кванто-
вой точки была получена с использованием модифицированного метода
эффективной массы.
Key words: exciton states, quantum dot, quantum-dimensional level, band
gap.
Наносистеми, наноматеріали, нанотехнології
Nanosistemi, Nanomateriali, Nanotehnologii
2017, т. 15, № 1, сс. 37–46
2017 ІÌÔ (Іíñòèòóò ìåòàëîôіçèêè
іì. Ã. Â. Êóðäþìîâà ÍÀÍ Óêðàїíи)
Надруковано в Óкраїні.
Ôотокопіювання дозволено
тільки відповідно до ліцензії
38 S. I. POKUTNYI, P. P. GORBYK, S. M. MAKHNO et al.
Ключові слова: екситонні стани, квантова точка, квантово-розмірний
рівень, ширина забороненої зони.
Ключевые слова: экситонные состояния, квантовая точка, квантово-
размерный уровень, ширина запрещённой зоны.
(Received 1 December 2016; after revision, 26 March 2017)
1. INTRODUCTION
The solid-state technology developments lead to the fabrication of
the quasi-zero-dimensional nanostructures, which are a semiconduc-
tor quantum dots (QDs) of spherical shape with a radius a 1–10
nm grown in the transparent dielectric (or semiconductor) matrix
[1–4]. Such linear dimensions of the QD are comparable with the de
Broglie wavelength of an electron and a hole, or (and) their Bohr
radius.
The mentioned characteristics lead to the fact that the phenome-
non of spatial size quantization of charge carriers plays an im-
portant role in the optical and electrooptical processes in these
nanosystems [5–12].
Optical and electrooptical properties of those quasi-zero-
dimensional nanostructures are largely determined by the energy
spectrum of a spatially bounded electron-hole pair (exciton) [1–11].
The energy spectrum of charge carriers in the QD, since the size a
is of the order of the Bohr radius of the electron ae or the hole ah
and less, is fully discrete. Therefore, those QDs are called ‘super-
atoms’ [1, 3, 4, 11]. In these conditions, the effect of a spherical
surface of interface (QD–dielectric matrix) can cause the size quan-
tization of the energy spectrum of electron and hole in the QD,
which is associated with a purely spatial limitation of field quanti-
zation, and polarization interaction of charge carriers with the sur-
face of the QD [1, 3–11].
A novel modified method of the effective mass approximation
was recently suggested [5] to describe the exciton energy spectrum
in semiconductor QDs with radii 0
ex
a a ( 0
ex
a —the Bohr radius of
exciton in the semiconductor material of the bulk of QDs.). As
shown, within the framework of the QD model, in which QD was
simulated by limitlessly deep potential well, the effective mass ap-
proximation is liable in description of exciton with QD with radius
a comparable with the Bohr radius of exciton 0
ex
a , considering an
adduced effective mass of exciton ( )a as a function of QD ra-
dius a.
The optical properties of the samples containing of QDs of zinc
selenide placed in air was reported earlier [12]. The average radii a
EXCITONS IN NANOSYSTEMS CONSISTING OF SEMICONDUCTOR QUANTUM DOTS 39
of such QDs were not exceeding 21 nm. At low concentrations of
QDs, when the optical properties of samples are mainly determined
by the optical properties of the single QD in the air, the decrease of
the band gap (Eg2.61–2.68 eV) was detected in comparison with
the band gap for bulk single crystal of zinc selenide (Eg2.7 eV).
The mechanism of such a decreasing in the band gap of zinc sele-
nide QDs is not clear yet.
Therefore, in this paper, we show that a decrease in the band gap
within such nanosystem detected under the experimental conditions
[12] was stipulated by transition of an electron from the quantum-
level located in the valence band of the QD on the level of the exci-
ton state zone. The energy of the base state of the exciton, which
are moving in volume of QDs of zinc selenide, as a function of ra-
dius a of the QD is obtained utilizing the variational method in a
context of the modified effective mass approximation [5]. As found,
in the band gap of QDs of zinc selenide, a zone of exciton states,
which is located at the bottom of the conduction band, appears.
