Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor
Represented in this paper is the method and results of theoretical investigations aimed at the influence of spatial confinement effects, self-polarization of heterojunction planes as well as exciton-phonon interaction on the position and shape of the band corresponding to exciton absorption in na...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2014
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irk-123456789-1183722017-05-31T03:07:03Z Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor Derevyanchuk, A.V. Pugantseva, O.V. Kramar, V.M. Represented in this paper is the method and results of theoretical investigations aimed at the influence of spatial confinement effects, self-polarization of heterojunction planes as well as exciton-phonon interaction on the position and shape of the band corresponding to exciton absorption in nanofilms of layered semiconductor in a dielectric matrix. The heterojunction is considered as unloaded, the nanosystem is modeled by an infinitely deep quantum well and characterized by an essential difference between dielectric permittivities on both sides of the heterojunction. Calculated in this work are the dependences for the form-function of the absorption band on the thickness of lead iodide nanofilm embedded into polymer E-MAA or glass, and on its temperature. The results of calculations are in good accordance with known data of experimental measurements. 2014 Article Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor / A.V. Derevyanchuk, O.V. Pugantseva, V.M. Kramar // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 2. — С. 188-192. — Бібліогр.: 21 назв. — англ. 1560-8034 PACS 73.21.Fg http://dspace.nbuv.gov.ua/handle/123456789/118372 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
Represented in this paper is the method and results of theoretical
investigations aimed at the influence of spatial confinement effects, self-polarization of
heterojunction planes as well as exciton-phonon interaction on the position and shape of
the band corresponding to exciton absorption in nanofilms of layered semiconductor in a
dielectric matrix. The heterojunction is considered as unloaded, the nanosystem is
modeled by an infinitely deep quantum well and characterized by an essential difference
between dielectric permittivities on both sides of the heterojunction. Calculated in this
work are the dependences for the form-function of the absorption band on the thickness
of lead iodide nanofilm embedded into polymer E-MAA or glass, and on its temperature.
The results of calculations are in good accordance with known data of experimental
measurements. |
| format |
Article |
| author |
Derevyanchuk, A.V. Pugantseva, O.V. Kramar, V.M. |
| spellingShingle |
Derevyanchuk, A.V. Pugantseva, O.V. Kramar, V.M. Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Derevyanchuk, A.V. Pugantseva, O.V. Kramar, V.M. |
| author_sort |
Derevyanchuk, A.V. |
| title |
Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor |
| title_short |
Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor |
| title_full |
Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor |
| title_fullStr |
Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor |
| title_full_unstemmed |
Temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor |
| title_sort |
temperature changes in the function of the shape inherent to the band of exciton absorption in nanofilm of layered semiconductor |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118372 |
| citation_txt |
Temperature changes in the function of the shape inherent
to the band of exciton absorption in nanofilm
of layered semiconductor / A.V. Derevyanchuk, O.V. Pugantseva, V.M. Kramar // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 2. — С. 188-192. — Бібліогр.: 21 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT derevyanchukav temperaturechangesinthefunctionoftheshapeinherenttothebandofexcitonabsorptioninnanofilmoflayeredsemiconductor AT pugantsevaov temperaturechangesinthefunctionoftheshapeinherenttothebandofexcitonabsorptioninnanofilmoflayeredsemiconductor AT kramarvm temperaturechangesinthefunctionoftheshapeinherenttothebandofexcitonabsorptioninnanofilmoflayeredsemiconductor |
| first_indexed |
2025-07-08T13:51:40Z |
| last_indexed |
2025-07-08T13:51:40Z |
| _version_ |
1837087013992398848 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 188-192.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
188
PACS 73.21.Fg
Temperature changes in the function of the shape inherent
to the band of exciton absorption in nanofilm
of layered semiconductor
A.V. Derevyanchuk, O.V. Pugantseva, V.M. Kramar
Yuriy Fedkovych Chernivtsi National University,
2, Kotsyubynsky str., 58012 Chernivtsi, Ukraine;
E-mail: v.kramar@chnu.edu.ua
Abstract. Represented in this paper is the method and results of theoretical
investigations aimed at the influence of spatial confinement effects, self-polarization of
heterojunction planes as well as exciton-phonon interaction on the position and shape of
the band corresponding to exciton absorption in nanofilms of layered semiconductor in a
dielectric matrix. The heterojunction is considered as unloaded, the nanosystem is
modeled by an infinitely deep quantum well and characterized by an essential difference
between dielectric permittivities on both sides of the heterojunction. Calculated in this
work are the dependences for the form-function of the absorption band on the thickness
of lead iodide nanofilm embedded into polymer E-MAA or glass, and on its temperature.
The results of calculations are in good accordance with known data of experimental
measurements.
Keywords: nanoheterostructure, quantum well, nanofilm, exciton, absorption, layered
semiconductor, lead iodide.
Manuscript received 23.01.14; revised version received 29.04.14; accepted for
publication 12.06.14; published online 30.06.14.
1. Introduction
Intense development of technologies to obtain and study
physical properties of low-dimensional heterostructures
based on lead iodide is related with the possibility to use
them for creation of X-ray and γ-radiation detectors as
well as other optoelectronic devices [1-9]. Spatial
confinement of quasi-particles in low-dimensional
structures cause additional quantization of their motion
characteristics, which is pronounced in experimentally
observed changes in their optical properties, including
those in the excitonic range of the spectrum [ 63 ].
Theoretical investigations of exciton states in
nanostructures with single quantum wells (QW) were
performed by many authors with various methods and
approaches (see, for instance, [10-13]), however, the
nanostructures based on layered crystals were not
considered in them.
Adduced in [14] are the results of theoretical
investigations devoted to temperature changes in the
energy of transition to the ground exciton state in a plain
nano-film (NF) of 2H-polytype of lead iodide (2H-PbI2)
embedded into dielectric surroundings – borosilicate
glass or ethylene-methacryl acid (E-MAA) copolymer.
Availability of the method to calculate the exciton
energy in NF enables to study temperature
transformations in the shape of the exciton absorption
band in this nanosystem. The method and results of this
investigation are enlightened in this paper below.
2. Choosing the model of investigations
and setting the task
Investigation of the character of temperature changes in
the bands corresponding to exciton transitions in
semiconductor crystals can be performed using the
model of Wannier-Mott exciton created as a result of the
direct electron phototransition. The spectral dependence
of the absorption coefficient related with these
transitions can be expressed using the following function
TSD
TTE
TD
T
,2
,,
,2
,
2
0
22
2
0
(1)
that depends on temperature as a parameter. Here,
TTE
T
TS
,,
,
,
22
(2)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 188-192.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
189
is the function of the shape inherent to the exciton
absorption band related with the transition to the exciton
state; D0 and E are matrix element of the electric dipole
moment and the energy of this transition, respectively;
ħω is the energy of exciting wave; T, and T,
are real and imaginary parts of the mass operator
),(),(),( TiTTM (3)
for the Green function of the exciton-phonon system
TME
TG
,
1
,
. (4)
The model for the Wannier-Mott exciton in NF of
layered semiconductor (medium 0) embedded into
isotropic dielectric (medium 1) was considered in [14].
The heterojunction is considered here as unstrained,
while QW is rectangular and, with account of dielectric
nature of barrier medium, the infinitely deep one.
If the origin of the Cartesian coordinate system is
placed into the NF center in such a manner that XOY plane
is parallel to its surface, and OZ axis coincides with the
crystallographic axis C of the layered crystal, then the NF
thickness (i.e., the QW width) can be considered as equal
to a = Nc, where c is the lattice parameter along C-axis,
while N is the natural number equal to the amount of PbI2
layers in NF well material (medium 0).
The Hamiltonian for this exciton-phonon system
necessary to find the mass operator (3) should be
constructed using the approach of effective masses for
the electron sub-system and in the form of dielectric
continuum – for the phonon one [15], and takes the
following look:
int
ˆˆˆˆ HHHH phex . (5)
The operators exĤ and phĤ describe,
respectively, the state of free electron and free phonon,
while intĤ – exciton-phonon interaction (EPI); their
apparent look and peculiarities in the case of NF
consisting of layered semiconductor are described in
detail in [14, 16], while offered in [13] is the method for
finding the energy of quasi-two-dimensional exciton in
NF as well as the function of its state. In the first
approximation of the perturbulation theory, the exciton
wave-function can be represented in the form of
production
)()()()( ,,
nmhnenhe zzzz
he
(6)
of functions )( pn z
p
that describe quantized states of
non-interacting electron (p = e) and hole (p = h), which
corresponds to their motion along the direction of the
film growth, as well as the function )(
nm of their state
bound by Coulomb interaction – their relative motion in
the plane of heterojunction ( ),( hehe yyxx
).
Here, ν = (ne, nh, n, m), where n and m are quantum
numbers that determine the states of 2D-exciton [17], ne
and nh – states of electron and hole that create it. The
energy of transition to the exciton state (without account
of interaction with phonons) can be represented as the
sum
b
he
g EEEEE )()((0)
ex , (7)
where Eg is the width of the forbidden band in well
material; E(e) are E(h), respectively, energies of electron
and hole in QW; Eb is the binding energy, the value of
which can be determined using one of versions among
variation methods (see [13]).
The Hamiltonians for free phonons and electron-
phonon interaction in double heterostructures (in
coordinate representation by electron variables) were
obtained in [18, 19]. In the case of the system studied here,
the former acquires the following look:
,2/1ˆˆ
2/1ˆˆˆˆˆ
,
,
LOILOph
AS q
qq
q
qq
bbq
bbHHH
(8)
where ΩLO is the energy of longitudinal optical (LO)
polarization phonons confined in QW medium;
qb ˆ and
qb ˆ are operators of creation and annihilation of the LO-
phonon state with the wave vector ),(
||
qqq
, the
longitudinal component of which (q||) is quantized and
acquires discrete values from the set λπ/Nc (λ = 1, 2, ...,
N ); )( q
is the energy of interface (I) phonons for
symmetric (σ = S) or anti-symmetric (σ = A) modes;
qq bb ˆˆ – operator of creation (annihilation) of the
respective I-phonon state.
The Hamiltonian for EPI [18, 19] expressed by the
authors of [13] in the secondary quantization
representation by all the variables of this electron-phonon
system is as follows:
IexLOexint
ˆˆˆ HHH
,ˆˆˆ
ˆˆˆ
,
,, ,,
q
qknqknnn
knn q
qknqknnn
hep
p
BaaqF
BaaqF
pppp
pp
pppp
(9)
and is determined by the functions of electron-phonon
coupling )(
qF nn
, the apparent look of which for QW of
the finite depth was adduced in [13], and of the infinite one
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 188-192.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
190
– in [16]. Here, η
e
= 1, η
h
= –1; n and n′ – numbers of
electron sub-bands in the state ( kn
, ); α – index of the
phonon branch equal to λ in the case of LO or σ – in the
case of I-phonons;
qqq bbB ˆˆˆ – phonon
operators, while
kn
a ˆ and
kn
a ˆ – electron operators of
secondary quantization.