2. VARIATIONAL CALCULATION OF THE EXCITON GROUND-
STATE ENERGY IN THE NANOSYSTEM
A model of a quasi-zero-dimensional system as a neutral spherical
semiconductor QD of radius a, which contains a semiconductor ma-
terial in the bulk with a dielectric constant 2, surrounded by a me-
dia with dielectric permittivity 1, is considered. Within the volume
of the described QD, electron (e) and hole (h) with the effective
masses me and mh, respectively (re and rh—distance of electron and
hole from the centre of the QD), are moving. Assuming the electron
and hole bands as parabolic, the characteristic quantities for a prob-
lem,
2 2 2
02 2 2
2 2 2
0
, ,
e h ex
e h
a a a
m e m e e
, (1)
are the Bohr radii of the electron, hole, and exciton, respectively, in
a semiconductor with the permittivity 2 (e is the electron charge,
and 0 ( )
e h e h
m m m m is the reduced exciton mass).
The energy of the polarization interaction U(re,rh,a) with a rela-
tive permittivity 2/11 can be represented as the algebraic
sum of energies of the interactions of an electron and a hole with
‘themselves’, ,
hh h
V r a , ,
ee e
V r a , and with ‘strangers’,
, , , ,
eh e h he e h
V r r a V r r a , respectively [5–7]:
, , , , , , , ,
e h hh h ee e eh e h he e h
U r r a V r a V r a V r r a V r r a , (2)
40 S. I. POKUTNYI, P. P. GORBYK, S. M. MAKHNO et al.
2 2
2
2 2
2 1
,
2hh h
h
e a
V r a
a a r
, (3)
2 2
2
2 2
2 1
,
2ee e
e
e a
V r a
a a r
, (4)
2
1 22 22
, , , ,
2 / 2 cos
eh e h he e h
e h e h
e a
V r r a V r r a
a
r r a r r a
, (5)
where 2 1 2 1 —parameter, ,
e h
r r —angle.
Within the studied simple model of quasi-zero-structures within
the framework of the above-mentioned approximations as well as
the effective mass approximation, using a triangular coordinate
system , ,
e e h h e h
r r r r r r r with initial point in the centre
of the QD, the Hamiltonian of the exciton moving in the QD vol-
ume, transforms into [5, 6]:
2 2 22 2 2
2
2 2 22 2 2
2
2 2
0
2
0
2
, ,
2
2
2
2
( ) , , , ,
2
e h
e h
e e e e ee
h e
h h h h hh
eh e h e h g
r r r
H r r a
m r r r r r rr
r r r
m r r r r r rr
V r U r r a V r r E
r rr
(6)
where the first three terms are the operators of the kinetic energy
of electron, hole and excitons; 0
g
E —band gap in the unlimited semi-
conductor with dielectric permittivity 2. In Hamiltonian H(re,rh,a)
(6), the energy of polarization interaction U(re,rh,a) is determined
by formulas (2)–(5), and the energy of the Coulomb interaction be-
tween the electron and the hole ( )
eh
V r is described by the formula:
2
2
( )
eh
e
V r
r
. (7)
In Hamiltonian, the exciton potential (6),
, 0 at , and , at ,
e h e h e h e h
V r r r r a V r r r r a , (8)
describes a motion of quasi-particles in the QD volume via model of
the infinitely deep potential well.
The variation of radial wave function of the base state of an exci-
ton (1s-electron state and 1s-hole state) in the QD with radius a can
EXCITONS IN NANOSYSTEMS CONSISTING OF SEMICONDUCTOR QUANTUM DOTS 41
be written as follows [5, 6]:
0
0
2
2 2 2 2
2 2
sin / sin /( )
, , exp
/
.
e h
e h
e h
e h he h
r a r aa
r r r A
r r
r a r ra r a r r
a aa a
(9)
To determine the energy of the base state of the exciton
1,0,0;1,0,0( )E a within the variational method for the QD radius a, the
average value of Hamiltonian of an exciton (6) with wave functions
(9) can be written as follows:
1,0,0;1,0,0 0
0
0 0
( , ( )) , , | , , | , ,
, , , , , , .
e h
e h e h e h
r ra a
e h e h e h e h e h
r
E a a r r r H r r a r r r
dr dr dr r r r r r r H r r a r r r
(10)
The calculation of the energy–radius dependence, 1,0,0;1,0,0( )E a , for
the QD with radius a in the base state of the exciton (ne1,
leme 0; nh 1, lh mh 0 where ne, le, me and nh, lh, mh are prin-
cipal, orbital and magnetic quantum numbers of the electron and a
hole, respectively) was performed via minimization of function
1,0,0;1,0,0( , ( ))E a a (10).
The results of the variational calculation of the base state of the
exciton energy 1,0,0;1,0,0( )E a (10) of the QD with radius a are shown in
Fig. 1. The obtained values of the energy of the base state of the
Fig. 1. Energy of a ground state of the exciton, Е1,0,0;1,0,0(а,) (10), as func-
tion of radius a of the quantum dot of the zinc selenide, where Eg is a
band gap of zinc selenide.