The Hamiltonian (9) was obtained in assumption that
EPI is realized via individual interaction between electron
and hole, which create exciton, with phonons. This
interaction re-normalizes the energy spectrum of quasi-
particles causing the shift of their states along the energy
scale. In accord with (7), it is pronounced in the change of
the exciton transition energy and can be observed as a
long-wave shift of the correspondent absorption band. The
task of investigations described in this paper was in
studying the character of temperature changes in the
position and shape of the exciton absorption band in 2H-
PbI2 nanofilms embedded into dielectric medium.
3. Results and discussion
To create a function describing the shape of the exciton
absorption band, the Hamiltonian for EPI can be
represented in the following comfortable look [20]:
k q
qkqk
BссqH
,, ,
,
)(
int
ˆˆˆˆ , (10)
where
k
с ˆ and
k
с
̂ are the operators of creation and
annihilation for the exciton state in NF ,| k
;
he kkk
is the transverse component of the exciton
wave vector;
qJqFqJqFq h
mnnmnn
e
mnnmnn hhee
)(
,
)(
,
)(Φ
(11)
– function of the exciton-phonon coupling;
hepdeqI nm
qi
mn
p
mnnm
p ,2),(*)(
,
; (12)
q
m
m
q h
e
ex
,
q
m
m
q e
h
ex
; (13)
he mmmex – effective mass of exciton in the
plane of the layered semiconductor layer.
Choosing )(
nm as a function of the ground
(n = 0, m = 0) state for 2D-exciton [17] in the following
form
ex
2
ex
00
2
exp
8)(
aa
, (14)
the expression (12) can be reduced to the look
12
ex
)(
0 4/1
aq pq
p
I
. (15)
Then
,
4/1
1
4/1
1
Φ
2
exex
2
exex
)(
mqammqam
qFq
eh
nnnn
(16)
where n = 1, 2, … is the number of the level
corresponding to dimensional quantization of carriers in
QW.
Using (16), let us represent MO for exciton-phonon
interaction (in the case of the ground exciton state) in the
form
,
,,1
Φ),,(
exex
,,
2)(
1
qqkE
Tq
qqkE
Tq
qTkM
qn
n
(17)
where ħω is the energy of the incident wave, and
)(ex kE
– energy of exciton from the ground exciton
band.
In NF of the studied type with the thickness
exceeding four layers of lead iodide, interaction with
confined phonons is dominant [14]. In this case, with
account of the quadratic dispersion law and changing
summation by the variable
q
in (17) by integration,
and separating real and imaginary parts in it, one can
obtain respective analytical expression. The real part of
MO
),(2),(),()(
),(2),(),()(1
)(
4
),(
),(),(),(
),(),(),(
1
2
1)0(
0
22
awJawJawJT
awJawJawJT
X
a
e
Tw
ehhe
ehhe
n
N
n
(18)
where )0(
0 is the static dielectric permittivity in the
plane of layer package for layered semiconductor
(superscript here indicates medium);
nX1 – quantity
adduced in [14] that defines interaction of electron (in its
ground state in QW (ne = 1)) with confined phonon
being in the state with a quantum number λ with
participation of n-th (ne = n) minilevel of carrier; ν(T)
are the numbers of filling the phonon states:
)()(1
),()(
1ln
),()(
1
1
1ln
),(
),()(
)(2)(
2
0
)(),(
),(
wDaA
awGaA
a
a
awGwD
w
awJ
pp
p
p
p
p
p
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 188-192.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
191
)(
1
)(
1
)()(
1ln
),()(),()( wDaAwDaA pppp
pp
;
),()(
1ln
)(
)1ln(
)(
1
),(
)()(
0
),(
2
)(
),(
awGaA
a
a
wD
aA
awJ
he
eh
ee
p
eh
eh
h
hh
ph aAawGwD
w
D )(
1ln
),()(
1
1
1ln
1
)(
2
)(),(),(
.
Here, a0 is the lattice parameter in the plane of the
layer package,
2
0)( 1)(
a
a
aA p
p , )1(1)(),( wwD p
p ,
1),(
2
0)( w
a
a
awG
– auxiliary functions,
2
0ex
ex
4
am
am p
p , p = (e, h) –
index of carriers, and
LO
(0)
ex
E
w
– shift of the
maximum inherent to the band of exciton absorption
relatively to its position )0(ex
(0)
ex EE . This shift is
normalized by the energy of confined phonon and Eex(0)
calculated without taking into account the EPI.
The imaginary part is reduced to the look
,
)/(/)(
)/(/)(
)(1
4
),(
2
0
2
0
2
)(
)(
0)(
)(
0
2
0
2
0
2
)(
)(
0)(
)(
0
1
2
1)0(
0
23
0
T
aawqa
II
aawqa
II
T
X
a
e
Tw
wq
h
wq
e
wq
h
wq
e
n
N
n
(19)
where 0/1)( awwq , and Γ0 is a phenomenologic
constant that takes into account exciton relaxation via
processes not accounted in this model.
Using 2H-PbI2 parameters adduced in [14] and
those of ambient medium as well as experimentally
determined value Γ0 = 10 meV, the authors calculated
spectral dependences for the normalized function
describing the shape of exciton absorption band in NF of
various thickness at T = 0 K with account of spatial
confinement, self-polarization of NF surfaces and EPI.
The results of these calculations showed that interaction
with virtual phonons shifts the peak of exciton
absorption band to the long-wave spectral range, and the
larger is the NF thickness, the stronger is this shift. At
the NF thickness equal to 4…7 layers of 2H-PbI2 in E-
MAA matrix, the shift value is 22…35 meV, while that
in glass – 18…35 meV. With increasing the NF
thickness, the shift magnitude approaches to the value
typical to bulk 2H-PbI2 crystal (i.e., 40 meV). The band
width remains practically unchanged, since at T = 0
processes of absorption and emission of real phonons
that cause widening are absent.
The character of temperature changes observed in
the shape of absorption band, which is calculated for the
case of NF PbI2/E-MAA with the thickness of 7 layers
from lead iodide and with account of interaction
exclusively with LO-phonons, shows that essential
temperature changes in the band position and shape
begin at T ≥ 75 K. The increase in temperature causes a
monotonic shift of the exciton peak to the long-wave
range, and at 300 K this shift reaches 80 meV (Fig. 1),
which is in good accordance with the data of
experimental measurements [4]. In this case, the band
width grows insignificantly as a consequence of exciton
scattering by LO-phonons (Fig. 2). The height of the
peak is respectively lowered, which is in good
accordance with the implication of the theory developed
for exciton absorption in the case of optical phonon
participation – changes in temperature do not change the
area of the band [21].
-110 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10
0.0
0.2
0.4
0.6
0.8
1.0
T = 300 K
T = 200 K
T = 150 K
T = 100 K
T = 0 K
S ex
(
)
(0)
exE , meV
Fig. 1. Temperature changes in the shape of exciton absorption
band.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 188-192.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
192
0 50 100 150 200 250 300
3
4
5
6
7
8
9
10
H
/2
, m
eV
T, K
Fig. 2. Temperature changes in the halfwidth of exciton
absorption band.
4. Conclusions
Obtained in this work are the expressions for calculations
of temperature dependences of the shape inherent to the
band of exciton absorption in plain nanoheterostructures
with a single quantum well. Calculations performed using
as an example the nanofilm 2H-PbI2 embedded into glass
or polymer E-MAA matrix have shown that essential
temperature changes in the position and shape of the
absorption band begin at T ≥ 75 K. Growth of the NF
temperature causes a monotonic shift of the exciton peak
to the long-wave spectral range, and this shift reaches
80 meV at 300 K. In this case, the width of the band is
increased to some extent and becomes 4…10 meV. The
results of these calculations are in good accordance with
the data of experimental measurements [3–5].
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9. I. Saikumar, Sh. Ahmad, J.J. Baumberg, G. Vijaya
Prakash, Fabrication of excitonic luminescent
inorganic-organic hybrid nano- and microcrystals //
Scr. Mater. 67, p. 834-837 (2012).
10. R. Zheng, M. Matsuura, T. Taguchi, Exciton-LO-
phonon interaction in zinc-compound quantum wells
// Phys. Rev. B, 61(15), p. 9960-9963 (2000).
11. R.T. Senger, K.K. Bajaj, Binding energy of
excitons in II-VI compound-semiconductor
based quantum well structures // Phys. status
solidi (b), 241(8), p. 1896-1900 (2004).
12. I.V. Ponomarev, L.I. Deych, V.A. Shuvayev,
A.A. Lisyansky, Self-consistent approach for
calculations of exciton binding energy in quantum
wells // Physica E, 25, p. 539-553 (2005).
13. V.M. Kramar, M.V. Tkach, Exciton-phonon
interaction and exciton energy in semiconductor nano-
films // Ukr. J. Phys. 54(10), p. 1027-1035 (2009).
14. O.V. Pugantseva, V.M. Kramar, I.V. Fesiv,
O.O. Kudryavtsev, Temperature changes of the
exciton transition energy in lead di-iodide nanofilms
// Semiconductor Physics, Quantum Electronics and
Optoelectronics, 16(2), p. 170-176 (2013).
15. M.V. Tkach, Quasi-particles in Nanoheterosystems.
Quantum Dots and Wires: A Manual. Chernivtsi
Univ. Press, Chernivtsi, 2003 (in Ukrainian).
16. O.V. Pugantseva, V.M. Kramar, Self-polarization
effect and electron-phonon interaction contributions
in forming of electron energy spectrum of PbI2
nanofilm embed in E-MAA copolymer // J. Nano-
Electron. Phys. 4(4), 04021(6 pp.) (2012).
17. M. Shinada, S. Sugano, Interband optical
transitions in extremly anisotropic semiconductors
// J. Phys. Soc. Jpn. 21(10), p. 1936-1946 (1966).
18. L. Wendler, R. Pechstedt, Dynamical screening,
collective excitations, and electron-phonon interaction
in heterostructures and semiconductor quantum wells
// Phys. status solidi (b) 141(1), p. 129-150 (1987).
19. N. Mori, T. Ando, Electron–optical-phonon
interaction in single and double heterostructures //
Phys. Rev. B, 40(9), p. 6175-6188 (1989).