42 S. I. POKUTNYI, P. P. GORBYK, S. M. MAKHNO et al.
exciton 1,0,0;1,0,0( )E a are valid only for values of the exciton energy,
which are governed by the inequality: 0
1,0,0;1,0,0( ) ( )
g
E a E V a ,
where ( )V a is a depth of the potential well of the electron in the
quantum dot. For a wide class of the semiconducting A2B6 QDs in
size range of 0
ex
a a , the value of ( )V a is of 2.3–2.5 eV [5, 6].
The Coulomb attraction between the electron and the hole within
the unlimited semiconductor volume facilitates’ formation of an ex-
citon with large radius. In Hamiltonian of the exciton , ,
e h
H r r a
(6), which moves in a volume of the QD, the Coulomb attraction
Veh(r) (7) is also reinforced by a certain effective attraction between
the electron and the hole caused by the repulsion of the electron
,
ee e
V r a (4) and the hole ,
hh h
V r a (3) from their own images (see
Fig. 1). Under this conditions, energy of the effective repulsion be-
tween the electron and the hole described by terms , ,
eh e h
V r r a and
, ,
he e h
V r r a (5), which are inducing an attraction of quasi-particles
to the surface of the QD (to the ‘foreign’ images), will be less than
the energy of additional effective attraction [5–9].
As a result, with decreasing of the QD radius 0
ex
a a , the value
of the additional effective attraction between the electron and the
hole will increase 1a [5–9]. This effective polarization attraction
leads to the fact that the exciton moves in the volume of the QD
with an effective mass ( )a , which is greater than the mass of
the exciton 0 in the bulk crystal with a dielectric constant 2. With
an increase of the QD radius 0
ex
a a , the effective attraction be-
tween the electron and the hole will decrease a1. Starting with
some values of the QD radius a, which is equal to ac, the energy of
such effective attraction between the electron and the hole is be-
come smaller when compared with the binding energy of the volu-
metric exciton [5, 6]:
2
20
02
ex ex
ex
E Ry
a
. (11)
The volumetric exciton in the QD was meant as the exciton whose
structure (reduced effective mass, Bohr radius, and binding energy)
is not differ from the exciton in the QD in an unlimited semicon-
ductor material. Consequently, the volumetric exciton will appears
only at the QD size of 0
ex
a a . Moreover, the formation of the vol-
umetric exciton has a threshold character, and it is only possible in
the QD whereas its size exceeds a certain value of critical radius of
QD 0
ex
a a [5, 6]. The behaviour of (a) indicates that, with an in-
crease in radius of the QD 0
ex
a a , the effective mass of the exciton
( )a decreases and, at the critical radius of the QD (i.e.,
03.90, 12.2с exa a nm), reaches the value of the effective mass of
the exciton 0 00.137m in a bulk crystal of ZnSe. Thus, the volu-
EXCITONS IN NANOSYSTEMS CONSISTING OF SEMICONDUCTOR QUANTUM DOTS 43
metric exciton occurs in QDs of the zinc selenide when the radius of
QD reaches 03.90, 12.2с exa a a nm.
Figure 1 displays the dependences of energies Е1,0,0;1,0,0(а,) (10) of
the base state of exciton in nanosystems containing of zinc selenide
QDs with radius a and shows that bound states of the electron–hole
pair occur near to the spherical surface of the QD starting with the
critical radius of QD а ас
(1)4.4 nm. The states of the electron–
hole pair, starting with radius of the QD а ас
(1), are in the region
of negative energies (measured from the top limit of the band gap
Eg of a bulk crystal of zinc selenide), which corresponds to a bound
state of the electron and the hole. In this case, the energy of the
Coulomb interaction Veh(r) (8) between the electron and the hole and
the polarization interaction energy U(re,rh,r,a,) (3) for the electron
and the hole with section of the spherical surface (QD–dielectric
matrix interaction) are prevail over the dimensional quantization
energy of the electron and the hole in nanosystems. With an in-
crease in the radius a of the QD, an increase of energy of the base
state of the exciton Е1,0,0;1,0,0(а,) (10) was observed. Starting with
radius of the QD of 03,90, 12.2с exa a a nm, the values (10) of
energy of the base state of the exciton approaches asymptotically to
the value of the binding energy of the volumetric exciton
(Eex28.41 meV) (11) (see Fig. 1).