20. A.S. Davydov, Theory of Solids. Nauka, Moscow,
1976 (in Russian).
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Layered Crystals. Naukova Dumka, Kiev, 1986 (in
Russian).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 188-192.
PACS 73.21.Fg
Temperature changes in the function of the shape inherent
to the band of exciton absorption in nanofilm
of layered semiconductor
A.V. Derevyanchuk, O.V. Pugantseva, V.M. Kramar
Yuriy Fedkovych Chernivtsi National University,
2, Kotsyubynsky str., 58012 Chernivtsi, Ukraine;
E-mail: v.kramar@chnu.edu.ua
Abstract. Represented in this paper is the method and results of theoretical investigations aimed at the influence of spatial confinement effects, self-polarization of heterojunction planes as well as exciton-phonon interaction on the position and shape of the band corresponding to exciton absorption in nanofilms of layered semiconductor in a dielectric matrix. The heterojunction is considered as unloaded, the nanosystem is modeled by an infinitely deep quantum well and characterized by an essential difference between dielectric permittivities on both sides of the heterojunction. Calculated in this work are the dependences for the form-function of the absorption band on the thickness of lead iodide nanofilm embedded into polymer E-MAA or glass, and on its temperature. The results of calculations are in good accordance with known data of experimental measurements.
Keywords: nanoheterostructure, quantum well, nanofilm, exciton, absorption, layered semiconductor, lead iodide.
Manuscript received 23.01.14; revised version received 29.04.14; accepted for publication 12.06.14; published online 30.06.14.
1. Introduction
Intense development of technologies to obtain and study physical properties of low-dimensional heterostructures based on lead iodide is related with the possibility to use them for creation of X-ray and γ-radiation detectors as well as other optoelectronic devices [1-9]. Spatial confinement of quasi-particles in low-dimensional structures cause additional quantization of their motion characteristics, which is pronounced in experimentally observed changes in their optical properties, including those in the excitonic range of the spectrum [
6
3
-
].
Theoretical investigations of exciton states in nanostructures with single quantum wells (QW) were performed by many authors with various methods and approaches (see, for instance, [10-13]), however, the nanostructures based on layered crystals were not considered in them.
Adduced in [14] are the results of theoretical investigations devoted to temperature changes in the energy of transition to the ground exciton state in a plain nano-film (NF) of 2H-polytype of lead iodide (2H-PbI2) embedded into dielectric surroundings – borosilicate glass or ethylene-methacryl acid (E-MAA) copolymer. Availability of the method to calculate the exciton energy in NF enables to study temperature transformations in the shape of the exciton absorption band in this nanosystem. The method and results of this investigation are enlightened in this paper below.
2. Choosing the model of investigations
and setting the task
Investigation of the character of temperature changes in the bands corresponding to exciton transitions in semiconductor crystals can be performed using the model of Wannier-Mott exciton created as a result of the direct electron phototransition. The spectral dependence of the absorption coefficient related with these transitions can be expressed using the following function
(
)
(
)
(
)
(
)
(
)
(
)
(
)
T
S
D
T
T
E
T
D
T
,
2
,
,
,
2
,
2
0
2
2
2
0
w
w
p
=
=
w
G
+
w
D
-
-
w
w
G
w
p
=
w
a
h
(1)
that depends on temperature as a parameter. Here,
(
)
(
)
(
)
(
)
T
T
E
T
T
S
,
,
,
,
2
2
w
G
+
w
D
-
-
w
w
G
=
w
h
(2)
is the function of the shape inherent to the exciton absorption band related with the transition to the exciton state; D0 and E are matrix element of the electric dipole moment and the energy of this transition, respectively; ħω is the energy of exciting wave;
(
)
T
,
w
D
and
(
)
T
,
w
G
are real and imaginary parts of the mass operator
)
,
(
)
,
(
)
,
(
T
i
T
T
M
w
G
-
w
D
=
w
(3)
for the Green function of the exciton-phonon system
(
)
(
)
T
M
E
T
G
,
1
,
w
+
-
w
=
w
h
.
(4)
The model for the Wannier-Mott exciton in NF of layered semiconductor (medium 0) embedded into isotropic dielectric (medium 1) was considered in [14]. The heterojunction is considered here as unstrained, while QW is rectangular and, with account of dielectric nature of barrier medium, the infinitely deep one.
If the origin of the Cartesian coordinate system is placed into the NF center in such a manner that XOY plane is parallel to its surface, and OZ axis coincides with the crystallographic axis C of the layered crystal, then the NF thickness (i.e., the QW width) can be considered as equal to a = Nc, where c is the lattice parameter along C-axis, while N is the natural number equal to the amount of PbI2 layers in NF well material (medium 0).
The Hamiltonian for this exciton-phonon system necessary to find the mass operator (3) should be constructed using the approach of effective masses for the electron sub-system and in the form of dielectric continuum – for the phonon one [15], and takes the following look:
int
ˆ
ˆ
ˆ
ˆ
H
H
H
H
ph
ex
+
+
=
.
(5)
The operators
ex
H
ˆ
and
ph
H
ˆ
describe, respectively, the state of free electron and free phonon, while
int
ˆ
H
– exciton-phonon interaction (EPI); their apparent look and peculiarities in the case of NF consisting of layered semiconductor are described in detail in [14, 16], while offered in [13] is the method for finding the energy of quasi-two-dimensional exciton in NF as well as the function of its state. In the first approximation of the perturbulation theory, the exciton wave-function can be represented in the form of production
)
(
)
(
)
(
)
(
,
,
r
j
y
y
=
r
Y
n
r
r
nm
h
n
e
n
h
e
z
z
z
z
h
e
(6)
of functions
)
(
p
n
z
p
y
that describe quantized states of non-interacting electron (p = e) and hole (p = h), which corresponds to their motion along the direction of the film growth, as well as the function
)
(
r
j
r
nm
of their state bound by Coulomb interaction – their relative motion in the plane of heterojunction (
)
,
(
h
e
h
e
y
y
x
x
-
-
=
r
r
). Here, ν = (ne, nh, n, m), where n and m are quantum numbers that determine the states of 2D-exciton [17], ne and nh – states of electron and hole that create it. The energy of transition to the exciton state (without account of interaction with phonons) can be represented as the sum
b
h
e
g
E
E
E
E
E
-
+
+
=
)
(
)
(
(0)
ex
,
(7)
where Eg is the width of the forbidden band in well material; E(e) are E(h), respectively, energies of electron and hole in QW; Eb is the binding energy, the value of which can be determined using one of versions among variation methods (see [13]).
The Hamiltonians for free phonons and electron-phonon interaction in double heterostructures (in coordinate representation by electron variables) were obtained in [18, 19]. In the case of the system studied here, the former acquires the following look:
(
)
(
)
(
)
,
2
/
1
ˆ
ˆ
2
/
1
ˆ
ˆ
ˆ
ˆ
ˆ
,
,
LO
I
LO
ph
å
å
å
=
s
s
+
s
^
s
l
l
+
l
^
^
^
^
^
^
+
W
+
+
+
W
=
+
=
A
S
q
q
q
q
q
q
b
b
q
b
b
H
H
H
r
r
r
r
r
r
r
(8)
where ΩLO is the energy of longitudinal optical (LO) polarization phonons confined in QW medium;
+
l
^
q
b
r
ˆ
and
^
l
q
b
r
ˆ
are operators of creation and annihilation of the LO-phonon state with the wave vector
)
,
(
||
^
=
q
q
q
r
r
, the longitudinal component of which (q||) is quantized and acquires discrete values from the set λπ/Nc (λ = 1, 2, ..., N );
)
(
^
s
W
q
r
is the energy of interface (I) phonons for symmetric (σ = S) or anti-symmetric (σ = A) modes;
÷
ø
ö
ç
è
æ
^
^
s
+
s
q
q
b
b
r
r
ˆ
ˆ
– operator of creation (annihilation) of the respective I-phonon state.
The Hamiltonian for EPI [18, 19] expressed by the authors of [13] in the secondary quantization representation by all the variables of this electron-phonon system is as follows:
=
+
=
-
-
I
ex
LO
ex
int
ˆ
ˆ
ˆ
H
H
H
(
)
(
)
,
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
,
,
,
,
,
ï
þ
ï
ý
ü
ú
ú
ú
û
ù
ï
î
ï
í
ì
ê
ê
ê
ë
é
+
h
=
å
å
å
å
^
^
^
^
^
^
^
^
^
^
^
^
s
s
+
+
¢
s
¢
¢
l
l
^
+
+
¢
l
¢
=
q
q
k
n
q
k
n
n
n
k
n
n
q
q
k
n
q
k
n
n
n
h
e
p
p
B
a
a
q
F
B
a
a
q
F
p
p
p
p
p
p
p
p
p
p
r
r
r
r
r
r
r
r
r
r
r
r
r
(9)
and is determined by the functions of electron-phonon coupling
)
(
^
a
¢
q
F
n
n
r
, the apparent look of which for QW of the finite depth was adduced in [13], and of the infinite one – in [16]. Here, ηe = 1, ηh = –1; n and n′ – numbers of electron sub-bands in the state (
^
k
n
r
,
); α – index of the phonon branch equal to λ in the case of LO or σ – in the case of I-phonons;
+
-
a
a
a
^
^
^
+
=
q
q
q
b
b
B
r
r
r
ˆ
ˆ
ˆ
– phonon operators, while
^
k
n
a
r
ˆ
and
+
^
k
n
a
r
ˆ
– electron operators of secondary quantization.
The Hamiltonian (9) was obtained in assumption that EPI is realized via individual interaction between electron and hole, which create exciton, with phonons. This interaction re-normalizes the energy spectrum of quasi-particles causing the shift of their states along the energy scale. In accord with (7), it is pronounced in the change of the exciton transition energy and can be observed as a long-wave shift of the correspondent absorption band. The task of investigations described in this paper was in studying the character of temperature changes in the position and shape of the exciton absorption band in 2H-PbI2 nanofilms embedded into dielectric medium.