3. SPECTROSCOPY OF EXCITON STATES IN QUANTUM DOTS
From results of the variational calculation of the base state of an
exciton Е1,0,0;1,0,0(а,) (10) in nanosystems, which contain QDs of the
zinc selenide with change of an average radii a of QD in interval
а ас
(1)4.4 nm, it is follows that, in the band gap of such QDs, a
zone of the exciton mode with width of
EexEex28.41 meV (12)
appears and is located under the bottom of the conductive zone.
The optical properties of the samples of zinc selenide containing
QDs located in air (with dielectric permeability 28.1 and an effec-
tive mass of the electron and the hole (me/m0)0.17 and
(mh/m0)0.7, respectively, where m0 is a mass of free electron) was
reported earlier [7]. For interpretation of the experimental results
[12], let us assume that the QDs have a spherical shape. The aver-
age radius of those QDs is in the range
14 21a nm. (13)
At low experimental concentrations of QDs (x0.003% and
44 S. I. POKUTNYI, P. P. GORBYK, S. M. MAKHNO et al.
x0.03%) [12], the mutual interactions of QDs are nonsignificant.
The optical properties of these samples were defined by energy spec-
tra of the electron and the hole, which are localized near the spheri-
cal surface of the singular QDs immersed in air. At such low con-
centrations of QDs, whereas optical properties are characterized by
optical properties of the singular QD in air, a narrowing of the
band gap zone were detected,
Eg 2.61–2.68 eV, (14)
comparing to the zinc selenide single-crystal band-gap energy (i.e.,
0
g
E 2.7 eV).
In Ref. [12], nanodimensional particles of the zinc selenide were
synthesized via hydrothermal method; 4 mmol of ZnSO4 was dis-
solved in DI water, and then, the ammonia hydroxide was added un-
til complete dissolution of sediment of the zinc hydroxide. Then, a
sodium selenide (Na2SeO3, 4 mmol in DI water) was added. The solu-
tion of hydrazine sulphate of pH 8–9 (adjusted by NaOH) was added
to the reaction at the vigorous stirring. Resulting mixture was
placed into the Teflon lined autoclave and kept at 433 K for 24
hours. Precipitate was washed with DI water and, therefore, dried
at 333 K. Results of XRD confirm the cubic phase of the zinc sele-
nide (ZnSe (JCPDS 37-1463)) with crystallite size of about 27 nm
(see Fig. 2).
For samples treated by ammonium hydroxide, which are partially
dissolve ZnSe, particles aggregate, and the size of crystallite in-
creases up to 42 nm. A width of the band gap zone is determined
via transforming the spectra into the Kubelka–Munk coordinates. A
width of the band gap for synthesized semiconductor ZnSe is
Fig. 2. Diffraction patterns for 1—ammonia treated ZnSe and 2—pristine
ZnSe.
EXCITONS IN NANOSYSTEMS CONSISTING OF SEMICONDUCTOR QUANTUM DOTS 45
Eg 2.61–2.68 eV, and Eg2.56 eV for ammonia-treated sample
(see Fig. 3). A width of the band gap for samples is narrower than
for bulk of zinc selenide ( 0
g
E 2.7eV) by
0
g g g
E E E 20–90 meV. (15)
The exciton mode zone Eex reaches a maximal width (12) start-
ing from radius a of QDs, 12.2
c
a a nm< which is lesser than
the average radius a of QDs from interval (13) in research reported
earlier [12]. Therefore, in regards to (14) and (15), the narrowing in
the width of band gap comparing to the same in bulk crystal of
ZnSe for value (15) is conditioned by transfer of the non-
equilibrium electron from quantum size level within the valence
zone of the QD to the level of the exciton mode with width Eex
(12). The electron transition within the zone of the exciton mode
invokes the significant absorption of irradiation in visible and near-
infrared wavelengths and causes a significant blurring of the ab-
sorption edge, which is experimentally observed. The origin of the
band gap value (Eg2.56 eV) [12] in the framework of the consid-
ered model of the exciton, which moves in a volume of QD of the
zinc selenide, is not clear. The origination of this value needs a fur-
ther investigation.
REFERENCES
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3. S. I. Pokutnyi, J. Nanophotonics, 10, No. 3: 033501 (2016).
Fig. 3. Reflection spectra in the Kubelka–Munk coordinates for 1—
ammonia treated ZnSe and 2—pristine ZnSe.
46 S. I. POKUTNYI, P. P. GORBYK, S. M. MAKHNO et al.
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