3. Results and discussion
To create a function describing the shape of the exciton absorption band, the Hamiltonian for EPI can be represented in the following comfortable look [20]:
(
)
å
å
F
=
^
^
^
^
^
^
^
n
¢
n
a
a
n
+
+
n
¢
¢
a
n
¢
n
k
q
q
k
q
k
B
с
с
q
H
r
r
r
r
r
r
r
,
,
,
,
)
(
int
ˆ
ˆ
ˆ
ˆ
,
(10)
where
+
n
k
с
r
ˆ
and
k
с
r
n
ˆ
are the operators of creation and annihilation for the exciton state in NF
>
n
^
,
|
k
r
;
^
^
^
-
=
h
e
k
k
k
r
r
r
is the transverse component of the exciton wave vector;
(
)
(
)
(
)
(
)
(
)
^
^
^
^
^
¢
¢
a
¢
¢
¢
a
¢
a
n
¢
n
-
=
q
J
q
F
q
J
q
F
q
h
m
n
nm
n
n
e
m
n
nm
n
n
h
h
e
e
r
r
r
r
r
)
(
,
)
(
,
)
(
Φ
(11)
– function of the exciton-phonon coupling;
(
)
(
)
(
)
(
)
h
e
p
d
e
q
I
nm
q
i
m
n
p
m
n
nm
p
,
2
)
,
(
*
)
(
,
=
r
r
j
r
j
=
r
¢
¢
¢
¢
ò
^
r
r
r
r
r
;
(12)
^
^
=
q
m
m
q
h
e
r
r
ex
,
^
^
=
q
m
m
q
e
h
r
r
ex
;
(13)
^
^
+
=
h
e
m
m
m
ex
– effective mass of exciton in the plane of the layered semiconductor layer.
Choosing
)
(
r
j
r
nm
as a function of the ground
(n = 0, m = 0) state for 2D-exciton [17] in the following form
÷
÷
ø
ö
ç
ç
è
æ
r
p
-
=
r
j
ex
2
ex
00
2
exp
8
)
(
a
a
r
,
(14)
the expression (12) can be reduced to the look
(
)
(
)
[
]
1
2
ex
)
(
0
4
/
1
-
+
=
^
a
q
p
q
p
I
r
.
(15)
Then
(
)
(
)
(
)
(
)
,
4
/
1
1
4
/
1
1
Φ
2
ex
ex
2
ex
ex
)
(
ú
ú
û
ù
ê
ê
ë
é
+
-
+
´
´
=
^
^
^
^
a
¢
a
¢
^
^
m
q
a
m
m
q
a
m
q
F
q
e
h
n
n
n
n
r
r
(16)
where n = 1, 2, … is the number of the level corresponding to dimensional quantization of carriers in QW.
Using (16), let us represent MO for exciton-phonon interaction (in the case of the ground exciton state) in the form
(
)
(
)
(
)
(
)
(
)
(
)
(
)
,
,
,
1
Φ
)
,
,
(
ex
ex
,
,
2
)
(
1
ú
ú
û
ù
ê
ê
ë
é
W
+
+
-
w
n
+
W
-
-
-
w
n
+
´
=
w
^
^
^
^
^
^
^
^
a
^
a
a
^
a
a
a
^
å
q
q
k
E
T
q
q
q
k
E
T
q
q
T
k
M
q
n
n
r
r
r
h
r
r
r
r
h
r
r
r
r
(17)
where ħω is the energy of the incident wave, and
)
(
ex
^
k
E
r
– energy of exciton from the ground exciton band.
In NF of the studied type with the thickness exceeding four layers of lead iodide, interaction with confined phonons is dominant [14]. In this case, with account of the quadratic dispersion law and changing summation by the variable
^
q
r
in (17) by integration, and separating real and imaginary parts in it, one can obtain respective analytical expression. The real part of MO
{
[
]
[
]
[
]
}
)
,
(
2
)
,
(
)
,
(
)
(
)
,
(
2
)
,
(
)
,
(
)
(
1
)
(
4
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
1
2
1
)
0
(
0
2
2
a
w
J
a
w
J
a
w
J
T
a
w
J
a
w
J
a
w
J
T
X
a
e
T
w
eh
h
e
eh
h
e
n
N
n
-
l
-
l
-
l
+
l
+
l
+
l
=
l
l
^
-
+
n
+
+
-
+
n
+
´
´
l
e
p
-
=
D
å
å
(18)
where
)
0
(
0
^
e
is the static dielectric permittivity in the plane of layer package for layered semiconductor (superscript here indicates medium);
l
n
X
1
– quantity adduced in [14] that defines interaction of electron (in its ground state in QW (ne = 1)) with confined phonon being in the state with a quantum number λ with participation of n-th (ne = n) minilevel of carrier; ν(T) are the numbers of filling the phonon states:
[
]
(
)
+
a
+
a
+
ú
ú
û
ù
ê
ê
ë
é
÷
÷
ø
ö
ç
ç
è
æ
l
+
-
-
-
=
±
l
±
l
l
±
l
±
±
l
)
(
)
(
1
)
,
(
)
(
1
ln
)
,
(
)
(
1
1
1
ln
)
,
(
)
,
(
)
(
)
(
2
)
(
2
0
)
(
)
,
(
)
,
(
w
D
a
A
a
w
G
a
A
a
a
a
w
G
w
D
w
a
w
J
p
p
p
p
p
p
p
m
(
)
ú
ú
û
ù
+
ê
ê
ë
é
a
+
a
+
±
l
±
l
)
(
1
)
(
1
)
(
)
(
1
ln
)
,
(
)
(
)
,
(
)
(
w
D
a
A
w
D
a
A
p
p
p
p
p
p
;
(
)
+
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ë
é
ú
ú
û
ù
ê
ê
ë
é
÷
÷
ø
ö
ç
ç
è
æ
l
+
+
a
-
a
a
+
a
´
´
-
=
±
l
l
±
l
±
l
)
,
(
)
(
1
ln
)
(
)
1
ln(
)
(
1
)
,
(
)
(
)
(
0
)
,
(
2
)
(
)
,
(
a
w
G
a
A
a
a
w
D
a
A
a
w
J
h
e
e
h
e
e
p
eh
(
)
(
)
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
a
-
a
a
+
a
+
-
+
l
±
l
±
±
e
h
h
h
h
p
h
a
A
a
w
G
w
D
w
D
)
(
1
ln
)
,
(
)
(
1
1
1
ln
1
)
(
2
)
(
)
,
(
)
,
(
m
.
Here, a0 is the lattice parameter in the plane of the layer package,
2
0
)
(
1
)
(
÷
ø
ö
ç
è
æ
l
a
-
=
l
a
a
a
A
p
p
,
)
1
(
1
)
(
)
,
(
m
w
w
D
p
p
a
+
=
±
,
1
)
,
(
2
0
)
(
m
w
a
a
a
w
G
+
÷
ø
ö
ç
è
æ
l
=
±
l
– auxiliary functions,
2
0
ex
ex
4
÷
÷
ø
ö
ç
ç
è
æ
p
=
a
^
a
m
a
m
p
p
, p = (e, h) – index of carriers, and
LO
(0)
ex
W
-
w
=
E
w
h
– shift of the maximum inherent to the band of exciton absorption relatively to its position
)
0
(
ex
(0)
ex
E
E
=
. This shift is normalized by the energy of confined phonon and Eex(0) calculated without taking into account the EPI.
The imaginary part is reduced to the look
(
)
[
]
(
)
(
)
[
]
(
)
(
)
(
)
[
]
(
)
(
)
,
)
/
(
/
)
(
)
/
(
/
)
(
)
(
1
4
)
,
(
2
0
2
0
2
)
(
)
(
0
)
(
)
(
0
2
0
2
0
2
)
(
)
(
0
)
(
)
(
0
1
2
1
)
0
(
0
2
3
0
ï
þ
ï
ý
ü
n
l
+
p
-
+
ï
î
ï
í
ì
+
l
+
p
-
n
+
´
´
l
e
p
+
G
=
G
+
-
=
l
l
^
+
+
-
-
å
å
T
a
a
w
q
a
I
I
a
a
w
q
a
I
I
T
X
a
e
T
w
w
q
h
w
q
e
w
q
h
w
q
e
n
N
n
(19)
where
0
/
1
)
(
a
w
w
q
±
p
=
±
, and Γ0 is a phenomenologic constant that takes into account exciton relaxation via processes not accounted in this model.
Using 2H-PbI2 parameters adduced in [14] and those of ambient medium as well as experimentally determined value Γ0 = 10 meV, the authors calculated spectral dependences for the normalized function describing the shape of exciton absorption band in NF of various thickness at T = 0 K with account of spatial confinement, self-polarization of NF surfaces and EPI. The results of these calculations showed that interaction with virtual phonons shifts the peak of exciton absorption band to the long-wave spectral range, and the larger is the NF thickness, the stronger is this shift. At the NF thickness equal to 4…7 layers of 2H-PbI2 in E-MAA matrix, the shift value is 22…35 meV, while that in glass – 18…35 meV. With increasing the NF thickness, the shift magnitude approaches to the value typical to bulk 2H-PbI2 crystal (i.e., 40 meV). The band width remains practically unchanged, since at T = 0 processes of absorption and emission of real phonons that cause widening are absent.
The character of temperature changes observed in the shape of absorption band, which is calculated for the case of NF PbI2/E-MAA with the thickness of 7 layers from lead iodide and with account of interaction exclusively with LO-phonons, shows that essential temperature changes in the band position and shape begin at T ≥ 75 K. The increase in temperature causes a monotonic shift of the exciton peak to the long-wave range, and at 300 K this shift reaches 80 meV (Fig. 1), which is in good accordance with the data of experimental measurements [4]. In this case, the band width grows insignificantly as a consequence of exciton scattering by LO-phonons (Fig. 2). The height of the peak is respectively lowered, which is in good accordance with the implication of the theory developed for exciton absorption in the case of optical phonon participation – changes in temperature do not change the area of the band [21].
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0.0
0.2
0.4
0.6
0.8
1.0
T
= 300 K
T
= 200 K
T
= 150 K
T
= 100 K
T
= 0 K
S
ex
(
w
)
(0)
ex
E
-
w
h
, meV
Fig. 1. Temperature changes in the shape of exciton absorption band.
0
50
100
150
200
250
300
3
4
5
6
7
8
9
10
H/2,
meV
T
, K
Fig. 2. Temperature changes in the halfwidth of exciton absorption band.
4. Conclusions
Obtained in this work are the expressions for calculations of temperature dependences of the shape inherent to the band of exciton absorption in plain nanoheterostructures with a single quantum well. Calculations performed using as an example the nanofilm 2H-PbI2 embedded into glass or polymer E-MAA matrix have shown that essential temperature changes in the position and shape of the absorption band begin at T ≥ 75 K. Growth of the NF temperature causes a monotonic shift of the exciton peak to the long-wave spectral range, and this shift reaches 80 meV at 300 K. In this case, the width of the band is increased to some extent and becomes 4…10 meV. The results of these calculations are in good accordance with the data of experimental measurements [3–5].
References
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15. M.V. Tkach, Quasi-particles in Nanoheterosystems. Quantum Dots and Wires: A Manual. Chernivtsi Univ. Press, Chernivtsi, 2003 (in Ukrainian).
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© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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